Explicit lower bounds on strong quantum simulation

Huang, Cupjin; Newman, Michael; Szegedy, Márió

doi:10.48550/arxiv.1804.10368

Cited by 9 publications

(11 citation statements)

References 7 publications

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“…Dalzell et al showed that under some fine-grained complexity conjectures, output probability distributions of the IQP model, the Boson sampling model, and the QAOA model cannot be classically sampled within a constant multiplicative error in certain exponential time [25,26]. Huang et al showed impossibilities of strong simulations (i.e., exact computations of output probability distributions) in some exponential time assuming the exponential time hypothesis (ETH) [27,28]. (The definition of ETH is given later.…”

Section: Introductionmentioning

confidence: 99%

Fine-grained quantum computational supremacy

Morimae

Tamaki²

2019

Preprint

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Output probability distributions of several sub-universal quantum computing models cannot be classically efficiently sampled unless some unlikely consequences occur in classical complexity theory, such as the collapse of the polynomial-time hierarchy. These results, so called quantum supremacy, however, do not rule out possibilities of super-polynomial-time classical simulations. In this paper, we study "fine-grained" version of quantum supremacy that excludes some exponential-time classical simulations. First, we focus on two sub-universal models, namely, the one-clean-qubit model (or the DQC1 model) and the HC1Q model. Assuming certain conjectures in fine-grained complexity theory, we show that for any a > 0 output probability distributions of these models cannot be classically sampled within a constant multiplicative error and in 2 (1−a)N+o(N) time, where N is the number of qubits. Next, we consider universal quantum computing. For example, we consider quantum computing over Clifford and T gates, and show that under another fine-grained complexity conjecture, output probability distributions of Clifford-T quantum computing cannot be classically sampled in 2 o(t) time within a constant multiplicative error, where t is the number of T gates.

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Section: Introductionmentioning

confidence: 99%

Fine-grained quantum computational supremacy

Morimae

Tamaki²

2019

Preprint

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“…For example, output probability distributions of the depth-four model [1], the Boson Sampling model [2], the IQP model [3,4], the one-clean qubit model [5][6][7][8][9], and the random circuit model [10,11] cannot be classically sampled in polynomial time unless some conjectures in classical complexity theory (such as the infiniteness of the polynomialtime hierarchy) are refuted. Impossibilities of exponential-time classical simulations of subuniversal quantum computing models have also been shown recently based on classical finegrained complexity theory [12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Sampling of globally depolarized random quantum circuit

Morimae,

Takeuchi,

Tani

2019

Preprint

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The recent paper [F. Arute et al. Nature 574, 505 (2019)] considered exact classical sampling of the output probability distribution of the globally depolarized random quantum circuit. In this paper, we show three results. First, we consider the case when the fidelity F is constant. We show that if the distribution is classically sampled in polynomial time within a constant multiplicative error, then BQP ⊆ SBP, which means that BQP is in the second level of the polynomial-time hierarchy. We next show that for any F ≤ 1/2, the distribution is classically trivially sampled by the uniform distribution within the multiplicative error F 2 n+2 , where n is the number of qubits.We finally show that for any F , the distribution is classically trivially sampled by the uniform distribution within the additive error 2F . These last two results show that if we consider realistic cases, both F ∼ 2 −m and m ≫ n, or at least F ∼ 2 −m , where m is the number of gates, quantum supremacy does not exist for approximate sampling even with the exponentially-small errors. We also argue that if F ∼ 2 −m and m ≫ n, the standard approach will not work to show quantum supremacy even for exact sampling.

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“…The strong exponential-time hypothesis also directly implies many interesting exponential lower bounds within NP, giving structure to problems within the complexity class. A wide range of problems (even outside of just NP-complete problems) can be shown to require strong exponential time assuming SETH: for instance, recent work shows that, conditioned on SETH, classical computers require exponential time for so-called strong simulation of several models of quantum computation [HNS18,MT19].…”

Section: Introductionmentioning

confidence: 99%

The Quantum Strong Exponential-Time Hypothesis

Buhrman,

Patro,

Speelman

2019

Preprint

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The strong exponential-time hypothesis (SETH) is a commonly used conjecture in the field of complexity theory. It states that CNF formulas cannot be analyzed for satisfiability with a speedup over exhaustive search. This hypothesis and its variants gave rise to a fruitful field of research, fine-grained complexity, obtaining (mostly tight) lower bounds for many problems in P whose unconditional lower bounds are hard to find. In this work, we introduce a framework of Quantum Strong Exponential-Time Hypotheses, as quantum analogues to SETH.Using the QSETH framework, we are able to translate quantum query lower bounds on black-box problems to conditional quantum time lower bounds for many problems in BQP. As an example, we illustrate the use of the QSETH by providing a conditional quantum time lower bound of Ω(n 1.5 ) for the Edit Distance problem. We also show that the n 2 SETH-based lower bound for a recent scheme for Proofs of Useful Work, based on the Orthogonal Vectors problem holds for quantum computation assuming QSETH, maintaining a quadratic gap between verifier and prover.Paturi, and, Zane [IP01, IPZ01] studied two ways in which this can be conjectured to be optimal. The first of which is called the Exponential-Time Hypothesis (ETH). Conjecture 1 (Exponential-Time Hypothesis).There exists a constant α > 0 such that CNF-SAT on n variables and m can not be solved in time O(m2 αn ) by a (classical) Turing machine.This conjecture can be directly used to give lower bounds for many natural NP-complete problems, showing that if ETH holds then these problems also require exponential time to solve. The second conjecture, most importantly for the current work, is the Strong Exponential-Time Hypothesis (SETH).Conjecture 2 (Strong Exponential-Time Hypothesis). There does not exist δ > 0 such that CNF-SAT on n variables and m clauses can be solved in O(m2 n(1−δ) ) time by a (classical) Turing machine.2 Lower bounds for the restricted Dyck language were recently independently proven by Ambainis, Balodis, Iraids, Prūsis, and Smotrovs [ABI + 19], and Frédéric Magniez [Mag19].3 We use O to denote asymptotic behavior up to polylogarithmic factors.

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Explicit lower bounds on strong quantum simulation

Cited by 9 publications

References 7 publications

Fine-grained quantum computational supremacy

Fine-grained quantum computational supremacy

Sampling of globally depolarized random quantum circuit

The Quantum Strong Exponential-Time Hypothesis

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