Nonadaptive fault-tolerant verification of quantum supremacy with noise

Theodoros Kapourniotis and Animesh Datta

Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom

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Quantum samplers are believed capable of sampling efficiently from distributions that are classically hard to sample from. We consider a sampler inspired by the classical Ising model. It is nonadaptive and therefore experimentally amenable. Under a plausible conjecture, classical sampling upto additive errors from this model is known to be hard. We present a trap-based verification scheme for quantum supremacy that only requires the verifier to prepare single-qubit states. The verification is done on the same model as the original sampler, a square lattice, with only a constant overhead. We next revamp our verification scheme in two distinct ways using fault tolerance that preserves the nonadaptivity. The first has a lower overhead based on error correction with the same threshold as universal quantum computation. The second has a higher overhead but an improved threshold ($1.97\%$) based on error detection. We show that classically sampling upto additive errors is likely hard in both these schemes. Our results are applicable to other sampling problems such as the Instantaneous Quantum Polynomial-time (IQP) computation model. They should also assist near-term attempts at experimentally demonstrating quantum supremacy and guide long-term ones.

The considerable experimental efforts being directed towards building quantum computers rest on the belief that they are more powerful than classical computers. Formal mathematical proof for this belief has, however, been rather limited. `Quantum computational supremacy' (QCS) is the ability of a quantum computer to perform a task beyond the capability of any classical computer. Crucially, QCS has been proven for solving easier problems that do not require all the powers of a universal quantum computer.

Two challenges plague experimental demonstration of QCS. The first is our limited trust in the experimental devices to produce the correct output, especially in the regime where predictions via classical simulation of the same system are impossible. The second is the presence of noise in each device which, if not dealt with, accumulates and overwhelms the computation, introducing errors. In both cases the output can be statistically far from the correct one, which prohibits the demonstration of QCS.

Our paper provides a scheme which resolves both challenges by verifying that the outputs of the experiment are sufficiently close to the desired distribution. The distribution we sample from comes from the 2D Ising model, a mathematical model of magnetism that has many applications in statistical physics and is an instance of a QCS task. Our analysis only requires that single qubit states can be prepared reliably, and the experimentalist possesses devices with noise rates below 1.97%. The latter permits larger noise levels than required for universal quantum computing. Our scheme is based on building trust in the correctness of a large computation by checking the correctness of small computations and takes a constant amount of time independent of the size of the lattice in the Ising model. It can therefore be used to prove the correct operation of a QCS demonstrator before building a universal quantum computer becomes possible.

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Cited by

[1] Samuele Ferracin, Theodoros Kapourniotis, and Animesh Datta, "Accrediting outputs of noisy intermediate-scale quantum computing devices", New Journal of Physics 21 11, 113038 (2019).

[2] Sergio Boixo, Sergei V. Isakov, Vadim N. Smelyanskiy, and Hartmut Neven, "Simulation of low-depth quantum circuits as complex undirected graphical models", arXiv:1712.05384.

[3] Yuki Takeuchi and Tomoyuki Morimae, "Verification of Many-Qubit States", Physical Review X 8 2, 021060 (2018).

[4] Alexandru Gheorghiu, Theodoros Kapourniotis, and Elham Kashefi, "Verification of quantum computation: An overview of existing approaches", arXiv:1709.06984.

[5] Alexandru Gheorghiu, Matty J. Hoban, and Elham Kashefi, "A simple protocol for fault tolerant verification of quantum computation", Quantum Science and Technology 4 1, 015009 (2019).

[6] Dominik Hangleiter, Juan Bermejo-Vega, Martin Schwarz, and Jens Eisert, "Anticoncentration theorems for schemes showing a quantum speedup", arXiv:1706.03786.

[7] Jacob Miller, Stephen Sanders, and Akimasa Miyake, "Quantum supremacy in constant-time measurement-based computation: A unified architecture for sampling and verification", Physical Review A 96 6, 062320 (2017).

[8] Samuele Ferracin, Theodoros Kapourniotis, and Animesh Datta, "Reducing resources for verification of quantum computations", Physical Review A 98 2, 022323 (2018).

[9] Daniel Mills, Anna Pappa, Theodoros Kapourniotis, and Elham Kashefi, "Information Theoretically Secure Hypothesis Test for Temporally Unstructured Quantum Computation (Extended Abstract)", arXiv:1803.00706.

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