# Mapping of Quantum Systems to the Probability Simplex

D. D. Yavuz and A. Yadav  Department of Physics, 1150 University Avenue,
University of Wisconsin at Madison, Madison, WI, 53706

###### Abstract

We start with the simplest quantum system (a two-level system, i.e., a qubit)
and discuss a one-to-one mapping of the quantum state in a two-dimensional
Hilbert space to a vector in an eight dimensional probability space
(probability simplex). We then show how the usual transformations of the
quantum state, specifically the Hadamard gate and the single-qubit phase gate,
can be accomplished with appropriate transformations of the mapped vector in
the probability simplex. One key defining feature of both the mapping to the
simplex and the transformations in the simplex is that they are not linear.
These results show that both the initial state and the time evolution of a
qubit can be fully captured in an eight dimensional probability simplex (or
equivalently using three classical probabilistic bits). We then discuss multi-
partite quantum systems and their mapping to the probability simplex. Here,
the key tool is the identical tensor product structure of combining multiple
quantum systems as well as multiple probability spaces. Specifically, we
explicitly show how to implement an analog of the two-qubit controlled-not
(CNOT) gate in the simplex. We leave it an open problem how much the quantum
dynamics of $N$ qubits can be captured in a probability simplex with $3N$
classical probabilistic bits. Finally, we also discuss the equivalent of the
Schrodinger’s equation for the wavefunction (in a Hilbert space of arbitrary
dimension), which dictates the time evolution of the vectors in the simplex.

##  I I. Introduction

The birth of quantum mechanics has fundamentally changed our understanding and
interpretation of the natural laws of the microscopic world. The mathematical
structure and the experimental predictions of the theory have been quite
clear; yet, there has been a wide-ranging and still ongoing rigorous debate
regarding the precise meaning and interpretation of various key components of
the theory, such as the wavefunction and the measurement bell1 ; fuchs1 .
According to the traditional Copenhagen interpretation, the evolution of the
quantum wavefunction is described by the Schrodinger’s equation, which is
deterministic. However, when a measurement is performed, the deterministic
evolution comes to an abrupt end with probabilities entering the picture. The
experimenter observes probabilistic outcomes, with the likelihood of each
outcome determined by the overlap of the wavefunction (at the time of the
measurement) with the eigenstates of the basis that the measurement is
performed at.

This probabilistic nature of the quantum theory is now very well understood,
and has been confirmed by countless experiments. Yet, the mathematical
structure of quantum mechanics is quite different compared to a traditional
theory of probability. This has been emphasized in detail by the recent
pioneering work of Fuchs and colleagues fuchs1 . Specifically, quantum
mechanics invokes the mathematical structure of wavefunctions with complex
coefficients that live in a Hilbert space. A key question is whether this
unique mathematical structure is strictly necessary. Can quantum mechanics be
interpreted under the general umbrella of classical probabilistic theories?
This question is not only important from a foundational point of view, but
also has practical implications. Recent achievements in quantum computing and
quantum information science have shown us that quantum devices have the
potential to make a large impact on society nielsen ; divincenzo1 ;
divincenzo2 ; vazirani ; shor ; ekert . However, to be able to understand the
true power of quantum devices, we need to better understand the boundary
between quantum mechanics, classical physics, and traditional probabilistic
theories tsang ; watrous ; wetterich1 ; wetterich2 . To answer this great
need, over the last two decades, there has been an impressive body of research
that has investigated the relationship between quantum mechanics and classical
probabilistic theories. One line of research has argued for a completely
subjective interpretation of the wavefunction fuchs1 ; fuchs2 . One of the
main tools in these investigations is fine-tuned operator classes that allow
Symmetric Informationally Complete (SIC) measurements fuchs3 ; boyd . Other
related research has tried to place quantum mechanics under the umbrella of
probability theories that are more general than classical, sometimes referred
to as post-classical theories of probability hardy ; barrett1 ; barnum ;
masanes ; rau . This research has identified a rich landscape and the goal is
to place quantum mechanics properly in this landscape in order to better
understand its unique properties.

The main goal of this paper is to point out another direction in investigating
the relationship between quantum mechanics and classical probabilistic
theories: maps from the Hilbert space to a probability space and
transformations in the same space, with the unique feature that the maps and
transformations are not necessarily linear. In this work, we focus on
capturing the initial state and the time evolution of quantum systems in a
classical probability space (i.e., we will not consider the measurement aspect
of quantum systems realq1 ; realq2 ). Specifically, we start with the simplest
quantum system (a two-level system, i.e., a qubit) and discuss a mapping of
the quantum state to a vector in a probability space (Fig. 1). The mapping is
one-to-one and preserves all the information encoded in the wavefunction. Not
surprisingly, to be able to store all the information encoded in the complex
coefficients, we need to increase the dimension of the system: the mapping is
to an eight dimensional probabilistic space (from the two dimensional Hilbert
space).

Once a quantum state is mapped, the next key question is whether the evolution
of the state can be captured in the probability simplex. It is well known that
an arbitrary evolution of a single qubit can be achieved using combinations of
Hadamard gates and phase rotations nielsen . We will show how these two main
operations can be implemented with appropriate transformations of the mapped
vector in the probability space. The transformations in the simplex are
affine, but not linear.

These results show that both the initial state and the time evolution of a
single qubit can be captured in an eight dimensional probability space (or
equivalently using three classical probabilistic bits). We then investigate
this mapping for multiple qubits. Here, the key tool is the identical tensor
product structure of combining multiple quantum systems and combining multiple
probability spaces. Specifically, we will explicitly discuss how to implement
an analog of the two-qubit controlled-not (CNOT) gate in the simplex. Within
our formulation, $N$ qubits will map to a tensor product of $N$ 8-dimensional
probability spaces (i.e., a Hilbert space of dimension $2^{N}$ will map to a
probability simplex of dimension $8^{N}$). We leave it as an open question how
much of the dynamics of $N$ qubits can be captured in the simplex with $3N$
classical probabilistic bits. Finally, we will also discuss an analogue of the
Schrodinger’s equation for the wavefunction which lives in a Hilbert space of
arbitrary dimension. This is a continuous differential equation that describes
the evolution of the simplex vector under an effective “Hamiltonian”.

Our work has been heavily influenced by the recent investigations of quantum
mechanics within the operational framework of probability theories; in
particular the pioneering work of Hardy hardy  and Barrett barrett1 . There is
also a relation of this work to a large body of literature who have attempted
to derive some features of quantum mechanics using classical “toy” theories. A
good summary of various toy theories is discussed in, for example, Ref.
rudolph1 . This work is also related to the mapping of quantum states to
probability-like distributions, typically referred to as quasiprobabilities
bartlett1 ; bartlett2 ; bartlett3 ; raussendorf1 ; raussendorf2 ; eisert ; zhu
; wootters . The most well-known example of a quasiprobability distribution is
the Wigner function. It is well-known that quasi-probabilities can have
negative values; in fact, the true quantum mechanical nature of the
wavefunction is expressed in these negative regions. We argue that when one
allows for maps and transformations that are not necessarily linear, one can
capture a quantum state (as well as its’ evolution) using only probabilities
(i.e., negative values are not needed). We will comment on these connections
more thoroughly below, in Section XIV of the paper.

Figure 1:  The schematic of the basic idea that we will study in this work. We
start with a qubit with wavefunction, $|\psi\rangle$, and discuss a one-to-one
mapping, $\varphi$ from the Hilbert space $H$ to a vector $\vec{s}$ in an
eight dimensional probability space $S$ (which is a real Euclidean space). The
vectors $\vec{s}$ have their values add up to unity (i.e., they are valid
probability distributions), and therefore lie on a hypersurface (the simplex).
We then show how various elementary gates that are applied to the
wavefunction, $|\psi\rangle$, can be mimicked in the probability simplex
(i.e., not just the initial quantum state, but also its evolution can be
mapped to the simplex).

We note that there is nothing new in the idea that a quantum state can be
represented as a vector of probabilities. Consider a specific set-of basis
states, $|\phi_{1}\rangle$, $|\phi_{2}\rangle$, $|\phi_{3}\rangle$, ….. Given
an initial wavefunction $|\psi\rangle$, we can view the measurement outcomes
in the mentioned basis as a probabilistic vector of the form,
$\left(|\langle\phi_{1}|\psi\rangle|^{2},|\langle\phi_{2}|\psi\rangle|^{2},|\langle\phi_{3}|\psi\rangle|^{2},...\right)$.
When the quantum state evolves, these probability values would change, and we
can always view this as mapping the initial probability vector to another
output probability vector (by multiplying with an appropriate matrix, for
example). Such straightforward mapping, however, always results in matrices
that depend on the initial wavefunction $|\psi\rangle$. This is, for example,
clearly discussed in the ontological model of Aaronson, in Ref. aaronson

What we will be discussing is something quite different. We will encode both
the amplitude and phase information of the complex wavefunction in
probabilities. The analogs of the quantum gates in the probability space (for
example the Hadamard gate and the single-qubit phase gate), will involve
affine transformations, with constant translations and constant matrices (that
will be independent of the initial input state). This essentially means that
all the interferences in the complex coefficients can be captured in the
probability space. To further emphasize the analogy, consider a single qubit
that evolves through any number of gates: for example, a Hadamard gate,
followed by a phase-gate, followed by $\pi/3$ Rabi rotation, and so forth. One
can construct circuits that would perform exact analogs of these gates in the
probability simplex, and exactly track the quantum evolution in the simplex.
Once the circuits are fixed, if the initial condition for the quantum state
changes, one only needs to modify the initial condition for the simplex
vector.

##  II II. Preliminaries

As we mentioned above, in traditional formulation of quantum mechanics, a
quantum state is described by a complex wavefunction, $|\psi\rangle$, in a
Hilbert space, $H$. This state will evolve according to Schrodinger’s
equation, which conserves the norm of the wavefunction. This time evolution of
the quantum state can be described using an appropriate unitary matrix,
$\hat{U}$, that satisfies,
$\hat{U}^{\dagger}\hat{U}=\hat{U}\hat{U}^{\dagger}=\hat{I}$. With this
evolution, the state is mapped to
$|\psi\rangle\longrightarrow\hat{U}|\psi\rangle$.

