[1]Shikha Bangar

[1]Department of Physics and Astronomy, University of Tennessee, 1408 Circle
Drive, Knoxville, Tennessee, 37996-1200, USA

2]Emerging Technologies and Physical Sciences, MITRE Corporation, 7515
Colshire Drive, McLean, Virginia, 22102-7539, USA

# Continuous-variable Quantum Boltzmann Machine

<EMAIL_ADDRESS>Leanto Sunny<EMAIL_ADDRESS>Kübra Yeter-Aydeniz
<EMAIL_ADDRESS>George Siopsis<EMAIL_ADDRESS>*  [

###### Abstract

We propose a continuous-variable quantum Boltzmann machine (CVQBM) using a
powerful energy-based neural network. It can be realized experimentally on a
continuous-variable (CV) photonic quantum computer. We used a CV quantum
imaginary time evolution (QITE) algorithm to prepare the essential thermal
state and then designed the CVQBM to proficiently generate continuous
probability distributions. We applied our method to both classical and quantum
data. Using real-world classical data, such as synthetic aperture radar (SAR)
images, we generated probability distributions. For quantum data, we used the
output of CV quantum circuits. We obtained high fidelity and low Kuller-
Leibler (KL) divergence showing that our CVQBM learns distributions from given
data well and generates data sampling from that distribution efficiently. We
also discussed the experimental feasibility of our proposed CVQBM. Our method
can be applied to a wide range of real-world problems by choosing an
appropriate target distribution (corresponding to, e.g., SAR images, medical
images, and risk management in finance). Moreover, our CVQBM is versatile and
could be programmed to perform tasks beyond generation, such as anomaly
detection.

###### keywords:

quantum Boltzmann machine, continuous variable, quantum machine learning,
photonic quantum computer, SAR images

##  1 Introduction

Boltzmann machines (BMs) are robust machine learning models which provide
modeling of probability distributions. BMs consist of a probabilistic network
that is represented as an undirected graph with hidden and visible nodes. In
this machine learning model, the probability distribution of the data is
approximated based on a finite set of samples. The information is encoded into
the bias coefficients and weights of the neural network. After a successful
training process, the learned distribution resembles the actual distribution
of the data that can make correct predictions about unseen instances. However,
the generalization of the model suffers from a growing number of parameters
and the training of a classical BM can become impractical. For this reason, it
is worth exploring engineering quantum Hamiltonian Ansätze to construct
quantum BMs [1]. In QBMs, the goal is to approximate the thermal state of the
Hamiltonian Ansatz to the target density matrix of the distribution through
training of the weights of the Hamiltonian Ansatz. One of the advantages of
QBMs over their classical counterparts is that QBMs have higher expressive
power, i.e., can fit a wider range of complex data patterns, since the
Hamiltonian can include more non-commuting terms [2].

Preparing a thermal state and computing its properties is a challenging task.
Hence, various heuristic methods that are compatible with noisy intermediate
scale quantum (NISQ) hardware have been proposed [3]. These methods include
quantum imaginary time evolution (QITE) [4], variational QITE (varQITE) [5],
and the variational quantum eigensolver ($\beta-$VQE) [6]. The thermal states
prepared utilizing these tools have also been used in realizing QBMs. For
instance, in Ref. [7], a variational QBM was proposed and utilized for a
discriminative learning task, specifically fraud detection, as well as a
generative learning task, specifically learning a Bell state probability
distribution. The method was implemented on IBM Quantum hardware. This work
was followed by several studies including [3, 8]. In Ref. [8], similar to [7],
they utilized variational QITE evolution to study QBMs and tested their model
using the bars and stripes (BAS) dataset. They focused on the classical cases
when the Hamiltonian for the QBM has only diagonal terms in the computational
basis, and reduced the required number of qubits by half by preparing an
initial pure state which was a superposition of all states. On the other hand,
in Ref. [3], $\beta-$VQE was used as a subroutine for preparing the Gibbs
state needed for QBM. In their QBM model, free energy, or equivalently quantum
relative entropy, was used as the loss function. They tested their model both
on quantum and classical data. It was proposed that, once the classical
dataset is embedded into the quantum circuit, a lower rank $\beta-$VQE can be
utilized for training of QBM. A similar feature was also observed for low
temperature quantum data. However, for higher temperature quantum data, a
higher rank $\beta-$VQE is required. They also tested their QBM model on a
real-world dataset, namely salamander retina, and demonstrated that
$\beta-$VQE QBM achieves lower Kuller-Leibler (KL) divergence value compared
to a classical Boltzmann machine. In Ref. [2], the sample complexity of QBM
machine learning was studied. It was shown that QBMs do not possess barren
plateaus (optimization landscape becoming flat due to various reasons [9]) and
that pre-training on a subset of the QBM parameters can only lower the sample
complexity bounds. In addition to gate based quantum computing, QBMs have been
demonstrated using quantum annealers. In Ref. [10], a restricted QBM was
utilized for balancing the imbalanced credit card transactions data through
synthetic data generation that can be utilized for fraud detection and other
cyber security applications.

Studies of quantum generative models have mainly focused on the generation of
discrete probability distribution instances. One example is Ref. [11], where a
variational quantum generative adversarial network (GAN) model was utilized
for the generation of the probability distribution of a real dataset obtained
using a variational quantum generator with fixed parameters. In Ref. [12],
this method was shown to be robust to quantum hardware noise by implementing
it on Rigetti quantum hardware. However, most real-world applications, such as
image and sound generation, computational fluid dynamics, study continuous
probability distributions. Use of discrete-variable (DV) quantum computing
(based on qubits) is only partially suited for continuous-valued data. One can
avoid the burden of encoding continuous valued classical data into DV by using
CV quantum machine learning tools. For instance, in Ref. [13], they studied
time series forecasting for continuous datasets, such as energy consumption
and stock price data. It is more natural to extend generative models to
continuous-variable quantum computing (CVQC) models for the study of
continuous data. In CVQC, quantum information is encoded into infinite-
dimensional qumodes, such as the electromagnetic field of photons in photonic
quantum hardware. In Ref. [14], a CV quantum Born machine was developed which
offered an efficient way of modeling continuous probability distributions,
such as a Gaussian distribution. By choosing a Gaussian kernel, it was
demonstrated that a CV quantum Born machine naturally provides a much better
fit to a Gaussian distribution with a single qumode compared to a 10-qubit
quantum circuit Born machine model. It was also shown that a 3-qumode Gaussian
and cubic kernel CV quantum Born machine learns a quantum target distribution
much faster than by using a classical Gaussian kernel method.

In this work, we discuss a proposal for a continuous-variable quantum
Boltzmann machine (CVQBM). Our CV model is a natural fit for studying
continuous probability distributions. We utilize our method for generative
modeling in real-world applications, such as SAR image probability
distribution generation, as well as probability distribution of data obtained
from prepared quantum states. To the best of our knowledge, this is the first
demonstration of generation of various real-world applications’ continuous
probability distributions using a CV quantum generative model. Even though
experiments were conducted on a simulator, we discuss experimental feasibility
of our simulations and demonstrate that problems studied in this work can be
implemented on experimentally feasible photonic quantum computing
architectures. In our model, the CV quantum circuit we employ demonstrates a
CV quantum generative model, i.e., CVQBM, and also presents a variational CV
QITE algorithm utilized for finite temperature state preparation. Our
variational CV QITE algorithm can be leveraged in other use cases, such as
ground state preparation in quantum field theories [15].

Our discussion is organized as follows. In Section 2, we provide the details
of the proposed CVQBM model. We design the CVQBM quantum circuit and discuss
the procedure for setting it up as a continuous probability distribution
generator. In Section 3, we present various case studies using our CVQBM,
focusing on classical data from the SAR images. We generate data for forest
and sea monitoring. In Section 4, we discuss quantum data, including both
Gaussian and non-Gaussian cases. An essential aspect of our proposed CVQBM is
that it is experimentally feasible on CV photonic quantum computers, which is
discussed in Section 5. We introduce noise to analyze the robustness of our
CVQBM, and discuss the success rate and gate parameters chosen in our case
studies. Finally, in Section 6, we present our conclusions and indicate future
directions.

