Generalized Quantum Arthur--Merlin Games

Hirotada Kobayashi,François Le Gall,Harumichi Nishimura

doi:10.1137/17m1160173

Abstract

This paper investigates the role of interaction and coins in public-coin quantum interactive proof systems (also called quantum Arthur-Merlin games). While prior works focused on classical public coins even in the quantum setting, the present work introduces a generalized version of quantum Arthur-Merlin games where the public coins can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs. First, it is proved that the class of two-turn quantum Arthur-Merlin games with quantum public coins, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interactions prior to the communications of the qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper provides a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking the properties related to quantum circuits, and is of independent interest. It is further proved that the class qq-QAM_1 of two-turn quantum-public-coin quantum Arthur-Merlin proof systems with perfect completeness gives new bounds for standard well-studied classes of two-turn interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of interaction and public coins in quantum Arthur-Merlin games: it is proved that, for any constant m>1, the class of problems having an m-turn quantum Arthur-Merlin proof system is either equal to PSPACE or equal to the class of problems having a two-turn quantum Arthur-Merlin game of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.

Highlights

Let qq-QAM be the class of problems having two-turn “fully quantum” Arthur-Merlin proof systems, i.e., two-turn quantum interactive proof systems in which the first message from the verifier consists only of polynomially many halves of EPR pairs
The main goal of this paper is to investigate the computational power of this class qq-QAM in order to figure out the advantages offered by sharing EPR pairs rather than classical randomness, and more generally, to make a step forward in the understanding of two-turn quantum interactive proof systems
This paper has introduced the generalized model of quantum Arthur-Merlin proof systems to provide some new insights on the power of two-turn quantum interactive proofs

Summary

Introduction

Interactive proof systems [9, 4] play a central role in computational complexity and have many applications such as probabilistically checkable proofs and zero-knowledge proofs The aim of such a system is the verification of an assertion (e.g., verifying if an input is in a language) by a party implementing a polynomial-time probabilistic computation, called the verifier, interacting with another party with unlimited power, called the prover, in polynomially many turns. The class of problems having interactive proof systems of a constant number of turns is equal to AM(2) (regardless of definitions with public coins or private coins), and this class is nowadays called AM. The class of problems having more general interactive proof systems of polynomially many turns, called IP, does coincide with PSPACE [26, 23, 28] (again regardless of definitions with public coins or private coins [10, 29])

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