How to Verify a Quantum Computation

TL;DR

This paper presents a new theoretical method for verifying quantum computations with high complexity, using a quantum interactive proof system that requires minimal overhead and relies on encryption and entanglement-based techniques.

Contribution

It introduces a verification protocol for quantum computations that does not require encoding the input, using encryption and a novel entanglement-based proof technique.

Findings

Verification protocol has linear overhead in circuit size

Unconditional soundness for unbounded provers

Minimal additional cost compared to performing the computation

Abstract

We give a new theoretical solution to a leading-edge experimental challenge, namely to the verification of quantum computations in the regime of high computational complexity. Our results are given in the language of quantum interactive proof systems. Specifically, we show that any language in $\mathsf{BQP}$ has a quantum interactive proof system with a polynomial-time classical verifier (who can also prepare random single-qubit pure states), and a quantum polynomial-time prover. Here, soundness is unconditional--i.e., it holds even for computationally unbounded provers. Compared to prior work achieving similar results, our technique does not require the encoding of the input or of the computation; instead, we rely on encryption of the input (together with a method to perform computations on encrypted inputs), and show that the random choice between three types of input (defining a computational run, versus two types of test runs) suffices. Because the overhead is very low for each run (it is linear in the size of the circuit), this shows that verification could be achieved at minimal cost compared to performing the computation. As a proof technique, we use a reduction to an entanglement-based protocol; to the best of our knowledge, this is the first time this technique has been used in the context of verification of quantum computations, and it enables a relatively straightforward analysis.