##  2 The Method

In this Section, we outline the design of a quantum circuit for the CVQBM and
set up the machine as a continuous distribution generator. We show that the
output can be thought of as a thermal state of an effective Hamiltonian
defined by the quantum circuit. This is similar to the approximate Gibbs state
that was prepared using a variational quantum imaginary time evolution
(varQITE) algorithm with discrete variables [7].

The CVQBM circuit is designed to prepare a thermal state,
$\frac{1}{\mathcal{Z}}e^{-\beta H(\vec{\zeta})}$, where
$\mathcal{Z}=\mathrm{Tr}e^{-\beta H(\vec{\zeta})}$, for a Hamiltonian that
depends on parameters collectively denoted as the vector $\vec{\zeta}$. For
simplicity, we set the inverse temperature $\beta=1$. This state represents
quantum imaginary time evolution (QITE) which is not unitary. The parameters
$\vec{\zeta}$ control the quantum gates of the CVQBM circuit, which are
unitary, and are trained according to the problem at hand. We achieve non-
unitary evolution by performing measurements on ancilla qumodes. Here, we
concentrate on a Hamiltonian that depends on a single (visible) qumode of
quadratures $(q_{v},p_{v})$. Extension of our method to include multiple
qumodes is straightforward.

Figure 1: CVQBM framework: Step 1 is the training of the CVQBM model by
training the parameters of the quantum circuit, which prepares an entangled
state between the visible and hidden modes of the CVQBM, and then performs
imaginary time evolution, minimizing the cost function (15), calculated using
the density matrix of the data and the generated density matrix. In Step 2,
the optimal quantum circuit parameters are used in the same quantum circuit
for CVQBM to generate a probability distribution. Step 3 involves verification
of how close the generated probability distribution is to the target
distribution by using metric KL divergence. For embedding quantum data we also
propose a gadget $\mathcal{G}$ (Figure 6) that is implemented on the visible
node following the QITE steps.

(a)

(b)

Figure 2: Designing the CVQBM circuit, detailed structure of (a) $U_{0}$ where
two modes are entangled by using squeezed gates and then applying the CX gate
and (b) $\mathcal{T}_{s}(\vec{\zeta}_{s})$ where the gates have trainable
parameters collectively represented by $\zeta$. An ancillary mode is added,
which introduces non-linearity into the network by entanglement (CX gate) and
photon number measurement.

The CVQBM circuit is depicted in Figure 1. In addition to the visible qumode,
it uses an ancilla qumode of quadratures $(q_{h},p_{h})$ representing a hidden
mode. Both modes are initially in the vacuum state. They are entangled with
the two-mode unitary $U_{0}$. No further action is taken on the hidden mode.
The visible mode is taken through a series of non-unitary quantum gates
$\mathcal{T}_{s}(\vec{\zeta}_{s})$, where $s=1,2,\dots$, that depend on
parameters $\vec{\zeta}_{s}$ to be trained. Next, we proceed to describe the
various quantum gates used in detail.

We start by engineering the entangled state:

$\ket{\psi_{0}}_{vh}=\frac{1}{\sqrt{\mathcal{N}}}\sum_{n=0}^{\infty}e^{-\delta
n}\ket{n}_{v}\ket{n}_{h}\ ,\ \ \mathcal{N}=\frac{1}{1-e^{-2\delta}}$ (1)

where $\delta>0$ is a fixed constant, and $\ket{n}$ is the eigenstate of the
number operator $N$ with $n$ photons. How this is implemented is shown in
Figure 2a. We use two squeezed modes, one squeezed in the $q$ quadrature with
squeezing parameter $r_{v}=-\frac{1}{2}\log\delta$, and the other one squeezed
in the $p$ quadrature with $r_{h}=\frac{1}{2}\log\delta$. We entangle them
with the CX gate $e^{iq_{v}p_{h}}$ to bring them in the state

$\frac{1}{\sqrt{\pi}}\int
dq_{v}dq_{h}e^{-\frac{\delta}{2}q_{v}^{2}}e^{-\frac{1}{2\delta}q_{h}^{2}}\ket{q_{v}}_{v}\ket{q_{v}+q_{h}}_{h}~{}.$
(2)

For small $\delta$, this is approximately $\ket{\psi_{0}}$ (Eq. (1)). This is
easily seen by switching to the number operator basis, in which this state can
be written as

$\frac{1}{\sqrt{\pi}}\sum_{n_{v},n_{h}}\int
dq_{v}dq_{h}e^{-\frac{\delta}{2}q_{v}^{2}}e^{-\frac{1}{2\delta}q_{h}^{2}}\braket{q_{v}}{n_{v}}\braket{n_{h}}{q_{v}+q_{h}}\ket{n_{v}}_{v}\ket{n_{h}}_{h}~{},$
(3)

expressing the inner products (harmonic oscillator wavefunctions in the
position representation) as

$\Psi_{n}(q)=\braket{q}{n}=\frac{1}{\sqrt{2^{n}n!\sqrt{\pi}}}H_{n}(q)e^{-\frac{1}{2}q^{2}}$
(4)

where $H_{n}$ is the Hermite polynomial:

$H_{n}(q)=(-)^{n}e^{q^{2}}\frac{d^{n}}{dq^{n}}e^{-q^{2}}$ (5)

and performing the integrals over the $q$ quadratures.

Next, we evolve the state $\ket{\psi_{0}}$ by acting with a string of non-
unitary gates $\mathcal{T}_{s}(\vec{\zeta}_{s})$ ($s=1,2,\dots,S$) acting on
the visible mode only. The total number of gates $S$ depends on the problem at
hand. The quantum circuit implementing a gate
$\mathcal{T}_{s}(\vec{\zeta}_{s})$ is depicted in Figure 2b. It consists of
Gaussian gates only that act on the visible qumode and an ancilla qumode of
quadratures $(q_{A},p_{A})$. Non-unitarity is introduced by a measurement of
the ancilla qumode, projecting it onto the state $\ket{n_{A}=1}_{A}$. Let
$\ket{\psi_{s-1}}_{vh}$ be the input at the $s$th step and denote by
$\mathcal{U}(\vec{\zeta}_{s})$ the two-mode unitary representing the
collective action of the Gaussian gates on the visible and ancilla qumodes,

$\mathcal{U}(\vec{\zeta}_{s})=\mathcal{R}(\chi_{s})\mathcal{S}(r_{s})\mathcal{R}(\chi^{\prime}_{s})\mathcal{D}(\alpha_{s})X(\kappa_{s})$
(6)

where $\mathcal{R}(\chi_{s})$ is phase space rotation operator,
$\mathcal{S}(r_{s})$ represents squeeze gate affecting position and momentum,
$\mathcal{D}(\alpha_{s})$, is displacement operator and $X(\kappa_{s})$ is
controlled-X gate, a controlled displacement in position. These gates are
defined by the following expressions:

$\mathcal{R}(\chi)=e^{i\chi\hat{a}^{\dagger}\hat{a}}~{},\ \
\mathcal{S}(r)=e^{\frac{r}{2}\left(\hat{a}^{2}-\hat{a}^{{\dagger}^{2}}\right)},\
\ \mathcal{D}(\alpha)=e^{\alpha(\hat{a}^{\dagger}-\hat{a})},\ \
X(\kappa)=e^{-i\kappa\hat{x}\otimes\hat{p}}~{},$ (7)

where $\chi,r,\alpha,\kappa\in\mathbb{R}$,
$\hat{x}=\frac{1}{\sqrt{2}}(\hat{a}+\hat{a}^{\dagger})$,
$\hat{p}=-i\frac{1}{\sqrt{2}}(\hat{a}-\hat{a}^{\dagger})$, and $\hat{a}$,
$\hat{a}^{\dagger}$ are annihilation and creation operators, respectively.
Then the output of the $s$th step is

$\ket{\psi_{s}}_{vh}\propto\mathcal{T}_{s}(\vec{\zeta}_{s})\ket{\psi_{s-1}}_{vh}\
,\ \
\mathcal{T}_{s}(\vec{\zeta}_{s})={}_{A}\braket{1}{\mathcal{U}(\vec{\zeta}_{s})}{0}_{A}~{}.$
(8)

The success probability of obtaining the desired output is
$P_{s}=\|\mathcal{T}_{s}(\vec{\zeta}_{s})\ket{\psi_{s-1}}_{vh}\|^{2}$. The
circuit must be designed so as to maximize the probability of success.

