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arXiv:1811.09905 · Physical Review A

Generative model benchmarks for superconducting qubits

Kathleen E. Hamilton, Eugene Dumitrescu, Raphael C. Pooser

Published Jun 18, 2019• 57 citations
arXivHTMLPDF

Abstract

In this work we demonstrate experimentally how generative model training can be used as a benchmark for small (fewer than five qubits) quantum devices. Performance is quantified using three data analytic metrics: the Kullback-Leibler divergence and two adaptations of the ${F}_{1}$ score. Using the $2\ifmmode\times\else\texttimes\fi{}2$ bars and stripes data set, we train several different circuit constructions for generative modeling with superconducting qubits. By taking hardware connectivity constraints into consideration, we show that sparsely connected shallow circuits outperform denser counterparts on noisy hardware.

DOI
10.1103/physreva.99.062323

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In this work we demonstrate experimentally how generative model training can be used as a benchmark for small (fewer than five qubits) quantum devices. Performance is quantified using three data analytic metrics: the Kullback-Leibler divergence and two adaptations of the F1{F}_{1}F1​ score. Using the 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 bars and stripes data set, we train several different circuit constructions for generative modeling with superconducting qubits. By taking hardware connectivity constraints into consideration, we show that sparsely connected shallow circuits outperform denser counterparts on noisy hardware.
Abstract 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.
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Quantum generative models use the intrinsic probabilistic nature of quantum mechanics to learn and reproduce complex probability distributions. In this paper, we present an implementation of a 3-qubit quantum circuit Born machine trained to model a 3-bit Gaussian distribution using a Kullback-Leibler (KL) divergence loss and parameter-shift gradient optimization. The variational quantum circuit consists of layers of parameterized rotations and entangling gates, and is optimized such that the Born rule output distribution closely matches the target distribution. We detail the mathematical formulation of the model distribution, the KL divergence cost function, and the parameter-shift rule for gradient evaluation. Training results on a statevector simulator show that the KL divergence is minimized to near zero, and the final generated distribution aligns quantitatively with the target probabilities. We analyze the convergence behavior and discuss the implications for scalability and quantum advantage. Our results demonstrate the feasibility of small-scale quantum generative learning and provide insight into the training dynamics of quantum circuit models.
The combination of quantum and classical computational resources towards more effective algorithms is one of the most promising research directions in computer science. In such a hybrid framework, existing quantum computers can be used to their fullest extent and for practical applications. Generative modeling is one of the applications that could benefit the most, either by speeding up the underlying sampling methods or by unlocking more general models. In this work, we design a number of hybrid generative models and validate them on real hardware and datasets. The quantum-assisted Boltzmann machine is trained to generate realistic artificial images on quantum annealers. Several challenges in state-of-the-art annealers shall be overcome before one can assess their actual performance. We attack some of the most pressing challenges such as the sparse qubit-to-qubit connectivity, the unknown effective-temperature, and the noise on the control parameters. In order to handle datasets of realistic size and complexity, we include latent variables and obtain a more general model called the quantum-assisted Helmholtz machine. In the context of gate-based computers, the quantum circuit Born machine is trained to encode a target probability distribution in the wavefunction of a set of qubits. We implement this model on a trapped ion computer using low-depth circuits and native gates. We use the generative modeling performance on the canonical Bars-and-Stripes dataset to design a benchmark for hybrid systems. It is reasonable to expect that quantum data, i.e., datasets of wavefunctions, will become available in the future. We derive a quantum generative adversarial network that works with quantum data. Here, two circuits are optimized in tandem: one tries to generate suitable quantum states, the other tries to distinguish between target and generated states.
Generative modelling is an important unsupervised task in machine learning. In this work, we study a hybrid quantum-classical approach to this task, based on the use of a quantum circuit born machine. In particular, we consider training a quantum circuit born machine using f-divergences. We first discuss the adversarial framework for generative modelling, which enables the estimation of any f-divergence in the near term. Based on this capability, we introduce two heuristics which demonstrably improve the training of the born machine. The first is based on f-divergence switching during training. The second introduces locality to the divergence, a strategy which has proved important in similar applications in terms of mitigating barren plateaus. Finally, we discuss the long-term implications of quantum devices for computing f-divergences, including algorithms which provide quadratic speedups to their estimation. In particular, we generalise existing algorithms for estimating the Kullback–Leibler divergence and the total variation distance to obtain a fault-tolerant quantum algorithm for estimating another f-divergence, namely, the Pearson divergence.
Quantum generative modeling is a growing area of interest for industry-relevant applications. This work systematically compares a broad range of techniques to guide quantum computing practitioners when deciding which models and methods to use in their applications. We compare fundamentally different architectural ansatzes of parametric quantum circuits: (1) A continuous architecture, which produces continuous-valued data samples, and (2) a discrete architecture, which samples on a discrete grid. We also compare the performance of different data transformations: the min-max and the probability integral transforms. We use two popular training methods: (1) quantum circuit Born machines (QCBM), and (2) quantum generative adversarial networks (QGAN). We study their performance and tradeoffs as the number of model parameters increases, with a baseline comparison of similarly trained classical neural networks. The study is performed on six low-dimensional synthetic and two real financial data sets. Our two key findings are that: (1) For all data sets, our quantum models require similar or fewer parameters than their classical counterparts. In the extreme case, the quantum models require two orders of magnitude less parameters. (2) We empirically find that a variant of the discrete architecture, which learns the copula of the probability distribution, outperforms all other methods.