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\title{Path integrals, complex probabilities and the discrete Weyl representation} %Title of paper

%\author{W.~N.~Polyzou}
%\email{polyzou@uiowa.edu}
\thanks{This work supported by the U.S. Department of Energy,
  Office of Science, Grant \#DE-SC16457}
\affiliation{Department of Physics and Astronomy\\
  The University of Iowa\\
  Iowa City, IA 52242, USA}



\begin{abstract}

A discrete formulation of the real-time path integral as the
expectation value of a functional of paths with respect to a complex
probability on a sample space of discrete valued paths is explored.
The formulation in terms of complex probabilities is motivated by a
recent reinterpretation of the real-time path integral as the
expectation value of a potential functional with respect to a complex
probability distribution on cylinder sets of paths.  The discrete
formulation in this work is based on a discrete version of Weyl
algebra that can be applied to any observable with a finite number of
outcomes.  The origin of the complex probability in this work is the
completeness relation.  In the discrete formulation the complex
probability exactly factors into products of conditional probabilities
and exact unitarity is maintained at each level of approximation.  The
approximation of infinite dimensional quantum systems by discrete
systems is discussed.  The method is illustrated by applying it to
scattering theory and quantum field theory.  The implications of these
applications for quantum computing is discussed.

\end{abstract}

\maketitle 

\bigskip

\section{Introduction}\label{intro}

Quantum computing has become a topic of current research because of
its potential for solving problems that are not accessible using
digital computers.  Quantum computations involve preparing an initial
state, evolving it in real time, and performing a measurement at a
later time; repeating this for a statistically significant number of
measurements.  An important step in this process is evolving the
prepared initial state using real-time evolution. Real-time path
integrals represent unitary time evolution as an integral over a
functional of classical paths.   

Path integrals are formally derived using the Trotter product formula
\cite{Simon}, which expresses real-time evolution as the limit of a
product of time evolution over small time steps, making approximations
that preserve unitarity and become exact in the limit of small time
steps.  These are the same steps that quantum computing algorithms are
designed to replicate.  Real quantum computers are discrete quantum
systems with a finite number of qubits.  Path integral treatments of
real-time evolution in discrete systems can be interpreted as models of
quantum computers.

Feynman's interpretation of time evolution as an integral over paths
follows by inserting integrals over complete sets of intermediate
states at each time step and performing integrals over the
intermediate momentum variables.  What remains is the limit of
products finite-dimensional Fresnel integrals that are interpreted as integrals
of a functional of classical paths.  This interpretation has great
intuitive appeal and is useful for generating perturbative expansions
and other results.  There are well-known difficulties with the
interpretation of time evolution as an integral over paths.  The
integrand involves oscillations that are not absolutely integrable,
and the volume element involves an infinite product of complex
quantities.  In addition the ``action'' functional that appears in the
integrand has finite difference approximations of derivatives of the
paths, where $\Delta x$ does not get small as $\Delta t \to 0$.  The
problem with the integral interpretation of the path integral is that
there is no positive countably additive measure on the space on paths.
This is only a problem with interpretation since the Trotter limit
exists independent of the absence of an interpretation as an integral
over paths.

An alternative interpretation
\cite{Muldowney}\cite{Katya_1}\cite{Katya_2} was recently introduced
that replaces the integral over paths by the expectation value of a
functional of paths with respect to a complex probability on cylinder
sets of paths.  The concept of complex probabilities is neither
intuitive or familiar, however in the Euclidean (imaginary time) case
there is a path measure which can alternatively be interpreted as a
probability measure.  When imaginary-time evolution is replaced by
real-time evolution there is no countably additive positive measure on
the space of paths and the probabilities become complex.  The result
of \cite{Muldowney}\cite{Katya_1}\cite{Katya_2} is that by replacing
Lebesgue integration over intermediate states by Henstock
\cite{Henstock}\cite{bartle} integration, which can be used to treat
products of the Fresnel integrals, it is possible to make mathematical
sense of the complex probability interpretation, which results in a
global solution of the Schr\"odinger equation and a unitary
one-parameter time-evolution group.  This provides a rigorous
reinterpretation of real-time path integrals as an average over a
collection of classical paths, as suggested by Feynman.

In the real-time case the Gaussian integrals over momenta are replaced
by Fresnel integrals which are not absolutely integrable.  The
Henstock integral is an adaptive generalization of the Riemann
integral that performs the cancellation of oscillating quantities in
Fresnel integrals before adding them, resulting in finite integrals.
For Gaussian Fresnel integrals the Henstock integral gives the same
result that is obtained using contour integration.  When used in
real-time path integrals the integrals over the intermediate spatial
coordinates are approximated by the generalized Riemann sums, and
paths that have values in sequences of these intervals at different
time slices define the cylinder sets of paths.  In this interpretation
a complex probability is associated with free time evolution of the
paths in each cylinder set.  The resulting complex probability has
most of the properties of a real probability, except that it is not
positive and countable additivity is replaced finite additivity.

The purpose of this paper is to further explore application based on
the complex probability interpretation of the path integral.  While
reference \cite{polyzou} was focused on quantum systems that act on
infinite dimensional Hilbert spaces, this work develops the complex
probability interpretation on finite ($M$) dimensional Hilbert spaces.
The finite dimensional case is more closely related to quantum
algorithms. In the finite dimensional case the complex probabilities
exactly factor into products of conditional probabilities at each time 
step.  Paths are identified with ordered sequences of discrete
eigenvalues of quantum observables at different intermediate times.
This has the property that there are a finite number of cylinder sets
associated with a finite sequence of time slices.  An alternative
quantum mechanical interpretation is to treat the sequence of
eigenvalues as labels for an ordered sequence of quantum mechanical
transition probability amplitudes that result from the dynamics
between each time step.  The complex probabilities used in this
application are due to the completeness sums over intermediate states.
Systems with continuous degrees of freedom, which are treated in
\cite{polyzou}, can be approximated as limits of discrete systems.

Below is a brief outline of the structure of this paper.  The ultimate
goal is to illustrate by example how discrete methods can in principle
be used to perform calculations of scattering observables and the
dynamics of quantum fields on a quantum computer.  The approach of
this work is bottom up in the sense that the continuum dynamics is
treated as the limiting case of a sequence of discrete models.  Real
time evolution of systems on finite dimensional Hilbert spaces is
formulated using a path integral representation.  Even for these
finite dimensional systems there is no countably additive positive
measure on the space of paths. In order to address this situation and
retain the interpretation of the dynamics as an average over a space
of paths, the formal path integral is replaced by the expectation
value of a ``potential functional'' of paths with respect to a complex
probability distribution on cylinder sets of paths.  This
reinterpretation of the path integral was developed in
\cite{Muldowney}\cite{Katya_1}\cite{Katya_2}.  In the continuum case
complex probability densities are related to products of free particle
transfer matrices.  While the Fresnel integrals that appear in these
transfer matrices are not Riemann integrable, they can be computed
using an adaptive generalization of the Riemann integral due to
Henstock, which is used to define the complex probabilities.  The
advantage of the application to discrete systems is that the Fresnel
integrals are replaced by finite sums. Since the complex probabilities
play a prominent role in this work, their essential properties are
discussed briefly in the next section.  The general treatment of
discrete systems is in discussed in section three.  The starting point
is a general observable with a finite number of outcomes.  It is used,
following a method due to Schwinger, to construct an irreducible pair
of unitary operators; one commuting with the original observable and
one complementary operator with the property that any operator in the
Hilbert space can be expressed as a finite degree polynomial in both
operators.  The path integral representation of the time evolution
operator for these discrete systems as the expectation value of a
potential functional on cylinder sets of discrete valued paths is
developed in section five.  The discrete form has the advantage that
the complex probability associated with $N$ time steps exactly factors
into a product of conditional probabilities for single time steps.
This factorization is used to exactly replace the sum over a large
number of paths by powers of a finite dimensional matrix.  While this
factorization is already useful, quantum computers are normally
formulated on a Hilbert space that is generated by tensor products of
qubits.  After a short discussion contrasting the difference between
classical and quantum computers, the formulation of the discrete path
integral in section five is expressed in terms of qubits.  This is
achieved by starting with an observable on a $N=2^L$ dimensional
Hilbert space.  The irreducible pair of unitary operators is expressed
in terms of a collection of L pairs of complementary unitary operators
that act on individual qubits.  The elementary unitary operators can
be identified with standard quantum gates.

Most problems of interest for quantum computing, like
scattering and the dynamics of quantum fields, involve operators with a
continuous rather than discrete spectrum.  The approximation of the of
quantum systems on infinite dimensional Hilbert spaces is treated as
the limit of finite dimensional systems discussed in sections five and
seven.  The resulting formulation of the real-time path
integral is given in section eight.  The application to a simple model of
a particle scattering off of a repulsive Gaussian potential is
discussed in the first appendix.  The calculations utilize
time-dependent methods which involve strong limits.  Narrow wave
packets in momentum are used to extract sharp momentum transition
matrix elements. The second appendix develops the application of
these methods to local field theories.  For field theories an
additional discretization is needed for computations.  In this
appendix a wavelet basis is used to represent the field in terms of
an infinite number of discrete modes.  The wavelet representation
is an exact representation of the field that 
replaces fields as operator valued distributions by an infinite
collection of well-defined, almost local, canonically conjugate
pairs of operators.  In this representation singular products of fields
are replaced by infinite sums of products of well-defined operators.
The
representation has natural volume and resolution truncations.
Computations require truncations to a finite number of degrees of
freedom.  The discrete path integral methods are used to compute the
time evolution of a $\phi^4(x)$ field with a Hamiltonian truncated to
two modes.  While this is a drastic truncation, it illustrates how
these discrete methods can used to model the dynamics of fields.


\section{Complex probabilities}\label{complexp}
    
A complex probability system is defined
by a sample set $S$ and a complex valued function $P$ on subsets of
$S$ with the
properties
\beq
P(S) =1 .
\label{cp1}
\eeq
\beq
P(S_i)+P(S_i^c)= 1
\label{cp2}
\eeq
where $P(S_i)$ is the complex probability assigned
to the subset $S_i$ of $S$ 
and  $S_i^c$ is the complement of $S_i$ in $S$. 
For a finite set of non-intersecting subsets of $S$
\beq
S_i\cap S_j = \mbox{$\emptyset$ }, \qquad  i \not=j  \qquad P(\cup S_i) =\sum_i P(S_i).
\label{cp3}
\eeq
In the applications that follow the sample set will be a collection of
``paths'' that take on discrete values at different times.  The
relevant subsets $S_i$ are the sets of paths whose values take on
specific discrete values at finite collections of times between $0$
and $t$.  Since $P(S_i)$ is complex, equation (\ref{cp3}) cannot be
extended to countable non-intersecting subsets, which is where complex
probabilities differ from ordinary probabilities
\cite{Muldowney2}\cite{Muldowney}.

The extension of the notion of complex probabilities to paths with
continuous values based on the Henstock integral
\cite{Henstock}\cite{bartle}\cite{Gill}, was used in
\cite{Katya_1}\cite{Katya_2}.  The extension of the integral to
continuously infinite dimensional path spaces is discussed in
\cite{Muldowney}.

\section{Schwinger's discrete Weyl algebra}\label{weyl}

This section reviews Schwinger's \cite{schwinger} method of
constructing an irreducible algebra of complementary unitary operators
for quantum systems of a finite number of degrees of freedom.  This
construction generates a finite degree of freedom version of the Weyl
(exponential) form of the canonical commutations relations.  This
algebra can be used to build discrete models of any finite quantum system.
%This construction is essentially the same as the treatment of the
%quantum Fourier transform discussed in \cite{nielsen} and elsewhere.

Let $X$ be a quantum observable with $M$ orthonormal eigenvectors
$\vert m \rangle$ associated with measurement outcomes $x_m$.  $X$
acts on an $M$ dimensional complex Hilbert space ${\cal H}$.
The eigenvectors of $X$ are a basis on ${\cal H}$:
:
\beq
X \vert m \rangle =x_m \vert m \rangle  \qquad m=1,\cdots, M .
\label{s1}
\eeq
Schwinger defines a unitary operator $U$ on ${\cal H}$ that cyclically shifts the
eigenvectors of $X$:
\beq
U\vert m \rangle = \vert {m+1} \rangle \qquad m&lt;M
\qquad
U\vert M \rangle = \vert 1 \rangle .
\label{s2}
\eeq
The labels $m$ on the eigenvectors are treated
as integers mod $M$ so 0 is identified with $M$, $1$ with $M+1$ etc..
%$U$ defined by (\ref{s2}) is unitary since
%\beq
%UU^{\dagger}= \sum_{m=1}^M U\vert m \rangle \langle m \vert U^{\dagger}=
%\sum_{m=1}^{M-1} \vert {m+1} \rangle \langle {m+1} \vert 
%+ \vert {1} \rangle \langle {1} \vert =
%\sum_{m'=1}^M \vert {m'} \rangle \langle {m'} \vert = I .
%\label{s3}
%\eeq
Since $M$ applications of $U$ leaves all $M$
basis vectors, $\vert m \rangle$,  unchanged, it follows that $U^M=I$.
Since $U^k\vert m \rangle$ are independent for all $k&lt;M$, there are no
lower degree polynomials in $U$ that vanish, so 
$P(\lambda) = \lambda^M-1 =0$ is the characteristic polynomial of $U$.
The eigenvalues $\lambda$ 
of $U$ are the $M$ roots of $1$:
\beq
\lambda = u_m = e^{2\pi  m i\over M}
\label{s3}
\eeq
with orthonormal eigenvectors $\vert u_m \rangle$:
\beq
U \vert u_m \rangle = u_m \vert u_m \rangle
\label{s4}
\eeq
\beq
\langle u_m \vert u_n \rangle = \delta_{mn}.
\label{s5}
\eeq
The normalization (\ref{s5}) does not fix the phase of the $\vert u_n \rangle$ 
which will be chosen later.  Both $U^M=I$  and $u_n^M=1$
imply that
\[
0=
(U^M-I ) = {1 \over u_n^M}  (U^M-I  )=
({U\over u_n})^M-I =
\]
\beq
%\prod_{m=1}^M ({U\over u_n}- {u_m \over u_n}) =
({U\over u_n}-I)(I +{U\over u_n} + ({U\over u_n})^2 +\cdots + ({U\over u_n})^{M-1}).
\label{s6}
\eeq
Since this expression is identically zero and $({u_m\over u_n}-1)\not=0$
for $m\not= n$ it follows that
\beq
I +{U\over u_n} + ({U\over u_n})^2 +\cdots ({U\over u_n})^{M-1}
= c \vert u_n \rangle \langle u_n \vert 
\label{s7}
\eeq
for some constant $c$.
Applying (\ref{s7}) to $\vert u_n \rangle$ implies that the constant $c=M$.
This results in an expression for the projection operator on each eigenstate
of $U$ as a degree $M-1$ polynomial in $U$
\beq
%\boxed{
\vert u_n \rangle \langle u_n \vert =
{1 \over M} \sum_{m=1}^M ({U\over u_n})^m =
{1 \over M} \sum_{m=0}^{M-1} ({U\over u_n})^m .
%}
\label{s8}
\eeq
Using (\ref{s8}) it follows that
\[
\langle k \vert u_n \rangle \langle u_n \vert k \rangle  =
\]
\[
{1 \over M} \sum_{m=0}^{M-1} \langle k \vert ({U\over u_n})^m \vert k \rangle 
=
\]
\[
{1 \over M} \sum_{m=0}^{M-1} ({1\over u_n})^m\langle k \vert {k+m} \rangle
\]
\beq
= {1 \over M} .
\label{s9}
\eeq
This means that for any $k$ and $n$ that
\beq
\vert \langle k \vert u_n \rangle \vert = {1 \over \sqrt{M}} .
\label{s10}
\eeq
The interpretation is that if the system is prepared in any eigenstate
of $U$ and $X$ is subsequently measured, then the probability of
measuring any of the eigenvalues of $X$ is the same, (1/M).  This means that
all of the information about the identity of the initial eigenstate of
$U$ is lost after measuring $X$.  This is the condition for the
observables $X$ and $U$ to be complementary.