It should be pointed out that the introduction of an ancilla qumode is similar
to the training of CV Quantum Neural Networks (QNN) that are experimentally
realizable, as we discussed in our previous work [16]. Here we aim at a
similar construction that would yield experimentally realizable CVQBM.

After $S$ steps, we obtain the output of the CVQBM,

$\ket{\mathrm{out}}_{vh}=\ket{\psi_{S}}\propto\mathcal{T}(\vec{\zeta})\ket{\psi_{0}}_{vh}\
,\ \
\mathcal{T}(\vec{\zeta})\equiv\mathcal{T}_{S}(\vec{\zeta}_{S})\cdots\mathcal{T}_{1}(\vec{\zeta}_{1})$
(9)

where $\vec{\zeta}\equiv\\{\vec{\zeta}_{1},\dots,\vec{\zeta}_{S}\\}$, and
after tracing over the hidden qumode, we obtain the state of the visible mode,

$\rho_{v}=\mathrm{Tr}_{h}\ket{\text{out}}\bra{\text{out}}\propto\mathrm{Tr}_{h}\mathcal{T}(\vec{\zeta})\ket{\psi_{0}}\bra{\psi_{0}}\mathcal{T}^{\dagger}(\vec{\zeta})\propto\mathcal{T}(\vec{\zeta})e^{-2\delta
N}\mathcal{T}^{\dagger}(\vec{\zeta})$ (10)

where we used Eq. (4) and the fact that $\mathcal{T}(\vec{\zeta})$ only acts
on the visible qumode. Evidently, this is a Hermitian operator. Hence we can
define an effective Hamiltonian $H_{\text{eff}}(\vec{\zeta})$ (a Hermitian
operator) by

$\rho_{v}=e^{-H_{\text{eff}}(\vec{\zeta})}$ (11)

which depends on the set of parameters $\vec{\zeta}$, showing that the output
state of the visible qumode in the CVQBM can be viewed as a thermal state. The
action of each non-unitary operator $\mathcal{T}_{s}(\vec{\zeta}_{s})$ can be
thought of as a QITE step [7, 15]. Thus, our quantum circuit _defines_ the
Hamiltonian and its parameters to be trained.

Having designed the CVQBM circuit, we may proceed to use the machine to solve
a variety of problems, such as generative learning tasks [7, 17, 18],
discriminative tasks [7, 1], reinforcement learning [19], quantum state
tomography [17], and anomaly detection [20, 21]. Here, we will concentrate on
using the CVQBM as a continuous distribution generator. Our goal is to make
the machine learn a continuous distribution and then use the trained machine
to generate samples that belong to that distribution. This type of machine can
be utilized as a synthetic data generation tool. There have been various
attempts to use QBM as a generation tool, such as solving a toy problem of the
phase of a state at 10 spatial points [22], generating probability
distributions for a state [7, 17] and Bernoulli distribution [1]. Generating a
continuous distribution in a CV setting provides a natural advantage.

We simulated the CVQBM using the Strawberry Fields software platform [23]. The
quantum machine learning toolbox application is built on top of it, with
TensorFlow [24] features. The detailed architecture for our CVQBM is shown in
Fig. 2. We started with two modes (visible and hidden), which were then
entangled to prepare the two-mode initial state using the quantum gate $U_{0}$
depicted in Fig. 2a. The visible mode was then acted upon with a series of
non-unitary gates $\mathcal{T}_{s}(\vec{\zeta}_{s})$ ($s=1,\dots,S$) whose
detailed architecture is shown in Fig. 2b. The parameters of these gates are
training parameters. At each step, an ancilla coherent mode was introduced,
which was entangled with the visible mode by using Gaussian gates and then
measured, thus projecting it onto the $n=1$ state. The success probability was
calculated, ensuring that the setup corresponded to a reasonable rate for
experimental implementation. Finally, we performed homodyne measurement on the
visible mode.

In the training stage of our CVQBM, our objective was to minimize the distance
between the target and output distributions. However, comparing two
distributions at every training epoch is computationally intensive. We found
it advantageous to work with density matrices. For a given set of data of
probability distribution $P_{\mathrm{data}}(q)$, where $\int
dqP_{\mathrm{data}}(q)=1$, we introduced the state

$\ket{\Psi_{\mathrm{data}}}\approx\sum_{n=0}^{n_{\mathrm{max}}}a_{n}\ket{n}\
,\ \ a_{n}=\int dq\Psi_{n}(q)\sqrt{P_{\mathrm{data}}(q)}$ (12)

where $\Psi_{n}(q)$ is the $n$th state of a harmonic oscillator in the
position representation given by Eq. (4). This can be written in a density
matrix form as

$\rho_{\mathrm{data}}=\ket{\Psi_{\mathrm{data}}}\bra{\Psi_{\mathrm{data}}}\ ,\
\ \rho_{\mathrm{data},mn}=a_{m}a_{n}$ (13)

which can be compared with the output state $\rho_{v}$ given in Eq. (10). The
fidelity is defined as:

$F(\rho_{\mathrm{data}},\rho_{v})=\left(\textup{Tr}\sqrt{\sqrt{\rho_{\mathrm{data}}}\rho_{v}\sqrt{\rho_{\mathrm{data}}}}\right)^{2}$
(14)

Our goal was to maximize this fidelity, and hence minimize the cost:

$C=1-F$ (15)

To this end, we used the Adam Optimizer.

Once the CVQBM was trained, the next step was to generate more samples by
loading the trained weights into the circuits. Since, ultimately, our goal is
to generate the learned distribution, we had to extract the output from the
circuit that corresponded to the output probability distribution,

$P_{v}(q)=\braket{q}{\rho_{v}}{q}\ .$ (16)

Switching to eigenstates of the photon number, the output probability
distribution can be written as

$P(q)=\sum_{m,n}\rho_{v,mn}\Psi_{m}(q)\Psi_{n}(q)\ ,\ \
\rho_{v,mn}=\bra{m}\rho_{v}\ket{n}$ (17)

where $\Psi_{n}(q)$ is given by Eq. (4). Although this is an infinite set of
basis states, in practice we need to introduce a cutoff $n_{\text{max}}$ in
the dimension of the Hilbert space. Using this result, we can generate various
samples from the trained CVQBM.

A metric to compare the target with a generated distribution is provided by
the Kuller-Leibler (KL) divergence,

$\textup{KL}(P_{\text{gen}}\|P_{\text{target}})=\sum_{q}P_{\text{gen}}(q)\log\frac{P_{\text{gen}}(q)}{P_{\text{target}}(q)}$
(18)

which indicates how much the generated probability distribution,
$P_{\text{gen}}$, diverges from the target probability distribution,
$P_{\text{target}}$, and takes values between 0 and $\infty$, with the value
of 0 when $P_{\text{gen}}(q)=P_{\text{target}}(q)$. We compared the two
distributions and obtained how much the generated distribution diverged from
the target distribution by calculating the KL divergence.

##  3 Classical Data Probability Distribution Generation

Our CVQBM can be used for generating different kinds of distribution involving
classical or quantum data. In this Section, we report on probability
distributions from synthetic-aperture radar (SAR) images which are classical
data. We used the SAR dataset available on the ICEYE website [25].

We selected a forest monitoring strip from Acre, Brazil. The dataset contains
the time-series stack of strip SAR images acquired within 12 days in Single
Look Complex (SLC) in HDF5 format and Ground Range Detected (GRD) in GeoTiff
format. For simplicity, we worked with the PNG image in the dataset, shown in
Fig. 3a. The image belongs to the data collected on 22 July 2021.

(a)

(b)

Figure 3: Studying SAR image for forest monitoring in Brazil, (a) The original
SAR image of the forest in Acre Brazil taken on 22 July 21. (b) The final plot
corresponds to the forest monitoring. The blue histogram is created from the
SAR image. The target distribution (red line) is fitted Gaussian on the
histogram. Finally, the generated distribution (black dashed line) comes from
the trained QBM, which has the KL divergence between the target and generated
distribution is 0.006.