In any classical probabilistic experiment, we will have a set of
probabilities, which we can also think of as constituting a vector, in a
probabilistic space, $S$. We will denote such a probabilistic vector with
$\vec{s}$. Each of the entries of this vector has to be between 0 and 1, i.e.,
$0<s_{i}<1$, and furthermore, the entries need to sum to unity,
$\sum_{i}s_{i}=1$. Because the entries add up to unity, such a vector lies on
certain surface in the probabilistic space, and this surface is called the
simplex definetti . Similar to a quantum state, such a probabilistic vector
can also evolve in time (perhaps the experimental conditions change, or our
knowledge of the experiment evolves). We can view such evolution as mapping a
vector in space $S$, to another vector. We will denote such mapping with
$T:S\longrightarrow S$. Usually, such evolution is described by multiplying
the vector $\vec{s}$ with a Stochastic matrix, $\tilde{\mathcal{M}}$, i.e.,
$T(\vec{s})=\tilde{\mathcal{M}}\cdot\vec{s}$. A stochastic matrix is a matrix
whose columns sum up to 1. This assures that the resultant vector also is
normalized; i.e., its’ components add up to unity. As we will discuss below in
detail, probabilistic vectors can also transform using maps that are not
linear, but instead affine. Throughout this paper, all quantum mechanical
operators will be presented by a hat (for example $\hat{U}$), whereas all the
transformations in the simplex will be presented by a tilde (for example,
$\tilde{\mathcal{M}}$).

##  III III. Two-level quantum system (qubit) and mapping

We now specifically focus on a single qubit wavefunction
$|\psi\rangle=c_{0}|0\rangle+c_{1}|1\rangle$. Here, the states $|0\rangle$ and
$|1\rangle$ are the logical states, and $c_{0}$ and $c_{1}$ are the complex
coefficients satisfying the usual normalization condition,
$|c_{0}|^{2}+|c_{1}|^{2}=1$. The quantum state can be represented as a vector
in the logical state basis:

$\displaystyle|\psi\rangle=\left(\begin{array}[]{c}c_{0}\\\
c_{1}\end{array}\right)\equiv\left(\begin{array}[]{c}x_{0}+iy_{0}\\\
x_{1}+iy_{1}\end{array}\right)\quad.$ (5)

Here, we have defined the real and imaginary parts of each complex
coefficient, i.e., $c_{0}\equiv x_{0}+iy_{0}$ and $c_{1}\equiv x_{1}+iy_{1}$.
In what follows, instead of the complex coefficients, we will work with these
real and imaginary parts. This is necessary since the probabilities are real
and positive, and therefore, components of a vector in the probability simplex
must only contain real coefficients.

We propose the following mapping of the quantum state $|\psi\rangle$ (in
Hilbert space $H$) to a vector $\vec{s}$ in the probability space,
$\varphi:H\rightarrow S$ :

$\displaystyle|\psi\rangle=\left(\begin{array}[]{c}x_{0}+iy_{0}\\\
x_{1}+iy_{1}\end{array}\right)\xrightarrow{\varphi}\vec{s}=\frac{1}{8}\left(\begin{array}[]{c}1+x_{0}\\\
1+x_{1}\\\ 1-x_{0}\\\ 1-x_{1}\\\ 1+y_{0}\\\ 1+y_{1}\\\ 1-y_{0}\\\
1-y_{1}\end{array}\right)\quad.$ (16)

We note that the vector $\vec{s}$ represents a valid probability distribution.
That is, each of the entries is between 0 and 1 (i. e., $0<s_{i}<1$), and
these entries sum up to unity, $\sum_{i}s_{i}=1$. The fact that we need to
increase the dimension from 2 to 8 is intuitive. For each complex coefficient,
we need to store two real numbers, the real part and the imaginary part.
Furthermore, for each real number, we need to store the quantity with both
signs. This is because, in order to map the transformations of the quantum
state, we will need access to both signs of these coefficients. Hence, the
factor of 4 increase in the dimension. The map is injective (i.e., one-to-
one), but not surjective. The main insight in the mapping of Eq. (2) is that
the phase and the amplitude information (for the real and imaginary parts of
the complex coefficients) can be stored in how much the probabilities deviate
from purely random quantity (hence the initial “1” in all the entries of
$\vec{s}$).

A key property of the mapping of Eq. (2) is that it is not linear. By
inspection, a superposition of two wavefunctions do not map to the same
superposition of their mapped vectors: $\varphi(a\ket{\psi}+b\ket{\phi})\neq
a\varphi(\ket{\psi})+b\varphi(\ket{\phi})$ for $\ket{\psi},\ket{\phi}\in
H;a,b\in\mathbb{C}$.

We note that the set of states $\vec{s}$ defined in the simplex by Eq. (2)
form a convex surface. That is, for two different states $\vec{s}$ and
$\vec{s}^{\prime}$, and for coefficients $\lambda$ and $\lambda^{\prime}$ such
that $\lambda+\lambda^{\prime}=1$, any combination
$\lambda\vec{s}+\lambda^{\prime}\vec{s}^{\prime}$ is also an allowed mapped
state. This is similar to what is discussed in Refs. barrett1 ; hardy . We
also note, however, that, differing from the prior work, the simplex vector
with all of its’ entries equal to 0 (which we can denote by $\vec{0}$) is not
a valid mapped vector. Even if we were to include not-normalized quantum
states (where the probabilities leek out of the system, for example), in the
limit, $x_{i}\rightarrow 0,y_{i}\rightarrow 0$, all of the entries for the
vector in the simplex would approach $\frac{1}{8}$, i.e.,
$\vec{s}\rightarrow\frac{1}{8}\vec{1}$.

##  IV iV. Transformations in the simplex

As we mentioned above, prior work has focused on linear transformations of
probability vectors of the form $\tilde{\mathcal{M}}\cdot\vec{s}$, using a
stochastic matrix $\tilde{\mathcal{M}}$. There is an important reason for the
consideration of only linear transformations in earlier work (for example, in
Refs. hardy  and barrett1 . From the construction of the probability space,
these authors included the null vector, $\vec{0}$, in their allowed states and
further posed the constraint that a reasonable map, $f$, should leave the null
vector unchanged, i.e., $f(\vec{0})\rightarrow\vec{0}$. However, as we
discussed above, the null vector, $\vec{0}$, in the probability simplex is not
a valid mapped vector in our formalism.

Furthermore, operations of probabilities that are not linear are encountered
frequently and we do not see an apriori reason to exclude such maps. One well-
known example is the nonlinearity due to the logical OR operation. Consider
two independent events, $A$ and $B$, with event probabilities, $p(A)$ and
$p(B)$, respectively. The probability of $A$ OR $B$ would be given by,
$p(A\lor B)=p(A)+p(B)-p(A\land B)=p(A)+p(B)-p(A)p(B)$, which is a nonlinear
function in each of the event probabilities.

The central question is what type of maps, $T:S\rightarrow S$, that we should
be looking for. Motivated by the mapping of Eq. (2), we look for affine
transformations of the simplex vector of the form a translation added on
linear combinations of the simplex vector entries. A key insight of this work
is that the mapped vectors [described by Eq.(2)] can be written in the
following form, which makes the correspondence with the Hilbert space more
clear.

$\displaystyle\vec{s}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\left(\begin{array}[]{c}1\\\ 1\\\ 1\\\ 1\\\ 1\\\ 1\\\
1\\\ 1\end{array}\right)+\frac{1}{8}\left(\begin{array}[]{r}x_{0}\\\ x_{1}\\\
-x_{0}\\\ -x_{1}\\\ y_{0}\\\ y_{1}\\\ -y_{0}\\\ -y_{1}\end{array}\right)$ (33)
$\displaystyle\Rightarrow\vec{s}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{8}\left(\vec{1}+\vec{p}\right)\quad.$ (34)

Here, we have defined another 8-dimensional vector, $\vec{p}\equiv
8\vec{s}-\vec{1}$. Note that the entries of $\vec{p}$ sum up to zero; i.e.,
$\sum_{i}p_{i}=0$. Furthermore, the Euclidian norm of $\vec{p}$ is a constant
$||\vec{p}||=\sqrt{2}$, since we have
$x_{0}^{2}+y_{0}^{2}+x_{1}^{2}+y_{1}^{2}=1$ (this is because of the
normalization of the wavefunction). We also note that the two vectors that
form the simplex vector $\vec{s}$ are orthogonal to each other,
$\vec{1}\cdot\vec{p}=0$. As a result, we have
$||\vec{s}||=\sqrt{||\vec{1}||^{2}+||\vec{p}||^{2}}/8=\sqrt{10}/8$, which is
also constant. This shows that $\vec{s}$ lies on the intersection of a seven-
dimensional hypersphere, with four seven-dimensional hyperplanes, resulting in
a three dimensional hypersurface $S$.

As it will be clear below, because the quantum gates form linear combinations
of the entries of $\vec{p}$, we first view the mapping of the simplex vector
$\vec{s}$, as instead mapping $\vec{p}$ to another vector. We will call the
matrix for this mapping to be $\tilde{M}$:

$\displaystyle\vec{p}\longrightarrow\tilde{M}\cdot\vec{p}\quad.$ (35)

Expressed as acting on the full simplex state $\vec{s}$, the transformation of
Eq. (4) is,
$T(\vec{s})=\frac{1}{8}\left(\vec{1}+\tilde{M}\cdot\vec{p}\right)$, which
gives
$T(\vec{s})=\frac{1}{8}\left[\vec{1}+\tilde{M}\cdot\left(8\vec{s}-\vec{1}\right)\right]$,
or writing it slightly differently,

$\displaystyle
T(\vec{s})=\frac{1}{8}\left(\tilde{I}-\tilde{M}\right)\cdot\vec{1}+\tilde{M}\cdot\vec{s}\quad.$
(36)

Here, the quantity $\tilde{I}$ is $8\times 8$ identity matrix. Below, we will
give explicit expressions for the $8\times 8$ matrices, $\tilde{M}$, for
specific evolution of the quantum state, such as the application of the
Hadamard gate. With the matrix $\tilde{M}$ given, Eq. (5) describes the
explicit transformation of the probability vector, with the map
$T:S\longrightarrow S$ in the simplex.