It is convenient to choose the phase of each $\vert u_n \rangle$
by
\beq
\langle M \vert u_n \rangle = \langle u_n \vert M \rangle
= {1 \over \sqrt{M}} .
\label{s11}
\eeq
It follows from (\ref{s11}) and (\ref{s8})  that
\[
\langle k \vert u_n \rangle \langle u_n \vert M \rangle =
\langle k \vert u_n \rangle {1 \over \sqrt{M}} =
\]
\[
{1 \over M} \langle k \vert \sum_{m=1}^M u_n^{-m} \vert m \rangle =
{1 \over M} u_n^{-k} =
\]
\beq
{1 \over M} e^{-2\pi i nk/M}.
\label{s12}
\eeq
Multiplying (\ref{s12}) by $ \sqrt{M}$ shows that the phase convention
(\ref{s11}) fixes the inner product $\langle k \vert u_n \rangle$
for all $k\not=M$:
\beq
\langle k \vert u_n \rangle =
{1 \over \sqrt{M}} e^{-2\pi i nk/M}.
\label{s13}
\eeq
Schwinger defines another unitary operator, $V$, that shifts the eigenvectors of
$U$ cyclically, but in the opposite direction
\beq
V \vert u_n \rangle = \vert u_{n-1} \rangle, \qquad n \not=1,
\qquad
V \vert u_1 \rangle = \vert u_{M} \rangle . 
\label{s14}
\eeq
The same methods, with $U$ replaced by $V$, give
\beq
V^M=I
\label{s15}
\eeq
\beq
V \vert v_m \rangle = v_m \vert v_m \rangle
\qquad v_m = e^{2 \pi i m \over M}
\label{s16}
\eeq
\beq
\vert v_n \rangle \langle v_n \vert
=
{1 \over M} \sum_{m=0}^{M-1} ({V \over v_n})^m =
{1 \over M} \sum_{m=1}^{M} ({V \over v_n})^m
\label{s17}
\eeq
and for unit normalized $\vert v_n \rangle$
\beq
\vert \langle u_k \vert v_n \rangle\vert = {1 \over \sqrt{M}}.
\label{s18}
\eeq
It is convenient to choose the  
phase of  $\vert v_n \rangle$ by the condition 
\beq
\langle u_M \vert v_n \rangle = {1 \over\sqrt{M}}. 
\label{s19}
\eeq
then (\ref{s17}) and the orthonormality of the
$\vert u_k \rangle$'s give
\[
\langle u_M \vert v_n \rangle \langle v_n \vert u_k \rangle =
\langle v_n \vert u_k \rangle {1 \over \sqrt{M}} =
\]
\beq
{1 \over M} \sum_{m=0}^{M-1} v_n^{-m} \langle u_m\vert u_k \rangle =
{1 \over M} v_n^{-k}. 
\label{s20}
\eeq
Multiplying (\ref{s20}) by $\sqrt{M}$ gives
\beq
%\boxed{
\langle v_k \vert u_n \rangle = {1 \over \sqrt{M}} e^{-2\pi i nk/M}.
%}
\label{s21}
\eeq
Comparing (\ref{s13}) and (\ref{s21}) it follows that
\[
\vert v_k \rangle =\sum_{m=0}^{M-1} \vert u_m \rangle \langle u_m \vert v_k \rangle=
\]
\[
\sum_{m=0}^{M-1} \vert u_m \rangle {e^{2 \pi i mk/M}\over \sqrt{M}} = 
\]
\beq
\sum_{m=0}^{M-1} \vert u_m \rangle \langle u_m \vert k \rangle =
\vert k \rangle ,
\label{s22}
\eeq
so, because of the phase choices,  the operators $X$ and $V$ have the same eigenvectors.

The spectral expansion of $V$, the identification (\ref{s22}) of
$\vert v_k \rangle$ with $\vert k\rangle$ and the definition
(\ref{s2}) of $U$ imply that the 
unitary operators $U$ and $V$ satisfy
\[
UV = U \sum_{m=0}^{M=-1} \vert v_m \rangle e^{i2 \pi m \over M}
\langle v_m \vert =
\]
\[
\sum_{m=0}^{M-1} \vert v_{m+1} \rangle e^{i2 \pi m \over M}
\langle v_m \vert =
\sum_{m=0}^{M-1} \vert v_{m+1} \rangle e^{i2 \pi m \over M}
\langle v_{m+1} \vert U=
\]
\[
\sum_{m=0}^{M-1} \vert v_{m+1} \rangle e^{i2 \pi (m+1) \over M}
e^{-{2 \pi i \over M}}
\langle v_{m+1} \vert U=
\]
\beq
e^{-{2 \pi i \over M}} VU .
\label{s23}
\eeq
%or
%\beq
%\boxed{
%UV = VU e^{-{2\pi i \over M}}.
%%}
%\label{s24}
%\eeq
$U$ and $V$ generate a complete set of operators in the sense that 
that any
operator on the Hilbert space can be expressed as a degree $(M-1) \times (M-1)$ polynomial
these two operators.  Note that $m-k$ applications of (\ref{s2})
give $U^{m-k} \vert v_k\rangle = \vert v_m\rangle$.  Using this
with (\ref{s17}) gives
\[
\vert v_m \rangle \langle v_k \vert   =
U^{m-k} \vert v_k \rangle \langle v_k \vert   =
\]
\beq
{1 \over M} \sum_{n=0}^{M-1} e^{-2 \pi i nk  /M} U^{m-k} V^n .
\label{s25}
\eeq
The order of the $U$ and $V$ operators can be changed using
multiple applications of (\ref{s23}):
\[
U^m V^n = U^{m-1}V^n U e^{-{2n \pi i \over M}} = 
U^{m-2}V^n U^2 e^{-{4n \pi i \over M}} =
\]
\beq
\cdots =
V^n U^m e^{-{2mn \pi i \over M}}.
\label{s25a}
\eeq
Using (\ref{s25a}) in (\ref{s25}) gives
\[
\vert v_m \rangle \langle v_k \vert
\]
\[
{1 \over M} \sum_{n=0}^{M-1} e^{-2 \pi i nk /M} V^n U^{m-k}
e^{-{2(m-k)n \pi i \over M}} =
\]
\beq
{1 \over M} \sum_{n=0}^{M-1} e^{-2 \pi i mk /M} V^n U^{m-k} .
\eeq
A general operator $O$ can be expressed in terms of its matrix elements
in a basis
\[
O = \sum_{m,k=0}^{M-1} \vert v_m \rangle \langle v_m \vert O \vert v_k \rangle
\langle v_k \vert =
\]
\[
\sum_{m,k=0}^{M-1} \langle v_m \vert O \vert v_k \rangle
\vert v_m \rangle  \langle v_k \vert =
\]
\[
{1 \over M} \sum_{m,n,k=0}^{M-1} e^{-2 \pi i nk /M} \langle v_m \vert O \vert v_k \rangle
U^{m-k} V^n =
\]
\beq
{1 \over M} \sum_{m,n,k=0}^{M-1} e^{-2 \pi i mn /M}\langle v_m \vert O \vert v_k \rangle V^n U^{m-k}.
\label{s26}
\eeq
These equations have the form
\beq
%\boxed{
O = \sum_{m,n=0}^{M-1} c_{mn} U^m V^n = \sum_{m,n=0}^{M-1} d_{mn} V^m U^n 
%}
\label{s27}
\eeq
which is the discrete Weyl representation of $O$. 
If $O$ commutes with $U$ then, using (\ref{s25a}), 
\[
0=  \sum_{mn=0}^{M-1} c_{mn} [U^m V^n,U] =
\]
\beq
\sum_{mn=0}^{M-1} c_{mn} U^{m+1} V^n
(e^{2\pi i n/M}-1)
\label{s28}
\eeq
which requires $n=M$ or $0$. This means $O$ is independent of $V$.
Similarly if $O$ commutes with $V$ it must be independent of $U$.  It
follows that any operator that commutes with both $U$ and $V$ is a
constant multiple of the identity. This means that the operators $U$
and $V$ are an irreducible set of unitary operators.

\section{Path integrals for finite dimensional systems}\label{findime}

This section considers a general system of a finite number, $M$, of
degrees of freedom.  $X$ is the observable of
interest.  It evolves under the influence of a Hamiltonian $H$ which
is a Hermitian matrix on the $M$-dimensional Hilbert space.  A measurement of $X$
gives one of its eigenvalues, $x_n$.  The $M$ eigenvectors of $X$ are chosen 
to be orthonormal.  The notation, $\vert n \rangle$, $1 \leq n
\leq M$ is used to label these eigenvectors. $X$ is analogous to a 
``coordinate'' operator in this setting, but in principle it can
be any Hermitian operator on ${\cal H}$.

The quantity of interest is the probability amplitude for a transition
from an initial eigenstate of  $\vert n_i \rangle$ of $X$ to
a final eigenstate $\vert n_f \rangle$ after time $t$.  This
is given by the matrix element of the unitary time evolution operator
\beq
\langle n_f \vert e^{-iHt } \vert n_i \rangle .
\label{f1}
\eeq
In this section (\ref{f1}) is expressed as the expectation value of
a ``functional of paths'' with respect to a complex probability on a
space of paths. 

Following the construction in the previous section irreducible pairs of
unitary operators $U$ and $V$ can be constructed from $X$.  The
eigenvectors $\vert n \rangle$ of $V$ are identical to the
eigenvectors of $X$, while the eigenvectors $\vert\bar{m} \rangle$ of
$U$ are complementary in the sense that
\beq
\vert \langle u \vert \bar{m} \rangle\vert^2 = {1 \over M}.
\label{f2}
\eeq
$U$ satisfies
\beq
U \vert n \rangle = \vert n+1 \rangle
\qquad
U \vert \bar{n} \rangle = \vert \bar{n} \rangle e^{i {2\pi n \over M}},
\label{f3}
\eeq
$V$ satisfies
\beq
V \vert \bar{n} \rangle = \vert \bar{n}-1 \rangle
\qquad
V \vert {n} \rangle = \vert {n} \rangle e^{i {2\pi n \over M}}
\label{f4}
\eeq
and
\beq
VU = UV e^{i{2 \pi \over M} }.
\label{f5}
\eeq
The inner product of the eigenvectors of these complementary operators is
\beq
\langle n \vert \bar{m} \rangle ={e^{-i mn\phi} \over \sqrt{M}}.
\label{f6}
\eeq
%\[
%\sum_m \langle n \vert \bar{m} \rangle = \delta_{n0}\sqrt{M}.
%\]

The time-evolution operator can be expressed as
the $N^{th}$ power of a unitary transfer matrix $T$:
\beq
e^{-iHt } = T^N \qquad T=e^{-iH \Delta t } \qquad \mbox{where} \qquad \Delta t= t/N .
\label{f7}
\eeq
Since  the pair of operators $U$ and $V$ is irreducible, 
$T$ or $H$ can be expressed as degree $(M-1)\times (M-1)$
polynomials in $U$ and $V$ using (\ref{s27}):
\beq
T= \sum_{mn} t_{mn} U^m V^n
\qquad H= \sum_{mn} h_{mn} U^m V^n .
\label{f8}
\eeq
The quantities $t_{mn}$ and $h_{mn}$ are complex valued functions of two
discrete variables.  The eigenvalues of $X$
can be taken as one of the variables.
It is useful to define another Hermitian operator $P$ that has the same eigenvectors
as $U$ with eigenvalues $p_n$ analogous to the relation between $X$ and $V$:
\beq
P \vert \bar{m} \rangle = p_m \vert \bar{m} \rangle .
\label{f9}
\eeq
The eigenvalues $x_n$ and $p_m$ will be referred to as
``coordinates'' and ``momenta'' for the purpose of illustration, although
in general they have no relation to coordinates and momenta.
In this section the choice of the eigenvalues, $p_m$, is not important
- all that is used are the eigenvectors, $\vert \bar{m} \rangle$.
Matrix elements of the transfer matrix in a mixed $(x,p)$ basis
can be computed using (\ref{f3}), (\ref{f4}) in (\ref{f8}):
\beq
\langle \bar{n} \vert T \vert {m} \rangle =
\langle \bar{n} \vert {m} \rangle \tilde{T}_{nm}
\label{f10}
\eeq
where the numerical coefficients $\tilde{T}_{nm}$ are
\beq
\tilde{T}_{nm} =\sum_{kl}t_{kl} e^{- 2 \pi i {(nk-ml)\over M}}.
\label{f11}
\eeq
Mixed matrix elements of the Hamiltonian have a similar form 
\beq
\langle \bar{n} \vert H \vert {m} \rangle =
\langle \bar{n} \vert {m} \rangle \tilde{H}_{nm}
\qquad
\tilde{H}_{nm} =\sum_{kl}h_{kl} e^{- 2 \pi i {(nk-ml)\over M}}.
\label{f11a}
\eeq
Changing the ``momentum''
$\vert \bar{m} \rangle$ basis back to the  ``coordinate'' $\vert n \rangle$ basis gives
\beq
\langle k \vert T \vert m \rangle =
\sum_n\langle k \vert \bar{n} \rangle
\langle \bar{n} \vert {m} \rangle \tilde{T}_{nm} \qquad
\label{f12a}
\eeq
and
\beq
\langle k \vert H \vert m \rangle =
\sum_n\langle k \vert \bar{n} \rangle
\langle \bar{n} \vert {m} \rangle \tilde{H}_{nm} .
\label{f12b}
\eeq
Matrix elements of 
the time evolution
operator in the coordinate basis can be expressed,
using representation (\ref{f12a}-\ref{f12b}) in (\ref{f7}), as 
\[
\langle k_f \vert e^{-iHt} \vert k_i \rangle =\langle k_f \vert [e^{-iHt/N}]^N \vert k_i \rangle =
\]
\[
\sum_{k_1 \cdots k_N} \langle k_f \vert T \vert k_N \rangle
\langle k_N \vert T \vert k_{N-1} \rangle \cdots
\langle k_2 \vert T \vert k_1 \rangle
\langle k_1 \vert T \vert k_i \rangle =
\]
\[
\sum_{k_1 \cdots k_N}\langle k_f  \vert \bar{n}_N \rangle
\langle \bar{n}_N \vert {k}_N \rangle
\langle k_N  \vert \bar{n}_{N-1} \rangle
\langle \bar{n}_{N-1} \vert {k_{N-1}} \rangle \cdots 
\]
\[
\langle k_3  \vert \bar{n}_2 \rangle
\langle \bar{n}_2 \vert {k}_2 \rangle
\langle k_2 \vert \bar{n}_1 \rangle
\langle \bar{n}_1 \vert {k}_i \rangle \times
\]
\beq
\tilde{T}_{n_Nk_N}\tilde{T}_{n_{N-1}k_{N-1}}\cdots \tilde{T}_{n_2k_2}
\tilde{T}_{n_1k_i}.
\label{f13}
\eeq
The time step, $t/N$, appears in the coefficients $\tilde{T}_{n_mk_m}$.  These are complex numbers.