We analyzed the data by plotting the image intensity probability distribution
as a function of the pixel intensity. To this end, we converted the image into
a Python Numpy array, and then created the histogram for the intensity values.
The histogram was divided into 256 bins (for each possible intensity value in
an 8-bit grayscale image) and counted how many pixels fell into each bin. The
range was set from 0 to 255, including all possible intensity values. The
histogram was then normalized to yield a probability distribution. The bar
graph was plotted to visualize the probability distribution of the pixel
intensities. Next, we applied re-scaling to both plot axes to make the problem
suitable for a CV quantum computer. By doing so, the range of axes became
approachable, and normalization was maintained. Because of the limitations of
the simulation, it was hard to use the discrete data as this would require a
higher cutoff value. Hence, the next step was to make the distribution
continuous, and for that, we used the gaussian kde from the SciPy library [26]
of Python. Once we obtained the distribution, we used Eq. (13) to deduce the
density matrix. Using the density matrix for cost calculation provided a
computational advantage over using the distribution directly.

As discussed in Section 2, the cost function depends on the fidelity of the
circuit output density matrix compared with the target density matrix. For an
optimal entangled initial state (Eq. (1)), we set $\delta=1.5$. We used the
Adam optimizer with the learning schedule of exponential decay to minimize the
cost. The initial learning rate was 0.05; the decay steps were 100, with a
decay rate of 0.96. The number of steps in this process was $S=2$. After
training for 100 epochs with a batch size of 32 and a cutoff of
$n_{\text{max}}=12$, the best fidelity we obtained was 99.8%, corresponding to
the cost value of 0.002. The low cost made us confident that the model had
been trained well.

(a)

(b)

Figure 4: Studying the SAR image for sea ice monitoring, (a) the original SAR
image of Gulf of Bothnia, taken on 25 Feb 2022, (b) The final plot corresponds
to the sea monitoring. The blue histogram is created from the SAR image. The
target distribution (red line) is fitted Gaussian on the histogram. Finally,
the generated distribution (black dashed line) comes from the trained QBM,
which has the KL divergence between the target and generated distribution is
0.013

After training, the model can be used to generate more data corresponding to
the target distribution. For this process, we loaded the trained weights and
ran the circuit several times to get samples for the density matrix. We
generated 256 samples corresponding to the original shape of the data. Then
using Eq. (17), we generated the distribution. The generated and target
distributions, as well as the original discrete probability distribution, are
shown in Fig. 3b. Notice that the generated distribution lies on top of the
target distribution, showing that we achieved high fidelity while training. We
also calculated the KL divergence between the generated and target
distributions, which came out to be 0.006, showing that the generated
distribution is close to the target distribution.

Generating the distribution for the forest was not hard, as we could fit the
probability distribution with a Gaussian. A more challenging example we
studied was provided by an image that belonged to the Gulf of Bothnia between
the west coast of Finland and the east coast of Sweden [25], shown in Fig. 4a.
The dataset had a pair of scan mode images, which covered an area of 10,000
km2 in the Gulf of Bothnia. Again, we randomly picked one of the images and
plotted the probability distribution vs. pixel intensity. This probability
distribution was best fitted with a Weibull probability distribution:

$W(x;\lambda,k)=\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}$ (19)

where $x\geq 0$, $k>0$ is the shape parameter, and $\lambda>0$ is the scale
parameter of the distribution. For this case, the fitted Weibull distribution
had $\lambda=161.2$ and $k=5.4$.

We adopted the same setup as in the forest case. First, we found the density
matrix corresponding to the probability distribution. Then, the cost function
was defined in terms of the fidelity between the target density matrix and the
circuit output density matrix. Again, we used the Adam optimizer with the same
learning schedule as the previous case. The number of steps used in this
process was $S=2$, with a cutoff dimension of $n_{\text{max}}=15$. After
training for 150 epochs with a batch size of 32, the best fidelity we obtained
was 98.9%. The final plot is shown in Fig. 4b. The generated distribution in
the dashed black line matches the target Weibull distribution (shown in red
solid line). We obtained KL divergence of 0.013.

(a)

(b)

(c)

Figure 5: Generating continuous probability distributions implementing some
commonly used distribution (a) Rayleigh, (b) Gamma and (c) ET-IB distribution

Our model works with various probability distributions requiring an increasing
amount of computational resources as their complexity increases. We
demonstrated this with several distributions commonly associated with real-
world data, including Rayleigh and Gamma distributions which are often used
for SAR and medical images, like ultrasound and PET scans, as well as
exponentiated transmuted-inverted beta (ET-IB) distributions that were
recently shown [27] to outperform (in terms of goodness-of-fit measures) its
competitors when it comes to fitting SAR images. We studied the performance of
our model on these distributions as follows:

  * •

Rayleigh distribution, defined as

$R(x;\sigma)=\frac{x}{\sigma^{2}}\textup{exp}\left(-\frac{x^{2}}{2\sigma^{2}}\right),\
\ x\geq 0$ (20)

where $\sigma$ is the scaling parameter. In Fig. 5a, we chose $\sigma=1$ and
obtained a fidelity of 96.1% after training and KL divergence of 0.03. We set
$\delta=2.5$ for an optimal entangled initial state (Eq. (1)). A real-world
application where the Rayleigh distribution is observed is the wind speed in a
city, studied over a time period in Ref. [28]. It was found that $\sigma=4$
(see Table 1).

  * •

Gamma distribution, defined as

$G(x;k,\theta)=\frac{x^{k-1}}{\Gamma(k)\theta^{k}}\textup{exp}\left(-\frac{x}{\theta}\right),\
\ x\geq 0$ (21)

where $k$ is the shape parameter, and $\theta$ is the scaling parameter. In
Fig. 5b, we chose $k=2.5$ and $\theta=0.5$ and obtained a fidelity of 98.1%
after training and KL divergence of 0.083. We set $\delta=2.5$ for an optimal
entangled initial state (Eq. (1)). There are many applications of the Gamma
distribution. As an example, to study the ultrasound images displaying breast
cancer, various parameter sets of distribution are used:
$(k,\theta)\in\\{(1,2),(2,0.5),(2,1),(2,1),(2,4),(4,2)\\}$ [29] (see Table 1).

  * •

ET-IB distribution, defined as

$\displaystyle f(x;\alpha,\beta,\lambda,\phi)$ $\displaystyle=$
$\displaystyle\frac{\phi
x^{\alpha-1}I_{x/(1+x)}(\alpha,\beta)^{\phi-1}}{(1+x)^{\alpha+\beta}B(\alpha,\beta)}$
(22) $\displaystyle\times\left(1+\lambda-2\lambda
I_{x/(1+x)}(\alpha,\beta)\right)\left(1+\lambda-\lambda
I_{x/(1+x)}(\alpha,\beta)\right)^{\phi-1}\ ,\ \ \ \ \ $

where

$I_{z}(\alpha,\beta)=\frac{1}{B(\alpha,\beta)}\int_{0}^{z}t^{\alpha-1}(1-t)^{\beta-1}dt\
,\ \ B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}$
(23)

The parameters are constrained as $\alpha,\beta,\phi>0$ and $|\lambda|<1$. For
Fig. 5c, we chose the four parameters
$(\alpha,\beta,\lambda,\phi)=(4.0,5.0,0.1,2.0)$, and obtained a fidelity of
91.4% after training, and KL divergence of 0.229. We set $\delta=4.0$ for an
optimal entangled initial state (Eq. (1)). In Ref. [27], the Monte Carlo
technique was used to perform parameter estimation for SAR images using
different terrains. For example, the parameters $(\alpha,\beta,\lambda,\phi)$
were found to be (6.3, 36.2, -0.9, 0.4) for forest images, (3, 37.5, 0.8, 3.7)
for the ocean, and (0.4, 5.3, 0.9, 66.9) for urban (see Table 1).

