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Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture

Sevag Gharibian, François Le Gall

TL;DR

This work studies when the Quantum Singular Value Transformation (QSVT) can be efficiently simulated classically, focusing on sparse matrices. It provides a dequantization framework that, for constant-degree polynomials $P$, yields a classical algorithm to approximate $v^ op P(\,\sqrt{A^ A})u$ from query-access to $A$ and sampling-access to $u,v$, with complexity poly$(\log N,s,1/\epsilon)$. As an application, the authors show that estimating the ground-state energy of local Hamiltonians given a good guiding state can be done classically with constant precision, while inverse-polynomial precision remains $ extsf{BQP}$-hard, giving theoretical evidence for quantum advantages in quantum chemistry. They also develop a broader dequantization of the QSVT for sparse matrices, with a concrete singular-value-estimation routine and a reduction to $ extsf{GLH}^{\text{est}}$, thereby linking these techniques to the Quantum PCP conjecture via notions of samplable states and the NLSS/MA-QMA landscape. The results illuminate how precision, data-access models, and problem structure govern quantum-classical gaps and suggest concrete pathways for both quantum advantages in chemistry and avenues to address core questions in quantum Hamiltonian complexity.

Abstract

The Quantum Singular Value Transformation (QSVT) is a recent technique that gives a unified framework to describe most quantum algorithms discovered so far, and may lead to the development of novel quantum algorithms. In this paper we investigate the hardness of classically simulating the QSVT. A recent result by Chia, Gilyén, Li, Lin, Tang and Wang (STOC 2020) showed that the QSVT can be efficiently "dequantized" for low-rank matrices, and discussed its implication to quantum machine learning. In this work, motivated by establishing the superiority of quantum algorithms for quantum chemistry and making progress on the quantum PCP conjecture, we focus on the other main class of matrices considered in applications of the QSVT, sparse matrices. We first show how to efficiently "dequantize", with arbitrarily small constant precision, the QSVT associated with a low-degree polynomial. We apply this technique to design classical algorithms that estimate, with constant precision, the singular values of a sparse matrix. We show in particular that a central computational problem considered by quantum algorithms for quantum chemistry (estimating the ground state energy of a local Hamiltonian when given, as an additional input, a state sufficiently close to the ground state) can be solved efficiently with constant precision on a classical computer. As a complementary result, we prove that with inverse-polynomial precision, the same problem becomes BQP-complete. This gives theoretical evidence for the superiority of quantum algorithms for chemistry, and strongly suggests that said superiority stems from the improved precision achievable in the quantum setting. We also discuss how this dequantization technique may help make progress on the central quantum PCP conjecture.

Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture

TL;DR

This work studies when the Quantum Singular Value Transformation (QSVT) can be efficiently simulated classically, focusing on sparse matrices. It provides a dequantization framework that, for constant-degree polynomials , yields a classical algorithm to approximate from query-access to and sampling-access to , with complexity poly. As an application, the authors show that estimating the ground-state energy of local Hamiltonians given a good guiding state can be done classically with constant precision, while inverse-polynomial precision remains -hard, giving theoretical evidence for quantum advantages in quantum chemistry. They also develop a broader dequantization of the QSVT for sparse matrices, with a concrete singular-value-estimation routine and a reduction to , thereby linking these techniques to the Quantum PCP conjecture via notions of samplable states and the NLSS/MA-QMA landscape. The results illuminate how precision, data-access models, and problem structure govern quantum-classical gaps and suggest concrete pathways for both quantum advantages in chemistry and avenues to address core questions in quantum Hamiltonian complexity.

Abstract

The Quantum Singular Value Transformation (QSVT) is a recent technique that gives a unified framework to describe most quantum algorithms discovered so far, and may lead to the development of novel quantum algorithms. In this paper we investigate the hardness of classically simulating the QSVT. A recent result by Chia, Gilyén, Li, Lin, Tang and Wang (STOC 2020) showed that the QSVT can be efficiently "dequantized" for low-rank matrices, and discussed its implication to quantum machine learning. In this work, motivated by establishing the superiority of quantum algorithms for quantum chemistry and making progress on the quantum PCP conjecture, we focus on the other main class of matrices considered in applications of the QSVT, sparse matrices. We first show how to efficiently "dequantize", with arbitrarily small constant precision, the QSVT associated with a low-degree polynomial. We apply this technique to design classical algorithms that estimate, with constant precision, the singular values of a sparse matrix. We show in particular that a central computational problem considered by quantum algorithms for quantum chemistry (estimating the ground state energy of a local Hamiltonian when given, as an additional input, a state sufficiently close to the ground state) can be solved efficiently with constant precision on a classical computer. As a complementary result, we prove that with inverse-polynomial precision, the same problem becomes BQP-complete. This gives theoretical evidence for the superiority of quantum algorithms for chemistry, and strongly suggests that said superiority stems from the improved precision achievable in the quantum setting. We also discuss how this dequantization technique may help make progress on the central quantum PCP conjecture.
Paper Structure (14 sections, 13 theorems, 58 equations)

This paper contains 14 sections, 13 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Main results
  4. Overview of our techniques
  5. Implications to quantum chemistry and the quantum PCP conjecture
  6. Preliminaries
  7. Query-access and sampling-access to vectors and matrices
  8. The Singular Value Transformation
  9. $\mathsf{BQP}$-Hardness of Polynomial-Precision Eigenvalue Estimation
  10. Dequantizing the Quantum Singular Value Transformation
  11. Main technique
  12. Singular Value Estimation by dequantizing the QSVT
  13. Classical estimation of the ground state energy of local Hamiltonians
  14. Implications to the Quantum PCP Conjecture

Key Result

Theorem 1.1

For any constants $\epsilon,\delta\in(0,1]$ and any $k=O(\log n)$, the problem $\mathsf{GLH}^{\mathsf{est}}(k,\epsilon,\delta)$ can be solved classically with probability at least $1-1/\exp(n)$ in $O(\mathrm{poly}(n))$ time.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Informal version of \ref{['th:classical-est-formal']}
  • Theorem 1.4: Informal version of \ref{['th:SVT-formal']}
  • Conjecture 1: quantum PCP conjecture
  • Theorem 1.5: Informal version of \ref{['th:qPCP-formal']}
  • Conjecture 2: NLTS conjecture
  • Conjecture 3: NLSS conjecture --- informal version of \ref{['conj:variant-formal']}
  • Lemma 2.1: Lemma 6.1 in Jerrum+86
  • Definition 2.2
  • ...and 21 more