Quantum Attacks on Classical Proof Systems: The Hardness of Quantum Rewinding

Computational and Proof Complexity of Partial String Avoidability

ACM Transactions on Computation Theory ◽

10.1145/3442365 ◽

2021 ◽

Vol 13 (1) ◽

pp. 1-25

Author(s):

Dmitry Itsykson ◽

Alexander Okhotin ◽

Vsevolod Oparin

Keyword(s):

Lower Bounds ◽

Circuit Complexity ◽

Conjunctive Normal Form ◽

Proof Complexity ◽

Proof Systems ◽

Finite Set ◽

Propositional Proof Systems ◽

Classical Proof ◽

Exponential Size ◽

Infinite String

The partial string avoidability problem is stated as follows: given a finite set of strings with possible “holes” (wildcard symbols), determine whether there exists a two-sided infinite string containing no substrings from this set, assuming that a hole matches every symbol. The problem is known to be NP-hard and in PSPACE, and this article establishes its PSPACE-completeness. Next, string avoidability over the binary alphabet is interpreted as a version of conjunctive normal form satisfiability problem, where each clause has infinitely many shifted variants. Non-satisfiability of these formulas can be proved using variants of classical propositional proof systems, augmented with derivation rules for shifting proof lines (such as clauses, inequalities, polynomials, etc.). First, it is proved that there is a particular formula that has a short refutation in Resolution with a shift rule but requires classical proofs of exponential size. At the same time, it is shown that exponential lower bounds for classical proof systems can be translated for their shifted versions. Finally, it is shown that superpolynomial lower bounds on the size of shifted proofs would separate NP from PSPACE; a connection to lower bounds on circuit complexity is also established.

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QMA variants with polynomially many provers

Quantum Information and Computation ◽

10.26421/qic13.1-2-8 ◽

2013 ◽

Vol 13 (1&2) ◽

pp. 135-157

Author(s):

Sevag Gharibian ◽

Jamie Sikora ◽

Sarvagya Upadhyay

Keyword(s):

Bounded Function ◽

Expressive Power ◽

Proof System ◽

Alternate Proof ◽

Cone Programming ◽

Parallel Repetition ◽

Proof Systems ◽

Classical Proof

We study three variants of multi-prover quantum Merlin-Arthur proof systems. We first show that the class of problems that can be efficiently verified using polynomially many quantum proofs, each of logarithmic-size, is exactly \class{MQA} (also known as QCMA), the class of problems which can be efficiently verified via a classical proof and a quantum verifier. We then study the class $\class{BellQMA}(\poly)$, characterized by a verifier who first applies unentangled, nonadaptive measurements to each of the polynomially many proofs, followed by an arbitrary but efficient quantum verification circuit on the resulting measurement outcomes. We show that if the number of outcomes per nonadaptive measurement is a polynomially-bounded function, then the expressive power of the proof system is exactly \class{QMA}. Finally, we study a class equivalent to \class{QMA}($m$), denoted $\class{SepQMA}(m)$, where the verifier's measurement operator corresponding to outcome {\it accept} is a fully separable operator across the $m$ quantum proofs. Using cone programming duality, we give an alternate proof of a result of Harrow and Montanaro [FOCS, pp. 633--642 (2010)] that shows a perfect parallel repetition theorem for $\class{SepQMA}(m)$ for any $m$.

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Secure Cloud Quantum Computing with Verification Based on Quantum Interactive Proof

Impact ◽

10.21820/23987073.2019.10.30 ◽

2019 ◽

Vol 2019 (10) ◽

pp. 30-32

Author(s):

Tomoyuki Morimae

Keyword(s):

Quantum Computing ◽

Quantum Information ◽

Theoretical Physics ◽

View Point ◽

Research Subjects ◽

Interactive Proof ◽

Open Problems ◽

Main Research ◽

Proof Systems ◽

Information Group

In cloud quantum computing, a classical client delegate quantum computing to a remote quantum server. An important property of cloud quantum computing is the verifiability: the client can check the integrity of the server. Whether such a classical verification of quantum computing is possible or not is one of the most important open problems in quantum computing. We tackle this problem from the view point of quantum interactive proof systems. Dr Tomoyuki Morimae is part of the Quantum Information Group at the Yukawa Institute for Theoretical Physics at Kyoto University, Japan. He leads a team which is concerned with two main research subjects: quantum supremacy and the verification of quantum computing.

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