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The problem of reliably certifying the outcome of a computation performed bya quantum device is rapidly gaining relevance. We present two protocols for aclassical verifier to verifiably delegate a quantum computation to twonon-communicating but entangled quantum provers. Our protocols havenear-optimal complexity in terms of the total resources employed by theverifier and the honest provers, with the total number of operations of eachparty, including the number of entangled pairs of qubits required of the honestprovers, scaling as $O(g\log g)$ for delegating a circuit of size $g$. This isin contrast to previous protocols, which all require a prohibitively largepolynomial overhead. Our first protocol requires a number of rounds that islinear in the depth of the circuit being delegated, and is blind, meaningneither prover can learn the circuit being delegated. The second protocol isnot blind, but requires only a constant number of rounds of interaction. Ourmain technical innovation is an efficient rigidity theorem which allows averifier to test that two entangled provers perform measurements specified byan arbitrary $m$-qubit tensor product of single-qubit Clifford observables ontheir respective halves of $m$ shared EPR pairs, with a robustness that isindependent of $m$. Our two-prover classical-verifier delegation protocols areobtained by combining this rigidity theorem with a single-proverquantum-verifier protocol for the verifiable delegation of a quantumcomputation, introduced by Broadbent (Theory of Computing, 2018).

Verifier-on-a-Leash: new schemes for verifiable delegated quantum computation, with quasilinear resources