If the $ \tilde{T}_{n_mk_m}$ are all set to 1 then what remains after
performing the completeness sums is the overlap of the final
``coordinate'' with the initial ``coordinate'', $\langle k_f \vert k_i
\rangle =\delta_{k_f,k_i}$.  An additional sum over $k_i$ (resp $k_f$)
or gives 1, independent of $k_f$ (resp $k_i$). This motivates the
definition of the complex probability 
\[
P_N(k_f; n_N,k_N,\cdots n_1,k_1):=
\]
\[
\langle k_f \vert \bar{n}_N \rangle
\langle \bar{n}_N \vert {k}_N \rangle
\langle k_N \vert  \bar{n}_{N_1} \rangle
\langle \bar{n}_{N-1} \vert {k_{N-1}} \rangle
\]
\[
\vdots
\]
\beq
\langle k_3  \vert \bar{n}_2 \rangle
\langle \bar{n}_2 \vert {k}_2 \rangle
\langle k_2  \vert \bar{n}_1 \rangle
\langle \bar{n}_1 \vert {k}_1 \rangle 
\label{f14}
\eeq
which by completeness satisfies
\beq
\sum_{n_1,k_1\cdots n_N k_N} 
P_N(k_f; n_N,k_N,\cdots n_1,k_1) =1
\label{f15}
\eeq
for any $\vert k_f \rangle $.
$P_N$ assigns a complex weight to a pair of eigenvalues of $X$ and $P$ at
each of $N$ time steps.
%for the possible
%values of the two non-commuting operators $U$ and
%$V$ (resp. $P$ and $X$) at each time step.

The exact time evolution can be expressed using the Trotter product formula
\[
e^{-iHt} = \lim_{N \to \infty}  (e^{-iHt/N})^N = 
\lim_{N \to \infty}  (I- iHt/N)^N
\]
where only the first order term in the exponent contributes in the
limit $N \to \infty$.   
In the small $\Delta t= t/N$ limit 
\[
\langle \bar{m} \vert T \vert n \rangle \approx
\langle \bar{m} \vert I -i H \Delta t \vert n \rangle =
\]
\beq
\langle \bar{m} \vert n \rangle (1 -i \tilde{H}_{mn} \Delta t) \approx 
\langle \bar{m} \vert n \rangle e^{ -i \tilde{H}_{mn} \Delta t}.
\label{f16}
\eeq
In this expression $\tilde{H}_{mn}$ is an ordinary function of $p_m$ and $x_n$.
If the quantum Hamiltonian $H$ is expressed with the $X$ operators
to the right of the $P$ operators, this would be the Hamiltonian with
these operators replaced by the ``coordinates'' and ``momenta''.
It is the analog of a classical phase space Hamiltonian.

%It is instructive to make the following analogy with an ordinary path integral
%in the limit of small $\Delta t$.  In this case 
%$n_m\to p_m,k_m\to q_m$ are labels for
%classical coordinates and momenta at time
%$t$ (in this case they are discrete).
%\[
%\langle p_m \vert e^{-iH\Delta t} \vert q_m \rangle  \approx
%\]
%\[
%\langle p_m \vert (I -iH \Delta t \vert q_m \rangle \approx
%\]
%\[
%\langle p_m \vert q_m \rangle (1 -i h(p_m,q_m) \Delta t ) \approx
%\]
%\[
%\langle p_m \vert q_m \rangle (e^{-i h(p_m,q_m) \Delta t}
%\]
%where we assumed that commutation relations are used to order all of the
%momentum operators to the left of the coordinates.
It follows from (\ref{f10}) and (\ref{f16}) that
\[
\tilde{T}_{nk}\approx e^{-i \tilde{H}(p_{n},x_{k}) \Delta t}
\]
and 
\beq
\tilde{T}_{n_Nk_N}\tilde{T}_{n_{N-1}k_{N-1}}\cdots \tilde{T}_{n_2k_2}
\tilde{T}_{n_1k_i} \approx
e^{-i \sum_m \tilde{H}(p_{n_{m}},x_{k_{m}}) \Delta t}
\label{f17}
\eeq
which is a functional of a classical ``phase space'' path
$p(t),x(t)$ with endpoints at $x(0)$ and $x(t)$, where $p(t),x(t)$ can take
on $M$ discrete values, $p_n$ and $x_m$ at each time step.  This is classical in the sense
that even though $P$ and $X$ do not commute, $\tilde{H}(p_n,x_m)$ is an ordinary
function of the eigenvalues $p_n$ and $x_m$.  
%
%\[
%e^{-i \int_0^t h(p(t'),q(t')) dt'}
%\]
%along a path with endpoint at $q(0)$ and $q(t)$.  With this interpretation
%the trotter product formula can be interpreted as the expectation of 
%\[
%e^{-t \int_0^t h(p(t'),q(t')dt'}
%\]
A {\it cylinder set of paths} is a set of M-valued functions
$p(t)$ and $x(t)$ that take on one of the $M$ possible eigenvalues
of $P$ and $X$ at each of $N$ time slices, subject to initial and final
values of the $x$ variable.  With this definition
of cylinder set
\beq
P_N(k_f; n_N,k_N,\cdots n_1,k_1)
\label{f18}
\eeq
can be interpreted as a complex probability associated with the cylinder set
of ``phase space'' paths that start at $x_{k_1}$, end at $x_{k_f}$
and have values $p_{n_m},x_{k_m}$ at the $m^{th}$ time
slice.

From a quantum mechanical point of view an $M$-valued path can be
thought of as an ordered sequence of quantum transition amplitudes
that alternate between states of irreducible pairs of observables.


Combining (\ref{f13},\ref{f14} and \ref{f16}) gives 
\[
%\boxed{
\langle k_f \vert e^{-iHt} \vert k_i \rangle \approx
\]
\beq
\sum_{n_N,k_N,\cdots n_1,k_1}
P_N(k_f; n_N,k_N,\cdots n_1,k_1)
e^{-i \sum_m \tilde{H}(x_{n_m} p_{k_m}) \Delta t}\delta_{k_1k_i}
%}
\label{f19}
\eeq
which approximates the transition probability amplitude as the
expectation value of a functional of the ``classical'' Hamiltonian
$\tilde{H}(x,p)$ over cylinder sets of ``paths in phase space''.  Here
the ``coordinate'' paths have fixed endpoints at $0$ and $t$, while
the ``momentum'' paths are unconstrained.  In this case the complex
probability interpretation is a consequence of the completeness relation.

In general the number of cylinder sets that must be summed over is prohibitively large, $(\sim M^{2N})$.
The definition (\ref{f14}) implies that the complex probability can be factored
into products of one-step probabilities
\[
P_N(k_f; n_N,k_N,\cdots n_1,k_1)=
\prod_{i=1}^N \langle k_{i+1} \vert  \bar{n}_{i} \rangle
\langle \bar{n}_{i} \vert {k_{i}} \rangle :=
\]
\beq
\prod_i P(k_{i+1};n_i,k_i), \qquad k_{N+1}=k_f .
\label{f19a}
\eeq
In this case
\[
\langle k_f \vert e^{-iHt} \vert k_i \rangle =
\]
\beq
\sum_{n_N,k_N,\cdots n_1,k_1}
\prod_i \left (P(k_{i+1}; n_i,k_i)
e^{-i \tilde{H}(x_{n_i} p_{k_i}) \Delta t}\right )\delta_{k_1k_i}
\label{f19b}
\eeq
where $k_{N+1}=k_f$.
This reduces the computation of (\ref{f19})  to computing the $N^{th}$ power of an $M\times M$ matrix.

The formalism discussed above can be applied to any Hamiltonian on the $M$-dimensional
Hilbert space.  If the Hamiltonian is a sum of the form
$\tilde{H}(p,x)=\tilde{H}_1(p)+\tilde{H}_2(x)$ then it is possible to sum
over the intermediate ``momentum'' variables resulting in a complex
probability on a space of paths in the coordinate variable, $X$.

In this case
\[
\sum_n \langle k \vert \bar{n}\rangle \langle \bar{n}  \vert H \vert {m} \rangle =
\]
\[
\sum_n \langle k \vert \bar{n}\rangle \langle \bar{n} \vert m \rangle (H_1(p_n)+H_2(x_m))=
\]
\beq
\sum_n \langle k \vert \bar{n}\rangle \langle \bar{n} \vert m \rangle \tilde{H}_{nm}
\label{f19b}
\eeq
where $\tilde{H}_{nm}$ is just the ``classical'' Hamiltonian as a function
of the eigenvalues.

A new complex probability can be defined by 
\[
P_X(k_f; k_N,\cdots ,k_1)
:=
\]
\beq
e^{i \tilde{H}_1(p_0) t}
\sum_{n_1 \cdots n_N}
P(k_f; n_N,k_N,\cdots n_1,k_1)
e^{-i \sum \tilde{H}_1(p_{n_m}) \Delta t} . 
\label{f20}
\eeq
To show that this is normalized like a complex probability,
interchange the order of the sum over the intermediate
$k$ indices and $n$ indices (both sums are finite).  The
$k_2 \cdots k_N$ sums are just expressions for the identity.
After eliminating $k_2 \cdots k_N$ what remains is a product of
Kronecker delta functions in the $n_i$ variables.
Since the operator
$H_1$ is a multiplication operator in the $n$ variables, 
all but one of the $n$ sums can be performed giving:
\[
\sum_{n_1 \cdots n_N} \sum_{k_1 \cdots k_N} 
P(k_f; n_N,k_N,\cdots n_1,k_1)
e^{-i \sum \tilde{H}_1(p_{n_m}) \Delta t} =
\]
\[
\sum_{n=0}^{M-1}\sum_{k_1=0}^{M-1}\langle k_f \vert \bar{n} \rangle e^{-i N \tilde{H}_1(p_{n}) \Delta t}
\langle \bar{n} \vert k_1 \rangle .
\]
The $k_1$ sum can be evaluated using  
\[
\sum_{n=0}^{M-1}\sum_{k_1=0}^{M-1} \langle k_f \vert \bar{n} \rangle
e^{-i \tilde{H}_1(p_n) t}\langle \bar{n} \vert k_1 \rangle
=
\]
%\sum_{mn=0}^M \langle n_f \vert n \rangle =1 
%\]
%sum_{m,n=0}^M \langle n_f \vert \bar{m} \rangle \langle \bar{m} \vert n \rangle
\[
{1 \over M} \sum_{n=0}^{M-1} e^{-i 2 \pi  k_f n/M} \sum_{k_1=0}^{M-1}
e^{i 2 \pi  k_1 n/M} e^{-i \tilde{H}_1(p_n) t} =
\]
\[
{1 \over M} \sum_{n=0}^{M-1}  e^{-i 2 \pi  k_f n/M} e^{-i \tilde{H}_1(p_n) t}
\times
\]
\[
\left \{
\begin{array}{cc}
  (1 - e^{i 2 \pi  M n/M})=0 &amp; 1&lt;n&lt;M-1\\
  M &amp; n=0
\end{array}
\right .
\]
\beq
= \delta_{n0} e^{-i H_1(p_0) t}.
\label{f21}
\eeq
%xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
%Using this in (\ref{f20}), summing over the right most $k$ sets the next $n$ to% 0, then the process repeats, moving right to left resulting in 
%xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Including the factor $e^{i H_1(p_0) t}$ gives 
\[
\sum_{k_1 \cdots k_N} P_X(k_f;,k_N,\cdots ,k_1)=
\]
\beq
e^{i \tilde{H}_1(p_0) t}
\sum_{n=0}^{M-1}\sum_{k_1=0}^{M-1}
P(k_f; n_N,k_N,\cdots n_1,k_1)
e^{-i \sum_m \tilde{H}_1(p_{m}) \Delta t} =1.
\label{f22}
\eeq
The expression for the evolution operator becomes
\[
\langle k_f \vert e^{-iHt} \vert k_i \rangle =
\]
\beq
\sum_{k_1 \cdots k_N} P_X(k_f;,k_N,\cdots ,k_1)
e^{-i \sum_m \tilde{H}_{2}( x_{n_m})  \Delta t}\delta_{k_1k_i}
\label{f23}
\eeq
which is expressed as the expectation value of the functional
$e^{-i \sum_m \tilde{H}_{2}( x_{n_m})  \Delta t}$ on cylinder sets of discrete paths.
The finite sums over the discrete intermediate ``momentum'' variables
replace the Gaussian Fresnel integrals in the continuum case.  

In this case the complex probability over cylinder sets of paths in
$x$ also factors into products of one-step probabilities with 
\beq
P_{X}(k_{n+1};k_n)=
\sum_{m}
P(k_{n+1}; m,k_n)
e^{-i (\tilde{H}_1(p_{m})-\tilde{H}_1(p_{0})) \Delta t} 
\label{f24}
\eeq
giving
\[
\langle k_f \vert e^{-iHt} \vert k_i \rangle =
\]
\beq
\lim_{N\to \infty}
\sum \left (\prod_n P_{X}(k_{n+1};k_n) e^{-i \sum_m \tilde{H}_2(p_{n}) \Delta t}
\right ) \delta_{k_1,k_i} \qquad k_f = k_{N+1}
\label{f25}
\eeq
which has the structure of the $N^{th}$ power of an $M \times M$ matrix.
In this discrete case the result becomes exact in the Trotter limit.

In the discrete case time evolution can be solved exactly by diagonalizing
the $M\times M$ Hamiltonian matrix in the $x$ basis, however the appeal of the
discrete path integral is that it is a model of a quantum circuit.
Accuracy and be improved by using a higher order approximation to the
transfer matrix at each step.   The complex probability interpretation is a
natural way to think about real-time path integrals.

The advantage of this approach is that the time evolution is
approximated by calculating the expectation value functional of a
finite set of ``classical'' paths with respect to a complex
probability distribution.  The ability of exactly factor the complex
probability into products of one step probabilities for each time
step facilitates the computation.  Finally exact unitarity is
maintained at each step.  While the limit for continuous time
evolution exists \cite{Katya_1}\cite{Katya_2}, real computations are
truncated at a finite number of time steps.

\section{Quantum logic}\label{logic}

The interest in discrete path integrals is that they serve as models
of quantum computers.  The difference between classical and quantum computers
is related to the difference between classical and quantum logic.
Classically if a system is prepared in a state
$A$ and a later measurement tests if it will be detected in state $B$,
there are two possible outcomes, true or false.  This leads to a two
valued system of logic that is encoded in the bits used in digital
computing.  In quantum mechanics there are three possibilities - the
final system will always be measured to be in the state $B$, it will
never be measured to be in the state $B$, or there is a finite
probability $P$, with $0&lt;P&lt;1$, that it will be measured to be in state
$B$. This leads to a three-valued logic or quantum logic.

The three valued logic of quantum mechanics \cite{birkhoff} has a
straightforward geometrical interpretation.  If state $A$ is
represented by a one-dimensional subspace of a Hilbert space and state
$B$ is represented by another one-dimensional subspace then there are
three possibilities - (1) the subspace $B$ is the
subspace $A$, (2) the subspace $B$ is orthogonal to the subspace $A$
or (3) any non-0 vector in $A$ has a non-zero projection on the subspace
$B$.  

