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A generative modeling approach for benchmarking and training shallow quantum circuits

Marcello Benedetti, Delfina Garcia-Pintos, Oscar Perdomo, Vicente Leyton-Ortega, Yunseong Nam, Alejandro Perdomo-Ortiz

arXiv:1801.07686v4quant-ph

TL;DR

NISQ devices need benchmarks and generative methods that reflect their hardware constraints, while generative training lacks a universally clear-cut objective. The paper introduces DDQCL, a quantum circuit Born machine trained from data and proposes qBAS for Bars-and-Stripes benchmarking; experiments demonstrate learned state preparation and hardware-sensitive evaluation.

  • Problem

    NISQ benchmarking needs metrics that capture device and hybrid-system performance, while generative modeling lacks a universally clear-cut training objective.

  • Method

    DDQCL trains quantum circuits as Born machines using data-driven sampling, native hardware-aware circuit designs, and classical optimization; qBAS evaluates Bars-and-Stripes generation.

  • Results

    The approach learns GHZ-state preparation, approximates coherent thermal states, and yields qBAS evaluations of shallow-circuit architectures and depths on an ion-trap quantum computer.

  • Takeaways & Limitations

    DDQCL can assist NISQ-device characterization and implement simple generative models, while qBAS provides a complementary performance measure for hybrid systems.

Abstract

from arXiv · show

Hybrid quantum-classical algorithms provide ways to use noisy intermediate-scale quantum computers for practical applications. Expanding the portfolio of such techniques, we propose a quantum circuit learning algorithm that can be used to assist the characterization of quantum devices and to train shallow circuits for generative tasks. The procedure leverages quantum hardware capabilities to its fullest extent by using native gates and their qubit connectivity. We demonstrate that our approach can learn an optimal preparation of the Greenberger-Horne-Zeilinger states, also known as "cat states". We further demonstrate that our approach can efficiently prepare approximate representations of coherent thermal states, wave functions that encode Boltzmann probabilities in their amplitudes. Finally, complementing proposals to characterize the power or usefulness of near-term quantum devices, such as IBM's quantum volume, we provide a new hardware-independent metric called the qBAS score. It is based on the performance yield in a specific sampling task on one of the canonical machine learning data sets known as Bars and Stripes. We show how entanglement is a key ingredient in encoding the patterns of this data set; an ideal benchmark for testing hardware starting at four qubits and up. We provide experimental results and evaluation of this metric to probe the trade off between several architectural circuit designs and circuit depths on an ion-trap quantum computer.

I. INTRODUCTION

The paper introduces DDQCL, a hybrid quantum-classical framework for generative modeling and NISQ-device benchmarking. It uses quantum circuits as Born machines, trains them from data-driven sampling objectives, and proposes qBAS as a task-specific hardware metric.

  • Motivation: DDQCL addresses how to benchmark NISQ computational power and usefulness while capturing effects of circuit depth, gate fidelity, and qubit connectivity.
  • Motivation: Generative modeling is challenging because sampling tasks lack a universally clear-cut objective function and may require choosing among competing cost functions.
  • Quantum circuit Born machines: The model uses quantum-circuit amplitudes and Born’s rule to represent data distributions, forming a quantum circuit Born machine executed directly on NISQ hardware.
  • Quantum circuit Born machines: Unlike approaches that assume Boltzmann sampling or require fault-tolerant compilation, the method does not assume a specific distribution and exploits native gates and hardware connectivity.
  • Benchmarking NISQ devices: The qBAS score complements quantum volume by measuring generative performance on Bars and Stripes, incorporating circuit capacity, hardware effects, and classical training choices.
  • The learning pipeline: Training maps binary data vectors to computational-basis states, minimizes KL divergence between data and circuit distributions, and updates parameters using sampled quantum measurements.
  • The learning pipeline: The circuit uses alternating single-qubit rotation and Mølmer-Sørensen XX entangling layers, while particle swarm optimization provides gradient-free parameter updates.
  • The learning pipeline: Because parameter count is independent of data-set size, approximation quality depends on circuit flexibility; deeper or differently connected circuits may improve expressivity but complicate optimization.

