Interactive Proofs for Quantum Computation

Jiang Zhang, Yu Yu, Dengguo Feng +3 more

Rahul Jain, Zhengfeng Ji, Sarvagya Upadhyay +1 more

This work considers the quantum interactive proof system model of computation, which is the (classical) interactive proof system model’s natural quantum computational analogue. An exact characterization of the expressive power of quantum interactive proof systems is obtained: the collection of computational problems having quantum interactive proof systems consists precisely of those problems solvable by deterministic Turing machines that use at most a polynomial amount of space (or, more succinctly, QIP = PSPACE). This characterization is proved through the use of a parallelized form of the matrix multiplicative weights update method, applied to a class of semidefinite programs that captures the computational power of quantum interactive proof systems. One striking implication of this characterization is that quantum computing provides no increase in computational power whatsoever over classical computing in the context of interactive proof systems, for it is well known that the collection of computational problems having classical interactive proof systems coincides with those problems solvable by polynomial-space computations.

shahab razavi

Gorjan Alagic, Andrew M. Childs, Alex B. Grilo +1 more

Rahul Jain, Zhengfeng Ji, Sarvagya Upadhyay +1 more

The interactive proof system model of computation has been studied extensively in computational complexity theory and theoretical cryptography for more than 25 years, and has driven the development of interesting new techniques and insights in those fields. This work considers the quantum interactive proof system model, which is the classical model's natural quantum computational analog. An exact characterization of the expressive power of quantum interactive proof systems is obtained: the collection of computational problems having quantum interactive proof systems consists precisely of those problems solvable with an ordinary classical computer using at most a polynomial amount of memory (or QIP = PSPACE in complexity-theoretic terminology). One striking implication of this characterization is that it implies quantum computing provides no increase in computational power whatsoever over classical computing in the context of interactive proof systems.

Thomas Vidick, John Watrous

Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class NP, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class QSZK, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.