Protocols for classically training quantum generative models on probability distributions Sachin Kasture1Oleksandr Kyriienko2and Vincent E. Elfving1

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Protocols for classically training quantum generative models on probability

distributions

Sachin Kasture,1Oleksandr Kyriienko,2and Vincent E. Elfving1

1PASQAL, 7 rue L´eonard de Vinci, 91300 Massy, France

2Department of Physics and Astronomy, University of Exeter,

Stocker Road, Exeter EX4 4QL, United Kingdom

(Dated: October 9, 2023)

Quantum Generative Modelling (QGM) relies on preparing quantum states and generating sam-

ples from these states as hidden — or known — probability distributions. As distributions from

some classes of quantum states (circuits) are inherently hard to sample classically, QGM represents

an excellent testbed for quantum supremacy experiments. Furthermore, generative tasks are increas-

ingly relevant for industrial machine learning applications, and thus QGM is a strong candidate for

demonstrating a practical quantum advantage. However, this requires that quantum circuits are

trained to represent industrially relevant distributions, and the corresponding training stage has an

extensive training cost for current quantum hardware in practice. In this work, we propose protocols

for classical training of QGMs based on circuits of the speciﬁc type that admit an eﬃcient gradient

computation, while remaining hard to sample. In particular, we consider Instantaneous Quantum

Polynomial (IQP) circuits and their extensions. Showing their classical simulability in terms of the

time complexity, sparsity and anti-concentration properties, we develop a classically tractable way of

simulating their output probability distributions, allowing classical training to a target probability

distribution. The corresponding quantum sampling from IQPs can be performed eﬃciently, unlike

when using classical sampling. We numerically demonstrate the end-to-end training of IQP circuits

using probability distributions for up to 30 qubits on a regular desktop computer. When applied

to industrially relevant distributions this combination of classical training with quantum sampling

represents an avenue for reaching advantage in the noisy intermediate-scale quantum (NISQ) era.

I. INTRODUCTION

Recent breakthrough works in quantum computing

demonstrated an improved scaling for sampling prob-

lems, leading to so-called the quantum supremacy [1].

While an exact boundary of classical simulation is still to

be established [2], for carefully selected task of random

circuits sampling and boson sampling [3,4] one achieves

an exponential separation for the task of generating sam-

ples (bit strings from measurement read-out). Generally,

the task of generating samples from underlying probabil-

ity distributions is the basis of generative modelling, and

represents a highly important part of classical machine

learning. Its quantum version — Quantum Generative

Modelling (QGM) — relies on training quantum circuits

as adjustable probability distributions that can model

sampling from some particular desired distribution [5–

10]. QGM exploits the inherent superposition properties

of a quantum state generated from a parameterized uni-

tary, along with the probabilistic nature of quantum mea-

surements, to eﬃciently sample from a trainable model.

QGM have potential applications in generating samples

from solutions of stochastic diﬀerential equations (SDEs)

for simulating ﬁnancial and diﬀusive physical processes

[11,12], scrambling data using a quantum embedding

for anonymization [13–15], generating solutions to graph-

based problems like maximum independent set or maxi-

mum clique [16,17], among many others. Given the suc-

cesses of quantum sampling this makes QGM a promising

contender for achieving a quantum advantage. However,

to date demonstrating QGM of practical signiﬁcance has

input data

{Xj}

samples PDF est. p(x)

~PDF p(x)

circuit training

diff. constraints

(SDE, Fokker-

Planck eqs.)

x∂p(x)/∂x =

= αp + ∂2p/∂x2

QCBM (implicit)

{Xj}Uθ

QCBM (explicit)

p(x)

DQGM (explicit)

p(x)

Uφ(x)

UθUθ

QCBM

{Xj}

Uθ

DQGM

{Xj}

Uθ

QFT

quantum classical (CPU/GPU) or quantum

sample-based

loss (MMD etc)

distribution-based loss and gradients (MSE)

implicit explicit

sampling

FIG. 1. Schematic of diﬀerent steps in training a quantum

generative model using QCBM and DQGM architectures. In

a conventional QCBM setup, the loss and gradient estimation

is done using input data samples directly, while in our work we

focus on an explicit version of QCBM and DQGM, allowing

for classical training.

eluded the ﬁeld as training speciﬁc generative models is

arXiv:2210.13442v2 [quant-ph] 6 Oct 2023

a complex time-intensive task.