We note that, the first term in the right hand side of Eq. (5) is a
translation for each of the entries of the vector (an offset). Because of this
term, the map $T:S\longrightarrow S$ is not linear (i.e., the sum of two
vectors $\vec{s}$ and $\vec{s^{\prime}}$ would not transform as the sum of the
individual transforms). However, $T$ is an affine map. For two vectors,
$\vec{s}$ and $\vec{s}^{\prime}$, and for coefficients $\lambda$ and
$\lambda^{\prime}$ such that $\lambda+\lambda^{\prime}=1$, we have
$T(\lambda\vec{s}+\lambda^{\prime}\vec{s}^{\prime})=\lambda
T(\vec{s})+\lambda^{\prime}T(\vec{s}^{\prime})$.

The constraints on the matrix $\tilde{M}$ of above such that
$T:S\longrightarrow S$ is a valid map is different from stochasticity.
Specifically, the two necessary constraints are (1) $\tilde{M}$ should be such
that the norm of the resulting vector is preserved since we need to have:
$||\tilde{M}\cdot\vec{p}||=\sqrt{2}$. Because of the specific form for the
vector $\vec{p}$, this norm conservation does not imply orthogonality of the
matrix $\tilde{M}$. By inspection, the necessary constraint is that the sum of
the squares of the entries in each row must add up to unity: i.e.,
$\sum_{j}\tilde{M}_{{ij}}^{2}=1$ for each row $i$. (2) The rows of $\tilde{M}$
should be related to each other such that the entries of
$\tilde{M}\cdot\vec{p}$ sum up to zero. Specifically, $\tilde{M}\cdot\vec{p}$
should produce a column vector of the form shown in Eq. (2), with respective
entries having equal amplitude and opposite signs. This assures that the
resulting full simplex vector,
$\frac{1}{8}\left(\vec{1}+\tilde{M}\cdot\vec{p}\right)$ is a valid probability
distribution (i.e., its’ entries add up to unity).

We next discuss how using the above affine transformations, one can find the
equivalents of the Hadamard gate, Rabi rotations, and the single-qubit phase
gate in the probabilistic simplex.

##  V V. Hadamard gate

The Quantum Mechanical Hadamard gate on a single qubit is accomplished by
multiplying the state vector $|\psi\rangle$ with the following unitary matrix
nielsen :

$\displaystyle\hat{U}_{H}=\left(\begin{array}[]{rr}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array}\right)\quad.$ (39)

Note that as required by unitarity, we have
$\hat{U}_{H}^{\dagger}\hat{U}_{H}=1$. The effect of the Hadamard gate on the
quantum state, explicitly expressed in terms of the real and imaginary parts
of the complex coefficients, is:

$\displaystyle\hat{U}_{H}|\psi\rangle$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{rr}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array}\right)\left(\begin{array}[]{c}x_{0}+iy_{0}\\\
x_{1}+iy_{1}\end{array}\right)$ (44) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\frac{1}{\sqrt{2}}x_{0}-\frac{1}{\sqrt{2}}x_{1}+i\left(\frac{1}{\sqrt{2}}y_{0}-\frac{1}{\sqrt{2}}y_{1}\right)\\\
\frac{1}{\sqrt{2}}x_{0}+\frac{1}{\sqrt{2}}x_{1}+i\left(\frac{1}{\sqrt{2}}y_{0}+\frac{1}{\sqrt{2}}y_{1}\right)\end{array}\right)\quad.$
(47)

We, therefore, have the following transformation of the quantum state as a
result of the Hadamard gate:

$\displaystyle|\psi\rangle$ $\displaystyle\longrightarrow$
$\displaystyle\hat{U}_{H}|\psi\rangle$
$\displaystyle\left(\begin{array}[]{c}x_{0}+iy_{0}\\\
x_{1}+iy_{1}\end{array}\right)$ $\displaystyle\longrightarrow$
$\displaystyle\left(\begin{array}[]{c}\frac{1}{\sqrt{2}}x_{0}-\frac{1}{\sqrt{2}}x_{1}+i\left(\frac{1}{\sqrt{2}}y_{0}-\frac{1}{\sqrt{2}}y_{1}\right)\\\
\frac{1}{\sqrt{2}}x_{0}+\frac{1}{\sqrt{2}}x_{1}+i\left(\frac{1}{\sqrt{2}}y_{0}+\frac{1}{\sqrt{2}}y_{1}\right)\end{array}\right)\quad.$
(52)

The central question is if a similar transformation can be accomplished in the
simplex, using an appropriate matrix, which we will denote with
$\tilde{M}(\hat{U}_{H})$. Specifically, we are looking for the following
transformation:

$\displaystyle\vec{p}$ $\displaystyle\longrightarrow$
$\displaystyle\tilde{M}(\hat{U}_{H})\cdot\vec{p}$
$\displaystyle\left(\begin{array}[]{r}x_{0}\\\ x_{1}\\\ -x_{0}\\\ -x_{1}\\\
y_{0}\\\ y_{1}\\\ -y_{0}\\\ -y_{1}\end{array}\right)$
$\displaystyle\longrightarrow$
$\displaystyle\left(\begin{array}[]{r}\left(\frac{1}{\sqrt{2}}x_{0}-\frac{1}{\sqrt{2}}x_{1}\right)\\\
\left(\frac{1}{\sqrt{2}}x_{0}+\frac{1}{\sqrt{2}}x_{1}\right)\\\
-\left(\frac{1}{\sqrt{2}}x_{0}-\frac{1}{\sqrt{2}}x_{1}\right)\\\
-\left(\frac{1}{\sqrt{2}}x_{0}+\frac{1}{\sqrt{2}}x_{1}\right)\\\
\left(\frac{1}{\sqrt{2}}y_{0}-\frac{1}{\sqrt{2}}y_{1}\right)\\\
\left(\frac{1}{\sqrt{2}}y_{0}+\frac{1}{\sqrt{2}}y_{1}\right)\\\
-\left(\frac{1}{\sqrt{2}}y_{0}-\frac{1}{\sqrt{2}}y_{1}\right)\\\
-\left(\frac{1}{\sqrt{2}}y_{0}+\frac{1}{\sqrt{2}}y_{1}\right)\end{array}\right)\quad.$
(69)

By inspection, the required $8\times 8$ matrix for the transformation of Eq.
(9) is:

$\displaystyle\tilde{M}(\hat{U}_{H})=\left(\begin{array}[]{cccccccc}\frac{1}{\sqrt{2}}&0&0&\frac{1}{\sqrt{2}}&0&0&0&0\\\
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0&0&0&0&0&0\\\
0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0&0&0&0&0\\\
0&0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0&0&0&0\\\
0&0&0&0&\frac{1}{\sqrt{2}}&0&0&\frac{1}{\sqrt{2}}\\\
0&0&0&0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0&0\\\
0&0&0&0&0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\\
0&0&0&0&0&0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\ \end{array}\right)\quad.$
(78)

With the matrix $\tilde{M}(\hat{U}_{H})$ given as above, the full
transformation of the simplex vector is given by Eq. (5), i.e.,
$T_{\hat{U}_{H}}(\vec{s})=\frac{1}{8}\left[\tilde{I}-\tilde{M}(\hat{U}_{H})\right]\cdot\vec{1}+\tilde{M}(\hat{U}_{H})\cdot\vec{s}$.

##  VI VI. Rabi rotations

Above in Section V, we discussed the Hadamard gate. However, any arbitrary
qubit rotation in that plane can be accomplished in a similar way: i.e., one
can observe “Rabi flopping” of a single qubit by implementing the following
rotation in the simplex:

$\displaystyle\hat{U}_{\theta}=\left(\begin{array}[]{rr}\cos{\theta}&-\sin{\theta}\\\
\sin{\theta}&\cos{\theta}\end{array}\right)\quad.$ (81)

Here, the quantity $\theta$ is the Rabi rotation angle. For $\theta=\pi/4$, we
recover the Hadamard gate of above. The effect of this rotation on the quantum
state is:

$\displaystyle\hat{U}_{\theta}|\psi\rangle$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{rr}\cos{\theta}&-\sin{\theta}\\\
\sin{\theta}&\cos{\theta}\end{array}\right)\left(\begin{array}[]{c}x_{0}+iy_{0}\\\
x_{1}+iy_{1}\end{array}\right)$ (86) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}x_{0}\cos{\theta}-x_{1}\sin{\theta}+i\left(y_{0}\cos{\theta}-y_{1}\sin{\theta}\right)\\\
x_{0}\sin{\theta}+x_{1}\cos{\theta}+i\left(y_{0}\sin{\theta}+y_{1}\cos{\theta}\right)\end{array}\right)\quad.$
(89)

The question again is if a similar transformation can be accomplished in the
simplex, using an appropriate matrix, which we term
$\tilde{M}(\hat{U}_{\theta})$. By inspection, the required $8\times 8$ matrix
for this transformation is:

$\displaystyle\tilde{M}(\hat{U}_{\theta})=\left(\begin{array}[]{cccccccc}\cos{\theta}&0&0&\sin{\theta}&0&0&0&0\\\
\sin{\theta}&\cos{\theta}&0&0&0&0&0&0\\\
0&\sin{\theta}&\cos{\theta}&0&0&0&0&0\\\
0&0&\sin{\theta}&\cos{\theta}&0&0&0&0\\\
0&0&0&0&\cos{\theta}&0&0&\sin{\theta}\\\
0&0&0&0&\sin{\theta}&\cos{\theta}&0&0\\\
0&0&0&0&0&\sin{\theta}&\cos{\theta}&0\\\
0&0&0&0&0&0&\sin{\theta}&\cos{\theta}\\\ \end{array}\right)\quad.$ (98)

With this transformation matrix, the individual entries of the simplex vector
would oscillate as a function of the rotation angle $\theta$. An example is
shown in Fig. 2. Here we perform the transformation on the simplex vector,
$T_{\hat{U}_{\theta}}(\vec{s})=\frac{1}{8}\left[\tilde{I}-\tilde{M}(\hat{U}_{\theta})\right]\cdot\vec{1}+\tilde{M}(\hat{U}_{\theta})\cdot\vec{s}$,
with the Rabi rotation matrix $\tilde{M}(\hat{U}_{\theta})$ of Eq. (13), with
the initial condition of $x_{0}=1,y_{0}=0,x_{1}=0,y_{1}=0$ for the quantum
wavefunction (and therefore the simplex amplitudes). As the rotation angle
$\theta$ is varied, the first probability component of the simplex vector
$s_{1}=\frac{1}{8}(1+x_{0}\cos{\theta}-x_{1}\sin{\theta})$ oscillates between
the values of $0$ and $\frac{1}{4}$.