In the quantum case states are represented by vectors or rays, $\vert a
\rangle$ in a Hilbert space.  Quantum probabilities are expressed in
terms of the Hilbert space inner product:
\beq
P_{ab} := {\langle a \vert b \rangle \langle b \vert a \rangle \over \langle a \vert a \rangle\langle b \vert b \rangle }
\label{ql1}
\eeq
which is independent of the vectors in the rays.
The three possibilities correspond to
\beq
(1) P_{ab}=1 
\eeq
\beq
(2) 
P_{ab}=0
\eeq
\beq
(3) 
0&lt;P_{ab}&lt;1.
\label{ql2}
\eeq
When the Hilbert space is two dimensional the difference in these two
types of logic is encoded in bits or qubits respectively.  A bit can
only be in one of two states, true or false.  For a qubit in a spin
up state, the three possibilities are (1) the probability of measuring
it to be in a spin up state is 1 (true), (2) the probability of
measuring it to be in a spin down state is 0 (false) and (3) the
probability of measuring it to be in a non-trivial linear combination
up and down states is a positive number strictly between zero and one
(maybe).

The discrete path integral developed in the previous section represents
the transition probability amplitude as the expectation value of a
random variable on a space of sequences of transition amplitudes
labeled by paths in the space of eigenvalues of complementary
observables.

%In quantum mechanics observable quantities are represented by linear
%operators $A$ on a Hilbert space.  The only possible outcomes for
%measuring $A$ are one of its eigenvalues, $a_n$ (this assumes $A$ is a
%normal operator whose eigenvectors form a basis). In this case the
%Hilbert space has a decomposition ${\cal H} = \oplus_n {\cal H}_n$,
%where the ${\cal H}_n$ are $A$-invariant subspaces of ${\cal H}$.
%
%The mean value of a measurement of $A$ in state $\vert b\rangle$ is  
%\beq
%\langle b \vert A \vert b \rangle =
%\sum_n P_{a_nb} a_n
%\qquad a_n \mbox{ eigenvalue of A} 
%\label{ql3}
%\eeq
%which is the weighted average of the quantum probabilities
%for a  measurement of $b$ to be in one of the eigenstates of $A$.


\section{Qubits}\label{qubit}

One property of Schwinger's discrete Weyl algebra is that
it has a natural representation in terms of qubits.  When $M$ can be
factored into products of prime numbers the $U$ and $V$ operators
can be replaced
by an algebra of commuting pairs of operators with cycles the length
of each prime factor.  The case of most interest for quantum computing
is when $M=2^L$.  In that case the irreducible algebra is represented
by tensor products of qubit gates.  For systems where the number of degrees
of freedom $K$  is not a power of two, they can be embedded in a space of
dimension $M=2^L$, for $M&gt;K$. 

Consider the case where  $M=2^L$ for large $L$.  The indices $n= 0 \cdots 2^L-1$
can be labeled by $L$ numbers that can only take the values $0$ and $1$.  It has
a L-bit representation
$n \leftrightarrow (n_1,n_2,\cdots, n_L)$
\beq
n=\sum_{m=1}^L n_m 2^{m-1}. 
\label{qb1}
\eeq
This results in the identifications
\beq
\vert u_{n_1\cdots n_L} \rangle :=\vert u_n \rangle
\qquad
\vert v_{n_1\cdots n_L} \rangle :=\vert v_n \rangle .
\label{qb2}
\eeq
Define unitary operators $U_i$ and $V_i$ by
\beq
U_i \vert v_{n_1\cdots n_L} \rangle =
\vert v_{n_1\cdots [n_i+1]_{\mbox{mod\,2}} \cdots n_L} \rangle
\label{qb3}
\eeq
\beq
V_i \vert u_{n_1\cdots n_L} \rangle =
\vert u_{n_1\cdots [n_i-1]_{\mbox{mod\,2}} \cdots n_L} \rangle .
\label{qb4}
\eeq
Applying what was done in the general case to $M=2^L$ gives
\beq
U_i^2-1= V_i^2 -1=0,
\label{qb5}
\eeq
\beq
[U_i,U_j]=[V_i,V_j]=0 \qquad [U_i,V_j]=0 \qquad i \not=j, 
\eeq
\beq
V_i U_i = U_i V_i e^{i \pi},
\label{qb6}
\eeq
\beq
U^n = \prod_{m=1}^L U_m^{n_m},
\label{qb7}
\eeq
\beq
V^n = \prod_{m=1}^L V_m^{n_m}.
\label{qb8}
\eeq
Since $U$ and $V$ can be constructed from the $U_i$ and $V_i $ the
set of $\{ U_i\}$ and $\{ V_i\}$ is also irreducible.

A matrix representation of $U_i$ and $V_i$ acting on the $i$-th qubit is
\beq
V_i = \sigma_3 \qquad U_i = \sigma_1
\label{qb9}
\eeq
which are simple quantum gates.
In this representation, $v_0=u_0=1; v_1=u_1=-1$ and
\beq
\vert v_0 \rangle =
\left(
\begin{array}{c}
1 \\
0\\
\end{array}
\right )
\qquad
\vert v_1 \rangle =
\left (
\begin{array}{c}
0 \\
1\\
\end{array}
\right )
\label{qb10}
\eeq

\beq
\vert u_0 \rangle ={1 \over \sqrt{2}}
\left(
\begin{array}{c}
1 \\
1\\
\end{array}
\right )
\qquad
\vert u_1 \rangle = {1 \over \sqrt{2}}
\left (
\begin{array}{c}
1 \\
-1\\
\end{array}
\right ).
\label{qb10}
\eeq
The operators  $\sigma_1$ and $\sigma_3$ 
\beq
U_i=
\sigma_1 = \left (
\begin{array}{cc}
  0&amp;1\\
  1&amp;0\\
\end{array}
\right )
\qquad
V_i= \sigma_3 = \left (
\begin{array}{cc}
  1&amp;0\\
  0&amp;-1\\
\end{array}
\right )
\qquad
\label{qb11}
\eeq
satisfy (\ref{s2}) and (\ref{s14}) for $M=2$.
They also satisfy
\beq
\sigma_3 \sigma_1 = \sigma_1 \sigma_3 e^{{2\pi i \over 2}}
\qquad
(\sigma_1^2 -1)= (\sigma_3^2-1) =0 .
\label{qb12}
\eeq
Any linear operator $A$ on this 2-dimensional
vector space is a polynomial with constant coefficients $a_i$ in these
operators:
\beq
A = a_1I + a_2 \sigma_1 + a_3 \sigma_3 +a_4 \sigma_3\sigma_1 .
\label{qb13}
\eeq
In this case the Hilbert space is represented by $L$ qubits. The
irreducible set of operators, $U_i$ and $V_i$ are represented by $L$
pairs of Pauli matrices ($\sigma_1$ and $\sigma_3$) that act on each
qubit. This representation has the advantage that it is local in the
sense that the $U_i$ and $V_i$ operators act on a single qubit
and the operators that act on different qubits commute.
Equations (\ref{qb7}-\ref{qb8}) relate the operators that appear in
the discrete path integral to tensor products of single qubit operators
$\sigma_1$ and $\sigma_3$. 

%Note for $M$ odd the same construction works with
%$3\times 3$ matrices with
%\beq
%U_i = \left (
%\begin{array}{ccc}
%  0&amp;0&amp;1\\
%  1&amp;0&amp;0\\
%  0&amp;1&amp;0
%\end{array}
%\right )
%\qquad
%V_i = \left (
%\begin{array}{ccc}
%1&amp;0&amp;0\\
%0&amp;e^{2 \pi i/3}&amp;  0\\
%0 &amp; 0 &amp; e^{4 \pi i /3}
%\end{array}
%\right ).
%\label{qb14}
%\eeq

\section{Schwinger's continuum limit}\label{cont}

While quantum computers are discrete quantum systems, many problems of
interest involve observables like momenta, coordinates, and canonical
fields that have continuous spectra. Applications require discrete
approximations to these continuous systems.  It is possible to use the
discrete algebra generated by $U$ and $V$ to make a discrete
approximation to the continuum in the large $M$ limit. The limit is
designed to give a representation of the Weyl algebra.

For large $M$ Schwinger \cite{schwinger} defines the small quantity
$\epsilon$ by
\beq
%\boxed{
\epsilon^2 :=
{2 \pi /M}.
%}
\label{cl1}
\eeq
For the purpose of approximating the continuum it is convenient
(but not necessary) to
choose $M=2K+1$ odd and number the eigenvectors and eigenvalues from
$-K \leq n \leq K$ instead of $0$ to $M-1$ or $1$ to $M$ (For even $M$
the indices could be labeled $-M/2 \leq n \leq M/2$).
Discrete approximations to continuous
variables $p$ and $x$ are defined by
\beq
p_l = l \epsilon = l \sqrt{2 \pi \over M}   \qquad x_l = l \epsilon = 
l \sqrt{2 \pi \over M} \qquad  -K\epsilon \leq x_l,p_l \leq K\epsilon
\label{cl2}
\eeq
where
\beq
K\epsilon  = \sqrt{M\pi \over 2} - \sqrt{\pi \over 2M}.
\label{cl3}
\eeq
With these definitions the separation between successive values of $p_l$
and $x_l$, $p_{l+1}-p_l =x_{l+1}-x_l = \epsilon$
vanishes as $M \to \infty$ while at the
same time the maximum and minimum values of $p_l$ and $x_l$,
$p_{\pm K} = x_{\pm K} = \pm (
\sqrt{M\pi \over 2} - \sqrt{\pi \over 2M} )$ approach $\pm\infty$ in
same limit.

While for finite $M$ any vector with a finite number of elements has a
finite norm, in the continuum limit ($M\to \infty$) this is no longer
true so the limiting vectors with finite norm should be square
summable.  This means that components of vectors with large $\vert l
\vert$ should approach $0$ in the $M\to \infty$ limit.  

For unitary operators $U$ and $V$ given by (\ref{s2}) and (\ref{s15})
Hermitian operators $\tilde{p}$ and $\tilde{x}$ are defined by
\beq
%\boxed{
V = e^{i \epsilon \tilde{p}}
\qquad
U= e^{i \epsilon \tilde{x}}.
%} .
\label{cl4}
\eeq
These can be used to define
\beq
V(x_m) = e^{i\tilde{p} x_m} = e^{i\tilde{p} \epsilon m} = V^m
\label{cl5}
\eeq
\beq
U(p_n) = e^{i\tilde{x} p_n} = e^{i\tilde{x} \epsilon n} = U^n .
\label{cl6}
\eeq
With these definitions equation (\ref{s22}) becomes
\[
V(x_m)U(p_k)= U(p_k)V(x_m) e^{i2 \pi mk \over M}=
\]
\beq
U(p_k)V(x_m) e^{i \epsilon m \epsilon k}=
U(p_k)V(x_m)e^{ip_kx_m}
\label{cl7}
\eeq
%\beq
%\boxed{
%V(x_m)U(p_k)= 
%U(p_k)V(x_m)e^{ip_kx_m}
%}
%\label{cl8}
%\eeq
which is the Weyl \cite{Weyl} form of the canonical commutation
relations, where in this
case the variables are discrete.
In order to take the continuum limit define numbers $p:=\epsilon n$ and
$x= \epsilon m$.  This motivates the definitions 
%Equations (\ref{cl5}-\ref{cl6}) motivate the definitions
\beq
%\boxed{
dp = \epsilon dn = \sqrt{2 \pi \over M} dn
\qquad
dx = \epsilon dm = \sqrt{2 \pi \over M} dm .
%}
\label{cl9}
\eeq
It follows from (\ref{cl4}) that eigenvectors of $V$ are also
eigenvectors of $\tilde{p}$ and the eigenvectors of $U$ are also
eigenvectors of $\tilde{x}$.  Choosing the normalization of the states $\vert
p_n\rangle$ and $\vert x_n \rangle$ so
\beq
\int dp \approx \sum_{n=-K}^K {dp \over dn} = \epsilon \sum_{n=-K}^K
\eeq
\beq
\int dx \approx \sum_{n=-K}^K {dx \over dm} = \epsilon \sum_{m=-K}^K
\label{cl9}
\eeq
\beq
I = \sum_{l=-K}^K  \vert v_l \rangle \langle v_l \vert =
\sum_{l=-K}^K  \vert p_l \rangle dp_l \langle p_l \vert =
\sum_{l=-K}^K  \vert p_l \rangle \epsilon \langle p_l \vert
\label{cl10}
\eeq
\beq
I = \sum_{l=-K}^K  \vert u_l \rangle \langle u_l \vert =
\sum_{l=-K}^K  \vert x_l \rangle dx_l \langle x_l \vert =
\sum_{l=-K}^K  \vert x_l \rangle \epsilon \langle x_l \vert 
\label{cl11}
\eeq
which expresses the sum over small steps in $p$ or $x$ as integrals.
Equations (\ref{cl10}) and (\ref{cl11})
imply the relations
\beq
%\boxed{
\vert p_l \rangle := \vert v_l \rangle /\sqrt{\epsilon}
%}
\label{cl12}
\eeq
and 
\beq
%\boxed{
\vert x_l \rangle := \vert u_l \rangle /\sqrt{\epsilon} .
%}.
\label{cl13}
\eeq
Using these relations gives
\beq
\langle p_m\vert x_n \rangle = 
        {1 \over \epsilon} \langle v_m \vert u_n \rangle =
        {1 \over \epsilon \sqrt{M}}e^{-2 \pi i mn\over M}
        =
        {1 \over \sqrt{2 \pi}} e^{-i p_m x_n}
\eeq
\beq
\langle p_m\vert p_n \rangle = 
{1 \over \epsilon} \langle v_m \vert v_n \rangle =
{\delta_{mn} \over \epsilon}
\label{cl14}
\eeq
and
\beq
\langle x_m\vert x_n \rangle = 
{1 \over \epsilon} \langle u_m \vert u_n \rangle =
{\delta_{mn} \over \epsilon}.
\label{cl15}
\eeq
In this notation equations (\ref{cl5}) and (\ref{cl6})
with (\ref{s2}) and (\ref{s14}) give
imply 
\beq
U(x_m) \vert x_n \rangle = \vert x_m+x_n \rangle
\label{cl16}
\eeq
\beq
V(p_m) \vert p_n \rangle = \vert p_m-p_n \rangle
\label{cl16}
\eeq
which can be expressed in terms of the ``continuum variables'' as  
\beq
U(x) \vert x' \rangle = \vert x'+x \rangle
\label{cl16}
\eeq
\beq
V(p) \vert p' \rangle = \vert p'-p \rangle
\label{cl16}
\eeq

\section{Complex probabilities in real-time path integrals on infinite
dimensional Hilbert spaces}\label{path}

%The path integral for a system with one continuous degree of freedom is
%formulated using the discrete representation discussed section \ref{weyl}.
%Following references \cite{Muldowney}\cite{Katya_1}\cite{Katya_2},
%the path integral will be defined as the expectation value of a
%potential functional with respect to a complex probability
%distribution.
%
%To do this it is necessary to:
%\begin{itemize}
%\item[1.)] Define the space of paths
%\item[2.)] Define complex probabilities on the space of paths
%\item[3.)] Identify the path integral with the expectation value of a
%functional on the space of paths.
%\end{itemize}
%
For problems involving scattering or canonical field theories the
relevant operators have continuous spectra.  Systems with continuous
variables can be approximated by discrete systems following section
VII.  In the continuous case a general Hamiltonian, $H(p,x)$, can be
expressed as
\[
H = \int \tilde{H}(p,x) U(x) V(p) dxdp \to
\]
\beq
\sum_{ij}\tilde{H}(p_j,x_i) U(x_i)
V(p_j) dq_i dp_j
\label{cpi1}
\eeq
with a similar representation for the transfer matrix
\[
T := e^{-iH\Delta t}=\int \tilde{T}(x,p) U(x) V(p) dxdp \to
\]
\beq
\sum_{ij}\tilde{T}(x_i,p_j) U(x_i)V(p_j) dx_i dp_j
\label{cpi2}
\eeq

where
\beq
\langle p \vert T \vert x \rangle = \tilde{T}(p,x)\langle p \vert x \rangle
\qquad
\mbox{and}
\qquad
\langle p \vert H \vert x \rangle = \tilde{H}(p,x)\langle p \vert x \rangle .
\label{cpi3}
\eeq
Here the quantities with the ``tildes'' are matrix elements of operators
in a mixed basis 
where the canonical
commutation relations are used to order the momentum operator to the left of the coordinate operators.

