conference paper
Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture
published
Sevag
Gharibian
author 715410000-0002-9992-3379
François Le
Gall
author
623
department
7
department
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, Gily\'en, 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.
2022
eng
Proceedings of the 54th ACM Symposium on Theory of Computing (STOC)
2111.09079
19-32
Gharibian, Sevag, and François Le Gall. “Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture.” In <i>Proceedings of the 54th ACM Symposium on Theory of Computing (STOC)</i>, 19–32, 2022.
Gharibian, S., & Gall, F. L. (2022). Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture. <i>Proceedings of the 54th ACM Symposium on Theory of Computing (STOC)</i>, 19–32.
@inproceedings{Gharibian_Gall_2022, title={Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture}, booktitle={Proceedings of the 54th ACM Symposium on Theory of Computing (STOC)}, author={Gharibian, Sevag and Gall, François Le}, year={2022}, pages={19–32} }
S. Gharibian, F.L. Gall, in: Proceedings of the 54th ACM Symposium on Theory of Computing (STOC), 2022, pp. 19–32.
Gharibian, Sevag, and François Le Gall. “Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture.” <i>Proceedings of the 54th ACM Symposium on Theory of Computing (STOC)</i>, 2022, pp. 19–32.
Gharibian S, Gall FL. Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture. In: <i>Proceedings of the 54th ACM Symposium on Theory of Computing (STOC)</i>. ; 2022:19-32.
S. Gharibian and F. L. Gall, “Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture,” in <i>Proceedings of the 54th ACM Symposium on Theory of Computing (STOC)</i>, 2022, pp. 19–32.
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