We used the same Adam optimizer with the learning schedule for all cases as
before. The number of steps was $S=2$, the cutoff dimension was
$n_{\text{max}}=15$, and the number of epochs was 100. The final plots for all
three distributions are shown in Fig. 5. The colored solid lines correspond to
the different target distributions, and the black dashed lines correspond to
the generated distribution. In all three cases, we achieved high fidelities.
An important metric indicating how well the generated distribution compares to
the target distribution is the KL divergence. This is evident in the plots,
with the Rayleigh distribution having a lower KL divergence and better fitting
the histogram compared to the ET-IB distribution, which has a higher value of
KL divergence. As discussed above, these distributions have numerous real-
world applications. In Table 1, we compare the distribution parameters used in
this work with some of the real-world applications. The parameters used for
the Gamma distribution are similar to those used in studying ultrasound images
of breast cancer. However, this was not the case for Rayleigh and ET-IB
distributions because they are steeper and require a higher cutoff dimension
in simulations. The study of these real-world applications would require high-
performance computers, or more efficient simulations, or an actual CV quantum
computer which is not currently available.

Table 1: Description of parameters used for different distributions and their comparison with real-world applications. Names | Parameters |  | Parameters used
---
in this work
Applications |  | Parameters used
---
in the applications
Rayleigh | $\sigma$ | 1 | Wind speeds [28] | 4
Gamma | (k, $\phi$) | (2.5, 0.5) |  | Breast cancer
---
ultrasound images [29]
| (1, 2), (2, 0.5), (2, 1),
---
(2, 4), (4, 2)
ET-IB | ($\alpha$, $\beta$, $\lambda$, $\theta$) | (4, 5, 0.1, 2) |  | SAR images [27]:
---
forest
ocean
urban
| (6.3, 36.2, -0.9, 0.4)
---
(3.0, 37.5, 0.8, 3.7)
(0.4, 5.3, 0.9, 66.9)

##  4 Quantum Data Probability Distribution Generation

Figure 6: Gadget ($\mathcal{G}$) for quantum data to be attached to the CVQBM
during training. $\ket{\Psi}$ is the target state and $\ket{\Phi}$ is the
output of the CVQBM. The first two lines represent ancilla qumodes initialized
in states $\ket{0}$ and $\ket{1}$, respectively.

Next, we apply our CVQBM to quantum data. We consider two cases: a Gaussian
distribution and a non-Gaussian distribution (cat state) both of which are
generated approximately by a quantum neural network (QNN) [16].

To handle quantum data, we need an additional gadget, $\mathcal{G}$, to attach
to the CVQBM shown in Figure 6. The state $\ket{\Psi}$ represents the target
quantum state, generated by a quantum circuit, and $\ket{\Phi}$ represents the
output state of the visible mode of our CVQBM. We introduce two control
qumodes in the states $\ket{0}$ and $\ket{1}$, respectively, and put them in
the state

$\ket{\text{ctrl}}=\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})$ (24)

with the aid of a 50:50 beam splitter. Then we apply the three-mode CSWAP gate

$U_{\text{CSWAP}}\ket{\text{ctrl}}\ket{\Psi}\ket{\Phi}=\frac{1}{\sqrt{2}}(\ket{01}\ket{\Psi}\ket{\Phi}+\ket{10}\ket{\Phi}\ket{\Psi})$
(25)

which can be implemented with the aid of a non-Gaussian Kerr gate [30, 31].
After channeling the beams through another 50:50 beam splitter, we obtain

$U_{\text{BS}}\cdot
U_{\text{CSWAP}}\ket{\text{ctrl}}\ket{\Psi}\ket{\Phi}=\frac{1}{2}(\ket{01}(\ket{\Psi}\ket{\Phi}-\ket{\Phi}\ket{\Psi})+\ket{10}(\ket{\Psi}\ket{\Phi}+\ket{\Phi}\ket{\Psi}))~{}.$
(26)

Measurement of the first control qumode yields probabilities $P_{i}$ ($i=0,1$)
with

$P_{1}-P_{0}=|\braket{\Phi}{\Psi}|^{2}~{},$ (27)

which is the fidelity to be used in the definition of the cost function (15).

(a)  (b)

Figure 7: Generating continuous probability distributions from quantum data:
(a) Gaussian distribution and (b) non-Gaussian distribution resulted from a
cat state preparation. The target (solid blue lines) and generated (dashed red
lines) distributions match well.

For the preparation of quantum data, we relied on our previous work on
experimentally realizable CV QNNs [16]. More specifically, we used the
circuits for state preparation to prepare a Gaussian and non-Gaussian state
approximately.

For a Gaussian state, we constructed a target Gaussian state for our CVQBM
from a single qumode in the vaccum state on which we applied a squeezing gate
with a squeezing of 1.73 dB and a displacement gate with a displacement of
parameter $\alpha=0.2$. The resulting Gaussian state had a mean of 0.083 and a
standard deviation of 0.145. We trained the CVQBM circuit with 1 step of our
QITE algorithm and an initial entanglement of $\delta=1.5$. Our trained CVQBM
circuit attained a fidelity of 99.55% resulting in a Gaussian state with mean
0.083 and standard deviation 0.147.

For the non-Gaussian state, our target quantum state was created using an
8-layer CV QNN [16] which aimed to approximate an even squeezed cat state with
a displacement magnitude of 1.2 and squeezing of 2.60 dB. It achieved this
with a fidelity of 96.89%. Our CVQBM, in turn, learned this non-Gaussian state
with a fidelity of 98.22%, utilizing an architecture that contained Gaussian
gates only and an ancillary mode on which we performed a projective
measurement, utilizing 3 steps of QITE and entangling parameter $\delta=2.3$.

In the CV QNN simulation, we used a cutoff dimension of 10 and optimized the
fidelity of the target state using the Nelder-Mead optimizer. For the CVQBMs,
we used the Adam optimizer with a learning scheduler. The initial learning
rate was 0.04 for both Gaussian and non-Gaussian data and both the Gaussian
and the non-Gaussian states were trained for 400 epochs with a cutoff of 10.

The target and generated probability distributions are shown in Fig. 7. The
solid blue lines correspond to the target distributions, whereas the dashed
red lines correspond to the generated distributions. The respective
distributions match well, with KL divergence of 0.002 and 0.025 for the
Gaussian and non-Gaussian (cat state) cases, respectively.

##  5 Experimental Feasibility

A main goal in developing novel quantum algorithms is achieving experimental
feasibility with currently available technology. In this Section, we discuss
the experimental feasibility of the gates in our proposed CVQBM, the success
probability, which is an important metric, since our CVQBM relies on non-
deterministic processes (projective measurements on ancilla qumodes), and the
effects of noise.

For experimental feasibility, we restricted the parameters of the gates during
training within an experimentally attainable range with current technology.
This is especially challenging when dealing with real-world problems like
studying SAR images on quantum optical hardware. The range of parameters for
the gate set that were used in our case studies are shown on Table 2. In
addition to these gates, the CX gate ($e^{i\kappa q\otimes p}$), can be
decomposed into two single-mode squeezers and beam splitters. Particularly,
achieving high squeezing rates is one of the most challenging tasks in
photonic quantum computing. Experiments are being conducted to achieve higher
squeezing values. One of the highest squeezing value (15 dB) was reported in
Ref. [32]. Hence, the squeezing parameters required to realize the use cases
studied in this paper are within experimental range.

Table 2: Range of trained parameters of gates in the CVQBM circuit Gates | Parameter |  | Classical case
---
Forest
| Quantum case
---
Cat state
Squeeze Gate | r (in dB) | 2.60 - 5.21 | 6.05 - 15.41
Displacement Gate | $\alpha$ | 0.8 - 1.3 | 0.01 - 0.51
CX | $\kappa$ | 0.08 - 1.7 | 0.01 - 0.45

Our CVQBM design is best suited for photonic quantum hardware. Unfortunately,
currently available photonic quantum devices have limitations that prevent us
from realizing our CVQBM with cloud-accessed photonic quantum hardware on
chip. For example, the Xanadu X-series [33], which is a cloud-accessed quantum
chip that includes squeezers, beam splitters, and interferometers cannot
accommodate our CVQBM mainly due to the lack of online squeezers (the
parameters of the available offline squeezers are fixed). The limitations of
current photonic quantum hardware highlight the importance of co-designing
quantum hardware, software, and algorithms. We leave the experimental
realization of our CVQBM model on photonic quantum hardware to future work.