The probability amplitude can be expressed in terms of transfer matrices 
\beq
\langle x_f \vert e^{-iHt} \vert x_i \rangle
\label{cpi4}
\eeq
is
\beq
\langle x_f \vert e^{-iHt} \vert x_i \rangle =
\langle x_f \vert [e^{-iHt/N}]^N \vert x_i \rangle =
\langle x_f \vert T^N \vert x_i \rangle .
\label{cpi5}
\eeq
Inserting intermediate states gives
\[
\langle x_f \vert e^{-iHt} \vert x_i \rangle =
\]
\[
\int \prod_{i} dp_{i} dx_{i} \langle x_f \vert p_N \rangle \langle p_N \vert
T \vert x_N \rangle  \langle x_N \vert p_{N-1} \rangle \langle p_{N-1} \vert T \vert x_{N-1} \rangle \cdots
\]
\[
\langle x_3 \vert p_2 \rangle \langle p_2 \vert T \vert x_2 \rangle  \langle x_2 \vert p_{1} \rangle \langle p_{1} \vert T \vert x_{1} \rangle \langle x_1 \vert x_i \rangle = 
 \]
\[
\int \prod_{i} dp_{i} dx_{i} \langle x_f \vert p_N \rangle \langle p_N \vert x_N \rangle \tilde{T}(p_N,x_N) \langle x_N \vert p_{N-1} \rangle
\times
\]
\[
\langle p_{N-1} \vert q_{N-1} \rangle  \tilde{T}(p_{N-1},x_{N-1})\cdots
\]
\beq
\langle x_3 \vert p_2 \rangle \langle p_2  \vert x_2 \rangle \tilde{T}(p_2,x_2) \langle x_2 \vert p_{1} \rangle \langle p_{1} \vert x_{1} \rangle \tilde{T}(p_1,x_1) \langle x_1 \vert x_i \rangle .
\label{cpi6}
\eeq
This is exact; the Trotter limit justifies the replacement
\beq
\tilde{T}(p_i,x_i) \to e^{-i \tilde{H}(p_i,x_i) t/N}
\label{cpi7}
\eeq
as $N \to \infty$.

Completeness implies
\[
P(x_f;p_N,x_N, \cdots , p_1,x_1):=
\]
\[
\langle x_f \vert p_N \rangle
\langle p_N \vert x_N \rangle \langle x_N \vert p_{N-1} \rangle
\langle p_{N-1} \vert x_{N-1} \rangle \cdots
\]
\beq
\langle x_3 \vert p_2 \rangle \langle p_2 \vert x_2 \rangle  \langle x_2 \vert p_{1} \rangle \langle p_{1} \vert x_{1} \rangle
\label{cpi8}
\eeq
satisfies
\beq
\int \prod_{i=1}^N dp_i dx_i P(x_f;p_N,x_N, \cdots , p_1,x_1) =1
\label{cpi9}
\eeq
independent of $N$. With the replacement (\ref{cpi8}) equation (\ref{cpi6}) becomes
\[
\langle x_f \vert e^{-iHt} \vert x_i \rangle =
\]
\[
\lim_{N\to \infty} \int \prod_{i} dp_{i} dx_{i}
P(x_f;p_N,x_N, \cdots , p_1,x_1)\times
\]
\beq
e^{-i \sum \tilde{H}(p_j,x_j) t/N}\delta (x_1-x_i)
\label{cpi10}
\eeq
where the limit is interpreted as a strong limit;
the initial coordinate must be multiplied by a wave packet and integrated.
The complex probability
interpretation follows by considering the integrals as Henstock integrals,
which are limits of generalized Riemann sums. Cylinder sets are
defined by sets of paths that go through a generalized Riemann interval at each time step.
The complex probability
admits a
factorization 
\[
\prod_{i=1}^N dp_i dx_i
P(x_f;p_N,x_N, \cdots , p_1,x_1):=
\]
\[
P(x_f;x_N,p_N)dx_N dp_N 
P(x_N;x_{N-1},p_{N-1})dx_{N-1} dp_{N-1} 
\cdots
\]
\beq
P(x_2;x_{1},p_{1})dx_{1} dp_{1} 
\label{cpi11}
\eeq
which expresses the complex probability as a product of one step
probabilities for each time step.  Using this factorization gives
\[
\langle x_f \vert e^{-iHt} \vert x_i \rangle =
\]
\[
\lim_{N\to \infty} \int \prod_{i} dp_{i} dx_{i} P(x_{i+1};p_i,x_i,)e^{-i \tilde{H}(p_i,x_i) t/N}\delta (x_1-x_i),
\]
\beq
x_{N+1}=x_f
\label{cpi12}
\eeq

In the discrete approximation of section VII the integrals are
replaced by sums over the discrete values of $p_m= m\epsilon$,
$x_m=m\epsilon$, $dp_m=\epsilon=dp$ and $dx_m=\epsilon=dx$ where $-K
\leq m \leq K$, $M=2K+1$ and $\epsilon^2 = {2 \pi \over M}$ .
Equation (\ref{cpi9}) is still satisfied independent of the number $M$
of discrete values of $p_m$, $x_m$.  In the discrete case a cylinder
set is the set of paths that take on specific discrete values of $p=n\epsilon$
and $q=m\epsilon$ at each of the $N$ intermediate time steps. In that case the
complex probability for phase space paths becomes
\[
P(x_f;p_N,x_N, \cdots ,
p_1,x_1)\prod_i dp_idx_i  \to
\]
\[
P(x_f;p_{Ni_N},x_{Nj_N}, \cdots , p_{1i_1},x_{1j_1})\epsilon^{2N}:=
\]
\[
\langle x_f \vert p_{Ni_N} \rangle \epsilon
\langle p_{Ni_N} \vert x_{Nj_N} \rangle \epsilon\langle x_{Nj_N} \vert p_{N-1,i_{N-1}}
\rangle \epsilon
\langle p_{N-1i_{N-1}} \vert x_{N-1 j_{N-1}} \rangle \epsilon \cdots
\]
\beq
\epsilon
\langle x_{3j_3} \vert p_{2i_2} \rangle
\epsilon
\langle p_{2i_2} \vert x_{2j_2} \rangle \epsilon  \langle x_{2j_2} \vert p_{1i_1} \rangle
\epsilon\langle p_{1i_1} \vert x_{1j_1} \rangle \epsilon.
\label{cpi13}
\eeq
Here these indices represent the discrete momenta and coordinates that define a path
in phase space.
This has the property that the sum over all of the $M^N$ cylinder sets for
$N$ intermediate time steps gives 1 independent of $x_f$. In this case there
are cylinder sets of paths in both the $p$ and $x$ variables. These are considered as
discrete approximations
to the phase space paths, $p(t),x(t)$.

In the Trotter limit the discrete transfer matrix and discrete
Hamiltonian (see (\ref{cpi1})),
\beq
H= \sum \tilde{H}(p_m,x_n)U(x_n)V(p_m)
\label{cpi14a}
\eeq
\beq
T = \sum \tilde{T}(p_m,x_n)U(x_n)V(p_m)
\label{cpi14b}
\eeq
are related by
\beq
\tilde{T}(p_m,x_n) \approx e^{-i  \tilde{H}(p_m,x_n) \Delta t},
\label{cpi15}
\eeq
where this approximation preserves unitarity.

In this case 
\[
\langle
x_f \vert e^{-i H t} \vert  x_i \rangle =
\]
\[
\lim_{N\to \infty} \sum
P(x_f;p_{Ni_N},x_{Nj_N}, \cdots , p_{1i_1},x_{1j_1}) \times
\]
\beq
e^{-i \sum \tilde{H}( p_{1i_k},x_{1j_k})t/N} \delta_{x_1,x_i}
\label{cpi16}
\eeq
which expresses time evolution as the expectation of a complex probability on
cylinder sets of paths in phase space. As in the continuous case the
complex probability factors into a product of $N$ one step complex
probabilities:
\[
P(x_f;p_{=Ni_N},x_{Nj_N}, \cdots , p_{1i_1},x_{1j_1})=
\]
\beq
\prod_k [\langle x_{k+1i_{k+1}} \vert p_{ki_k} \rangle \epsilon 
\langle p_{ki_k} \vert x_{ki_k} \rangle \epsilon] \qquad x_{N+1}=x_f
\label{cpi17}
\eeq
and the probability amplitude for a transition from $x_i$ to $x_f$ after time $t$ becomes
\[
\langle
x_f \vert e^{-i H t} \vert  x_i \rangle =
\]
\beq
\lim_{N\to \infty}
\sum
\prod_k [\langle x_{k+1i_{k+1}} \vert p_{ki_k} \rangle \epsilon 
\langle p_{ki_k} \vert x_{ki_k} \rangle \epsilon e^{-i \tilde{H}(p_{ki_k} x_{ki_k})t/N}]
\delta_{x_1,x_i}
 \label{cpi18}
\eeq
where the sum is over cylinder sets.
This reduces the calculation to computing powers of an $M \times M$ matrix

When $H$ has the form 
\beq
H(p,x) = K(p) + V(x) \qquad K(0) =0
\label{cpi19}
\eeq
equation  (\ref{cpi18})  becomes
\[
\langle
x_f \vert e^{-i H t} \vert  x_i \rangle =
\]
\[
\lim_{N\to \infty}
\sum
\prod_k [\langle x_{k+1i_{k+1}} \vert p_{ki_k} \rangle \epsilon e^{-i \tilde{K}(p_{ki_k}) t/N}\langle p_{ki_k} \vert x_{ki_k} \rangle \times
\]
\beq
e^{-i \tilde{V}(x_{ki_k} t/N}\epsilon ]\delta_{q_1,q_i}
\label{cpi20}
\eeq
Since everything is finite the sums over the $p$ values can be computed
first, defining 
\[
P_X(x_f;x_{Ni_{N}},x_{N-1i_{N-1}}\cdots x_{1i_1}):=
\]
\beq
\sum_{p}
\prod_k [\langle x_{k+1i_{k+1}} \vert p_{ki_k} \rangle e^{-i \tilde{K}(p_{ki_k}) t/N}
\epsilon \langle p_{ki_k} \vert x_{ki_k} \rangle \epsilon 
\label{cp21}
\eeq
which is complex valued function on cylinder sets of paths.  It follows from
(\ref{f21}), that summing over
all of the cylinder sets, starting with the right most index and working to the left gives
\beq
\sum_{x}    P_X(x_f;x_{Ni_{N}},x_{N-1i_{N-1}}\cdots x_{1i_1})=
e^{-iK(0)t} =1.
\label{cpi22}
\eeq

With this definition
\[
\langle
x_f \vert e^{-i H t} \vert  x_i \rangle =
\]
\beq
\lim_{N\to \infty} \sum_q
P_X(x_f,x_{Nj_N}, \cdots ,x_{1j_1})
e^{-i \sum \tilde{V}(x_{1j_k})t/N} \delta_{x_1,x_i}.
\label{cpi23}
\eeq
As in the general case the complex probability factors into products of
conditional probabilities associated with each time step:
\beq
P_X(x_f;x_{Nj_N}, \cdots ,x_{1j_1})= \prod_k
P_X(x_{k+1i_{k+1}};x_{ki_k})
\label{cpi24}
\eeq
where
\[
P_X(x_{k+1i_{k+1}};x_{ki_k})=
\]
\beq
\sum_{p_ki_k} \langle x_{k+1i_{k+1}} \vert p_{ki_k} \rangle e^{-i \tilde{K}(p_{ki_k}) t/N}\epsilon
\langle p_{ki_k} \vert x_{ki_k} \rangle \epsilon
\label{cpi25}
\eeq
which gives the following expression for the probability amplitude
\[
\langle
x_f \vert e^{-i H t} \vert  x_i \rangle =
\]
\beq
\lim_{N\to \infty}
\sum
\prod_k [P_X(x_{k+1i_{k+1}};x_{ki_k})e^{-i \tilde{V}(x_{ki_k}t/N}]
\delta_{x_1 x_i}
\label{cpi26}
\eeq
This expresses the transition amplitude as the $N^{th}$ power of a
$M \times M$ matrix.

Note that treating an infinite dimensional system as the limit of
finite dimensional systems rather than discretizing the infinite
dimensional system has some advantages.  Discretizing was used in
\cite{polyzou} where the rate of convergence was sensitive to how the
points used to evaluate the interaction in each interval were chosen.
The factorization of the complex probability into products of
conditional probabilities becomes approximate upon
discretization. These choices could in principle affect unitarity
numerically.  These issues do not appear in the discrete case.  In
addition, the discrete analogs of the Fresnel integrals are
well-defined finite sums.  The complex probability interpretation arises
naturally from the completeness relation.

 
\section{Summary and conclusion}\label{sum}

This paper discusses an interpretation of the Feynman path integral
for finite dimensional systems as the expectation value of a
functional of paths with respect to a complex probability on cylinder
sets of paths.  This interpretation follows from the completeness
relation and the Trotter product formula.  While the passage from
imaginary time to real time is problematical for an interpretation of
the path integral an integral, replacing the real-time integral by the
expectation value of a functional on cylinder sets of paths with
respect to a complex probability agrees with the Trotter limit.
%In the finite dimensional case the complex probability
%arises naturally by using the completeness relation at each time slice
%in the Trotter product.
In the ``phase space'' form, based on Schwinger's
\cite{schwinger} discrete Weyl representation, it can be applied to any
Hamiltonian matrix.  Infinite dimensional systems can be approximated as
limits of finite dimensional systems. 

The complex probability approach provides an alternative way of
thinking about path integrals.  It is difficult to interpret the
meaning of a discrete-valued function of a continuous time variable.
One possible interpretation follows by noting that a sequence of
discrete eigenvalues at different time steps are labels for transition
amplitudes from the state at a given time step to another state at the
next time step.  All possible transition amplitudes are allowed at
intermediate times between these successive time steps.  In this sense
a cylinder set of paths can be thought of as an ordered sequence of
transition amplitudes between specific states at a sequence of time
steps.  The eigenvalues that label these amplitudes are the quantities
are normally associated with a classical path.  The path integral is a
complex weighted average over all ordered sequences of transition
amplitudes.  This provides a purely quantum mechanical interpretation
of the path integral.  This interpretation is natural when the complex
probabilities are derived from the completeness relations.

The discrete representation has an equivalent qubit representation in
terms of an irreducible set of elementary qubit gates.  This
representation may be more efficient for computational purposes since
the complementary operators, $U_i$ and $V_i$ act on single qubits.

Discrete approximations to infinite dimensional systems are discussed
in Appendix I and Appendix II.  These applications demonstrate how
this discrete formulation of the path integral can be applied to solve
problems in potential scattering and quantum field theory.  The path
integral approach is not the most efficient way to perform scattering
calculations, but the simple calculation presented in Appendix I, based
time-dependent scattering theory, 
illustrates some of the issues need to be considered in future
treatments of scattering formulated in terms of quantum computers.

