Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation

Williams, Richard Ryan

doi:10.4230/lipics.ccc.2016.2

Cited by 5 publications

(2 citation statements)

References 51 publications

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“…Using evaluation on n + 1 distinct points, we can deterministically solve UPIT in time Õ(n 2 ), while evaluating on Õ(1) random points yields a randomized solution in time Õ(n). Williams [47] proved that a O(n 2−ε )-time deterministic UPIT algorithm refutes the Nondeterministic Strong Exponential Time Hypothesis posed by Carmosino et al [11]. We establish that a sufficiently strong (nondeterministic) derandomization of UPIT also yields progress on MM-Verification.…”

Section: Upitmentioning

confidence: 57%

“…By evaluating and comparing p and q at n + 1 distinct points or Õ(1) random points, we can solve UPIT deterministically in time Õ(n 2 ) or with high probability in time Õ(n), respectively. A nondeterministic derandomization, more precisely, a O(n 2−ε )-time verifier, would have interesting consequences [47]: it would refute the Nondeterministic Strong Exponential Time Hypothesis posed by Carmosino et al [11], which in turn would prove novel circuit lower bounds, deemed difficult to prove. We observe that a sufficiently strong nondeterministic derandomization of UPIT would also give a faster matrix multiplication verifier.…”

Section: Structural Results: Avenues Via Other Problemsmentioning

confidence: 99%

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On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress

Künnemann

2018

Preprint

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Motivated by studying the power of randomness, certifying algorithms and barriers for fine-grained reductions, we investigate the question whether the multiplication of two n × n matrices can be performed in near-optimal nondeterministic time Õ(n 2 ). Since a classic algorithm due to Freivalds verifies correctness of matrix products probabilistically in time O(n 2 ), our question is a relaxation of the open problem of derandomizing Freivalds' algorithm.We discuss consequences of a positive or negative resolution of this problem and provide potential avenues towards resolving it. Particularly, we show that sufficiently fast deterministic verifiers for 3SUM or univariate polynomial identity testing yield faster deterministic verifiers for matrix multiplication. Furthermore, we present the partial algorithmic progress that distinguishing whether an integer matrix product is correct or contains between 1 and n erroneous entries can be performed in time Õ(n 2 ) -interestingly, the difficult case of deterministic matrix product verification is not a problem of "finding a needle in the haystack", but rather cancellation effects in the presence of many errors.Our main technical contribution is a deterministic algorithm that corrects an integer matrix product containing at most t errors in time Õ( √ tn 2 + t 2 ). To obtain this result, we show how to compute an integer matrix product with at most t nonzeroes in the same running time. This improves upon known deterministic output-sensitive integer matrix multiplication algorithms for t = Ω(n 2/3 ) nonzeroes, which is of independent interest.

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Section: Upitmentioning

confidence: 57%

Section: Structural Results: Avenues Via Other Problemsmentioning

confidence: 99%

On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress

Künnemann

2018

Preprint

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Subquadratic Algorithms for Succinct Stable Matching

et al. 2019

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How many qubits are needed for quantum computational supremacy?

Dalzell

Harrow

Koh

et al. 2020

Quantum

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Quantum computational supremacy arguments, which describe a way for a quantum computer to perform a task that cannot also be done by a classical computer, typically require some sort of computational assumption related to the limitations of classical computation. One common assumption is that the polynomial hierarchy (PH) does not collapse, a stronger version of the statement that P≠NP, which leads to the conclusion that any classical simulation of certain families of quantum circuits requires time scaling worse than any polynomial in the size of the circuits. However, the asymptotic nature of this conclusion prevents us from calculating exactly how many qubits these quantum circuits must have for their classical simulation to be intractable on modern classical supercomputers. We refine these quantum computational supremacy arguments and perform such a calculation by imposing fine-grained versions of the non-collapse conjecture. Our first two conjectures poly3-NSETH(a) and per-int-NSETH(b) take specific classical counting problems related to the number of zeros of a degree-3 polynomial in n variables over F2 or the permanent of an n×n integer-valued matrix, and assert that any non-deterministic algorithm that solves them requires 2cn time steps, where c∈{a,b}. A third conjecture poly3-ave-SBSETH(a′) asserts a similar statement about average-case algorithms living in the exponential-time version of the complexity class SBP. We analyze evidence for these conjectures and argue that they are plausible when a=1/2, b=0.999 and a′=1/2.Imposing poly3-NSETH(1/2) and per-int-NSETH(0.999), and assuming that the runtime of a hypothetical quantum circuit simulation algorithm would scale linearly with the number of gates/constraints/optical elements, we conclude that Instantaneous Quantum Polynomial-Time (IQP) circuits with 208 qubits and 500 gates, Quantum Approximate Optimization Algorithm (QAOA) circuits with 420 qubits and 500 constraints and boson sampling circuits (i.e. linear optical networks) with 98 photons and 500 optical elements are large enough for the task of producing samples from their output distributions up to constant multiplicative error to be intractable on current technology. Imposing poly3-ave-SBSETH(1/2), we additionally rule out simulations with constant additive error for IQP and QAOA circuits of the same size. Without the assumption of linearly increasing simulation time, we can make analogous statements for circuits with slightly fewer qubits but requiring 104 to 107 gates.

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Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation

Cited by 5 publications

References 51 publications

On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress

On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress

Subquadratic Algorithms for Succinct Stable Matching

How many qubits are needed for quantum computational supremacy?

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