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arXiv:2306.17026

Quantum Chebyshev Transform: Mapping, Embedding, Learning and Sampling Distributions

Chelsea A. Williams, Annie E. Paine, Hsin‐Yu Wu, Vincent E. Elfving, Oleksandr Kyriienko

Published Jun 29, 2023• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.2306.17026

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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.
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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.
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 is emerging as a powerful tool for advancing data analysis in high-energy physics, where complex multivariate distributions are common. However, efficiently learning and sampling these distributions remains challenging. We propose a quantum protocol for a bivariate probabilistic model based on shifted Chebyshev polynomials, trained as a circuit-based representation of two correlated variables, with sampling performed via quantum Chebyshev transforms. As a key application, we study fragmentation functions (FFs) of charged pions and kaons from single-inclusive hadron production in electron-positron annihilation. We learn the joint distribution of momentum fraction z and energy scale Q, and infer their correlations from the entanglement structure. Building on the generalization capabilities of the quantum model and extended register architecture, we perform fine-grid multivariate sampling for FF dataset augmentation. Our results highlight the growing potential of quantum generative modeling to advance data analysis and scientific discovery in high-energy physics. Quantum computing holds great promise for advancing data analysis in science, including high-energy physics. This work presents a quantum protocol for learning and sampling multivariate distributions, using Chebyshev polynomials and quantum Chebyshev transforms to study fragmentation functions and energy-momentum correlations in hadron production.
Many quantum algorithms can be represented in a form of a classical circuit positioned between quantum Fourier transformations. Motivated by the search for new quantum algorithms, we turn to circuits where the latter transformation is replaced by the SU(2) quantum Schur transform---a global transformation which maps the computational basis to a basis defined by angular momenta. We show that the output distributions of these circuits can be approximately classically sampled in polynomial time if they are sufficiently close to being sparse, thus isolating a regime in which they could lead to algorithms with exponential computational advantages. Our paper is primarily motivated by a conjecture that underpinned the hardness of permutational quantum computing, a restricted quantum computational model that has the above circuit structure in one of its computationally interesting regimes. The conjecture stated that approximating transition amplitudes of permutational quantum computing model to inverse-polynomial precision on a classical computer is computationally hard in this case. We disprove the extended version of this conjecture---even in the case when the hardness of approximation originated from a difficulty of finding the large elements in the output probability distributions. Finally, we present some evidence that output of the above permutational quantum computing circuits could be efficiently approximately sampled from on a classical computer.