Figure 2:  “Rabi flopping” in the simplex. Here we apply the Rabi rotation
matrix of Eq. (13) to the initial simplex vector with the initial condition of
$x_{0}=1,y_{0}=1,x_{1}=0,y_{1}=0$ for the quantum wavefunction (and therefore
the simplex amplitudes). As the rotation angle $\theta$ is varied, the first
probability component of the simplex vector
$s_{1}=\frac{1}{8}(1+x_{0}\cos{\theta}-x_{1}\sin{\theta})$ oscillates between
the values of $0$ and $\frac{1}{4}$.

##  VII VII. Single-qubit phase gate

The Quantum Mechanical single-qubit phase gate is accomplished by multiplying
the state vector $|\psi\rangle$ with the following unitary matrix:

$\displaystyle\hat{U}_{S}=\left(\begin{array}[]{cc}1&0\\\
0&\exp{(i\alpha)}\end{array}\right)=\left(\begin{array}[]{cc}1&0\\\
0&\cos{\alpha}+i\sin{\alpha}\end{array}\right)\quad.$ (103)

Note again that as required by unitarity, we have
$\hat{U}_{S}^{\dagger}\hat{U}_{S}=1$. The effect of the single-qubit phase
gate on the quantum state is:

$\displaystyle\hat{U}_{S}|\psi\rangle$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&0\\\
0&\cos{\alpha}+i\sin{\alpha}\end{array}\right)\left(\begin{array}[]{c}x_{0}+iy_{0}\\\
x_{1}+iy_{1}\end{array}\right)$ (108) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}x_{0}+iy_{0}\\\
\left(x_{1}\cos{\alpha}-y_{1}\sin{\alpha}\right)+i\left(x_{1}\sin{\alpha}+y_{1}\cos{\alpha}\right)\end{array}\right)\quad.$
(111)

By inspection, the required $8\times 8$ matrix for this transformation is:

$\displaystyle\tilde{M}(\hat{U}_{S})=\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\
0&\cos\alpha&0&0&0&0&0&\sin\alpha\\\ 0&0&1&0&0&0&0&0\\\
0&0&0&\cos\alpha&0&\sin\alpha&0&0\\\ 0&0&0&0&1&0&0&0\\\
0&\sin\alpha&0&0&0&\cos{\alpha}&0&0\\\ 0&0&0&0&0&0&1&0\\\
0&0&0&\sin\alpha&0&0&0&\cos{\alpha}\\\ \end{array}\right)\quad.$ (120)

As we discussed above, with the matrix $\tilde{M}(\hat{U}_{S})$ given, the
full transformation of the simplex vector which mimics single-qubit phase
rotation is
$T_{\hat{U}_{S}}(\vec{s})=\frac{1}{8}\left[\tilde{I}-\tilde{M}(\hat{U}_{S})\right]\cdot\vec{1}+\tilde{M}(\hat{U}_{S})\cdot\vec{s}$.

##  VIII VIII. Arbitrary evolution of the two-level quantum system

Above, we argued that one can perform operations identical to the Hadamard
gate and the single-qubit phase gate in the probability simplex. We also note
that the transformations in the simplex rely on forming linear combinations of
the vector, $\vec{p}$. As a result, a sequence of quantum gates can be
mimicked by cascading the appropriate transformations in the simplex
sequentially. More specifically, consider a quantum state that first evolves
according to the unitary operator $\hat{U}_{1}$ followed by evolution with
$\hat{U}_{2}$. We would find the corresponding transformation matrices
$\tilde{M}(\hat{U}_{1})$ and $\tilde{M}(\hat{U}_{2})$, and map the
corresponding vector as
$\vec{p}\longrightarrow\tilde{M}(\hat{U}_{2})\tilde{M}(\hat{U}_{1})\cdot\vec{p}$.
We, therefore, have
$\tilde{M}(\hat{U}_{2}\hat{U}_{1})=\tilde{M}(\hat{U}_{2})\tilde{M}(\hat{U}_{1})$.
We will provide a more rigorous proof for this cascading theorem later in
Section XII.

Because it is well-known that any arbitrary of evolution of a two-level
quantum system can be accomplished using an appropriate combination of a
Hadamard gate and a single-qubit phase gate nielsen , we conclude that any
arbitrary evolution can be achieved in the simplex. This result shows that
both the initial state and the full evolution dynamics of a single qubit can
be captured by using an 8-dimensional probabilistic simplex (or equivalently,
using three classical probabilistic bits, since the simplex dimension of three
classical probabilistic bits is 8).

Below, in Section XII, we will derive the differential equation for the time
evolution of the simplex state $\vec{s}(t)$ in time $t$, analogous to the
Schrodinger’s equation for the quantum state $|\psi(t)\rangle$. The existence
of a time evolution equation in the simplex also points to the same result;
that is, full evolution dynamics of the quantum state can be captured in the
simplex.

##  IX IX. Specific implementation for a Hadamard gate

A detailed discussion of how the above transformations can physically and
practically be accomplished using classical probabilistic bits is beyond the
scope of this work. It is likely that extensions of the recently demonstrated
probabilistic bits (the so-called p-bits) will be feasible datta1 ; datta2 .
In this section, we will discuss one specific approach to construct the
Hadamard transformation in the simplex, which is summarized by the matrix of
$\tilde{M}(\hat{U}_{H})$ of Eq. (10), with the full simplex transformation
given by:
$T_{\hat{U}_{H}}(\vec{s})=\frac{1}{8}\left[\tilde{I}-\tilde{M}(\hat{U}_{H})\right]\cdot\vec{1}+\tilde{M}(\hat{U}_{H})\cdot\vec{s}$.
We discuss on the first row of this transformation; other rows can be
implemented in a similar manner. Using this map, the first entry of the
simplex vector will need to transform as:

$\displaystyle\frac{1}{8}\left(1+x_{0}\right)\longrightarrow\frac{1}{8}\left[\frac{1+x_{0}}{\sqrt{2}}+\frac{1-x_{1}}{\sqrt{2}}+(1-\sqrt{2})\right]\quad.$
(121)

We note that this transformation involves a constant offset added to the
linear combination of the first and fourth entries in the initial simplex
vector: $\frac{1}{8}(1+x_{0})$ and $\frac{1}{8}(1-x_{1})$, respectively. Let’s
denote two random variables, $a$ and $b$, with event probabilities $p(a)$ and
$p(b)$, respectively. We choose the event probabilities $p(a)$ and $p(b)$ to
be exactly these first and fourth entries of the simplex vector:

$\displaystyle p(a)$ $\displaystyle=$
$\displaystyle\frac{1}{8}\left(1+x_{0}\right)\quad,$ $\displaystyle p(b)$
$\displaystyle=$ $\displaystyle\frac{1}{8}\left(1-x_{1}\right)\quad.$ (122)

To implement the first row of the Hadamard gate, we need to construct a
circuit, which will have two entries at its input, $p(a)$ and $p(b)$, and
produce the following output:

$\displaystyle\frac{p(a)}{\sqrt{2}}+\frac{p(b)}{\sqrt{2}}+\frac{1}{8}(1-\sqrt{2})\quad.$
(123)

One way to accomplish this would be to utilize the probability of the logical
OR operation. Note that, the probability of $p(a\lor b)$ is:

$\displaystyle p(a\lor b)=p(a)+p(b)-p(a\land b)\quad.$ (124)

Rewriting Eq. (19) in a slightly different way:

$\displaystyle\frac{p(a)}{\sqrt{2}}+\frac{p(b)}{\sqrt{2}}+\frac{1}{8}(1-\sqrt{2})=p(a)+p(b)+(\frac{1}{\sqrt{2}}-1)p(a)+(\frac{1}{\sqrt{2}}-1)p(b)+\frac{1}{8}(1-\sqrt{2})\quad.$
(125)

We see that if these two events are chosen to be such that their intersecting
set satisfies $p(a\land
b)=(1-\frac{1}{\sqrt{2}})p(a)+(1-\frac{1}{\sqrt{2}})p(b)+\frac{1}{8}(\sqrt{2}-1)$,
we have $p(a\lor b)$ giving the correct transformation. We note that the
appearance of the logical OR and logical AND operations in above is further
indication that we do not need to specifically focus on linear maps in the
simplex (since logical operations are inherently nonlinear).