The key asset of quantum generative modelling is that

quantum measurements (collapse to one of eigenstates of

a measurement operator) provide a new sample with each

shot. Depending on the hardware platform, generating

one sample can take on the order of a few hundreds of

microseconds to several milliseconds [18–24]. For large

probability distributions, represented by entangled 50+

qubit registers, performing the classical inversion is in-

deed much more costly. However, often the challenge

comes from the inference side, when quantum circuits

are required to match speciﬁc distributions. Typically,

training of quantum generative models utilizes gradient-

based parametric learning, similarly to training of deep

neural networks [25]. Parameterized quantum circuits

(also referred as quantum neural networks — QNNs) are

trained by estimating the gradient of gate parameters

θ. Moreover, the gradient of a full QGM loss has to

be estimated with respect to θ. For quantum devices

this can be done by the parameter-shift rule and its gen-

eralization [26–28], where number of circuit estimation

increases linearly with the number of parameters. The

overall training cost corresponds to the measurement of

the loss function at each iteration step. In the case of

quantum circuit Born machine the loss may correspond

to Kullback-Leibler(KL) divergence [9], Sinkhorn diver-

gence [29] or maximum mean discrepancy (MMD) [30],

and may require extensive sampling for resolving the loss

as an average. For quantum generative adversarial net-

works (QGAN) the loss minimization is substituted by

the minimax game [31,32] requiring multi-circuit esti-

mation. In all cases the convergence is not guaranteed

due to exponential reduction of gradients [33].

In this work we investigate the possibility of training

the parameters of quantum generative models classically,

while still retaining the quantum advantage in sampling

[34]. For instance, the ability of classical training for a

diﬀerent paradigm was shown for Gaussian Boson Sam-

pling (GBS) devices [35], but under certain conditions

of ﬁxing an initial set of samples and non-universal op-

eration. For the digital quantum computing operation,

results from previous works [36,37] motivate the possi-

bility that estimating probability density classically can

be feasible without losing the sampling complexity. We

explore this possibility in more detail and use this fur-

ther to develop methods to train circuits classically to

output a desired distribution using a gradient-based ap-

proach. We show that our method is feasible using nu-

merics for up to 30 qubits on a regular desktop computer.

We explore diﬀerent families of quantum circuits in detail

and perform numerical studies to study their sampling

complexity and expressivity. For expressivity studies in

particular, we look at training a Diﬀerentiable Quantum

Generative Model (DQGM) [38–40] architecture which

allows training in the latent (or ‘frequency’) space, and

sampling in the bit-basis. This presents a good test-

ing ground for applying the proposed method to explicit

quantum generative models. We also show that QCBMs

can be trained classically for certain distributions, while

still hard to sample. Our protocols contribute towards

tools for achieving the practical advantage in sampling

once the target distributions are chosen carefully. We

highlight the diﬀerences between well known strategies

for QGM and the method discussed in the paper in Fig. 1.

II. METHODS

A. Preliminaries: QCBM and DQGM as implicit

vs explicit generative models

Generally, there are two types of generative models.

Explicit generative models assume a direct access (or

‘inference’) to probability density functions (PDF). At

the same time, implicit generative models are described

by hidden parametric distributions, where samples are

produced by transforming randomness via inversion pro-

cedure. These two types of models have crucial diﬀer-

ences. For example, training explicit models involves

loss functions measuring distances between a given PDF,

ptarget(x) and a model PDF, pmodel(x), for example with

a mean square error (MSE) loss which is deﬁned as

LMSE =X

x|pmodel(x)−ptarget(x)|2(1)

where explicit knowledge of the ptarget(x) is used. On

the other hand, training implicit models involves com-

paring the samples generated by the model with given

data samples (e.g. with a MMD loss [30]). The MMD

loss is deﬁned as

LMMD =E

x∼pmodel ,y∼pmodel

K(x, y)

−2E

x∼pmodel ,y∼ptarget

K(x, y)

x∼ptarget,y∼ptarget

K(x, y)

(2)

where K(x, y) is an appropriate kernel function. The

MMD loss measures the distance between two probabil-

ity distributions using samples drawn from the respec-

tive distributions as shown in the above equation. In the

context of QGM, QCBM is an excellent example of im-

plicit training where typically a MMD like loss-function is

used. On the other hand, recent work showcases how ex-

plicit quantum models such as DQGM [38] and Quantum

Quantile Mechanics [11] beneﬁt from a functional access

to the model probability distributions, allowing input-

diﬀerentiable quantum models [39,40] to solve stochas-

tic diﬀerential equations or to model distributions with

diﬀerential constraints.

Let us consider a quantum state |Ψ⟩created by ap-

plying a quantum circuit ˆ

U(which can be parameter-

ized) to a zero basis state. For a general ˆ

U, simulat-

ing the output PDF values that follow the Born rule

pmodel(x) = |⟨x|Ψ⟩|2and producing samples from |Ψ⟩are

both classically hard. But what if estimating the PDF for

DQGM/QCBM

setting up...

Use optimized parameters in the

DQGM sampling circuit or QCBM

circuit and generate samples

If hard to sample continue...

Measure anti-concentration

and t-sparseness of probability

distributions

Measure cross-entropy

difference to Porter-Thomas

distribution using sampling

Number of qubits <20?