Turning to the success probability of our probabilistic algorithms, we note
that the odds of a full pass-through of our CVQBM circuit is dictated by the
probability of successfully measuring the required number of photons in the
ancilla qumodes. Since we are post-selecting runs with a specific number of
photons measured, the overall success probability is a product of success
probabilities at each step of the QITE algorithm. If $p_{s}$ represents the
probability of measuring 0 photons at the $s-$th QITE step, then the overall
success probability is given by $P_{\textrm{success}}=p_{1}p_{2}\cdots p_{S}$,
where $S$ is the total number of steps in the QITE algorithm. This
multiplicative nature arises because the success of each step is conditional
on the success of the previous steps. Therefore, to achieve a successful pass-
through of the entire circuit, each individual step must succeed, leading to
the cumulative success probability described above.

We studied the success probability of generating the forest and sea SAR images
using our CVQBM model. For the parameters indicated in Section 3, we obtained
a success probability of 16.3% (27.2%) with fidelity 99.7% (98.7%) for the
forest (sea) SAR images. It is possible to increase success probability at the
expense of fidelity. For forest (sea) data, we obtained a success probability
of 99.2% (99.7) with a fidelity of 95.1% (94.6%). Similar results were
obtained for the other distributions we studied. For the Rayleigh, Gamma, and
ET-IB distributions, we achieved fidelity of 96.2%, 98.1%, and 91.4% at a
success rate of 50.5%, 1.6%, and 49%, respectively. Similarly, in the case of
quantum data, we were able to train models with fidelity of 95.0% (98.2%) at a
success rate of 99.9% (21.3%) for Gaussian (non-Gaussian) states. Results on
achieved fidelity and corresponding success rates are shown in Table 3.

(a)  (b)

Figure 8: Studying various distributions under the loss channel. Each loss parameter ($T$) run was done 5 times to get the mean cost values and the filled region around it represents the standard deviation of the fidelity. (a) Forest SAR images, and (b) Cat state distributions under the noise loss channel Table 3: Achieved fidelity and success rates in use cases Use case | Fidelity | Success Rate |
---|---|---|---
Forest SAR Images | 99.7% | 16.3% |
| 95.1% | 99.2% |
Sea SAR Images | 98.7% | 27.2% |
| 94.6% | 98.7% |
Rayleigh Distribution | 96.2% | 50.5% |
Gamma Distribution | 98.1% | 1.6% |
ET-IB Distribution | 91.4% | 49% |
Quantum Gaussian State | 95.0% | 99.9% |
Quantum Non-Gaussian (Cat) State | 98.2% | 21.3% |

Finally, we added noise to our model to assess the robustness of the proposed
CVQBM. We chose the simple noise model, loss channel, available in Strawberry
Fields. This channel couples a given qumode in the circuit, with annihilation
operator $\hat{a}$, to an unwanted qumode representing noise, with
annihilation operator $\hat{b}$, prepared in the vacuum state. Thus,

$\hat{a}\to\sqrt{T}\hat{a}+\sqrt{1-T}\hat{b}~{},$ (28)

where $T\in[0,1]$ is the loss parameter, also known as energy transmissivity.
$T=1$ represents the noiseless case (identity map) and for $T=0$, the state is
mapped to the vacuum state. We introduced loss channels for the various
qumodes in our circuits.

Figure 9: KL divergence (Eq. (18)) against the loss parameter ($T$) for the
forest SAR image (classical) and cat state (quantum) data cases. Each data
point corresponds to 5 runs.

We analyzed the cost during training as a function of the energy
transmissivity $T$ for one classical (forest SAR image as seen in Fig. 3a) and
one quantum (non-Gaussian cat state Fig. 7(b)) case. Results are shown in Fig.
8. Each loss parameter run was done 5 times. The mean of each run is displayed
with a solid line with a shaded region around it representing the standard
deviation. As expected, we obtained the best fidelity for forest (cat) 99.7%
(98.2%) when $T=1$, corresponding to no loss. The fidelity decreases as we
decrease the loss parameter, which makes sense because increasing the noise
will make it harder for the machine to train. Eventually, at $T=0$, no
training occurs. With every 10% decrease in the loss, the fidelity changes by
around 5%. At 90% loss in the channel, the fidelity change is around 30%.

In order to study how the loss channel impacts the performance of our CVQBM,
we plotted the KL divergence vs. the loss parameter ($T$) in Fig. 9. In this
plot, each data point corresponds to 5 training runs. We generated the
distribution using the optimized parameters and hence calculated the KL
divergence between the target and generated distributions (Eq. (18)). The
lower value of KL divergence corresponds to the better match between the two
distributions. Hence, the KL divergence decreases as $T\to 1$. As expected,
best results are obtained at $T=1$ (no loss). The difference between the
classical and quantum cases is small. We note that, up to 40% loss, our CVQBM
performs better the quantum case, but with more loss, the performance in the
classical case is slightly better.

There is ongoing research for developing error mitigation strategies for
photon loss in photonic hardware. For example, in Ref. [34], techniques were
presented that can mitigate the effects of photon loss on both output
probabilities and expectation values in noisy linear optical circuits. These
techniques involve the creation of recycled probabilities, which are derived
from output statistics affected by loss and are designed to amplify the signal
of the ideal probabilities. These recycled probabilities are then processed to
produce a set of loss-mitigated probabilities or expectation values. These
technique were shown to perform better than post-selection and zero-noise-
extrapolation (ZNE) techniques. We leave study of photon loss mitigation
strategies as a future work.

##  6 Conclusion

We proposed a CVQBM which can be realized experimentally with a CV photonic
quantum computer. We introduced a novel method to prepare the thermal state
required by the BM setup using an entangled state followed by QITE steps. We
implemented the entire CVQBM design using Gaussian gates and photon number
resolving measurements, making it experimentally realizable.

We tested our CVQBM against both classical and quantum data. For classical
data, we worked with the SAR images. These images are not affected by
altitude, bad weather, or light, making them amenable to high-resolution
remote sensing. We focused on datasets obtained from forest and sea ice
monitoring. We generated distributions that were of very low KL divergence.
All of the distributions we studied have various real-life applications
ranging from SAR images [35] to medical images [36]. For example, SAR images
are obtained from a wide-range of applications, such as agriculture
monitoring, disaster management after damages done by a disaster like an
earthquake, flood, and volcanic eruptions, search and rescue operations for
the same, climate change studies, geology, and mining. For quantum data, we
focused on a Gaussian distribution generated by a coherent state as well as
one generated by a cat state, which have various applications, such as in
designing qubits which are resistant to errors [37], implementing entangling
gates [38], and quantum spectroscopy [39].

In this work, we focused on generating continuous probability distributions
that have many real-world applications. However, our proposed CVQBM can also
be further utilized for anomaly detection [40] and classification tasks, as
well as variational inference as discussed in [41] by optimizing a candidate
probability distribution to approximate a posterior distribution over
unobserved variables. This task has many real-world applications with
continuous valued data that range from identifying diseases from symptoms to
classifying economic regimes from price movements. Work in this direction is
in progress.

Acknowledgements

Research funded by the National Science Foundation under award DGE-2152168 and
by the Department of Energy under award DE-SC0024325. A portion of the
computation for this work was performed on the University of Tennessee
Infrastructure for Scientific Applications and Advanced Computing (ISAAC)
computational resources.

KYA was supported by MITRE’s Quantum Horizon Program and by the Department of
Energy under award DE-SC0024325. ©2024 The MITRE Corporation. ALL RIGHTS
RESERVED. Approved for public release. Distribution unlimited
PR_24$-$00320$-$3.

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$\displaystyle\leq\frac{\varepsilon}{(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+8)\cdot\nu^{*}-(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+7)\cdot\varepsilon}$
(15)

We now compare the bounds on $\mathcal{G}\left(K,\frac{\varepsilon}{4}\right)$
given by Eq. 12 and 15, and we show that, for sufficiently small
$\varepsilon$, the latter bound implies the former. Therefore, we seek for
$\varepsilon$ s.t.

