A lower bound on the space overhead of faulttolerant quantum computation
Abstract
The threshold theorem is a fundamental result in the theory of faulttolerant 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 nonunitary 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 $\max\left\{\mathrm{Q}(\mathcal{N})^{1}n,\alpha_\mathcal{N} \log T\right\}$ 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 $\alpha_\mathcal{N}>0$ 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 faulttolerant quantum computation with amplitude damping noise resolving a conjecture by BenOr, Gottesman, and Hassidim (2013).
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 arXiv:
 arXiv:2202.00119
 Bibcode:
 2022arXiv220200119F
 Keywords:

 Quantum Physics;
 Computer Science  Information Theory
 EPrint:
 23 pages, 2 figures, an earlier version of this paper appeared in proceedings of ITCS 2022. In the current version, we have extended our results to the model with noiseless classical computation for all nonunitary qubit noise channels