##  X X. Multiple systems: the tensor product

When we have more than one qubit, the Hilbert space is given by the tensor
product of the Hilbert space of the individual qubits; i.e., the wavefunctions
will be of the form (for $N$ qubits):

$\displaystyle|\psi\rangle=|\psi_{1}\rangle\otimes|\psi_{2}\rangle\otimes|\psi_{3}\rangle\cdots\otimes|\psi_{N}\rangle\quad.$
(126)

For probabilistic spaces, we combine multiple vectors in an identical way.
This has been discussed and rigorously proven in Ref. barrett1 ; it is also
implicit in the discussion of the mathematical structure of the probability
theory by de Finetti definetti . However, this feature of combining
probability spaces is not widely known. It is usually assumed that the tensor
product is a feature that is very special and specific to quantum mechanics.
For probabilistic spaces, the combined vector in the simplex is given by:

$\displaystyle\vec{s}=\vec{s_{1}}\otimes\vec{s}_{2}\otimes\vec{s}_{3}\cdots\otimes\vec{s}_{N}\quad.$
(127)

##  XI XI. Two-qubit controlled-not (CNOT) gate

In this section, we discuss how to map the two-qubit controlled-not (CNOT)
gate to the probabilistic simplex. When we have two qubits, the Hilbert space
of the quantum system is spanned by states $|00\rangle$, $|01\rangle$,
$|10\rangle$, and $|11\rangle$. We now have four complex coefficients that
describe the state, i.e., our wavefunction is

$\displaystyle|\psi\rangle=c_{00}|00\rangle+c_{01}|01\rangle+c_{10}|10\rangle+c_{11}|11\rangle\quad,$
(128)

or equivalently in vector notation, we can express the wavefunction as:

$\displaystyle|\psi\rangle=\left(\begin{array}[]{c}c_{00}\\\ c_{01}\\\
c_{10}\\\ c_{11}\end{array}\right)\quad.$ (133)

The two-qubit CNOT gate maps $|00\rangle\rightarrow|00\rangle$,
$|01\rangle\rightarrow|01\rangle$, $|10\rangle\rightarrow|11\rangle$, and
$|11\rangle\rightarrow|10\rangle$. Here, the first bit is the control bit and
the second bit is the target bit. In matrix notation, the gate is accomplished
using the following unitary matrix:

$\displaystyle\hat{U}_{CNOT}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&0&1\\\ 0&0&1&0\end{array}\right)\quad.$ (138)

Using our formalism, we map each qubit to a simplex of dimension 8: i.e., the
corresponding simplex dimension for the multi-partite system,
$\vec{s}=\vec{s}_{1}\otimes\vec{s}_{2}$, is $8\times 8=64$. To make the
notation analogous to the two-qubit Hilbert space, we write each simplex
vector as:

$\displaystyle\vec{s}_{i}=\tilde{S}_{8\times
8}\left(\begin{array}[]{c}\vec{0}_{i}\\\
\vec{1}_{i}\end{array}\right),\quad\tilde{S}_{8\times
8}=\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\
0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0\\\
0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&1\\\ \end{array}\right)\quad.$ (149)

Here, both $\vec{0}_{i}$ and $\vec{1}_{i}$ are column vectors with four
entries:
$\vec{0}_{i}=(1+x^{i}_{0}\,\,1-x^{i}_{0}\,\,1+y^{i}_{0}\,\,1-y^{i}_{0})^{T}/8$
stores the real and imaginary parts of the complex coefficient for state
$|0\rangle$ for the $i$’th qubit, while the vector
$\vec{1}_{i}=(1+x^{i}_{1}\,\,1-x^{i}_{1}\,\,1+y^{i}_{1}\,\,1-y^{i}_{1})^{T}/8$
stores the information for state $|1\rangle$. Because of the unique way the
mapping of Eq. (2) is defined, we need the $8\times 8$ “shuffling” matrix
$\tilde{S}_{8\times 8}$ in Eq. (27). The complete basis states for the two-
qubit Hilbert space mapped to the simplex can similarly be written as:

$\displaystyle\vec{s}=\vec{s}_{1}\otimes\vec{s}_{2}=\tilde{S}_{64\times
64}\left(\begin{array}[]{c}\vec{0}_{1}\otimes\vec{0}_{2}\\\
\vec{0}_{1}\otimes\vec{1}_{2}\\\ \vec{1}_{1}\otimes\vec{0}_{2}\\\
\vec{1}_{1}\otimes\vec{1}_{2}\end{array}\right)\quad.$ (154)

Here, we note that each of these entries in the column of Eq. (28) is of
dimension $4\times 4=16$, and the total dimension is, $64$, as required. We
have found that the new “shuffling” matrix $\tilde{S}_{64\times 64}$ can be
expressed as:

$\tilde{S}_{64\times 64}=(\tilde{S}_{8\times 8}\otimes\tilde{S}_{8\times
8})(\tilde{I}_{2\times 2}\otimes\tilde{S}_{8\times 8}\otimes\tilde{I}_{4\times
4})\quad.$ (155)

Here, the quantities $\tilde{I}_{2\times 2}$ and $\tilde{I}_{4\times 4}$ are
$2\times 2$ and $4\times 4$ identity matrices, respectively. The corresponding
matrix that would implement the equivalent of CNOT gate would be a $64\times
64$ matrix, acting on vectors with the form of above. In an analogous way to
the CNOT gate of Eq. (26), this $64\times 64$ matrix will be:

$\displaystyle\tilde{M}_{CNOT}=\left(\begin{array}[]{cccc}\tilde{I}_{16\times
16}&0&0&0\\\ 0&\tilde{I}_{16\times 16}&0&0\\\ 0&0&0&\tilde{I}_{16\times 16}\\\
0&0&\tilde{I}_{16\times 16}&0\end{array}\right)\quad.$ (160)

Here, the quantity $\tilde{I}_{16\times 16}$ is the $16\times 16$ identity
matrix. With the matrix of Eq. (30), we can perform an analogous operation on
the simplex vector using the transformation,
$\vec{s}\longrightarrow\tilde{S}_{64\times
64}\tilde{M}_{CNOT}\tilde{S}_{64\times 64}^{-1}\cdot\vec{s}$. We note that
differing from the single-qubit discussion of above, the “two-qubit”
transformation of the simplex state is a linear transformation, and is a
stochastic matrix.

We also note that, in general, with an initial separable simplex vector of the
form $\vec{s}=\vec{s}_{1}\otimes\vec{s}_{2}$, the transformation
$\vec{s}\longrightarrow\tilde{S}_{64\times
64}\tilde{M}_{CNOT}\tilde{S}_{64\times 64}^{-1}\cdot\vec{s}$, would produce a
simplex vector that can no longer be written as the tensor product of two
individual simplex vectors (i.e., $\vec{s}\neq\vec{s}_{1}\otimes\vec{s}_{2}$).
Mathematically, this is identical to a CNOT gate producing entanglement in a
two-qubit system.

In this section, we discussed the most straightforward way to map multiple
qubits to the probability space. Each qubit is mapped to an 8-dimensional
probabilistic vector, and the tensor product of qubits is mapped to a tensor
product of these 8 dimensional simplex vectors. For example, as we detailed
above, using this approach, two qubits are mapped to a probability space of
dimension $8\times 8=64$. However, it is clear that this manner of mapping
multiple qubits to the simplex is not efficient. This is because, two qubits
form a Hilbert space of dimension 4 (i.e., we have 4 complex coefficients),
and using an extension of the mapping of Eq. (2), we should need a $4\times
4=16$ dimensional simplex vector to store these 4 complex coefficients.
Mathematically, this means that while the approach that we used here is
intuitive and has a rigorous theoretical foundation as described in Ref.
barrett1 , it does not formally preserve the map $\varphi:H\rightarrow S$,
i.e.,
$\varphi(\otimes_{i=1}^{N}\ket{\psi_{i}})\neq\otimes_{i=1}^{n}\vec{s}_{i}$. We
have found a more efficient way for combining simplex vectors that resembles
the tensor product rule and simultaneously preserves the map $\varphi$. We
will discuss this approach and its’ specific application to the CNOT gate in
Appendices A and B.

##  XII XII. Infinitesimal evolution: the Schrodinger’s Equation for a single
qubit

To derive the Schrodinger’s equation for infinitesimal evolution in the
probability space, we will re-express both the mapping and transformations in
a more compact form. We first rewrite the mapping $\varphi$ of Eq. (2)
explicitly in terms of the real and imaginary components of the wavefunction.
Defining $\vec{x}\equiv\Re{\ket{\psi}}$ and $\vec{y}\equiv\Im{\ket{\psi}}$:

$\varphi(\ket{\psi})=\varphi(\vec{x}+i\vec{y})=\vec{s}=\frac{1}{8}\left(\begin{array}[]{c}1\\\
1\\\ 1\\\
\vdots\end{array}\right)+\frac{1}{8}\left(\begin{array}[]{r}\vec{x}\\\
-\vec{x}\\\ \vec{y}\\\
-\vec{y}\end{array}\right)=\frac{1}{8}(\vec{1}+\vec{p})\quad.$ (161)

and in algebraic form,

$\varphi(\ket{\psi})=\frac{1}{8}(\vec{1}+\vec{r}\otimes\Re{\ket{\psi}}+\vec{i}\otimes\Im{\ket{\psi}}),$
(162)

where, $\vec{r}=(1\,\,-1\,\,0\,\,0)^{T}$ and
$\vec{i}=(0\,\,0\,\,1\,\,-1)^{T}$.

