Generalized quantum circuit differentiation rules

Kyriienko, Oleksandr; Elfving, Vincent E.

doi:10.1103/physreva.104.052417

Cited by 31 publications

(18 citation statements)

References 77 publications

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“…In the next subsection, where we analyse the phase feature map, we will also show how to read out x derivatives of DQGM. While this can be done through the parameter shift rule [65,66] and generalizations [67], can be readout exactly and more efficiently by avoiding the regular parameter shift rule.…”

Section: Model Differentiation and Constrained Training From Stochast...mentioning

confidence: 99%

“…where the spectrum of frequencies Ω represent all possible differences of eigenvalues of Ĝ, and c ω,θ are θ-dependent coefficients associated to each frequency [67,70]. The important properties of the spectrum are that it includes zero frequency, pairs of equal-magnitude positive and negative frequencies, and coefficients obey c ω = c * −ω leading to real-valued models (as expected from an expectation value).…”

Section: Phase Feature Map Analysismentioning

confidence: 99%

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Protocols for Trainable and Differentiable Quantum Generative Modelling

Kyriienko¹,

Paine²,

Elfving³

2022

Preprint

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We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modelling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQCbased model, where data is encoded in a latent space with a phase feature map, followed by a variational quantum circuit. We then map the trained model to the bit basis using a fixed unitary transformation, coinciding with a quantum Fourier transform circuit in the simplest case. This 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. Finally, our approach opens a route to multidimensional generative modelling 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 qubit-wise training and feature map sparsification.

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Section: Model Differentiation and Constrained Training From Stochast...mentioning

confidence: 99%

Section: Phase Feature Map Analysismentioning

confidence: 99%

Protocols for Trainable and Differentiable Quantum Generative Modelling

Kyriienko¹,

Paine²,

Elfving³

2022

Preprint

Self Cite

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“…Using the parameter shift rule means we calculate the analytic derivative though it does place some requirements on the gates parametrized by x/y such as being involutory. Generalized parameter shift rules are possible, where such requirements are relaxed [69][70][71][72][73].…”

Section: Derivativesmentioning

confidence: 99%

Quantum Kernel Methods for Solving Differential Equations

Paine¹,

Elfving²,

Kyriienko³

2022

Preprint

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We propose several approaches for solving differential equations (DEs) with quantum kernel methods. We compose quantum models as weighted sums of kernel functions, where variables are encoded using feature maps and model derivatives are represented using automatic differentiation of quantum circuits. While previously quantum kernel methods primarily targeted classification tasks, here we consider their applicability to regression tasks, based on available data and differential constraints. We use two strategies to approach these problems. First, we devise a mixed model regression with a trial solution represented by kernel-based functions, which is trained to minimize a loss for specific differential constraints or datasets. Second, we use support vector regression that accounts for the structure of differential equations. The developed methods are capable of solving both linear and nonlinear systems. Contrary to prevailing hybrid variational approaches for parametrized quantum circuits, we perform training of the weights of the model classically. Under certain conditions this corresponds to a convex optimization problem, which can be solved with provable convergence to global optimum of the model. The proposed approaches also favor hardware implementations, as optimization only uses evaluated Gram matrices, but require quadratic number of function evaluations. We highlight trade-offs when comparing our methods to those based on variational quantum circuits such as the recently proposed differentiable quantum circuits (DQC) approach. The proposed methods offer potential quantum enhancement through the rich kernel representations using the power of quantum feature maps, and start the quest towards provably trainable quantum DE solvers.

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“…Furthermore, the example shows the regular parameter shift rule, which only applies to involutory quantum generators in the feature map. Generalized parameter shift rules exist for arbitrary generators, and hence for arbitrary feature map circuits [14]. In some cases, more intricate feature maps can be highly beneficial for increased expressivity of the quantum universal approximator.…”

Section: Differentiable Quantum Universal Function Approximatorsmentioning

confidence: 99%

“…Recent progress in automatic differentiation on quantum computers [6][7][8][9][10][11][12][13][14] introduce the prospect of extending recent classical results in scientific machine learning to a quantum setting. These techniques have already led to quantum methods of solving differential equations [15] which are analogous to methods involving classical neural networks [2].…”

Section: Introductionmentioning

confidence: 99%

Quantum Model-Discovery

Heim¹,

Ghosh²,

Kyriienko³

et al. 2021

Preprint

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Quantum computing promises to speed up some of the most challenging problems in science and engineering. Quantum algorithms have been proposed showing theoretical advantages in applications ranging from chemistry to logistics optimization. Many problems appearing in science and engineering can be rewritten as a set of differential equations. Quantum algorithms for solving differential equations have shown a provable advantage in the fault-tolerant quantum computing regime, where deep and wide quantum circuits can be used to solve large linear systems like partial differential equations (PDEs) efficiently. Recently, variational approaches to solving nonlinear PDEs also with near-term quantum devices were proposed. One of the most promising general approaches is based on recent developments in the field of scientific machine learning for solving PDEs. We extend the applicability of near-term quantum computers to more general scientific machine learning tasks, including the discovery of differential equations from a dataset of observations.We use differentiable quantum circuits (DQCs) to solve equations parameterized by a library of operators, and perform regression on a combination of data and equations. Our results show a promising path to quantum model discovery (QMoD), on the interface between classical and quantum machine learning approaches. We demonstrate successful parameter inference and equation discovery using QMoD on different systems including a second-order, ordinary differential equation and a non-linear, partial differential equation.

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Generalized quantum circuit differentiation rules

Abstract: The version presented here may differ from the published version. If citing, you are advised to consult the published version for pagination, volume/issue and date of publication

Cited by 31 publications

References 77 publications

Protocols for Trainable and Differentiable Quantum Generative Modelling

Protocols for Trainable and Differentiable Quantum Generative Modelling

Quantum Kernel Methods for Solving Differential Equations

Quantum Model-Discovery

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