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 $\max\left\{\mathrm{Q}(\mathcal{N})^{-1}n,\alpha_\mathcal{N} \log T\right\}$ 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 fault-tolerant quantum computation with amplitude damping noise resolving a conjecture by Ben-Or, Gottesman, and Hassidim (2013).

翻译：阈值定理是容错量子计算理论中的基本结果，该定理指出，当噪声水平低于某个常数时，任意长的量子计算可以通过多对数级别的开销实现。Fawzi、Grospellier 和 Leverrier（FOCS 2018）基于 Gottesman（QIC 2013）的结果，最新研究表明：若仅考虑电路长度被宽度多项式有界的情况，空间开销可渐近降低至与电路无关的常数。本研究采用量子容错的最小模型，建立了容错所需空间开销的一般性下界。针对任意非酉量子比特通道 $\mathcal{N}$ 以及由 $\mathcal{N}$ 建模的 $\mathrm{i.i.d.}$ 噪声下的量子容错方案，我们证明：对于长度为 $T$、宽度为 $n$ 的电路，所需物理量子比特数存在下界 $\max\left\{\mathrm{Q}(\mathcal{N})^{-1}n,\alpha_\mathcal{N} \log T\right\}$。其中，$\mathrm{Q}(\mathcal{N})$ 表示通道 $\mathcal{N}$ 的量子容量，$\alpha_\mathcal{N}>0$ 是仅取决于通道 $\mathcal{N}$ 的常数。在我们的模型中，允许在电路执行过程中用新鲜量子比特替换原有量子比特，并假设经典计算无噪声且完美执行。这改进了此前假设经典计算也受噪声影响且不允许添加新鲜量子比特的研究结果。在此过程中，我们以振幅阻尼噪声为例，证明了容错量子计算最大长度的指数上界，解决了 Ben-Or、Gottesman 和 Hassidim（2013）提出的猜想。