Given a general unitary matrix $\hat{U}$ acting on a quantum state vector
$\ket{\psi}$, we note that the real and imaginary parts of the wavefunction
will transform as:

$\hat{U}\ket{\psi}=\left[\real(\hat{U})+i\imaginary(\hat{U})\right](\vec{x}+i\vec{y})=\left[\real(\hat{U})\vec{x}-\imaginary(\hat{U})\vec{y}\right]+i\left[\real(\hat{U})\vec{y}+\imaginary(\hat{U})\vec{x}\right]\quad.$
(163)

Here, the quantities $\real(\hat{U})$ and $\imaginary(\hat{U})$ are the real
and imaginary components of the evolution operator $\hat{U}$, respectively.
This implies that, under general unitary evolution, the real and imaginary
parts of the wavefunction will evolve as:

$\displaystyle\vec{x}$ $\displaystyle\longrightarrow$
$\displaystyle\left[\real(\hat{U})\vec{x}-\imaginary(\hat{U})\vec{y}\right]\quad$
$\displaystyle\vec{y}$ $\displaystyle\longrightarrow$
$\displaystyle\left[\real(\hat{U})\vec{y}+\imaginary(\hat{U})\vec{x}\right]\quad.$
(164)

For the mapped vector $\vec{s}$ in the simplex, the above evolution of the
real and imaginary parts of the wavefunction implies the following
transformation of the vector $\vec{p}$:

$\vec{p}\longrightarrow\left(\begin{array}[]{c c c
c}\real(\hat{U})&O&O&\imaginary(\hat{U})\\\
O&\real(\hat{U})&\imaginary(\hat{U})&O\\\
\imaginary(\hat{U})&O&\real(\hat{U})&O\\\
O&\imaginary(\hat{U})&O&\real(\hat{U})\end{array}\right)\left(\begin{array}[]{r}\vec{x}\\\
-\vec{x}\\\ \vec{y}\\\
-\vec{y}\end{array}\right)=\tilde{M}(\hat{U})\cdot\vec{p}\quad.$ (165)

We also note that due to the structure of the $\vec{p}$ vector, the following
two transformations are equivalent:

$\left(\begin{array}[]{c c c c}\real(\hat{U})&O&O&\imaginary(\hat{U})\\\
O&\real(\hat{U})&\imaginary(\hat{U})&O\\\
\imaginary(\hat{U})&O&\real(\hat{U})&O\\\
O&\imaginary(\hat{U})&O&\real(\hat{U})\end{array}\right)\left(\begin{array}[]{r}\vec{x}\\\
-\vec{x}\\\ \vec{y}\\\ -\vec{y}\end{array}\right)=\left(\begin{array}[]{c c c
c}O&-\real(\hat{U})&O&\imaginary(\hat{U})\\\
-\real(\hat{U})&O&\imaginary(\hat{U})&O\\\
\imaginary(\hat{U})&O&O&-\real(\hat{U})\\\
O&\imaginary(\hat{U})&-\real(\hat{U})&O\end{array}\right)\left(\begin{array}[]{r}\vec{x}\\\
-\vec{x}\\\ \vec{y}\\\ -\vec{y}\end{array}\right)\quad.$ (166)

As a result of the above equivalence, we consider both of the matrices to be
equivalent definitions of the transformation $\tilde{M}(\hat{U})$ which is
associated with a general evolution of the wavefunction by the operator
$\hat{U}$:

$\tilde{M}(\hat{U})=\left(\begin{array}[]{c c c
c}\real(\hat{U})&O&O&\imaginary(\hat{U})\\\
O&\real(\hat{U})&\imaginary(\hat{U})&O\\\
\imaginary(\hat{U})&O&\real(\hat{U})&O\\\
O&\imaginary(\hat{U})&O&\real(\hat{U})\end{array}\right)=\tilde{\mathcal{R}}\otimes\real(\hat{U})+\tilde{\mathcal{I}}\otimes\imaginary(\hat{U})\quad,$
(167)

or,

$\tilde{M}(\hat{U})=\left(\begin{array}[]{c c c
c}O&-\real(\hat{U})&O&\imaginary(\hat{U})\\\
-\real(\hat{U})&O&\imaginary(\hat{U})&O\\\
\imaginary(\hat{U})&O&O&-\real(\hat{U})\\\
O&\imaginary(\hat{U})&-\real(\hat{U})&O\end{array}\right)=-\tilde{\mathcal{I}}^{2}\otimes\real(\hat{U})+\tilde{\mathcal{I}}\otimes\imaginary(\hat{U})\quad.$
(168)

Above we have used two $4\times 4$ matrices $\tilde{\mathcal{R}}$ and
$\tilde{\mathcal{I}}$ to simplify the notation:

$\tilde{\mathcal{R}}=\tilde{I}_{4\times 4}=\left(\begin{array}[]{c c c
c}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right),\quad\tilde{\mathcal{I}}=\left(\begin{array}[]{c c c
c}0&0&0&1\\\ 0&0&1&0\\\ 1&0&0&0\\\ 0&1&0&0\end{array}\right)\quad.$ (169)

We note again that, given the transformation matrix associated with arbitrary
evolution of the quantum state, $\tilde{M}(\hat{U})$, the full transformation
of the simplex vector is given by
$T_{\hat{U}}(\vec{s})=\frac{1}{8}\left[\tilde{I}-\tilde{M}(\hat{U})\right]\cdot\vec{1}+\tilde{M}(\hat{U})\cdot\vec{s}$.
We now rigorously prove the cascading theorem as follows:

$\displaystyle\tilde{M}(\hat{U}_{2})\tilde{M}(\hat{U}_{1})=[\tilde{\mathcal{R}}\otimes\real(\hat{U}_{2})+\tilde{\mathcal{I}}\otimes\imaginary(\hat{U}_{2})][\tilde{\mathcal{R}}\otimes\real(\hat{U}_{1})+\tilde{\mathcal{I}}\otimes\imaginary(\hat{U}_{1})]$
$\displaystyle=\tilde{\mathcal{R}}\otimes\real(\hat{U}_{2})\real(\hat{U}_{1})+\tilde{\mathcal{I}}\otimes(\real(\hat{U}_{2})\imaginary(\hat{U}_{1})+\imaginary(\hat{U}_{2})\real(\hat{U}_{1}))+\tilde{\mathcal{I}}^{2}\otimes\imaginary(\hat{U}_{2})\imaginary(\hat{U}_{1})$
$\displaystyle\equiv\tilde{\mathcal{R}}\otimes(\real(\hat{U}_{2})\real(\hat{U}_{1})-\imaginary(\hat{U}_{2})\imaginary(\hat{U}_{1}))+\tilde{\mathcal{I}}\otimes(\real(\hat{U}_{2})\imaginary(\hat{U}_{1})+\imaginary(\hat{U}_{2})\real(\hat{U}_{1}))$
$\displaystyle=\tilde{\mathcal{R}}\otimes\real(\hat{U}_{2}\hat{U}_{1})+\tilde{\mathcal{I}}\otimes\imaginary(\hat{U}_{2}\hat{U}_{1})=\tilde{M}(\hat{U}_{2}\hat{U}_{1})\quad.$
(170)

In this derivation, we have used the equivalence established above in Eqs.
(37) and (38).

In order to derive the analogue of Schrodinger’s equation in the simplex, we
follow the above formalism for infinitesimal evolution of the quantum state
$\ket{\psi(t)}$, under the presence of some general Hamiltonian $\hat{H}(t)$.
The time evolution of the wavefunction, which is determined by the
Schrodinger’s equation, is:

$i\hbar\frac{d}{dt}\ket{\psi(t)}=\hat{H}(t)\ket{\psi(t)}\quad.$ (171)

The transformation of the state from $\ket{\psi(t)}$ to $\ket{\psi(t+\delta
t)}$ is due to the action of the differential unitary operator
$\delta\hat{U}=\exp[-i\hat{H}(t)/\hbar\,\delta t]$. Correspondingly the
differential transformation $\tilde{M}(\delta\hat{U})$ must act on the vector
$\vec{p}(t)$, to produce a vector $\vec{p}(t+\delta t)$:

$\vec{p}(t+\delta
t)=\tilde{M}(\delta\hat{U})\cdot\vec{p}(t)=\tilde{M}[\hat{I}-i\hat{H}(t)/\hbar\,\delta
t]\cdot\vec{p}(t)=\vec{p}(t)-\frac{\delta
t}{\hbar}\tilde{M}(i\hat{H}(t))\cdot\vec{p}(t)\quad.$ (172)

We then evaluate $\tilde{M}(i\hat{H}(t))$ in Eq. (42) using Eq. (35) and take
the $\delta t\rightarrow 0$ limit to obtain the following differential
equation for $\vec{p}(t)$:

$\hbar\frac{d\vec{p}(t)}{dt}=[\tilde{\mathcal{R}}\otimes\imaginary(\hat{H}(t))-\tilde{\mathcal{I}}\otimes\real(\hat{H}(t))]\cdot\vec{p}(t)\equiv\tilde{H}_{eff}(t)\cdot\vec{p}(t)\quad.$
(173)

Here, we have defined
$\tilde{H}_{eff}(t)=\tilde{\mathcal{R}}\otimes\imaginary(\hat{H}(t))-\tilde{\mathcal{I}}\otimes\real(\hat{H}(t))$,
which can be viewed as the effective “Hamiltonian” operator in the simplex. We
next use Eq. (43), to find the differential equation for the full simplex
vector, $\vec{s}(t)$. Using
$\vec{s}(t)=\frac{1}{8}\left[\vec{1}+\vec{p}(t)\right]$, we have
$\dot{\vec{s}}=\frac{1}{8}\dot{\vec{p}}$ for the time derivative of the full
simplex vector. Using this time derivative in Eq. (43), we finally arrive at:

$\displaystyle\hbar\frac{d\vec{s}(t)}{dt}=\tilde{H}_{eff}(t)\cdot(\vec{s}-\frac{1}{8}\vec{1})\quad.$
(174)

##  XIII XIII. Schrodinger’s Equation for a general quantum state of dimension
K

Equation (44) is valid for the explicit mapping of a single qubit to an 8
dimensional probability simplex of Eq. (2). We note that it is straightforward
to generalize this result. For a Hilbert space of dimension $K$, we would have
$K$ complex coefficients to describe an any wavefunction that lies in this
space. We can then map the real and imaginary parts of these coefficients to a
probability simplex of dimension $4K$.