In contrast to scattering problems, path integrals are one of the more
direct methods to solve interacting field theories.  The illustrated
application to field theory in Appendix II uses a basis of wavelets to
replace the fields by an infinite collection of almost local
operators.  Volume and resolution truncations are used to replace the
field by a finite number of discrete modes.  The calculations
presented in Appendix II use a severe truncation of a $\phi(x)^4$
field theory to two modes, but the calculation was completely
non-perturbative.  Realistic calculations are still a long way off.

%\ack{{This work supported by the U.S. Department of Energy,
%  Office of Science, Grant \#DE-SC16457,
%The author would like to thank William Hester for pointing out some errors
%in the original version of this manuscript.
%}

\section{Appendix I: Scattering in the discrete representation}\label{scatt}

Formal scattering theory is an idealization.  A real scattering experiment
takes place in a finite volume during a finite time interval.  The relevant
physics is dominated by a finite number of degrees of freedom that are
limited by the energy and scattering volume.

The fundamental quantum mechanical observable is the probability
for a transition from a prepared initial state to a detected final state
\beq
P_{fi} = \vert \langle \psi_f (t)\vert \psi_i (t) \rangle \vert^2.
\label{sca1}
\eeq
While the individual states depend on time, the probability
(\ref{sca1}) is independent of $t$ due to the unitarity of the time
evolution operator.  The important constraint is that both states have to be
evaluated at the same time.  The problem of scattering theory is that
there is no common time when both the initial and final states are
simple.  On the other hand the initial state is simple before the
collision and the final state is simple after the collision.

The initial and final states at the time of collision can be
determined by evolving them from times where they behave like
non-interacting subsystems to the collision time.  Since localized
wave packets spread, the effects of spreading can be eliminated by
starting with localized wave packets at the collision time,
evolving them beyond the range of
interactions using free time evolution, and then evolving them back to the
interaction region using the full Hamiltonian.  The result is a
unitary mapping that transforms the free wave packet at the collision
time to the dynamical wave packet at the same time.

If $U_0(t)$ and $U(t)$ represent the free and dynamical unitary
time evolution operators,
then assuming the time of collision is approximately at time $t=0$ the scattering
asymptotic conditions have the form
\beq
\Vert  U(\pm \tau)\vert \psi_{\pm}(0) \rangle -
U_0(\pm \tau) \vert  \psi_{0\pm}(0) \rangle \Vert 
\approx 0
\label{sca2}
\eeq
where the time $\tau$ is sufficiently large for the interacting particles to be
separated beyond the range of their mutual interactions.  This
expression is independent of $\tau$ for sufficiently large $\tau$, but
the minimum value of $\tau$ depends on the range of the interaction
and the structure of $\vert \psi_{0\pm}(0) \rangle$.  Normally dependence
on these conditions is removed by taking the limit $\tau \to
\infty$.  In this work, for computational reasons, it is desirable to
choose $\tau$ as small as possible, which requires paying attention to
the range of the interaction and the structure of the initial and
final states.

The unitarity of the time evolution operator means that (\ref{sca2})
can be replaced by
\beq
\Vert  \psi_{\pm} (0)\rangle -
U(\mp \tau)U_0(\pm \tau) \vert \psi_{0\pm}(0) \rangle \Vert 
\approx 0 
\label{sca3}
\eeq
for sufficiently large $\tau$.
The operators
\beq
\Omega_{\pm}(\tau) := U(\pm \tau)U_0(\mp \tau)
\label{sca4}
\eeq  
are unitary mappings from $\vert \psi_{0\pm} (0)\rangle$ to
$\vert \psi_{\pm}(0) \rangle$.

Using these definitions the scattering probability can be expressed as
\beq
P_{fi} = \vert \langle \psi_{+}(0)\vert \psi_{-}(0) \rangle \vert^2
=
\vert \langle \psi_{0+}(0)\vert  S(\tau) \vert \psi_{0-}(0) \rangle \vert^2 \label{sca5}
\eeq
where
\beq
S(\tau) := \Omega^{\dagger}(\tau)\Omega (-\tau)
\label{sca6}
\eeq
is the scattering operator.
Since $S(\tau)$ is unitary it can be expressed in terms of a self-adjoint
phase shift operator
\beq
S(\tau) = e^{2i \delta (\tau)}
\label{sca6}
\eeq
where $S(\tau)$ should be independent of $\tau$ for sufficiently
large $\tau$.

In a real experimental measurement the probability (\ref{sca5})
depends on the structure of the initial and final wave packets, which
cannot be precisely controlled by experiment.  If the matrix elements
of $S(\tau)$ in sharp-momentum states are slowly varying functions of
momentum, then the dependence on the wave packet factors out
\cite{brenig} and can be eliminated to compute differential cross
sections.  In this case the sharp-momentum matrix elements can be
approximated from the matrix elements using Gaussian (minimal
uncertainty) wave packets with a ``delta-function normalization'' that
are sharply peaked about the desired momenta.

%In the large $\tau$ limit sharp momentum matrix elements of $S(\tau)$
%exactly conserves energy; in a calculation using finite $\tau$ energy
%conservation is only approximate - so states should be chosen so they
%have the same mean energy.

This formulation of scattering admits a path integral
treatment.  As previously discussed scattering reactions are dominated
by a finite number of degrees of freedom.  The use of the discrete
Weyl representation has the advantage that unitarity is exactly
preserved on truncation to a finite number of degrees of freedom.
Alternative path integral treatments of scattering appear in
\cite{Campbell}\cite{Rosenfelder1}\cite{Rosenfelder2}.

The advantage of the discrete representation is that 
$U_0(-\tau)U(2\tau)U^0(-\tau)$ can be expressed as the limit of
products of the transfer matrices defined section 8
(see (\ref{cpi23})-(\ref{cpi25}))
\beq
S(\tau) = \lim_{N\to \infty} Y^{-N}X^{2N} Y^{-N}
\label{sca7}
\eeq
where
\beq
Y_{ij}= P_X(x_i,x_j,\Delta t)
\qquad
(X)_{ij}= P_X(x_i,x_j,\Delta t) e^{-i V(x_j)\Delta t},
\label{sca8}
\eeq
$\Delta t = \tau/N$ and $N$ is the number of Trotter time slices. 
Note also that
\beq
Y^{-N} = P_X(x_f,x_i,-N\Delta t).
\eeq

Sharp-momentum matrix elements of the scattering operator can be expressed in
terms of the matrix elements of the transition operator, $T$, which is
easier to calculate in the discrete representation
\beq
S= I - 2 \pi i \delta (E_f -E_i) T
\label{sca9}
\eeq
where $T$ is approximately given by  
\beq
T \approx V \Omega (-\tau)
\label{sca10}
\eeq
when evaluated in normalizable states with sharply peaked momenta.
The advantage of this representation is that for scattering problems
$V$ is a short range operator that provides a volume cutoff.

In the discrete representation sharp-momentum eigenstates are
normalizable however {\it they cannot be used in scattering calculations}
because they are completely delocalized in space since the
discrete momenta and coordinates are complementary
- making it impossible to get to the asymptotic region.

The most straightforward way to construct suitable initial or final wave packets
in the discrete representation is to approximate the corresponding
minimal uncertainty states of the continuum theory.  The quantities
to control are the mean position,  momentum and the uncertainty
in both of these quantities defined for a given state $\vert \psi \rangle$
by:
\beq
\langle x \rangle_{\psi} := \sum_{n=-K}^K {\langle \psi \vert u_n \rangle
  n \epsilon \langle u_n \vert \psi \rangle \over
  \langle \psi \vert \psi \rangle }
\eeq
\beq
\langle p \rangle_{\psi} := \sum_{n=-K}^K {\langle \psi \vert v_n \rangle
n \epsilon \langle v_n \vert \psi \rangle \over
  \langle \psi \vert \psi \rangle }
%\label{sca12}
\eeq
\[
(\Delta x)^2 =\langle \psi \vert (x-\langle x \rangle)^2 \vert \psi \rangle =
\]
\beq
\sum_{n=-K}^K {\langle \psi \vert u_n \rangle
((n \epsilon)^2 - \langle x \rangle^2 )\langle u_n \vert \psi \rangle \over
  \langle \psi \vert \psi \rangle }
\label{sca13}
\eeq
 \[
(\Delta p)^2 = \langle \psi \vert (p-\langle p \rangle)^2 \vert \psi \rangle=
\]
\beq
\sum_{n=-K}^K {\langle \psi \vert v_n \rangle
((n \epsilon)^2 - \langle p \rangle^2 )\langle v_n \vert \psi \rangle \over
\langle \psi \vert \psi \rangle }.
\label{sca14}
\eeq
The continuum delta-function normalized minimal uncertainty states are
\beq
\langle p \vert \psi_{0}  (0) \rangle =
{1 \over 2 \sqrt{\pi} \Delta p} e^{- {(p-\langle p \rangle )^2 \over 4 (\Delta p)^2}} .
\label{sca15}
\eeq
where $\langle p \rangle$ is the mean momentum and
$\Delta p$ is the quantum mechanical uncertainty in
$p$ for this wave packet.  This wave packet needs to be evolved to
$-\tau$ using the free time evolution which adds a phase to (\ref{sca15}): 
\beq
\langle p \vert \psi_{0} (-\tau) \rangle =
{1 \over 2 \sqrt{\pi} \Delta p} e^{ - {(p-p_i)^2 \over 4 (\Delta p)^2}
  + i {p^2 \over 2 \mu} \tau}.
\label{sac16}
\eeq
In the discrete ``$p$'' representation this is replaced by 
\beq
\langle n \vert \psi_{0}(-\tau) \rangle = C e^{-{(\epsilon n - \langle p \rangle)^2 \over
4 (\Delta p)^2} + i {n^2 \epsilon^2 \over 2 \mu} \tau} .
\label{sac17}
\eeq
where $C$ is a normalization constant.
In the $x$ representation this becomes
\beq
\langle m \vert \psi_0 (-\tau) \rangle = {\epsilon \over \sqrt{2\pi}}
\sum_{n=-K}^K e^{i \epsilon^2 mn}
\langle n \vert \psi_{0}(-\tau) \rangle .
\label{sac18}
\eeq

To illustrate that this gives a good approximation to the continuum
results  $\langle p \rangle$,
$\langle x \rangle$, $\Delta p$ and $\Delta x$ were calculated starting
with $\langle p \rangle=2.5$, $\Delta p=.25$ and $K=300$ as input parameters in
(\ref{sac16}).  The results of the calculation
\begin{eqnarray*}
\mbox{mean}_{p-calc}&amp;=2.500000\\
\mbox{mean}_{x-calc}&amp;=0.000000\\
%\mbox{mean}_{x-calc}&amp;=-2.51 \times 10^{-17}\\
\Delta_{p-calc}&amp;=.3000000\\
\Delta_{x-calc}&amp;= 1.666667
\end{eqnarray*}
are consistent with the input parameters,  the minimal uncertainty condition,
$\Delta p \Delta x = 1/2$ 
and the continuum results.

With these states the sharp-momentum half-shell transition matrix elements are
\beq
\langle p_f \vert T(E_i) \vert p_i \rangle \approx
\langle \psi_{0f}(0) \vert V X^N  \vert \psi_{0i}(-\tau)\rangle.
\eeq
As a test
the discrete approximation
was applied to the problem of one-dimensional scattering of
particle of mass $m$ by a repulsive Gaussian potential of the form
\beq
V(q) = \lambda e^{-\alpha q^2}
\eeq
with $\lambda=.5$ and $\alpha =2.0$.
The potential is plotted in figure 1.  The particle's mass is taken to
be 1 in dimensionless units so the velocity and momentum can be
identified.  The initial wave packet is a Gaussian with a delta
function normalization in momentum space with mean momentum $p=2.5$
and width $\Delta p= .25 $. It is pictured in figure 2.  The Fourier
transform of the initial wave packet is given in figure 3.  The oscillations
are because the momentum space wave packet has a non-zero mean momentum.  Given the
size of the potential and wave packets, the wave packet needs to move
about 18 units to the left in order to be out of the range of the
potential.  This suggest that for $v=p/m=2.5$ that $\tau=7$ should
be sufficient to move the wave packet out of the range of the
potential.  The resulting
free wave packet at $\tau=-7$ is shown in figure 4.  The scattered
wave function with $K=300$ ($M=601$) after $N=100$ time steps is shown
in figure 5,
and that
result multiplied by the potential is shown in figure 6.
Compared to the wave function in figure 3, the wave function in figure 5
includes the effects of the interaction.
Figure 6 illustrates the cutoff due to the short range potential; it
shows how only the part of the wave function inside the range of the
interaction contributes to the scattering operator.
Figure 7 compares the result of the off-shell Born approximation $\langle p
\vert V \vert \psi(0) \rangle$ to the calculation of the real and
imaginary parts of $\langle p \vert T \vert \psi(0) \rangle$ while
figure 8 compares $\langle p \vert T \vert \psi(0) \rangle$ to
$\langle p \vert T (p_0)\vert p_0 \rangle$ obtained by numerically
solving the Lippmann-Schwinger equation using the method \cite{Rubtsova}.

Figure 8 shows that the path integral computation with an initial wave
packet with a width of 1/10 of the momentum converges to the numerical
solution of the integral equation.  In unrelated time-dependent
scattering calculations \cite{Kopp:2011vv} a $\Delta p$ of about a tenth of
$p$ gave good approximations to sharp momentum matrix elements of the
transition operator for a wide range of momenta.

Unlike the solution of
the Lippmann Schwinger equation, in the path integral approach for
each energy it is necessary to determine minimal values of
$M$,$N$,$\tau$ and $\Delta p$ that are needed for convergence.
In practice there are a number of trade offs.  Making
the wave packets narrow in momentum increases the scattering volume
in the coordinate representation.  This in turn requires a larger $\tau$
to get out of the range of the potential.  If $\tau$ gets too large
the wave packet can move past $q_{max}=K\epsilon$ and will reappear
at $q_{min}=-K\epsilon$.  As $p$ gets large the oscillations in the
$q$ space wave function have higher frequencies, which requires
smaller time steps, while when $p$ gets small it is necessary to
make the wave packet width in momentum small enough so the
coordinate space tail of the wave function
gets out of the interaction volume.

The computations require storing the initial vector.  It is not
necessary to store the one-step transfer matrix - it can be computed
efficiently on the fly.  This is important for realistic calculations
since the vectors will be significantly larger in higher dimensions.
The hope is that in the future qubits can be used to represent large
vectors.

This one-dimensional example approximated half-shell
sharp-momentum transition matrix elements.  The on-shell values can be
used to extract other observables such as phase shifts and in the
one-dimensional case transmission and reflection coefficients.  This
formulation of the one-dimensional problem in terms of transition
matrix elements has the advantage that the method can be formally
be used in higher dimensions and to treat complex reactions or scattering
in quantum field theory.

The formulation of the discrete path integral used the
discrete Schwinger representation based on a single pair of
complementary operators where the complex one time step
probability is represented by a dense matrix.  An equivalent
representation in terms of qubits involves tensor products of
matrices (\ref{qb7}-\ref{qb8}) that act on single qubits, which may
have computational advantages.

One observation from these calculations that will have an impact on
future computations using quantum computers is that width of the wave
packets, total time for scattering, the scattering energy, the number
of time steps and the resolution of the discretization all have to be
considered together in order ensure an efficient calculation.  This is
because each one of these approximations generates its own source of
errors that impact the errors in the other approximations.

The general strategy discussed above can also principle be utilized to
formulate scattering quantum field theory.  The Haag-Ruelle
formulation of scattering
\cite{Haag:1958vt}\cite{Brenig:1959}\cite{Ruelle:1962}\cite{jost}
is the natural field-theoretic generalization of the quantum
mechanical treatment of time dependent scattering.  Like ordinary
quantum mechanical scattering it uses strong limits and needs one-body
(i.e. bound state) solutions to formulate the scattering asymptotic
conditions.

