A Lower Bound on the Space Overhead of Fault-Tolerant Quantum Computation

Authors Omar Fawzi, Alexander Müller-Hermes, Ala Shayeghi

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Author Details

Omar Fawzi
  • Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France
Alexander Müller-Hermes
  • Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622 Villeurbanne cedex, France
  • Department of Mathematics, University of Oslo, Norway
Ala Shayeghi
  • Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France


We would like to thank Cambyse Rouzé for helpful discussions.

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Omar Fawzi, Alexander Müller-Hermes, and Ala Shayeghi. A Lower Bound on the Space Overhead of Fault-Tolerant Quantum Computation. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 68:1-68:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation stating that arbitrarily long quantum computations can be performed with a polylogarithmic overhead provided the noise level is below a constant level. A recent work by Fawzi, Grospellier and Leverrier (FOCS 2018) building on a result by Gottesman (QIC 2013) has shown that the space overhead can be asymptotically reduced to a constant independent of the circuit provided we only consider circuits with a length bounded by a polynomial in the width. In this work, using a minimal model for quantum fault tolerance, we establish a general lower bound on the space overhead required to achieve fault tolerance. For any non-unitary qubit channel 𝒩 and any quantum fault tolerance schemes against i.i.d. noise modeled by 𝒩, we prove a lower bound of max{Q(𝒩)^{-1}n,α_𝒩 log T} on the number of physical qubits, for circuits of length T and width n. Here, Q(𝒩) denotes the quantum capacity of 𝒩 and α_𝒩 > 0 is a constant only depending on the channel 𝒩. In our model, we allow for qubits to be replaced by fresh ones during the execution of the circuit and in the case of unital noise, we allow classical computation to be free and perfect. This improves upon results that assumed classical computations to be also affected by noise, and that sometimes did not allow for fresh qubits to be added. Along the way, we prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude damping noise resolving a conjecture by Ben-Or, Gottesman and Hassidim (2013).

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Hardware → Quantum error correction and fault tolerance
  • Fault-tolerant quantum computation
  • quantum error correction


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