Specifically, for a wavefunction of dimension $K$, we can define its’ real and
imaginary parts in exactly the same manner $\vec{x}_{K}\equiv\Re{\ket{\psi}}$
and $\vec{y}_{K}\equiv\Im{\ket{\psi}}$. The map $\varphi(\ket{\psi})$ is then
given by a similar expression as Eq. (31) of above:

$\varphi(\ket{\psi})=\varphi(\vec{x}_{K}+i\vec{y}_{K})=\vec{s}=\frac{1}{4K}\left(\begin{array}[]{c}1\\\
1\\\ 1\\\
\vdots\end{array}\right)+\frac{1}{4K}\left(\begin{array}[]{r}\vec{x}_{K}\\\
-\vec{x}_{K}\\\ \vec{y}_{K}\\\
-\vec{y}_{K}\end{array}\right)=\frac{1}{4K}(\vec{1}_{K}+\vec{p})\quad.$ (175)

Following the identical steps outlined above in Section XII, we can then
derive the time evolution equation for the $4K$ dimensional simplex state:

$\displaystyle\hbar\frac{d\vec{s}(t)}{dt}=\tilde{H}_{eff}(t)\cdot(\vec{s}-\frac{1}{4K}\vec{1}_{K})\quad.$
(176)

We note that the time evolution of Eq. (46) preserves the probabilistic norm
of the simplex vector, that is $\sum_{i}s_{i}(t)=1$ at all times. This can be
proven using:

$\displaystyle\frac{d\vec{s}}{dt}=\frac{1}{4K}\frac{d\vec{p}}{dt}\implies\vec{1}_{K}^{T}\cdot\frac{d\vec{s}}{dt}=\frac{1}{4K}\vec{1}_{K}^{T}\cdot\frac{d\vec{p}}{dt}=\frac{1}{4K\hbar}\vec{1}_{K}^{T}\cdot\tilde{H}_{eff}\cdot\vec{p}=0\quad,$
$\displaystyle\therefore\frac{d}{dt}(\sum_{i}s_{i}(t))=0\implies\sum_{i}s_{i}(t)=\sum_{i}s_{i}(0)=1\quad.$
(177)

##  XIV XIV. Relation to prior work

Since the inception of quantum mechanics, the precise meaning of the
wavefunction $|\psi\rangle$ has been a source of constant debate. Does the
wavefunction correspond to something real (the ontological interpretation)
montina ; rudolph2 ; rudolph3 , or is it simply a tool to summarize an
observer’s incomplete information about a system (the epistemological
interpretation) spekkens1 . This work firmly supports the epistemological
interpretation of the wavefunction. We have shown that the initial state and
the dynamics of a single qubit can be completely captured using an
8-dimensional probabilistic simplex. It is the thesis of this work the
traditional mathematical formulation of quantum theory, i.e., complex vectors
that live in a Hilbert space, is not needed. The problem can instead be
formulated as a real vector in a probability space and traditional features of
quantum theory, such as interference of complex amplitudes, can be captured in
this space. As also discussed by other authors barrett1 , the identical nature
of the tensor product for combining separate systems, also strongly favors the
epistemological interpretation.

Within this context, perhaps the closest relation to our work is the “toy”
theory of Spekkens, which captures some features of a quantum two-level system
using four epistemological states, based on the knowledge balance principle
spekkens1 . Ontological, hidden variable theories of quantum mechanics has
also been widely discussed rudolph1 . Several prominent examples of these are
due to Bell bell2 , Beltrametti-Bugajski bugajski , Kochen-Specker kochen ,
Aaronson aaronson , and Aerts aerts . Of particular importance to this work is
Aaronson’s model aaronson , which discusses representing the quantum state as
a vector of probabilities, and mapping this vector to another set of
probabilities using an appropriate matrix. However, as we mentioned above,
when only represented as a vector of projected probabilities, such a matrix
inevitably depends on the initial state of the wavefunction. We emphasize
again that this is very different from our approach and our results. The
analogs of the quantum gates that we have discussed in the probability space,
such as the Hadamard gate of Eq. (10) and the single-qubit phase gate of Eq.
(16), involve constant translations and constant matrices (i.e., the
information about the initial state, $x_{0},y_{0},x_{1},y_{1}$ does not appear
in these matrices). We believe this is crucial for the approach presented
here. Consider a quantum system that evolves through any number of gates. One
can construct probabilistic circuits that would perform exact analogs of these
gates in the probability simplex, and exactly track the quantum evolution in
the simplex. Once the circuits are fixed, if the initial condition for the
quantum state changes, one only needs to correspondingly modify the initial
simplex vector.

We also note that our work has connection to the work of Fuchs and colleagues,
who has argued for a completely subjective interpretation of the quantum
wavefunction fuchs1 ; fuchs2 ; fuchs3 . In this approach, the key ingredient
is a set of projection operators, which are called SICs (abbreviation for
Symmetic Informationally Complete). These operators have a number of fine-
tuned properties. Of particular importance, when one expresses the density
matrix operator for a quantum system as a summation of SICs, the expansion
coefficients explicitly have the measurement outcome probabilities. As a
result, these operators allow for a smooth relation between the measurement
outcome probabilities and the density matrix. There are no other operator
classes that allow such a simple and explicit connection between measurement
probabilities and the density matrix for a quantum system.

We note that here are several key differences between our work and the
approach of SIC operators. (1) The SIC approach focuses on the measurement
outcomes of a quantum mechanical system. As we mentioned above, our manuscript
focuses on capturing the initial state and the dynamics of a quantum system.
(2) To our knowledge, our approach is unique, in the sense that it allows
storing both the amplitude and phase information of the complex coefficients
of a quantum system. This is critical for capturing the single-qubit and two-
qubit gates in the probability simplex, as well as finding the equivalent of
the Schrodinger’s equation in the probability space.

Our work also has an important connection to the large body of literature that
have studied mapping of quantum states to quasiprobability distributions. The
most famous of these is the Wigner function which is in the continuous phase
space. Recent work has focused on negativity of these quasiprobability
distributions, their studies in discrete phase spaces, as well as their
relationship to quantum contextuality bartlett1 ; bartlett2 ; bartlett3 ;
raussendorf1 ; raussendorf2 ; eisert ; zhu ; wootters . It is known that such
quasiprobability distributions can have negative values, and the amount of
negativity is typically viewed as the “quantumness” of the system. The key
difference of our work is that we have discussed that if one considers
appropriate maps, then mapping to a classical probabilistic space is possible
(i.e., negative values are not needed).

We also note that in this work, we have focused on the initial state and the
time evolution of quantum systems and how we can capture these in the
probability space: i.e., we have not considered the measurement aspect of
quantum systems. More specifically, the focus of the current work is on
capturing the initial quantum states and the gates, largely within the context
of an $N$-qubit quantum computer. One key reason for this focus is that when
one analyzes a quantum algorithm in detail (for example, Shor factoring
algorithm nielsen ), the key advantage is in the interference of complex
amplitudes in an exponentially large Hilbert space (i.e., the key advantage is
not in the measurement; the measurement in the end is more like an
afterthought). We also have not gone into a detailed discussion of
entanglement bell1 , nonlocality barrett2 , or contextuality spekkens2 . A
rigorous discussion of these important concepts is beyond the scope of this
work.

##  XV XV. Conclusions and future directions

In conclusion, we have discussed mapping of the initial state and time
evolution of a qubit to the probability simplex. We have discussed analogs of
the single-qubit Hadamard and phase gates, as well as two-qubit CNOT gate in
the probabilistic simplex. We have also discussed how multi-partite quantum
systems can be mapped, as well as the analog of the Schrodinger’s equation
which dictates the time evolution of the simplex vector.

It is well known that the quantum gates that we have discussed here (single-
qubit Hadamard gate, single-qubit phase rotation, and the two-qubit CNOT gate)
form a universal gate set nielsen . That is, arbitrary evolution of an $N$
qubit quantum computer can be approximated with these gates. Due to the
similar nature of the mathematical formalism, perhaps a similar result can be
expected for $N$ simplex vectors: but this is an open question.

We believe that the approach presented here may provide a unique way to
simulate quantum systems that are more efficient than currently possible. For
example, for an $N$-qubit quantum computer, the initial state at the beginning
of a computation would be an unentangled product state of the individual
qubits. We would then map each qubit to an 8-dimensional probabilistic vector,
$\vec{s}$. This could be any 8-dimensional probabilistic system, whose
probabilities (individual entries in the vector) have appropriate values so
that the initial state of each qubit is accurately mapped. The tensor product
of $N$ qubits would then map to a tensor product of $N$ 8-dimensional
probabilistic vectors. The evolution of the quantum computer would proceed as
a sequence of single qubit and two-qubit gates. The simulation that we
envision then is to implement the analogs of these gates in the probabilistic
system, using the appropriate transformations in the simplex that we have
described above. It is an open question how much $N$ simplex vectors (forming
a probability space of dimension $8^{N}$) can efficiently simulate an $N$
qubit quantum computer (a Hilbert space of dimension $2^{N}$).

Within this context, an exciting immediate experimental direction is to
experimentally demonstrate the simplex transformations for a single qubit that
we have discussed. It may be possible to extend the recent experimental work
of Datta and colleagues on probabilistic bits (p-bits) datta1 ; datta2 . One
key goal would be to observe the “Rabi flopping” behavior of Fig. 2 in the
simplex using an appropriate circuit acting on 3 p-bits.

We think it is also possible that progress along the above posed questions
will help clarify the quantum/classical boundary niklas ; mucciolo ; pittenger
, as well as the quantum measurement problem zurek1 ; zurek2 .

##  XVI XVI. Acknowledgements

We would like to thank Ben Lemberger and Volkan Rodoplu for many helpful
discussions. D. D. Yavuz would also would like to thank Bin Yan for an early
discussion on the subject. This work was supported by the National Science
Foundation (NSF) Grant No. 2016136 for the QLCI center Hybrid Quantum
Architectures and Networks (HQAN), and also by the University of Wisconsin-
Madison, through the Vilas Associates award.

##  XVII Appendix A. Multiple systems: The tensor product revisited

Above in Section X, we discussed the most straightforward way to map multiple
qubits to the probability space. Each qubit is mapped to an 8-dimensional
probabilistic vector, and the tensor product of qubits is mapped to a tensor
product of these 8 dimensional simplex vectors. For example, as we detailed
above, using this approach, two qubits are mapped to a probability space of
dimension $8\times 8=64$. However, it is clear that this mapping is not
efficient. This is because, two qubits form a Hilbert space of dimension 4
(i.e., we have 4 complex coefficients), and using an extension of the mapping
of Eq. (2), we should need a $4\times 4=16$ dimensional simplex vector to
store these 4 complex coefficients. Mathematically, this means while the
approach that we used in Section X is intuitive, it does not formally preserve
the map $\varphi:H\rightarrow S$, i.e.,
$\varphi(\otimes_{i=1}^{N}\ket{\psi_{i}})\neq\otimes_{i=1}^{n}\vec{s}_{i}$.