The qBAS score

The qBAS score benchmarks shallow quantum circuits on the BAS sampling task by combining model quality with circuit, hardware, and classical-design factors.

  • Definition: qBAS evaluates how well a quantum circuit encodes and samples all BAS patterns in its wave function.DDQCL first learns the BAS state, then the best circuit is scored.
  • Scope: Entanglement, circuit depth, gate fidelities, qubit connectivity, native gates, optimizer choice, and cost functions all contribute to the benchmarked performance.The score is intended to capture both quantum architectural and classical hybrid-system factors.
  • Definition: qBAS is an F1-based metric combining precision, the fraction of BAS samples, and recall, the fraction of unique BAS patterns measured.Its F1 form is F1 = 2pr/(p + r).
  • Calibration: Nreads is fixed using the coupon-collector estimate Nreads = NBAS(n,m)HNBAS(n,m), balancing near-complete recall against score insensitivity.The chosen readout count is designed to keep recall near 1 for the ideal target without making deviations invisible.
  • Robustness: Unlike KL divergence, qBAS remains computable when finite measurements assign zero probability to BAS patterns and requires relatively few readouts for intermediate sizes.KL divergence can become undefined as the number of pixels grows.

Experiments

The experiments tested DDQCL on GHZ states, coherent thermal states, and BAS(2,2) using simulations and ion-trap experiments. DDQCL recovered compact GHZ preparation recipes and produced entangled BAS representations.

  • Experimental design: DDQCL was evaluated on GHZ preparation, coherent thermal-state preparation, and BAS(2,2) with simulations and experiments.The study varied qubit connectivity topology and circuit-layer count, evaluating KL divergence for the learned distributions.
  • GHZ states: DDQCL recovered two parity-dependent GHZ preparation recipes from classical samples, verified for larger odd and even qubit counts.Three- and five-qubit instances produced one recipe, while four- and six-qubit instances produced another.
  • GHZ states: The recovered GHZ recipes matched previously used ion-trap protocols and were identified as compact and efficient for XX-gate preparation.Single-layer chain-topology tests also produced approximate cascaded-entangling recipes for some small instances.
  • GHZ states: All independently learned GHZ parameters converged to the same value, although noisy hardware can make parameter interpretation more difficult.The simulations used noiseless hardware, and DDQCL may compensate certain systematic noise effects non-trivially.
  • State representations: DDQCL learned cat-state circuits from classical distributions, while mixed-state extensions could represent alternatives such as zero-temperature ferromagnets.The proposed extensions use an ancilla register or decoherence to obtain mixed states.

Coherent thermal states

DDQCL learns quantum-circuit approximations to coherent thermal states, with task difficulty increasing as temperature decreases. Deeper circuits outperform shallow ones on harder targets, while the inverse Bethe approximation performs best only on easier cases.

  • The inverse Bethe approximation excels on the easy task, matches the best quantum model at intermediate difficulty, and underperforms on the hard task.It produces a classical closed-form model rather than preparing a quantum state.
  • For BAS(2,2), DDQCL compares circuit topologies and depths using KL divergence and the qBAS score after training candidate circuits.
  • All-to-all topology outperforms sparse topologies, while increasing all-to-all depth from L = 2 to L = 4 does not always significantly improve performance.Possible explanations include optimizer difficulty from more parameters or near-optimal L = 2 performance on the small data set.
  • When T > Tc, shallow circuits perform very well, but depth-dependent performance gaps emerge near Tc.The figure tracks bootstrapped median KL divergence with 90% confidence intervals over 50 iterations.
  • When T < Tc, deeper circuits perform much better than shallow circuits on the difficult thermal-state task.
  • The qBAS score improves for star circuits from L = 2 to L = 4, but deeper experimental circuits accumulate errors and star L = 2 trails all-to-all L = 2.The score can compare device generations, architectures, or classical resources such as optimizers.

