[Paper Review] Quantum-inspired classical algorithm for molecular vibronic spectra

Changhun Oh, Youngrong Lim+3 more|arXiv (Cornell University)|Feb 3, 2022

Quantum Computing Algorithms and Architecture54 references4 citations

TL;DR

This paper proposes a quantum-inspired classical algorithm that efficiently computes molecular vibronic spectra using Fourier transformation and sparse fast Fourier transforms, demonstrating that both Fock-state and Gaussian boson sampling problems—commonly considered candidates for quantum advantage—can be classically solved with high accuracy. The key result is that these specific vibronic spectra problems do not provide quantum advantage, but a more general class of problems may still be classically hard and suitable for quantum speedup.

ABSTRACT

We have recently seen the first plausible claims for quantum advantage using sampling problems such as random circuit sampling and Gaussian boson sampling. The obvious next step is to channel the potential quantum advantage to solving practical applications rather than proof-of-principle experiments. Recently, a quantum simulator, specifically a Gaussian boson sampler, has been proposed to generate molecular vibronic spectra efficiently, which is an essential property of molecules and an important tool for analyzing chemical components and studying molecular structures. Computing molecular vibronic spectra has been a challenging task, and its best-known classical algorithm scales combinatorially in the system size. Thus, it is a candidate of tasks for which quantum devices provide computational advantages. In this work, we propose a quantum-inspired classical algorithm for molecular vibronic spectra for harmonic potential. We first show that the molecular vibronic spectra problem corresponding to Fock-state boson sampling can be efficiently solved using a classical algorithm as accurately as running a boson sampler. In particular, we generalize Gurvits's algorithm to approximate Fourier components of the spectra of Fock-state boson sampling and prove using Parseval's relation that the error of the spectra can be suppressed as long as that of the Fourier components are small. We also show that the molecular vibronic spectra problems of Gaussian boson sampling, which corresponds to the actual molecular vibronic spectra problem in chemistry, can be exactly solved even without Gurvits-type algorithms. Consequently, we demonstrate that those problems are not candidates of quantum advantage. We then provide a more general molecular vibronic spectra problem, which is also chemically well-motivated, for which we might be able to take advantage of a boson sampler.

Motivation & Objective

• To investigate whether molecular vibronic spectra computation, proposed as a candidate for quantum advantage, can be efficiently solved using classical algorithms.
• To determine the computational complexity of vibronic spectra problems corresponding to Fock-state and Gaussian boson sampling.
• To develop a classical algorithm that matches the accuracy of boson sampling for these problems.
• To identify whether there exists a chemically motivated vibronic spectra problem that remains classically hard and thus suitable for quantum advantage.

Proposed method

• Generalize Gurvits’s algorithm to approximate Fourier components of Fock-state boson sampling spectra with controlled error.
• Use Parseval’s relation to bound the error in the final spectra from the error in the Fourier components.
• Apply sparse fast Fourier transformation to efficiently compute spectra when the number of bins exceeds polynomial size.
• Show that for Gaussian boson sampling, the Fourier components can be computed exactly and efficiently using positive P-representation.
• Leverage the structure of the generalized P-representation to enable efficient sampling and computation of expectation values.
• Use random hashing and iterative peak detection in sparse FFT to handle superpolynomially large weight vectors.

Experimental results

Research questions

• RQ1 Can molecular vibronic spectra problems corresponding to Fock-state boson sampling be classically approximated with the same accuracy as a boson sampler?
• RQ2 Is the Gaussian boson sampling problem for molecular vibronic spectra classically solvable with exact or efficient approximation?
• RQ3 Does the proposed classical algorithm achieve the same error bounds as a quantum boson sampler?
• RQ4 Are there chemically motivated vibronic spectra problems that remain classically hard despite the proposed method?
• RQ5 Can the framework of Fourier transformation and sparse FFT be used to efficiently compute spectra with additive error guarantees?

Key findings

• The Fock-state boson sampling problem for molecular vibronic spectra can be classically approximated with error bounded by the error in Fourier component estimation, thanks to Parseval’s relation.
• The proposed algorithm achieves spectral error less than ε if the Fourier components are estimated within error ε, ensuring accuracy comparable to a boson sampler.
• For Gaussian boson sampling, the Fourier components can be computed exactly and efficiently using the positive P-representation, making the entire problem classically solvable.
• The sparse fast Fourier transform enables efficient computation even when the number of spectral bins grows superpolynomially in system size.
• The method fails to efficiently solve a more general class of vibronic spectra problems, indicating that such problems may still be candidates for quantum advantage.
• The paper concludes that Fock-state and Gaussian boson sampling-based vibronic spectra problems do not provide a quantum advantage, but broader classes of problems may.