Abstract
The problem of reliably certifying the outcome of a computation performed by a quantum device is rapidly gaining relevance. We present two protocols for a classical verifier to verifiably delegate a quantum computation to two non-communicating but entangled quantum provers. Our protocols have near-optimal complexity in terms of the total resources employed by the verifier and the honest provers, with the total number of operations of each party, including the number of entangled pairs of qubits required of the honest provers, scaling as \(O(g\log g)\) for delegating a circuit of size g. This is in contrast to previous protocols, whose overhead in terms of resources employed, while polynomial, is far beyond what is feasible in practice. Our first protocol requires a number of rounds that is linear in the depth of the circuit being delegated, and is blind, meaning neither prover can learn the circuit or its input. The second protocol is not blind, but requires only a constant number of rounds of interaction.
Our main technical innovation is an efficient rigidity theorem which allows a verifier to test that two entangled provers perform measurements specified by an arbitrary m-qubit tensor product of single-qubit Clifford observables on their respective halves of m shared EPR pairs, with a robustness that is independent of m. Our two-prover classical-verifier delegation protocols are obtained by combining this rigidity theorem with a single-prover quantum-verifier protocol for the verifiable delegation of a quantum computation, introduced by Broadbent.
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Notes
- 1.
Using results of Ji [Ji16], this allows the protocol to be single-round. Alternatively, the state can be created by a single prover and teleported to the others with the help of the verifier, resulting in a two-round protocol.
- 2.
Blindness is a property of delegation protocols, which informally states that the prover learns nothing about the verifier’s private circuit.
- 3.
The \(\log g\) overhead is due to the complexity of sampling from the right distribution in rigidity tests. We leave the possibility of removing this by derandomization for future work. Another source of overhead is in achieving blindness: in order to hide the circuit, we encode it as part of the input to a universal circuit, introducing a factor of \(O(\log g)\) overhead.
- 4.
Here, we consider the decomposition of the attack as a sum of tensors of Pauli \(A = \sum _k \sum _{Q \in \{I,X,Z,Y\}} \alpha _{k,Q} Q\).
- 5.
We make the assumption that the players employ a pure-state strategy for convenience, but it is easy to check that all proofs extend to the case of a mixed strategy. Moreover, it is always possible to consider (as we do) projective strategies only by applying Naimark’s dilation theorem, and adding an auxiliary local system to each player as necessary, since no bound is assumed on the dimension of their systems.
- 6.
See [RUV12, Appendix A] for an extended discussion of this issue, with a similar resolution to ours.
- 7.
One must ensure that a prover does not realize if the alternative protocol is executed instead of the original; this is easily enforced by only interacting with any of the provers at specific, publicly decided times.
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Acknowledgments
We thank Anne Broadbent for useful discussions in the early stages of this work. All authors acknowledge the IQIM, an NSF Physics Frontiers Center at the California Institute of Technology, where this research was initiated. AC is supported by AFOSR YIP award number FA9550-16-1-0495. AG is supported by ERC Consolidator Grant 615307-QPROGRESS and was previously supported by ERC QCC when AG was a member of IRIF (CNRS/Université Paris Diderot). SJ is supported by an NWO WISE Grant. TV is supported by NSF CAREER Grant CCF-1553477, MURI Grant FA9550-18-1-0161, AFOSR YIP award number FA9550-16-1-0495, and the IQIM, an NSF Physics Frontiers Center (NSF Grant PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028).
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Coladangelo, A., Grilo, A.B., Jeffery, S., Vidick, T. (2019). Verifier-on-a-Leash: New Schemes for Verifiable Delegated Quantum Computation, with Quasilinear Resources. In: Ishai, Y., Rijmen, V. (eds) Advances in Cryptology – EUROCRYPT 2019. EUROCRYPT 2019. Lecture Notes in Computer Science(), vol 11478. Springer, Cham. https://doi.org/10.1007/978-3-030-17659-4_9
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