Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A lower bound on the space overhead of fault-tolerant quantum computation

Omar Fawzi, Alexander Müller-Hermes, Ala Shayeghi

TL;DR

This work investigates fundamental limits on the space overhead required for fault-tolerant quantum computation under i.i.d. noise modeled by a non-unitary qubit channel. The authors introduce a minimal fault-tolerance model with free classical computation and the possibility of fresh ancillas, and prove a lower bound on the number of physical qubits needed: $\max\{\mathrm{Q}(\mathcal{N})^{-1} n, \alpha_{\mathcal{N}} \log T\}$ for circuit width $n$ and length $T$, with $\mathrm{Q}(\mathcal{N})$ the channel’s quantum capacity and $\alpha_{\mathcal{N}}>0$ depending only on $\mathcal{N}$. The bound shows that constant-space overhead is impossible for sufficiently long computations unless $\mathrm{Q}(\mathcal{N})>0$, and it recovers stronger infeasibility results when $\mathrm{Q}(\mathcal{N})=0$ (e.g., $p>1/3$ depolarizing noise). The analysis employs a quantum Shannon-theoretic viewpoint, leveraging contractions of the $\chi^2$-divergence under separable channels to bound information retention over time and to connect the overhead to both the quantum capacity and circuit depth, with implications for the practicality of fault-tolerant architectures in noisy devices.

Abstract

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 $\mathcal{N}$ and any quantum fault tolerance schemes against $\mathrm{i.i.d.}$ noise modeled by $\mathcal{N}$, we prove a lower bound of $\max\left\{\mathrm{Q}(\mathcal{N})^{-1}n,α_\mathcal{N} \log T\right\}$ on the number of physical qubits, for circuits of length $T$ and width $n$. Here, $\mathrm{Q}(\mathcal{N})$ denotes the quantum capacity of $\mathcal{N}$ and $α_\mathcal{N}>0$ is a constant only depending on the channel $\mathcal{N}$. In our model, we allow for qubits to be replaced by fresh ones during the execution of the circuit and 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).

A lower bound on the space overhead of fault-tolerant quantum computation

TL;DR

This work investigates fundamental limits on the space overhead required for fault-tolerant quantum computation under i.i.d. noise modeled by a non-unitary qubit channel. The authors introduce a minimal fault-tolerance model with free classical computation and the possibility of fresh ancillas, and prove a lower bound on the number of physical qubits needed: for circuit width and length , with the channel’s quantum capacity and depending only on . The bound shows that constant-space overhead is impossible for sufficiently long computations unless , and it recovers stronger infeasibility results when (e.g., depolarizing noise). The analysis employs a quantum Shannon-theoretic viewpoint, leveraging contractions of the -divergence under separable channels to bound information retention over time and to connect the overhead to both the quantum capacity and circuit depth, with implications for the practicality of fault-tolerant architectures in noisy devices.

Abstract

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 and any quantum fault tolerance schemes against noise modeled by , we prove a lower bound of on the number of physical qubits, for circuits of length and width . Here, denotes the quantum capacity of and is a constant only depending on the channel . In our model, we allow for qubits to be replaced by fresh ones during the execution of the circuit and 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).
Paper Structure (10 sections, 9 theorems, 26 equations, 2 figures)

This paper contains 10 sections, 9 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our model
  3. Discussion of our main results, proof techniques, and related work
  4. Related work on limitations of quantum circuits under strong noise
  5. Preliminaries
  6. Maximal length of noisy quantum computation
  7. Implications for fault-tolerant quantum computation
  8. A high-level definition of fault tolerance
  9. A lower bound on the space overhead for fault tolerance
  10. Appendix

Key Result

Theorem 1

Let $\mathcal{I}$ denote the qubit identity channel and $\mathcal{N}$ be a non-unitary qubit channel. Then there exists a constant $p\in(0,1]$ only depending on $\mathcal{N}$ such that the following holds: Let $C$ be an arbitrary circuit of length $T$ and width $n$ such that $T\geq \left( 2/p \righ for some universal constant $\epsilon_0\geq 1/128$.

Figures (2)

  • Figure 1: Illustration of the circuit model
  • Figure 2: An illustration of $\mathrm{i.i.d._\mathcal{N}}(C)$. In each time step, every qubit is subject to the noise channel $\mathcal{N}$. The classical subsystems are assumed to be noise-free. The registers are partitioned into disjoint subsets A and B such that the quantum channels $\mathcal{L}_i$ are separable with respect to this bipartition.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Lemma 4
  • Lemma 5
  • Lemma 7
  • Theorem 1: Restated
  • Definition 8
  • Theorem 2: Restated
  • Lemma 9
  • ...and 1 more