A Converse for Fault-tolerant Quantum Computation

Uthirakalyani G1, Anuj K. Nayak2, and Avhishek Chatterjee1

1Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai, India.
2Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, USA.

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As techniques for fault-tolerant quantum computation keep improving, it is natural to ask: what is the fundamental lower bound on space overhead? In this paper, we obtain a lower bound on the space overhead required for $\epsilon$-accurate implementation of a large class of operations that includes unitary operators. For the practically relevant case of sub-exponential depth and sub-linear gate size, our bound on space overhead is tighter than the known lower bounds. We obtain this bound by connecting fault-tolerant computation with a set of finite blocklength quantum communication problems whose accuracy requirements satisfy a joint constraint. The lower bound on space overhead obtained here leads to a strictly smaller upper bound on the noise threshold for noise that are not degradable. Our bound directly extends to the case where noise at the outputs of a gate are non-i.i.d. but noise across gates are i.i.d.

Quantum fault-tolerance mitigates noise in gate based quantum circuits by careful addition of redundant qubits or ancillas. It is well known that if sufficient number of ancillas are added carefully and the noise in the circuit is small, then quantum fault-tolerance can enable almost accurate quantum computation. Two questions that naturally arise here are: what is the minimum number of ancillas that is necessary for reasonably accurate quantum computation and what is the minimum noise beyond which fault-tolerance is not useful? In this paper, we address these questions by establishing a connection between quantum computation and finite resource (block-length) quantum communication. Our answers (bounds) to these questions are non-asymptotic and tighter than existing results, and are applicable to a broad class of noise models including correlated noise at a gate.

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Cited by

[1] Uthirakalyani G., Anuj K. Nayak, Avhishek Chatterjee, and Lav R. Varshney, "Limits of fault tolerance on resource-constrained quantum circuits for classical problems", Physical Review A 108 5, 052425 (2023).

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