Measure sparsity of distribution to verify quantum

advantage for sampling

Check time complexity and compare with other IQP-like circuits

Start with a family known to admit additive

polynomial estimation of probabilities

Identify feature map and variational

ansatz which allow additive polyno-

mial estimation of probabilities and

are classically hard to sample from

Classically simulate circuits to

estimate p(x) and vary circuit

parameters to minimize loss for

DQGM training circuit or QCBM circuit

FIG. 2. Workﬂow used for classical training of quantum samplers, both in the diﬀerentiable quantum generative modelling

(DQGM) and the quantum circuit Born machine (QCBM) setting. First, we identify circuits ˆ

Usuitable for classically tractable

probability estimation and hard sampling (see the right chart). For this we: 1) check that a circuit admits additive polynomial

estimation of probability; 2) compare the time-complexity for exact probability calculation with known circuit families; 3)

verify the sparseness of the probability distributions, depending on the number of qubits n; 4) measure anti-concentration/t-

sparseness or cross entropy diﬀerence using sampling. Analysing these properties we conﬁrm if ˆ

Uadmits classical training and

computationally hard sampling. After the classical training is performed variationally by minimizing DQGM/QCBM loss, we

use optimized parameters for quantum sampling circuits.

certain ˆ

Uto suﬃcient accuracy is classically tractable? In

this case one can use an explicit training, and at the infer-

ence stage have access not only to probabilities but also

the capacity to sample eﬃciently via quantum measure-

ments. This scenario describes a potential for classical

training of quantum generative models.

To enable the classical training, we propose a strategy

described in a schematic shown in Fig. 2. Here, our goal

is ﬁnding circuits satisfying ’classical training + quantum

sampling’ conditions.

B. π-simulable circuits that are hard to sample

We note that certain families of quantum circuits, in-

cluding Cliﬀord [41,42] and match-gate sequences [43–

45], admit a classical tractable estimation of probabili-

ties pmodel(x). At the same time, for these circuits the

generation of samples is also classically ‘easy’, thus lim-

iting the potential for achieving a quantum advantage.

However, there exist families of quantum circuits that

allow for complexity separation between the two tasks.

For instance, in Ref. [36] the authors show that one can

estimate probabilities for IQP (Instantaneous Quantum

Polynomial) circuits [46,47] up to an additive polynomial

error, while retaining a classical hardness for sampling.

For a typical IQP circuit with input |0⊗n⟩, the amplitude

to obtain a certain bit-string xat output is given by

Ψ(x) = ⟨x|ˆ

Hˆ

Uˆ

H|0⊗n⟩(3)

where ˆ

Uconsists of single and 2-qubit Z-basis rotations

and ˆ

Hrepresents the Hadamard gate applied to all the

qubits. Moreover, it is known to be classically hard to

even approximately sample from IQP circuits in an aver-

age case [48]. Therefore, such circuits oﬀer an opportu-

nity for explicit training of a quantum generative model,

and a potential for quantum advantage in sampling. IQP

circuits by its structure are also strongly related to the

so-called forrelation problem [49–51], they only diﬀer by

an additional ˆ

Hlayer in the middle. This corresponds to

calculating an overlap between states deﬁned as

Φ = ⟨0⊗n|ˆ

Hˆ

U2ˆ

Hˆ

U1ˆ

H|0⊗n⟩(4)

after the action of quantum circuit ˆ

UF:= ˆ

Hˆ

U2ˆ

Hˆ

U1ˆ

where ˆ

U1,ˆ

U2are circuits that consist of Z-basis rota-

tions ˆ

Rzand Ising-type propagators ˆ

Rzz. Hereafter, ˆ

is a layer of Hadamard gates applied to all nqubits. It

was shown that Φ can be calculated eﬃciently classically.

Therefore, from the squared forrelation |Φ|2one can esti-

mate the probability to obtain |0⊗n⟩at the output after

the action of ˆ

UF, provided the input is |0⊗n⟩. At the

same time, in the following we show that the variant of

forrelation can be used for achieving the sampling advan-

tage.

Classical estimation of probability using the forrelation

has been shown to be possible [52] under the condition

that the circuit’s entangling properties are constrained.

To understand this, we use the concept of connectivity in

graph theory. Consider a graph Gwith nnodes, where

each node represents a qubit in a quantum circuit. Two

nodes are connected by an edge if there is a 2-qubit entan-

gling gate between the corresponding qubits in the cir-

cuit. Typically IQP circuits have all-to-all connectivity,

usually by using a two-qubit entangling gate. However, if

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ProtocolsforclassicallytrainingquantumgenerativemodelsonprobabilitydistributionsSachinKasture,1OleksandrKyriienko,2andVincentE.Elfving11PASQAL,7rueL´eonarddeVinci,91300Massy,France2DepartmentofPhysicsandAstronomy,UniversityofExeter,StockerRoad,ExeterEX44QL,UnitedKingdom(Dated:October9,2023)QuantumGe...

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Protocols for classically training quantum generative models on probability distributions Sachin Kasture1Oleksandr Kyriienko2and Vincent E. Elfving1

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