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arXiv:2202.08253 · Physical Review Research

Protocols for trainable and differentiable quantum generative modeling

Oleksandr Kyriienko, Annie E. Paine, Vincent E. Elfving

Published Sep 12, 2024• 14 citations
arXivHTMLPDF

Abstract

We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modeling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQC-based model, where data is encoded in a latent space with the proposed phase feature map of exponential capacity. This is followed by a trainable quantum circuit, forming the model. We then map the trained model to the bit basis using a fixed unitary transformation, in this case corresponding to a quantum Fourier transform circuit. It allows fast sampling from parametrized distributions using a single-shot readout. Importantly, latent space training provides models that are automatically differentiable, and we show how samples from solutions of stochastic differential equations (SDEs) can be accessed by solving stationary and time-dependent Fokker-Planck equations with a quantum protocol. Our approach opens a route to multidimensional generative modeling with qubit registers explicitly correlated via a (fixed) entangling layer. In this case quantum computers can offer advantage as efficient samplers, which perform complex inverse transform sampling enabled by the fundamental laws of quantum mechanics. On a technical side the advances are multiple, as we introduce the phase feature map, analyze its properties, and develop frequency-taming techniques that include qubitwise training and feature map sparsification. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.033291

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We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modeling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQC-based model, where data is encoded in a latent space with the proposed phase feature map of exponential capacity. This is followed by a trainable quantum circuit, forming the model. We then map the trained model to the bit basis using a fixed unitary transformation, in this case corresponding to a quantum Fourier transform circuit. It allows fast sampling from parametrized distributions using a single-shot readout. Importantly, latent space training provides models that are automatically differentiable, and we show how samples from solutions of stochastic differential equations (SDEs) can be accessed by solving stationary and time-dependent Fokker-Planck equations with a quantum protocol. Our approach opens a route to multidimensional generative modeling with qubit registers explicitly correlated via a (fixed) entangling layer. In this case quantum computers can offer advantage as efficient samplers, which perform complex inverse transform sampling enabled by the fundamental laws of quantum mechanics. On a technical side the advances are multiple, as we introduce the phase feature map, analyze its properties, and develop frequency-taming techniques that include qubitwise training and feature map sparsification. Published by the American Physical Society 2024
We develop a paradigm for building quantum models in the orthonormal space of Chebyshev polynomials. We show how to encode data into quantum states with amplitudes being Chebyshev polynomials with degree growing exponentially in the system size. Similar to the quantum Fourier transform which maps computational basis space into the phase (Fourier) basis, we describe the quantum circuit for the mapping between computational and Chebyshev spaces. We propose an embedding circuit for generating the orthonormal Chebyshev basis of exponential capacity, represented by a continuously-parameterized shallow isometry. This enables automatic quantum model differentiation, and opens a route to solving stochastic differential equations. We apply the developed paradigm to generative modeling from physically- and financially-motivated distributions, and use the quantum Chebyshev transform for efficient sampling of these distributions in extended computational basis.
We develop and analyze generative models based on quantum solvers of stochastic differential equations. Unlike the common approach based on layered variational quantum circuits with parameterized rotations, we parameterize the potential of a Langevin process. The associated Fokker-Planck (FP) equations are Hermitianized and solved using a linear combination of cosine operators implemented through quantum singular value transformation. By encoding the square root of the stationary density, the model enables sampling through computational-basis measurements and is efficiently controlled by coefficients of the block-encoded diagonal potential. We train the proposed model to match empirical distributions within the family described by FP equations, demonstrating parameter recovery for multivariate probability distributions. Crucially, the model remains trainable at scale. Sample-efficient and convex optimization for in-distribution targets arises from a physics-informed inductive bias and holds regardless of the dimensionality of the problem. We provide circuit constructions and efficient block encodings of the main components of the algorithm and analyze resource requirements. Our approach frames quantum generative learning as quantum model discovery and provides a foundation for extending this framework to high-dimensional, strongly correlated scientific data.
We propose an algorithmic framework for building and training quantum generative models corresponding to multivariate probability distributions. Our model uses parallel Fourier feature maps for embedding continuous-valued variables combined with a forrelation-type quantum circuit for tuning Fourier coefficients of the quantum model. Crucially, we develop a distinct training strategy where training is enabled at large scale by log-likelihood loss with unbiased Monte Carlo estimator based on Parseval's identity. Unlike prior work that relied on maximal mean discrepancy (MMD) loss, our approach goes beyond matching just low frequency moments, while enabling efficient classical training. Once the model is trained, we use inverse quantum Fourier transforms to map it into a separate sampling circuit in the computational basis. We demonstrate the efficiency of the suggested framework by validating loss estimation at the scale of over 1000 qubits on a single GPU. We show that univariate and bivariate models with highly non-trivial structure can be trained to low total variation distance, while fine-tuned IQP models with MMD loss show poor performance. Comparing to classical baselines represented by normalizing flow and diffusion models, we show that our approach avoids oversmoothing and preserves multi-modal structure of the target. Finally, we have deployed the trained models on superconducting quantum devices, successfully sampling distributions with per-sample execution times of approximately 300 μs300\,μ\mathrm{s}300μs. Our work shows that quantum generative models with the train-on-classical deploy-on-quantum approach can provide both high-quality structure at increased scale and fast sampling access needed for inference.
Quantum generative modeling (QGM) relies on preparing quantum states and generating samples 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 test bed for quantum supremacy experiments. Furthermore, generative tasks are increasingly 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 anticoncentration 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.
Training the circuit of a quantum generative model is hampered by barren plateaus, where gradients vanish exponentially with the number of qubits. In the present work we propose a generative protocol based on the paradigm of quantum extreme learning machines, in which the circuit is never trained. A random latent vector is encoded into a quantum state, evolved with a fixed unitary and measured, and only a linear output layer is trained. The encoding of the latent variable determines the functions available to the model, while the unitary decides which combinations of them are seen by the trained layer. With the help of Pauli propagation we show that for a Clifford unitary each feature is a single product of sines and cosines of the encoded angles, so that a Clifford circuit, even a strongly entangling one, does not create new functions. Doping the circuit with TTT gates mixes these products and changes the nonstabilizerness of the state, but the space of functions remains the one fixed by the encoding. For brickworks of Haar-random two-qubit gates the features are no longer single products, and their variation with the latent variable decreases with the depth of the circuit. For the proposed encoding and for one- and two-dimensional distributions, generated both from expectation values and from single measurements, neither the entanglement produced by a Clifford reservoir nor the additional nonstabilizerness of Haar-random brickworks systematically improves the generator, and a reservoir with no gates at all is competitive with entangling circuits.
Abstract An approach for building quantum models based on the exponentially growing orthonormal basis of Hartley kernel functions is developed. First, a differentiable Hartley feature map parameterized by real‐valued argument that enables quantum models suitable for solving stochastic differential equations and regression problems is designed. Unlike the naturally complex Fourier encoding, the proposed Hartley feature map circuit leads to quantum states with real‐valued amplitudes, introducing an inductive bias and natural regularization. Next, a quantum Hartley transform circuit is proposed as a map between computational and Hartley basis. The developed paradigm is applied to generative modeling from solutions of stochastic differential equations, and utilize the quantum Hartley transform for fine sampling from parameterized distributions through an extended register. Finally, the capability of multivariate quantum generative modeling is demonstrated for both correlated and uncorrelated distributions. As a result, the developed quantum Hartley‐based generative models (QHGMs) offer a distinct quantum approach to generative AI at increasing scale.