\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.4]{fig_1.pdf}
\caption{\bf Potential}
\label{fig:1}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.4]{fig_2.pdf}
\caption{\bf Momentum space initial Gaussian wave packet}
\label{fig:2}
\end{minipage}
\end{figure}

\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.4]{fig_3.pdf}
\caption{\bf Coordinate space initial Gaussian wave packet}
\label{fig:3}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.4]{fig_4.pdf}
\caption{\bf Free Gaussian wave packet at $\tau=-4$}
\label{fig:4}
\end{minipage}
\end{figure}

\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.4]{fig_5.pdf}
\caption{\bf Initial scattering state at $t=0$ }
\label{fig:5}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.4]{fig_6.pdf}
\caption{\bf V $\times$ initial scattering state at $t=0$}
\label{fig:6}
\end{minipage}
\end{figure}


\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.4]{fig_7.pdf}
\caption{\bf $\langle p \vert V \vert \psi_{0i}(0) \rangle$}
\label{fig:7}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.4]{fig_8.pdf}
\caption{\bf $\langle p \vert T \vert \psi_{0i}(0) \rangle$}
\label{fig:8}
\end{minipage}
\end{figure}

\bigskip\bigskip

\section{Appendix: Discrete multi-resolution representation of quantum field theory}\label{qft}

One motivation for studying quantum computing in physics is that it
might provide a framework for a numerical treatment of problems in
strongly interacting quantum field theory.  Clearly this goal is a
long way off for realistic theories, but the state of quantum
computing is advancing rapidly.  Discrete formulations of field theory
naturally fit into the discrete framework discussed in this work and
should be relevant for future applications.  Discrete truncations to a
small number of degrees of freedom can be used as a laboratory to
explore how different field modes interact in a more realistic
truncation of the theory.

A numerical treatment of quantum field theory requires a truncation to
a system with a finite number of degrees of freedom.  For reactions
that take place in a finite space-time volume and involve a finite
energy it is natural to limit the number of degrees of freedom by
making volume and resolution truncations.  Degrees of freedom that are
outside of this volume or energetically inaccessible due to their
resolution are expected to be unimportant for the given reaction.
Daubechies wavelets
\cite{daubechies}\cite{jorgensen1}\cite{jorgensen2} and scaling
functions are a basis of square integrable functions and a natural
representation to perform both kinds of truncations.  The basis
consists of a complete orthonormal set of functions that have compact
support and a limited amount of smoothness.  They have the property
that in any small volume there are an infinite number of basis
functions supported entirely in that volume.  This means that they can
be used to construct ``local'' observables by smearing the fields with
basis functions.  All of the basis functions $\xi_n(x)$ are generated
from the solution of a linear renormalization group equation by
translations and dyadic scale transformations, which facilitates
computations.  Because they are complete they can be used to {\it
exactly} expand the canonical fields
\[
\Phi (\mathbf{x},t) = \sum \Phi_n (t) \xi_n (\mathbf{x})
\]
\[
%\qquad
\Pi (\mathbf{x},t) = \sum \Pi_n (t) \xi_n (\mathbf{x})
\]
where
\[
\Phi_n (t) := \int d\mathbf{x}\Phi (\mathbf{x},t)\xi_n (\mathbf{x}) 
\]
and
\[
\Pi_n (t) := \int d\mathbf{x}\Pi (\mathbf{x},t)\xi_n (\mathbf{x}) 
\]
are discrete field operators.
If the fields satisfy canonical equal-time commutation relations
\beq
[\Phi (\mathbf{x},t), \Pi(\mathbf{y},t)] = i \delta (\mathbf{x}-\mathbf{y})
\eeq
then the discrete fields $\Phi_n$ and $\Pi_n$ will satisfy discrete
versions of the canonical equal time commutation relations
\cite{Bulut:2013bg} \cite{Polyzou:2017wnj}
\cite{Polyzou:2020ifj}: 
\[
[\Phi_m(t),\Pi_n(t)]=i \delta_{mn} 
\]
\[
[\Phi_m(t),\Phi_n(t)] =0
\]
\beq
[\Pi_m(t), \Pi_n(t)] =0 .
\label{w1}
\eeq
In terms of these degrees of freedom the Hamiltonian
for a $\phi^4$ theory has the form
\[
H= {1 \over 2}\sum_n \Pi_n\Pi_n + {m^2 \over 2}\sum_n \Phi_n\Phi_n +
\sum_{mn} D_{mn} \Phi_m \Phi_n +
\]
\beq
\lambda
\sum_{klmn} \Gamma_{klmn} \Phi_k\Phi_l\Phi_m\Phi_n
\label{wav2}
\eeq
where the sum are all infinite.  Since $H$ commutes with itself the
discrete fields in $H$ can be evaluated at $t=0$.
The matrices in (\ref{wav2}) are constants defined
by the integrals
\beq
D_{mn} = 
{1 \over 2} \int \pmb{\nabla} \xi_n (\mathbf{x})\cdot
\pmb{\nabla} \xi_m (\mathbf{x})d\mathbf{x} 
\label{wav3}
\eeq
\beq
\Gamma_{klmn}= \int
\xi_k (\mathbf{x}) \xi_l (\mathbf{x}) \xi_m (\mathbf{x}) \xi_n (\mathbf{x})
d\mathbf{x} .
\label{wav4}
\eeq
For the wavelet basis
these constants vanish unless all of the functions appearing in the
integrals have a common support, which makes them almost local.  In
addition, because all of the functions in the integrand are related by
translations and scale transformations to a single function, the
integrals can all be expressed as linear combinations of solutions of
some small linear systems generated by a renormalization group
equation (see (\ref{w3})).  Unlike a lattice truncation,
the wavelet representation
of the field theory is (formally) exact (before truncation).  The
basis functions regularize the fields so local products of fields that
appear in the Hamiltonian are replaced by infinite sums of
well-defined products of discrete field operators.  The basis functions are
differentiable, so there are no finite difference approximations.

Wavelet representations of quantum field theories have been discussed
by a number of authors
\cite{best:1994}
\cite{federbush:1995}
\cite{1995NuPhB.436..414H}
\cite{Battle:1999}
\cite{best:2000}
\cite{Ismail1:2003}
\cite{Ismail2:2003}
\cite{altaisky:2007}
\cite{albeverio:2009}
\cite{altaisky:2010}
\cite{altaisky:2010}
\cite{altaisky:2013}
\cite{Bulut:2013bg}
\cite{altaisky:2013b}
\cite{PhysRevA.92.032315}
\cite{PhysRevLett.116.140403}
\cite{altaisky:2016b}
\cite{altaisky:2016c}
\cite{altaisky:2016}
\cite{altaisky:2017}
\cite{Polyzou:2017wnj}
\cite{Neuberger2018}
\cite{Tomboulis1}
\cite{Polyzou:2020ifj}
\cite{Altaisky:2021hbq}
\cite{brennan3}.
What is relevant is that the Hamiltonian (\ref{wav2}) has the form
$\tilde{H} = \tilde{H}_1(p)+\tilde{H}_2(x)\to  \tilde{H}_1(\Pi)+\tilde{H}_2(\Phi)$,
except it involves an infinite number of degrees of
freedom.  It is diagonal and quadratic in the discrete momentum
operators and has a non-trivial (almost local) dependence on the
$\Phi_n$ operators.
%Because all of the basis functions are constructed
%from the fixed point $s(x)$ of the renormalization group equation
%(\ref{w3}) the constant quantities
%$D_{mn}$ and $\Gamma_{klmn}$ can be expressed in terms of a finite
%set of elementary integrals which are solutions of small linear systems.

The advantage of this basis is that it has natural volume and
resolution truncations.  For reactions taking place in a finite volume
with a finite energy a finite number of these degrees of
freedom should provide a good approximation.  This reduces the problem
to a problem with a finite number of discrete degrees of freedom.  In
addition the truncated Hamiltonian still has the form (\ref{wav2}),
except the sums are only over the retained discrete modes. As the volume and
resolution are increased (i.e as more modes are added) the parameters of
the theory have to be adjusted to keep the some physical observables
constant.  

The truncated problem is a finite number of degree of freedom
generalization of the one degree of freedom problem discussed in the
section \ref{scatt}.
%For a quantum field theory the vector
%representing the state of the field will be much larger than in the
%one degree of freedom scattering case.

The construction of the wavelet basis used to construct the
discrete representation of the Hamiltonian (\ref{wav2}) is outlined below.
The starting point the solution of the linear renormalization group
equation
\beq
s(x) = \sum_{l=0}^{2L-1}h_l D T^l s(x)
\label{w3}
\eeq
%\label{wav.1}
where
\beq
Df(x):= \sqrt{2}f(2x) \qquad \mbox{and} \qquad Tf(x) := f(x-1)
%\label{wav.2}
\label{w4}
\eeq
are unitary discrete dyadic scale transformations and unit
translations.  The $h_l$ are constants that depend on the choice of
$L$. Generally as $L$ increases the solutions, $s(x)$, become smoother
but the support increases.  A useful case is $L=3$ where the
solution $s(x)$ of (\ref{w3}), called the scaling function, has
support on $[0,2L-1]=[0,5]$ and has one continuous derivative.  In
that case the coefficients $h_l$ for the Daubechies $L=3$ scaling
functions are
\[
h_0=(1+\sqrt{10}+\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}
\]
\[
h_1=(5+\sqrt{10}+3\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}
\]
\[
h_2=(10-2\sqrt{10}+2\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}
\]
\[
h_3= (10-2\sqrt{10}-2\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}
\]
\[
h_4=(5+\sqrt{10}-3\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}
\]
\beq
h_5 =(1+\sqrt{10}-\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2} . 
\label{w5}
\eeq
They are determined by the requirement that the solution of (\ref{w3}) and unit translates
are orthonormal and locally finite linear combinations of these
unit translates can
be used to locally pointwise represent degree 2
polynomials.
Given the solution, $s(x)$, of (\ref{w3})
new functions are constructed from $s(x)$
by rescaling and translating
\beq
s^k_n(x) := D^k T^n (x)s(x) = \sqrt{2^k} s(2^k x-n) .
%\label{wav.6}
\label{w6}
\eeq
Since (\ref{w3}) is homogeneous in $s(x)$
the starting scale can be fixed by requiring
\beq
\int s(x) dx=1 .
\label{w7}
\eeq
The functions $s^k_n(x)$ for fixed $k$ span a subspace of the square
integrable functions on the real line with a resolution $2^{-k}L$:
\beq
{\cal S}^k := \{ f(x) \vert f(x) = \sum_{n=-\infty}^\infty c_n s^k_n(x) \qquad
\sum_{n=-\infty}^\infty  \vert c_n\vert^2 &lt; \infty \}.
%\label{wav.5}
\label{w8}
\eeq
The renormalization group equation (\ref{w3}) implies
\beq
{\cal S}^k \subset {\cal S}^{k+1}.  
%\label{wav.11}
\label{w9}
\eeq
It follows that
\beq
{\cal S}^{k+1} = {\cal S}^{k} \oplus {\cal W}^{k}.    
%\label{wav.12}
\label{w10}
\eeq
where ${\cal W}^k$ is the orthogonal complement of
${\cal S}^k$ in ${\cal S}^{k+1}$.
An orthonormal basis for the subspace 
${\cal W}^k$ is the ``wavelet functions'':
\beq
w^k_n(x)=D^kT^n w(x)
\label{w11}
\eeq
where
\beq
w (x):= \sum_{l=0}^{2L-1} (-)^l h_{2L-1-l} D T^{l} s(x) 
%\label{wav.13}
\label{w12}
\eeq
is called the ``mother wavelet''.
This decomposition can be continued to generate a multi-resolution decomposition of $L^2(\mathbb{R})$
\beq
%\boxed{
L^2 (\mathbb{R}) = {\cal S}^{k} \oplus_{l=0}^\infty {\cal W}^{k+l}.
%=
%\oplus_{n=-\infty}^{\infty}   {\cal W}^{n}.
%\label{wav.17}
%}
\label{w13}
\eeq
This results in a multi-resolution  orthonormal basis for
$L^2 (\mathbb{R})$
\beq
\{\xi_n(x) \}_{n=-\infty}^{\infty} :=
\{ s^k_n(x)\}_{n=-\infty}^\infty \cup \{ w^m_n(x)\}_{n=-\infty,l=k}^\infty .  
%\label{wav.18}
\label{w14}
\eeq
For the choice $L=3$ the basis functions $s^k_n(x)$ and $w^k_n(x)$ have
compact support on
$[2^{-k}n,2^{-k}(n+5)]$.  All of the basis functions have one
continuous derivative so the coefficients (\ref{wav3}) are defined .  The functions $s^k_n(x)$ are like splines in
that linear combinations can be used to locally pointwise represent
degree 2 polynomials while the functions $w^l_n(x)$ are orthogonal to
the same polynomials on their support.  The Fourier transforms of the
basis functions are entire functions due to their compact support.
Orthonormal three dimensional basis functions are products of
one-dimensional basis functions.   In spite of these nice properties,
the basis functions are fractal valued (since they are related to fixed points
of a renormalization group equation) and cannot be written down in
closed form. 

In order to use this representation the constant coefficients 
$D_{mn}$ and $\Gamma_{n_1 \cdots n_k}$ that appear in the Hamiltonian (\ref{wav2})
need to be computed.
Using scale transformations (\ref{w4}) and the renormalization group
equation (\ref{w3}) they can all be expressed in terms of the integrals
\beq
d_{n} = 
\int {ds(x) \over dx} {ds(x-n) \over dx}dx \qquad -4\leq n \leq 4
\eeq
\beq
\gamma_{m,n,k}= \int s(x) s(x-m) s(x-n) s(x-k) dx \qquad -4 \leq m,n,k \leq 4 .
\eeq
These integrals for different values of $m,n,k$
are related to each other by finite linear equations
derived from the renormalization group equation (\ref{w3}) and the
scale fixing condition (\ref{w7}).  These linear systems can formally
be solved in terms of the coefficients $h_l$ (\ref{w5}).  The
coefficients $d_n$ are rational numbers and can be found in the
literature on wavelets \cite{beylkin1}.  To find the $\gamma_{mnk}$
requires finding eigenvalues of a $9^3 \times 9^3$ matrix.  This
eliminates the need be able to evaluate fractal valued functions.
Alternatively the integrals $\gamma_{mnk}$ can be approximated by
noting that the renormalization group equation (\ref{w3}) and the
scale fixing condition (\ref{w7}) can be used to exactly calculate the
basis functions and their derivatives at all dyadic rational points.
Since the functions and their derivatives are continuous and the
dyadic rationals are dense this can be used to estimate these
quantities and integrals involving these quantities to any desired
accuracy.

In order to illustrate a path integral treatment of this system
consider a truncation of the theory in 1+1 dimensions 
where only 2 adjacent modes of the Hamiltonian (\ref{wav2}) are retained.
In this case the overlap coefficients that appear in the Hamiltonian
are related to $d_n$ and $\gamma_{lmn}$ by
\beq
\qquad D_{mn} = d_{n-m} \qquad \mbox{and} \qquad 
\Gamma_{klmn} = \gamma_{l-k,m-k,n-k}
\eeq
The coefficients that couple adjacent modes can be expressed in
terms of the following quantities
\begin{eqnarray*}
\Gamma_{0000}&amp;= 0.9528539 \\
\Gamma_{0001}&amp;= 0.0670946 \\
\Gamma_{0011}&amp;= 0.0890895 \\
\Gamma_{0111}&amp;=-0.1424536 \\
D_{00}&amp;=295./56. \\
D_{01}&amp;=-356./105. \\
D_{10}&amp;=D_{01}\\
D_{11}&amp;=D_{00}.   
\end{eqnarray*}
where the $\Gamma$ coefficients were computed by numerical integration
using the trapezoidal rule with the basis functions evaluated
at 256 dyadic points on their support.  Convergence was verified 
using 512 dyadic points. 