III. DISCUSSION

The discussion presents DDQCL as a data-driven framework for both learning useful quantum states and benchmarking NISQ systems. Its demonstrations include GHZ preparation, thermal-state approximation, and the qBAS metric, while performance depends on modular design choices.

  • DDQCL assists NISQ-device characterization and implements simple generative models, with results demonstrated on three data sets.
  • DDQCL learned a GHZ preparation recipe for an ion-trap quantum computer that generalized to any number of qubits with minimal human intervention.Different noise conditions may yield different circuits realizing the same target probability distribution.
  • DDQCL trained circuits to approximate thermal states, illustrating Born machines' ability to approximate Boltzmann machines when data contain thermal-like features.
  • The qBAS score evaluates generative-modeling performance on Bars and Stripes, a classically simple data set whose modeling requires entanglement.Device errors affect the score, which can compare architectures, device generations, or hybrid-system optimizers.
  • The framework is modular: this work studies circuit layout and cost function, while later work examines differentiable costs, optimizers, and hardware noise.
  • DDQCL has bidirectional capability, using NISQ devices for machine learning and machine learning for NISQ-device characterization.
  • DDQCL uses native ion-trap gates and connectivity, including arbitrary single-qubit rotations and Mølmer-Sørensen XX gates between any two qubits.Simulations assume perfect gate fidelities and error-free measurements.
  • Circuit expressivity and parameter count depend on entangling topology, with all-to-all layers using N(N −1)/2 parameters and chain or star layers using N −1.

Gradient-free optimization

DDQCL fixes a circuit template and optimizes its parameters with a clipped negative log-likelihood using global-best particle swarm optimization. Particles represent candidate circuits and are updated through parameter-space velocities.

  • After fixing circuit depth and entangling topology, DDQCL varies circuit parameters to minimize a clipped negative log-likelihood cost.
  • Global-best particle swarm optimization treats each particle as a candidate circuit with a position and velocity in parameter space.Particles and velocities are randomly initialized and updated at each iteration.
  • The number of particles is set to twice the circuit's number of parameters, a conservative choice for the parameter-rich circuits studied.

Data sets details

The study uses synthetic ferromagnetic, thermal, and Bars-and-Stripes distributions to evaluate generative circuit learning across different preparation challenges.

  • The zero-temperature ferromagnet assigns equal probability to the all-zero and all-one states, requiring a GHZ-like pure state preparation.
  • Thermal instances sample ±1 spin configurations from a Boltzmann distribution with random coefficients and temperatures T ∈ {2Tc, Tc, Tc/1.5}.Lower temperatures are used to generate increasingly complex instances.
  • Bars and Stripes consists of n × m black-and-white images whose rows or columns are independently assigned stripe colors.The data set tests whether a generative model captures correlations and reconstructs unseen patterns.
  • The number of BAS patterns is obtained by counting stripe configurations across rows and columns, then subtracting the duplicated empty and full patterns.
  • The BAS data set is used to design a task-specific performance indicator for hybrid quantum-classical systems.Table I gives example experimental requirements, with Nreads denoting readouts required for each qBAS-score estimation.

Bootstrapping analysis

The analyses use bootstrap resampling to estimate central performance values and uncertainty intervals from repeated DDQCL runs and qBAS evaluations.

  • DDQCL was executed 25 times with random parameter initialization to estimate KL-divergence uncertainty.The repetitions were resampled into 10,000 bootstrap data sets of size 25.
  • The KL-divergence estimate is the median of the bootstrap medians, with the 5th and 95th percentiles defining a 90% confidence interval.
  • The qBAS score was computed 25 times from batches of Nreads samples before bootstrap resampling.The analysis sampled 10,000 data sets of size 25 with replacement.
  • The qBAS estimate is the mean of the bootstrap means, with error bars from two standard deviations representing a 95% confidence interval.

