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 et al. [ Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning, in Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2020), 2020, pp. 387–400] 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 [Formula: see text]-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.
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 et al. [ Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning, in Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2020), 2020, pp. 387–400] 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 [Formula: see text]-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.
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.
Several quantum algorithms for linear algebra problems, and in particular quantum machine learning problems, have been “dequantized” in the past few years. These dequantization results typically hold when classical algorithms can access the data via length-squared sampling. This assumption, which is standard in the field of randomized linear algebra, means that for a unit-norm vector u∈Cn , we can sample from the distribution pu:{1,…,n}→[0,1] defined as pu(i)=∣u(i)∣2 for each i∈{1,…,n} . Since this distribution corresponds to the distribution obtained by measuring the quantum state | u > in the computational basis, length-squared sampling access gives a reasonable classical analogue to the kind of quantum access considered in many quantum algorithms for linear algebra problems. In this work we investigate how robust these dequantization results are. We introduce the notion of approximate length-squared sampling, where classical algorithms are only able to sample from a distribution close to the ideal distribution in total variation distance. While quantum algorithms are natively robust against small perturbations, current techniques in dequantization are not. Our main technical contribution is showing how many techniques from randomized linear algebra can be adapted to work under this weaker assumption as well. We then use these techniques to show that the recent low-rank dequantization framework by Chia, Gilyén, Li, Lin, Tang and Wang (JACM 2022) and the dequantization framework for sparse matrices by Gharibian and Le Gall (STOC 2022), which are both based on the Quantum Singular Value Transformation, can be generalized to the case of approximate length-squared sampling access to the input. We also apply these results to obtain a robust dequantization of many quantum machine learning algorithms, including quantum algorithms for recommendation systems, supervised clustering and low-rank matrix inversion.
Several quantum algorithms for linear algebra problems, and in particular quantum machine learning problems, have been "dequantized" in the past few years. These dequantization results typically hold when classical algorithms can access the data via length-squared sampling. In this work we investigate how robust these dequantization results are. We introduce the notion of approximate length-squared sampling, where classical algorithms are only able to sample from a distribution close to the ideal distribution in total variation distance. While quantum algorithms are natively robust against small perturbations, current techniques in dequantization are not. Our main technical contribution is showing how many techniques from randomized linear algebra can be adapted to work under this weaker assumption as well. We then use these techniques to show that the recent low-rank dequantization framework by Chia, Gilyén, Li, Lin, Tang and Wang (JACM 2022) and the dequantization framework for sparse matrices by Gharibian and Le Gall (STOC 2022), which are both based on the Quantum Singular Value Transformation, can be generalized to the case of approximate length-squared sampling access to the input. We also apply these results to obtain a robust dequantization of many quantum machine learning algorithms, including quantum algorithms for recommendation systems, supervised clustering and low-rank matrix inversion.
We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffices to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ2-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.
Quantum-inspired classical algorithms have dequantized several quantum machine learning routines by replacing quantum linear-algebra subroutines with classical counterparts. However, the sampler based on quantum singular value transformation (QSVT) for learning with optimized random features is not covered by existing dequantization frameworks, because the matrix to be inverted is not itself available through sampling access. In this work, we develop a classical algorithm to address this type of quantum-advantage candidate. Our method samples heavy indices, reduces the transformation to a small principal block, and outputs a sparse classical representation with operator-norm guarantees. Applying this method dequantizes the sampler for optimized random features, giving a classical sampler with prescribed accuracy and polynomially related runtime. These results show that the factorization underlying a quantum block encoding can itself provide sufficient classical structure even when sampling-and-query access to the composite matrix is unavailable.
The field of quantum machine learning (QML) produces many proposals for attaining quantum speedups for tasks in machine learning and data analysis. Such speedups can only manifest if classical algorithms for these tasks perform significantly slower than quantum ones. We study quantum-classical gaps in QML through the quantum singular value transformation (QSVT) framework. QSVT, introduced by Gilyén, Su, Low and Wiebe [GSLW19], unifies all major types of quantum speedup [MRTC21]; in particular, a wide variety of QML proposals are applications of QSVT on low-rank classical data. We challenge these proposals by providing a classical algorithm that matches the performance of QSVT in this regime up to a small polynomial overhead.