$\displaystyle\frac{\varepsilon}{(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+8)\cdot\nu^{*}-(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+7)\cdot\varepsilon}$
$\displaystyle\leq\frac{1}{2}\cdot\frac{\varepsilon}{2\nu^{*}-\varepsilon}$ if
$\displaystyle\left(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+8\right)\cdot\nu^{*}-\left(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+7\right)\cdot\varepsilon$
$\displaystyle\geq 4\cdot\nu^{*}-2\cdot\varepsilon$ if
$\displaystyle\left(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+4\right)\cdot\nu^{*}$
$\displaystyle\geq\left(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+5\right)\cdot\varepsilon$
if
$\displaystyle\varepsilon\leq\frac{8(\mu^{*}+W)+4\delta}{8(\mu^{*}+W)+5\delta}\cdot\nu^{*}$

######  Assumption 7.

$\varepsilon\leq\varepsilon_{0}:=\frac{8(\mu^{*}+W)+4\delta}{8(\mu^{*}+W)+5\delta}\cdot\nu^{*}$.

Therefore, for sufficiently small $\varepsilon\leq\varepsilon_{0}<\nu^{*}$ it
suffices to solve Eq. 15. Recall that
$\mathcal{G}\left(K,\frac{\varepsilon}{4}\right)=a\cdot 2^{d}\cdot e^{-b\cdot
K\cdot\varepsilon^{2}/16}$. Thus, we seek for $K$ s.t.

$\displaystyle a\cdot 2^{d}\cdot e^{-b\cdot K\cdot\varepsilon^{2}/16}$
$\displaystyle\leq\frac{\varepsilon}{(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+8)\cdot\nu^{*}-(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+7)\cdot\varepsilon}$
if $\displaystyle e^{-b\cdot K\cdot\varepsilon^{2}/16}$
$\displaystyle\leq\frac{1}{a\cdot
2^{d}}\cdot\frac{\varepsilon}{(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+8)\cdot\nu^{*}-(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+7)\cdot\varepsilon}$
if $\displaystyle-b\cdot K\cdot\varepsilon^{2}/16$
$\displaystyle\leq\ln\left(\frac{1}{a\cdot
2^{d}}\cdot\frac{\varepsilon}{(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+8)\cdot\nu^{*}-(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+7)\cdot\varepsilon}\right)$
if $\displaystyle K$
$\displaystyle\geq\frac{16}{b\cdot\varepsilon^{2}}\cdot\ln\left(a\cdot
2^{d}\cdot\frac{(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+8)\cdot\nu^{*}-(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+7)\cdot\varepsilon}{\varepsilon}\right)$
if $\displaystyle K$
$\displaystyle\geq\frac{16}{b\cdot\varepsilon^{2}}\cdot\ln\left(a\cdot
2^{d}\cdot\frac{(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+8)\cdot\nu^{*}}{\varepsilon}\right)$
if $\displaystyle K$
$\displaystyle\geq\frac{16}{b\cdot\varepsilon^{2}}\cdot\left(\ln a+d\ln
2+\ln\left(\frac{8\mu^{*}}{\delta}+\frac{8W}{\delta}+8\right)+\ln\nu^{*}-\ln\varepsilon\right)=:K_{2}$
(16)

The constant $K_{2}$ is of the form $\frac{\alpha}{\varepsilon^{2}}$, with
$\alpha$ linear in $\varepsilon$ and $d$ (recall that $a$ is exponential in
$\varepsilon$), and polynomial in the characteristics of the Markov chain.
Thus, $K=O(\frac{1}{\varepsilon})$. ∎

###  B-B Infinite-memory synthesis

####  B-B1 Inside an EC

We complete the proof of Lemma 6 by showing the two claims. The first claim
relies on Lemma 3.9 in BrazdilBrozekEtessamiKucera:IC2013, while the second
claim relies on Lemma 18.

See 1

###### Proof.

Recall that $p_{i,K}$ is, by definition, the probability that the total payoff
goes below $\vec{N}_{i}$ in any component during phase $i$. Since at the
beginning of phase $i$ the total payoff is $\mathsf{TP}_{K\cdot
i}>2\cdot\vec{N}_{i}$ by definition, $p_{i,K}$ is upper bounded by the
probability that the total payoff decreases by at least $\vec{N}_{i}$ in any
component:

$\displaystyle
p_{i,K}\leq\mathbb{P}_{s_{0}}^{\mathcal{G}[f_{K}]}\left[\exists(K\cdot i\leq
h<K\cdot(i+1))\cdot\mathsf{TP}_{h}-\mathsf{TP}_{K\cdot
i}\not>-\vec{N}_{i}\right]$

For any state $s$ in the EC, and payoff threshold $N\in\mathbb{N}$, let
$p_{s,N,K}$ be the probability that the total payoff goes below $-N$ in any
component when starting from $\vec{0}$ within at most $K$ steps:

$\displaystyle p_{s,N,K}$
$\displaystyle=\mathbb{P}_{s}^{\mathcal{G}[f_{K}]}\left[\exists(0\leq
h<K)\cdot\exists(1\leq j\leq d)\cdot\mathsf{TP}_{h}[j]\leq-N\right]$

Thus, $p_{i,K}\leq\min_{s\in S}p_{s,\min\vec{N}_{i},K}$. We prove the claim
thanks to the following lemma. (It follows from Lemma 3.9 in
BrazdilBrozekEtessamiKucera:IC2013. For completeness, we provide a proof
below.)

######  Lemma 21.

There exists a rational constant $c<1$ and an integer $N^{*}\geq 0$ s.t., for
every $N\geq N^{*}$ and starting state $s$ in $\mathcal{G}[f_{K}]$ s.t.
$\mathbb{P}_{s}^{\mathcal{G}[f_{K}]}\left[\mathsf{MP}\geq\vec{0}\right]=1$,

$\displaystyle p_{s,N,K}\leq 2^{d}\cdot\frac{c^{N}}{1-c}$

By Lemma 1 we can assume that when playing according to $f^{exp}$ we obtain a
Markov chain $\mathcal{G}[f^{exp}]$ which is _unichain_. Since $f^{exp}$ has
strictly positive expected mean payoff and $\mathcal{G}[f^{exp}]$ is unichain,
the mean payoff is almost surely strictly positive, and thus we can apply
Lemma 21 to the Markov chain $\mathcal{G}[f^{exp}]$ and obtain constants
$N^{*}$ and $c$ s.t. for every $N\geq N^{*}$, $K\geq 0$, and state $s$,
$p_{s,N,K}\leq 2^{d}\frac{c^{N}}{1-c}$. Let $\nu^{*}=\min_{1\leq j\leq
d}\vec{\nu}[j]$, and let $K^{*}$ be large enough s.t., for any $K\geq K^{*}$,
$N_{i}^{min}\geq N^{*}$. (For example, take $K^{*}=\frac{2N^{*}}{\nu^{*}}$.)
Assume we are in any phase $i\geq 1$. Since $N_{i}^{min}\geq N^{*}$ by the
choice of $K^{*}$, by Lemma 21 we obtain, for every $K\geq K^{*}$,
$p_{s,N_{i}^{min},K}\leq
2^{d}\cdot\frac{c^{N_{i}^{min}}}{1-c}=2^{d}\cdot\frac{c^{\frac{\nu^{*}\cdot
i\cdot K}{2}}}{1-c}$. Since $p_{i,K}\leq\min_{s\in S}p_{s,\min\vec{N}_{i},K}$,
we have $p_{i,K}\leq 2^{d}\frac{c^{\frac{\nu^{*}\cdot i\cdot K}{2}}}{1-c}$.
The claim follows by taking $a=2^{d}/(1-c)$ and $b=c^{\frac{\nu^{*}}{2}}$,
which is $<1$ since $c$ is. ∎

###### Proof of Lemma 21.

We first prove the lemma in the unidimensional case. Let $T$ be the set of
states from where the total payoff is almost surely $-\infty$, and let $T^{*}$
be the set of states which can reach $T$ with positive probability. Lemma 3.9
in BrazdilBrozekEtessamiKucera:IC2013 when applied to the Markov chain
$\mathcal{G}[f_{K}]$ (in general, it applies to Markov decision processes),
guarantees that there exists a rational constant $c<1$ and an integer
$N^{*}\geq 0$ s.t. for every $N\geq N^{*}$