The truncated Hamiltonian in this case is
\[
H= {1 \over 2}\sum_{n=0}^1 \Pi_n\Pi_n + {m^2 \over 2}\sum_{n=0}^1 \Phi_n\Phi_n
+\sum_{m,n=0}^1 D_{mn} \Phi_m \Phi_n +
\]
\beq
\lambda
\sum_{k,l,m,n=0}^1 \Gamma_{klmn} \Phi_k\Phi_l\Phi_m\Phi_n
\label{wavxx}
\eeq
where
$\Gamma_{0000}=\Gamma_{1111}$,  $\Gamma_{0001}=\Gamma_{0010}=\Gamma_{0100}=
\Gamma_{1000}$, etc..
The path integral treatment of the field theory in the discrete representation
is a multi-dimensional generalization of the treatment for one degree of freedom
where each field mode represents an independent degree of freedom.

A general numerical treatment involves a truncation and
renormalization
followed by two approximations.  The truncation discards all but a
finite number, $F$, of discrete degrees of freedom.
\beq
H \to H_F .
\label{w22}
\eeq
Ideally physics at a given energy scale and in a given volume should
be dominated by a finite number of accessible degrees of freedom.  The
remaining degrees of freedom that are not expected to impact
calculations at that given scale and volume are discarded.  The
truncated theory is renormalized by adjusting the parameters of the
theory so a set of observables agree with experiment.  This gives the
parameters a dependence on the choice of retained degrees of freedom.
This is a truncation rather than an approximation. It assumes that no
additional parameters need to be introduced beyond what appears in the
truncated Hamiltonian and that there is a limit as the volume becomes
infinite and resolution becomes arbitrarily small.  This is followed
by two approximations.  The first approximation is to approximate the
unitary time evolution operator for the truncated theory using the
Trotter product formula with $N$ time slices:
\[
U_F(t) = e^{-i H_F t} =
\]
\beq
\lim_{N\to \infty} (e^{-i H_F(\Pi)\Delta t}e^{-iH_F(\Phi)\Delta t})^N 
\label{w23}
\eeq 
where $\Delta t = t/N$ and
\beq
H_F = H_F(\Pi) + H_F(\Phi)
\label{w24}
\eeq
with
\beq
H_F(\Pi) := {1 \over 2}\sum_n \Pi_n\Pi_n
\label{w24a}
\eeq
and
\[
H_F(\Phi):=
 {m^2 \over 2}\sum_n \Phi_n\Phi_n +
 \sum_{mn} D_{mn} \Phi_m \Phi_n +
 \]
 \beq
 \lambda
\sum_{klmn} \Gamma_{klmn} \Phi_k\Phi_l\Phi_m\Phi_n
\label{w24b}
\eeq
which expresses $H_F$ as the sum of a part with only the $\Pi_n$
fields and another part with only the $\Phi_n$ fields.  Since the
discrete canonical pairs of field operators $\Phi_n$ and $\Pi_n$
satisfy canonical commutation relations they have a continuous
spectrum on the real line.  This is because each one of these
complementary operators generates translations in the other operator.
The last step is to approximate the continuous spectrum of the
discrete field operators $\Phi_n$ and $\Pi_n$ by a collection of
$M=2K+1$ closely spaced eigenvalues $\phi_n,\pi_n = n \epsilon$ where
$-K \leq n \leq K$ and $\epsilon^2 = 2 \pi/M$.  This is exactly what
was done in the scattering example, except in this case there are
$F$ degrees of freedom where $F$ is the number of retained discrete
field modes.  Unlike the truncation, both of these steps are
mathematical approximations.

Let $\langle \pmb{\phi} \vert \chi \rangle =
\chi(n_1\epsilon, \cdots , n_F \epsilon) $ be a localized function of the
amplitudes of the $F$ discrete field modes that represent an initial free wave
packet.

The goal is to use path integrals to calculate the time evolution of
these coupled modes.  This gives a non-perturbative treatment of the
truncated problem.

For the field theory, before truncation, in the discrete
representation the path integral involves integrals over an infinite
number of modes.  The normalization of the complex probability $P_X$
is such that summing over all modes in the absence of interactions
gives 1.  This means that {\it the only modes that contribute
  non-trivially to the time evolution are the retained modes}.  The
discrete approximation results in a sample space with a finite number
of discrete paths.

The Trotter approximation is 
\[
\langle
n_1,n_2,\cdots n_F \vert U_F(t) \vert \chi (0) \rangle =
\]
\beq
\lim_{N\to \infty} \langle
n_1,n_2,\cdots n_F \vert  (e^{-i H_F(\Pi)\Delta t}e^{-iH_F(\Phi)\Delta t})^N
\vert \chi (0) \rangle .
\label{w25}
\eeq
This can be evaluated by inserting complete sets of eigenstates of
the complementary fields 
between each of the operators.  The following abbreviations
are used for sums over intermediate states:
\beq
\int d\pmb{\phi} = \epsilon^F \sum_{n_1=-K}^K \cdots  \sum_{n_F=-K}^K,
\label{w26}
\eeq
for vectors representing a value of the eigenvalues of each of the $F$
independent $\phi$
fields, 
\beq
\pmb{\phi}=(n_1 \epsilon, \cdots , n_F \epsilon) \qquad -K \leq n_i \leq K, 
\label{w27}
\eeq
for vectors representing the value of the eigenvalues of each of the $F$
independent $\pi$
fields
\beq
\pmb{\pi}=(n_1 \epsilon, \cdots , n_F \epsilon) \qquad -K \leq n_i \leq K 
\label{w28}
\eeq
and
\beq
\gamma = (\pmb{\phi}_0,\pmb{\phi}_1, \cdots, \pmb{\phi}_N)
\label{w29}
\eeq
for a ``path'' that ends at $\pmb{\phi}_0$ where
$\pmb{\phi}_j$ ($j&gt;0$) 
represents values of each of the $\phi_n$ fields at
each of $N$ time steps.

The following definitions are generalizations of the definitions in
section \ref{path}:
\beq
P_X(\pmb{\phi}',\pmb{\phi},\Delta t) := \sum_{\mathbf{n}''}
\langle \pmb{\phi}' \vert \pmb{\pi}'' \rangle e^{-i \pmb{\pi}'' \cdot\pmb{\pi}'' \Delta t}
\epsilon^F\langle \pmb{\pi} \vert \pmb{\phi} \rangle \epsilon^F .
\label{w30}
\eeq
It follows from (\ref{f22}) that $P_X(\pmb{\phi}',\pmb{\phi},\Delta t)$ has the property
\beq
\sum_{\mathbf{n}} P_X(\pmb{\phi}',\pmb{\phi},\Delta t) =1
\label{w31}
\eeq
and
\[
P_{X}(\pmb{\phi}_f,\pmb{\phi}_N, \cdots \pmb{\phi}_1) :=
\]
\[
P_X(\pmb{\phi}_f,\pmb{\phi}_N,\Delta t)
P_X(\pmb{\phi}_N,\pmb{\phi}_{N-1},\Delta t)
\cdots
\]
\beq
P_X(\pmb{\phi}_3,\pmb{\phi}_2,\Delta t)
P_X(\pmb{\phi}_2,\pmb{\phi}_1,\Delta t)
\label{w32}
\eeq
also satisfies
\beq
\sum_{\gamma \in \Gamma}  
P_{X}(\pmb{\phi}_f,\pmb{\phi}_N, \cdots , \pmb{\phi}_1) =1 .
\label{w33}
\eeq
Equation (\ref{w32}) represents the complex probability of a given path, where at
each time slice each of the $F$ $\phi$'s has one of the $M$ allowed values
between $-K\epsilon$  and $K\epsilon$.  Removing the last factor
of $\epsilon^F$ and only summing over $\pmb{\phi}_N \cdots \pmb{\phi}_2$
gives the evolution due to free propagation
\beq
\langle
\pmb{\phi}_f \vert e^{-{i\over 2} \pmb{\Pi}\cdot \pmb{\Pi}t}
\vert \pmb{\phi}_1 \rangle =
\sum_{\pmb{n}_n \cdots \pmb{n}_1} 
P_X(\pmb{\phi}_f,\pmb{\phi}_N, \cdots , \pmb{\phi}_1)\epsilon^{-F} .
\label{w34}
\eeq
The full path integral including the effects of the interaction can
be expressed as the expectation of the following potential functional
of the path $\gamma$ 
\beq
W[\gamma] :=
e^{i\sum_{n} H_F (\pmb{\phi}_n)   \Delta t}
\label{w36}
\eeq
with respect to the complex probability
distribution (\ref{w33}),
where
$H_F (\pmb{\phi}_n)$ represents the value of the $\pmb{\phi}$-dependent part
of the Hamiltonian evaluated at the value of the path $\gamma$ at the
$n$-th time slice.

This gives the path integral approximation
\[
\langle
n_{1f},n_{2f},\cdots n_{Ff} \vert U_F(t) \vert \chi (0) \rangle =
\]
\beq
\sum_{\gamma} 
P_X(\pmb{\phi}_f,\pmb{\phi}_N, \cdots \pmb{\phi}_1)W[\gamma]
\chi (\pmb{\phi}_1)
\label{w37}
\eeq
which again represents the path integral for fields as the expectation value
of
a potential functional with respect to a
complex probability distribution.  As in the one degree of freedom case
this can be exactly factored into a product of one-time step operators
\[
P_X(\pmb{\phi}_f,\pmb{\phi}_N, \cdots \pmb{\phi}_1)W[\gamma] =
\]
\[
X(\pmb{\phi}_f,\pmb{\phi}_N,\Delta t)
e^{iH_F (\pmb{\phi}_N)   \Delta t}
X(\pmb{\phi}_N,\pmb{\phi}_{N-1},\Delta t)
e^{iH_F (\pmb{\phi}_{N-1})   \Delta t}
\cdots
\]
\beq
X(\pmb{\phi}_3,\pmb{\phi}_2,\Delta t)
e^{iH_F (\pmb{\phi}_2)   \Delta t}  
X(\pmb{\phi}_2,\pmb{\phi}_1,\Delta t)
e^{iH_F (\pmb{\phi}_1)   \Delta t} .
\label{w38}
\eeq
This can be used to represent time evolution as the product of
large approximate transfer matrices.

Each stage the calculation uses finite mathematics.   The use
of the finite Weyl representation exactly preserves unitary at each
level of approximation.  Both the $\pmb{\phi}$ and $\pmb{\pi}$
transfer matrices are unitary and can be expressed exactly in the
truncated model.  This means that the discrete Trotter approximation to
time evolution is exactly unitary.

Figures nine and ten show calculations of the initial
real and
imaginary parts of the two field modes.  In this case the
initial modes are real and taken to be Gaussians of the form
\beq
\langle \phi_1,\phi_2 \vert
\psi \rangle =
N e^{-\sum_{i=0}^1 (\phi_i - \langle \phi_i \rangle)^2/(4 \delta \phi_i^2)}
\eeq 
Figures 11 and  12 show the real and imaginary parts of 
the time $t=.5$ evolved amplitudes of the two
discrete modes with $M=41$ field values using 
$N=20$ Trotter steps.

Figures 13 and 14 show plots of the real and imaginary parts of $\phi_1$ when
$\phi_2=0$ at $T=0$ and $T=.5$. 

In the initial calculations the initial mean displacement and
uncertainty of each mode was taken to be .5. The initial state has no
imaginary part but one develops due to the non-zero displacement of
the initial state.  This truncation is too crude to contain any real
physics, however it illustrates the application of the discrete path integral
to fields.

A more drastic truncation of the discretization of the continuum could
be used to explore the dynamics of fields with a larger number of modes.

The wavelet representation for the Hamiltonian satisfies a functional
renormalization group equation that could be used to reduce the number
of amplitudes.  This relates infinite volume truncated Hamiltonians at different
resolutions using a canonical transformation along with 
mass, wave function, and coupling constant renormalizations: 
\[  
H_k (\Pi,\Phi,\mu,\lambda ) = 2^k H_0 (2^{-k}\Pi,2^{k}\Phi,2^{-2k}\mu,2^{-2k}\lambda]. 
\]

Realistic calculations require a large number of field modes.  Time-
dependent scattering calculations of the type discussed in section 9
also require volumes sufficiently large for the scattered fragments to
become stable particles.  These calculations cannot be performed on a
classical computer and will still be very challenging on a quantum
computer.


% calculations performed using daub.c
% graphs daub_r.gp, daub_i.gp unicode greek letters
%
\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{fig_9.pdf}
\caption{\bf Two modes (real) at T=0.0}
\label{fig:9}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{fig_10.pdf}
\caption{\bf Two modes (imaginary) at T=0.0}
\label{fig:10}
\end{minipage}
\end{figure}

\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{fig_11.pdf}
\caption{\bf Two modes (real) at T=0.5}
\label{fig:11}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{fig_12.pdf}
\caption{\bf Two modes (imaginary) at T=0.5}
\label{fig:12}
\end{minipage}
\end{figure}

\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.4]{fig_13.pdf}
\caption{\bf One mode (real) at T=0.0 and 0.5}
\label{fig:13}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.4]{fig_14.pdf}
\caption{\bf One mode (imaginary) at T=0.0 and 0.5}
\label{fig:14}
\end{minipage}
\end{figure}
  
\bigskip\bigskip
  
\bibliographystyle{unsrt}
%\section*{References}
%\begin{thebibliography}
%\bibliography{collins.bib}
\bibliography{quant_comp.bib}
\end{document}




This is a rigorous reformulate on the real time path integral in the
sense that there is a sample space of paths, and there is complex
weight assigned to subsets of the sample space of paths.  Real time
evolution is expressed as the expectation value of a ``potential''
functional of the paths with respect to the complex probability.
In the discrete case the there are finite number of paths associated
with a finite collection of time slices.  
It the discrete formulation discussed in this paper these integral are
replaced by finite sums.  The next section introduces the discrete
method that is used in this work.  It is based on Schwinger's
discrete version of the Weyl representation.  It can be used to
express any observable on an $N$-dimensional Hilbert space as a finite
degree polynomials in an irreducible pair of operators.  This is used in
section four to introduce the complex probability formulation of the path
integral for finite dimensional systems.   While the formulation is
general,  the number of paths in sample subsets, even for a system with
a finite number of degrees of freedom,  is prohibitively large.
In addition, the  

If the Dimension of the Hilbert space in an integer power of two is can be
reformulated using tensor products of qubit operators.



This
will be used to replace coordinates and momenta in the discrete case.
Section four discusses the complex probability formulation of the path
integral for finite dimensional systems.  Section five gives a brief
discussion of quantum logic, bits and qubits and section six discusses
a factorization of Schwinger's Weyl representation into products of
elementary quantum gates.  The continuum limit is discussed in section
seven.  Section eight discusses the complex probability formulation of
the path integral for infinite dimensional systems.  Section nine
applies the method of section eight to potential scattering in one
dimension and section ten applies it to a two mode truncation of a
$\phi^4(x)$ field theory. The conclusion summarizes the implications of these
classical calculations for future quantum computations.
</pre></body></html>