In this section, we discuss an approach to combine multiple simplex vectors
that preserves the map and also closely resembles the tensor product
structure. For this purpose, we first look at the combination scheme of the
real and imaginary parts of the quantum states for the two qubits,
$\ket{\psi_{1}}$ and $\ket{\psi_{2}}$, since the definition of map
$\varphi:H\rightarrow S$ in Eq. (32) explicitly includes the real and
imaginary components. The real and imaginary parts for the wavefunction of the
tensor product is:

$\ket{\psi_{1}}\otimes\ket{\psi_{2}}=(\vec{x}_{1}+i\vec{y}_{1})\otimes(\vec{x}_{2}+i\vec{y}_{2})=(\vec{x}_{1}\otimes\vec{x}_{2}-\vec{y}_{1}\otimes\vec{y_{2}})+i(\vec{x_{1}}\otimes\vec{y}_{2}+\vec{y}_{1}\otimes\vec{x}_{2})\quad.$
(178)

Here, we have defined the real and imaginary parts of the wavefunctions for
each qubit: $\vec{x}_{1}\equiv\Re{\ket{\psi_{1}}}$,
$\vec{y}_{1}\equiv\Im{\ket{\psi_{1}}}$,
$\vec{x}_{2}\equiv\Re{\ket{\psi_{2}}}$, and
$\vec{y}_{2}\equiv\Re{\ket{\psi_{2}}}$. Using an extension of the map of Eq.
(32), this then implies:

$\displaystyle\varphi(\ket{\psi_{1}}\otimes\ket{\psi_{2}})=\frac{1}{16}(\vec{1}_{16}$
$\displaystyle+\vec{r}\otimes[\Re{\ket{\psi_{1}}}\otimes\Re{\ket{\psi_{2}}}-\Im{\ket{\psi_{1}}}\otimes\Im{\ket{\psi_{2}}}]$
$\displaystyle+\vec{i}\otimes[\Re{\ket{\psi_{1}}}\otimes\Im{\ket{\psi_{2}}}+\Im{\ket{\psi_{1}}}\otimes\Re{\ket{\psi_{2}}}])\quad.$
(179)

We note that the above equation is precisely equal to
$\frac{1}{16}(\vec{1}_{16}+\vec{r}\otimes\Re{\ket{\psi_{1}}\otimes\ket{\psi_{2}}}+\vec{i}\otimes\Im{\ket{\psi_{1}}\otimes\ket{\psi_{2}}})$.
Hence we introduce a closed operation which we refer to as the “box product”,
$\boxtimes$, over any two $4m$ ($m\geq 2,m\in\mathbb{N}$) dimensional vectors
that have the form of a typical $\vec{p}$ to preserve the map while combining
systems,

$\vec{p}_{1}\boxtimes\vec{p}_{2}=\left(\begin{array}[]{r}\vec{x}_{1}\\\
-\vec{x}_{1}\\\ \vec{y}_{1}\\\
-\vec{y}_{1}\end{array}\right)\boxtimes\left(\begin{array}[]{r}\vec{x}_{2}\\\
-\vec{x}_{2}\\\ \vec{y}_{2}\\\
-\vec{y}_{2}\end{array}\right)\equiv\left(\begin{array}[]{r}(\vec{x}_{1}\otimes\vec{x}_{2}-\vec{y}_{1}\otimes\vec{y_{2}})\\\
-(\vec{x}_{1}\otimes\vec{x}_{2}-\vec{y}_{1}\otimes\vec{y_{2}})\\\
(\vec{x_{1}}\otimes\vec{y}_{2}+\vec{y}_{1}\otimes\vec{x}_{2})\\\
-(\vec{x_{1}}\otimes\vec{y}_{2}+\vec{y}_{1}\otimes\vec{x}_{2})\end{array}\right)\quad.$
(180)

We note that this operation is closed because the resultant vector,
$\vec{p}_{1}\boxtimes\vec{p}_{2}$ follows the typical form of the $\vec{p}$
vector. We also note that the above defined box product operation remains
closed even if it is extended to the matrices of the usual form
$\tilde{M}(\hat{U})$ for any unitary operator $\hat{U}$. that acts on the
composite quantum system, $\hat{U}=\hat{U}_{1}\otimes\hat{U}_{2}$. To prove
this we start with:

$\tilde{M}(\hat{U}_{1}\otimes\hat{U}_{2})(\vec{p}_{1}\boxtimes\vec{p}_{2})=\tilde{M}(\hat{U}_{1})\boxtimes\tilde{M}(\hat{U}_{2})(\vec{p}_{1}\boxtimes\vec{p}_{2})=\tilde{M}(\hat{U}_{1})\vec{p}_{1}\boxtimes\tilde{M}(\hat{U}_{2})\vec{p}_{2}\quad.$
(181)

We then expand the rightmost expression and equate it term by term with the
middle term to obtain the following form for
$\tilde{M}(\hat{U}_{1})\boxtimes\tilde{M}(\hat{U}_{2})$:

$\displaystyle\tilde{M}(\hat{U}_{1})\boxtimes\tilde{M}(\hat{U}_{2})=\tilde{\mathcal{R}}\otimes[\real(\hat{U}_{1})\otimes\real(\hat{U}_{2})-$
$\displaystyle\imaginary(\hat{U}_{1})\otimes\imaginary(\hat{U}_{2})]+$
$\displaystyle\tilde{\mathcal{I}}\otimes[\real(\hat{U}_{1})\otimes\imaginary(\hat{U}_{2})+\imaginary(\hat{U}_{1})\otimes\real(\hat{U}_{2})]\quad.$
(182)

The expression in Eq. (52) is exactly equal to
$\tilde{M}(\hat{U}_{1}\otimes\hat{U}_{2})$. This is because
$\real(\hat{U}_{1}\otimes\hat{U}_{2})\neq\real(\hat{U}_{1})\otimes\real(\hat{U}_{2})$,
but instead
$\real(\hat{U}_{1}\otimes\hat{U}_{2})=\real(\hat{U}_{1})\otimes\real(\hat{U}_{2})-\imaginary(\hat{U}_{1})\otimes\imaginary(\hat{U}_{2})$.
Similarly, we also have
$\imaginary(\hat{U}_{1}\otimes\hat{U}_{2})=\real(\hat{U}_{1})\otimes\imaginary(\hat{U}_{2})+\imaginary(\hat{U}_{1})\otimes\real(\hat{U}_{2})$.
Hence, in the above defined probabilistic simplex space, $S$, the combined
vector in the simplex is given by:

$\displaystyle\vec{s}=\frac{1}{2^{N+2}}(\vec{1}_{2^{N+2}}+((((\vec{p}_{1}\boxtimes\vec{p}_{2})\boxtimes\vec{p}_{3})\cdots)\boxtimes\vec{p}_{N}))\quad.$
(183)

One important distinctive property of the box product operation versus the
tensor product operation is that it is distributive over addition but not
associative (just like the universal logical operations NAND and NOR).

##  XVIII Appendix B. Two-qubit controlled-not (CNOT) gate revisited

In this section, we will discuss how to map the two-qubit controlled-not gates
to the probability simplex, using the techniques introduced in the previous
section, in Appendix A. As we discussed above, when we have two qubits, the
Hilbert space of the quantum system is of dimension four and it is spanned by
states $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$. We can,
therefore, represent the wavefunction as $\ket{\psi}=\vec{x}+i\vec{y}$, where
$\vec{x}$ and $\vec{y}$ are each of vectors of dimension $4$. In the box
product formalism, the dimension of the simplex space would be $4\times 4=16$
and in that space the matrix of transformation would be
$\tilde{M}(\hat{U}_{CNOT})=\tilde{R}\otimes\hat{U}_{CNOT}$, since
$\hat{U}_{CNOT}$ is real,

$\tilde{M}(\hat{U}_{CNOT})=\left(\begin{array}[]{cccc}\hat{U}_{CNOT}&O&O&O\\\
O&\hat{U}_{CNOT}&O&O\\\ O&O&\hat{U}_{CNOT}&O\\\
O&O&O&\hat{U}_{CNOT}\end{array}\right)\quad.$ (184)

As our next example, we consider a general conditional unitary gate which can
be written as:
$\hat{C}_{U}=\hat{P}_{0}\otimes\hat{I}+\hat{P}_{1}\otimes\hat{U}$. The two-
qubit transformation matrix for this gate, $\tilde{M}(\hat{C}_{U})$ is given
by:

$\displaystyle\tilde{M}(\hat{C}_{U})=\tilde{M}(\hat{P}_{0}\otimes\hat{I}+\hat{P}_{1}\otimes\hat{U})=\tilde{M}(\hat{P}_{0}\otimes\hat{I})+\tilde{M}(\hat{P}_{1}\otimes\hat{U})$
$\displaystyle\Rightarrow$
$\displaystyle\tilde{M}(\hat{C}_{U})=\tilde{\mathcal{R}}\otimes[\hat{P}_{0}\otimes\hat{I}]+\tilde{\mathcal{R}}\otimes[\hat{P}_{1}\otimes\real(\hat{U})]+\tilde{\mathcal{I}}\otimes[\hat{P}_{1}\otimes\imaginary(\hat{U})]$
$\displaystyle\Rightarrow$
$\displaystyle\tilde{M}(\hat{C}_{U})=\tilde{\mathcal{R}}\otimes\hat{C}_{\real(U)}+\tilde{\mathcal{I}}\otimes\hat{P}_{1}\otimes\imaginary(\hat{U})\quad.$
(185)

In matrix form this can be represented as:

$\tilde{M}(\hat{C}_{U})=\left(\begin{array}[]{cccc}\hat{C}_{\real(U)}&O&O&\hat{P}_{1}\otimes\imaginary(\hat{U})\\\
O&\hat{C}_{\real(U)}&\hat{P}_{1}\otimes\imaginary(\hat{U})&O\\\
\hat{P}_{1}\otimes\imaginary(\hat{U})&O&\hat{C}_{\real(U)}&O\\\
O&\hat{P}_{1}\otimes\imaginary(\hat{U})&O&\hat{C}_{\real(U)}\end{array}\right)\quad.$
(186)

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