$\displaystyle\mathbb{P}_{s}^{\mathcal{G}[f_{K}]}\left[\exists
i\in\mathbb{N}\cdot\mathsf{TP}_{i}\leq-N\textrm{ and $T^{*}$ is not visited
}\right]\leq\frac{c^{N}}{1-c}$

Since we assume $\mathbb{P}_{s}^{\mathcal{G}[f_{K}]}\left[\mathsf{MP}\geq
0\right]=1$, by Lemma 12 the probability to have a total payoff equal to
$-\infty$ is $0$, and thus $T$ is empty. Therefore, the probability on the
left above is just $\mathbb{P}_{s}^{\mathcal{G}[f_{K}]}\left[\exists
i\in\mathbb{N}\cdot\mathsf{TP}_{i}\leq-N\right]$, which is the definition of
$p_{s,N,K}$, and thus

$\displaystyle p_{s,N,K}\leq\frac{c^{N}}{1-c}$

Now let $d>1$, and, for a component $1\leq j\leq d$, let $p_{s,N,K}^{j}$ be
the probability that we have a drop by $-N$ in the total payoff in component
$j$, i.e.,

$\displaystyle p_{s,N,K}^{j}=\mathbb{P}_{s}^{\mathcal{G}[f_{K}]}\left[\exists
i\in\mathbb{N}\cdot\mathsf{TP}_{i}[j]\leq-N\right]$

By looking at the Markov chain $\mathcal{G}[f_{K}]$ projected to component
$j$, we can apply the result from the first part and obtain that, for every
fixed $j$, there exist $c_{j}<1$ and $N_{j}^{*}\geq 0$ s.t., for every $N\geq
N_{j}^{*}$,

$\displaystyle p_{s,N,K}^{j}\leq\frac{c_{j}^{N}}{1-c_{j}}$

By taking $N^{*}=\max_{1\leq j\leq d}N_{j}^{*}$ and letting $c$ be the $c_{j}$
maximizing $\frac{c_{j}}{1-c_{j}}$, we have, for every $N\geq N^{*}$,

$\displaystyle p_{s,N,K}^{j}\leq\frac{c^{N}}{1-c}$

By definition, $1-p_{s,N,K}^{j}$ is the probability that the total payoff
never goes below $-N$ in component $j$, and thus
$\Pi_{j=1}^{d}(1-p_{s,N,K}^{j})$ is a lower bound on the probability that the
total payoff never goes below $-N$ in _any component_. Therefore
$p_{s,N,K}\leq 1-\Pi_{j=1}^{d}(1-p_{s,N,K}^{j})\leq
1-(1-\frac{c^{N}}{1-c})^{d}$. By a simple calculation (cf. the proof of Lemma
9), we derive $(1-\frac{c^{N}}{1-c})^{d}\geq 1-2^{d}\cdot\frac{c^{N}}{1-c}$,
and thus

$\displaystyle p_{s,N,K}\leq
1-\left(1-2^{d}\cdot\frac{c^{N}}{1-c}\right)=2^{d}\cdot\frac{c^{N}}{1-c}$

as required. ∎

See 2

###### Proof.

Recall the definition of
$q_{i,K}=\mathbb{P}_{s_{0}}^{\mathcal{G}[f_{K}]}\left[\mathsf{TP}_{K\cdot(i+1)}\not>2\cdot\vec{N}_{i+1}\right]$.
By the definition of $\vec{N}_{i+1}=\frac{\vec{\nu}(i+1)K}{2}$,

$\displaystyle
q_{i,K}=\mathbb{P}_{s_{0}}^{\mathcal{G}[f_{K}]}\left[\mathsf{MP}_{K\cdot(i+1)}\not>\vec{\nu}\right]$

Let $\vec{\nu}^{*}$ be the expected mean payoff of $f^{exp}$, and let
$\delta=\min_{1\leq j\leq d}(\vec{\nu}^{*}[j]-\vec{\nu}[j])>0$. Then,
$q_{i,K}$ is upper bounded by the probability that the mean payoff deviates
from its expected value $\vec{\nu}^{*}$ by more than $\delta$ in any
component. By Lemma 18, this is upper bounded by
$a^{\prime}2^{d}e^{-b^{\prime}(i+1)K\delta^{2}}$ for sufficiently large
$(i+1)K$, and thus for sufficiently large $K$. Take $a=a^{\prime}\cdot 2^{d}$
and $b=e^{-b\delta^{2}}<1$. ∎

###  B-C Inside a general MDPs

In this section, we prove the correctness of the reduction of the BWC problem
to solving system $T^{\prime}$.

See 7

The following proposition is the analogous of Proposition 3 where we consider
MEC (instead of MWEC) and arbitrary strategies (instead of finite-memory
ones). As before, it rests on the assumption that all states of $\mathcal{G}$
are reachable from $s_{0}$.

######  Proposition 6.

Let $\mathcal{G}$ be a pruned multidimensional mean-payoff MDP. If there
exists a strategy $h$ s.t.
$(\vec{0};\vec{\nu})\in\mathsf{BWCSol}^{+}_{\mathcal{G}}({s_{0},h})$, then
there exists a strategy $h^{*}$ with the same property, and such that, for
every MEC $U$, the set of states visited infinitely often by $h^{*}$ is a
subset of $U$ with positive probability.

The following proposition (and its proof) is analogous to Proposition 4, where
MWECs are replaced by MECs.

######  Proposition 7.

If $T^{\prime}$ has a non-negative solution, then there exists a (finite-
memory) strategy $\hat{h}$ s.t.

  1. 1.

$\vec{\nu}\in\mathsf{ExpSol}^{+}_{\mathcal{G}}({s_{0},\hat{h}})$.

  2. 2.

For every MEC $U$, there is a probability $y^{*}_{U}>0$ s.t. the set of states
visited infinitely often by $\hat{h}$ is inside $U$ with probability
$y^{*}_{U}$.

  3. 3.

The set of states visited infinitely often by $\hat{h}$ is almost surely an
EC. Consequently, $\sum_{\textrm{MEC }U}y^{*}_{U}=1$.

  4. 4.

Once inside a MEC $U$, $\hat{h}$ achieves expected mean payoff
$\vec{\nu}_{U}>\vec{0}$.

  5. 5.

$\sum_{\textrm{MEC }U}y^{*}_{U}\cdot\vec{\nu}_{U}>\vec{\nu}$.

###### Proof of Lemma 7.

The proof is similar to finite-memory case in Lemma 4. We sketch here the
crucial differences.

For one direction, let $h$ be a strategy s.t.
$(\vec{0};\vec{\nu})\in\mathsf{BWCSol}^{+}_{\mathcal{G}}({s_{0},h})$. By
Proposition 6, there exists a strategy $h^{*}$ that additionally visits each
MEC infinitely often with positive probability. By the construction in
Proposition 4.4 of [9] applied to strategy $h^{*}$, we obtain a solution to
system $T^{\prime}$. Equations (A1), (A1’), (A2), (C1), (C1’), and (C2) are
shown to be satisfied in the proof of Proposition 4.4. Eq. (B-bis) is
satisfied since, by Proposition 1, $h^{*}$ is eventually trapped in an EC
almost surely (not necessarily a WEC). Finally, Eq. (C3-bis) is satisfied: By
construction, $h^{*}$ visits each MEC infinitely often with positive
probability. For every MWEC $U$ there exist $s,t\in U$ s.t. $x^{*}_{st}>0$.
Since $h^{*}$ is winning for the worst-case, it achieves an expected mean
payoff $>\vec{0}$ in $U$, and thus Eq. (C3-bis) is satisfied.

For the other direction, we use Proposition 7 to obtain strategy $\hat{h}$,
and then we proceed as in the second part of the proof of Lemma 4 by replacing
MWEC with MEC, and by using Lemma 5 instead of Lemma 3 in the construction of
strategies $h_{U}$’s. ∎

IEEEtran bibliography