Comparison of cost functions

The paper compares three cost functions for learning thermal distributions, finding that alternatives to negative log-likelihood can remain effective while improving computational scalability.

  • The comparison evaluates clipped negative log-likelihood, earth mover’s distance, and moment matching using sampled probabilities and histograms.
  • Clipped negative log-likelihood avoids infinite cost when sampled target configurations receive zero circuit probability.A small ε provides the clipping threshold.
  • Earth mover’s distance uses Hamming distance between bitstrings to measure the minimum transport cost between data and circuit histograms.With normalized histograms, it is equivalent to the 1st Wasserstein distance.
  • For N = 5 thermal states with three all-topology layers, clipped negative log-likelihood outperforms the alternatives but is expected to degrade as problem size increases.
  • Moment matching closely tracks the other cost functions while requiring only O(N^2) terms per iteration.The authors expect it to be a useful heuristic for DDQCL on large systems.

Approximate preparation of coherent thermal states for N = 6

DDQCL is evaluated for six-qubit coherent thermal-state preparation across three circuit depths and three temperatures using all-to-all entangling connectivity.

  • Six-qubit coherent thermal states are evaluated at circuit depths L ∈ {1, 2, 3} and temperatures T ∈ {2Tc, Tc, Tc/1.5}.
  • The simulations use an all-to-all topology for the entangling gate layer l = 2.
  • At low temperature, the inverse Bethe approximation converged in only 7 of 25 instances, so no median value was extracted.
  • For all low-temperature instances, DDQCL outperformed the inverse Bethe approximation relative to the plotted KL-divergence reference of 2.0.

Details for theoretical and experimental results

The results compare simulated and experimental output states for DDQCL circuits encoding BAS(2,2) across all-to-all and star connectivity settings. The all-to-all two-layer circuit is close to optimal, while the displayed circuits vary in depth and topology.

  • Circuit comparisons: DDQCL identifies best BAS(2,2) circuits under all-to-all connectivity with L = 2 and star connectivity with L = 4 or L = 2.The comparison is based on KL divergence and examines both simulated circuits and their ion-trap experimental implementations.
  • Entanglement and circuit structure: The all-to-all L = 2 theoretical state reaches SBAS(2,2) = 1.69989 when entanglement entropy is averaged over all two-qubit subsets.The corresponding circuit diagrams for the three connectivity settings are provided in Figure 8.
  • Analytical solution: The theoretical all-to-all circuit with L = 2 achieves zero KL divergence for BAS(2,2) at α = π−1 arctan(2−1/2).All single-qubit rotations can be zero in the analytical solution.
  • Analytical solution: The DDQCL all-to-all solution uses α = 0.2 and two non-zero Rz rotations while remaining almost optimal.Those Rz gates act as the identity on the |0000⟩ state.

Entanglement entropy of BAS(2,2)

The paper characterizes BAS(2,2) representations through average two-qubit von Neumann entropy and finds that their maximum reaches the known four-qubit maximum. The entropy is parameterized by two variables, with extrema at specified values.

  • Definition: The entanglement measure is the average von Neumann entropy over all two-qubit subsets of a four-qubit pure state.For a 4-qubit cat state, the corresponding entropy is SGHZ = 1.
  • Parameterization: The BAS(2,2) entropy can be expressed using v1 = u2 −u3 −u4 + u5 and v2 = u1 −u3 −u4.Figure 9 plots SBAS(2,2) as a function of these variables.
  • Extrema: The minimum occurs at v1 = v2 = 0, obtainable with u1 = · · · = u5 = 0.These parameter values specify one entropy extremum of the BAS(2,2) representation.
  • Extrema: The maximum occurs at v1 = 4π/3 and v2 = 2π/3, obtainable with u1 = u2 = u3 = 0 and u4 = −u5 = 2π/3.The maximum value coincides with the maximum entanglement entropy known for any 4-qubit state.
  • Extrema: The maximum BAS(2,2) entropy is SBAS(2,2) = 1.79248.Figure 9 marks the maximizing states with black dots.
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