Verification of Quantum Computation: An Overview of Existing Approaches

Abstract

Quantum computers promise to efficiently solve not only problems believed to be intractable for classical computers, but also problems for which verifying the solution is also considered intractable. This raises the question of how one can check whether quantum computers are indeed producing correct results. This task, known as quantum verification, has been highlighted as a significant challenge on the road to scalable quantum computing technology. We review the most significant approaches to quantum verification and compare them in terms of structure, complexity and required resources. We also comment on the use of cryptographic techniques which, for many of the presented protocols, has proven extremely useful in performing verification. Finally, we discuss issues related to fault tolerance, experimental implementations and the outlook for future protocols. arXiv:1709.06984v2 [quant-ph] 9 Jul 2018Problem 1 (Verifiability of BQP computations). Does every problem in BQP admit an interactive-proof system in which the prover is restricted to BQP computations?As mentioned, this complexity theoretic formulation of the problem was considered by Gottesman, Aaronson and Vazirani [10,11] and, in fact, Aaronson has offered a 25$ prize for its resolution [10]. While, as of yet, the question remains open, one does arrive at a positive answer through slight alterations of the 4 BPP and MA are simply the probabilistic versions of the more familiar classes P and NP. Under plausible derandomization assumptions, BPP = P and MA = NP [13]. 5 Even if this were the case, i.e. BQP ⊆ MA, for this to be useful in practice one would require that computing the witness can also be done in BQP. In fact, there are candidate problems known to be in both BQP and MA, for which computing the witness is believed to not be in BQP (a conjectured example is [17]). 6 MA can be viewed as an interactive-proof system where only one message is sent from the prover (Merlin) to the verifier (Arthur).consideration for any realistic implementation of a verification protocol. Finally, in Subsection 5.3 we outline some of the existing experimental implementations of these protocols. Throughout the review, we are assuming familiarity with the basics of quantum information theory and some elements of complexity theory. However, we provide a brief overview of these topics as well as other notions that are used in this review (such as measurement-based quantum computing) in the appendix, Section 7. Note also, that we will be referencing complexity classes such as BQP, QMA, QPIP and MIP * . Definitions for all of these are provided in Subsection 7.3 of the appendix. We begin with a short overview of blind quantum computing. Blind quantum computingThe concept of blind computing is highly relevant to quantum verification. Here, we simply give a succinct outline of the subject. For more details, see this review of blind quantum computing protocols by Fitzsimons [34] as well as [35][36][37][38][39]. Note that, while the review of Fitzsimons covers all of the material ...

A simple protocol for fault tolerant verification of quantum computation

“…At this point, the protocol no longer satisfies the correctness nor the soundness criteria. In fact, this is common to all other verification protocols in the single-prover setting [3]. To address this issue we now give a fault tolerant version of the post hoc protocol that works in the presence of quantum devices subject to local noise having a constant error-rate.…”

Accrediting outputs of noisy intermediate-scale quantum computing devices

“…This calls for protocols able to test circuits as a whole rather than individual gates. Such protocols have been devised inspired by Interactive proof Systems [25]. In these protocols (which we call 'cryptographic protocols') the outputs of the target circuit are verified through an interaction between a trusted verifier and an untrusted prover ( figure 1(a)).…”

“…To compute soundness, we denote as out We conclude this appendix by showing how our protocol can be made blind. Blindness is a property exhibited by many cryptographic protocols [25] defined as follows:…”

Quantum leakage detection using a model-independent dimension witness

“…Leakage is of particular concern for quantum computer engineers, due to its impact on fault-tolerance thresholds [11]. For remote users, it may undermine the assumption of trusted operations -preparation, memory or measurement -that quantum verification [12] or quantum key distribution protocols [13] rely on to guarantee their security. Leakage has been studied in some detail in specific quantum computing platforms [14][15][16][17][18].…”