Fault-tolerant quantum input/output

TL;DR

This work extends Kitaev’s fault-tolerance framework to circuits with quantum inputs and outputs, enabling the fault-tolerant transformation of any QIO circuit into an implementation whose effective input/output is degraded only by a small quantum channel. It introduces a modular, composable scheme built from quantum codes, logic channels, and interfaces, and develops transformation rules that convert approximate code-level representations into exact ones, preserving logical information. The paper provides explicit constructions for fault-tolerant state preparation and quantum communication under two noise models (general circuit noise and circuit-level stochastic noise), showing that input/output corruption can be tightly controlled by Malus-type channels with parameters scaling as $O( abla)$ or $O((c\delta)^k)$. As an application, it shows how standard communication codes with linear or sub-linear minimum distance can be lifted to fault-tolerant communication codes for general circuit noise or local stochastic noise, with toric codes serving as a concrete example. Overall, the results enable reliable quantum information processing and communication across noisy, distributed quantum systems by modular, scalable fault-tolerant constructions.

Abstract

Usual scenarios of fault-tolerant computation are concerned with the fault-tolerant realization of quantum algorithms that compute classical functions, such as Shor's algorithm for factoring. In particular, this means that input and output to the quantum algorithm are classical. In contrast to stand-alone single-core quantum computers, in many distributed scenarios, quantum information might have to be passed on from one quantum information processing system to another one, possibly via noisy quantum communication channels with noise levels above fault-tolerant thresholds. In such situations, quantum information processing devices will have quantum inputs, quantum outputs or even both, which pass qubits among each other. Working in the fault-tolerant framework of [Kitaev, 1997], we show that any quantum circuit with quantum input and output can be transformed into a fault-tolerant circuit that produces the ideal circuit with some controlled noise applied at the input and output. The framework allows the direct composition of the statements, enabling versatile future applications. We illustrate this with a concrete application, namely, communication over a noisy channel with faulty encoding and decoding operations [Christandl and M{ü}ller-Hermes, 2024]. For communication codes with linear minimum distance, we construct fault-tolerant encoders and decoders for general noise (including coherent errors). For the weaker, but standard, model of local stochastic noise, we obtain fault-tolerant encoders and decoders for any communication code that can correct a constant fraction random errors.

Figures (10)

• Figure 1: The figure shows the channel $(\otimes_{i = 1}^{n'} {\cal W}_i) \circ ({\cal T}_{\Phi} \otimes {\cal F}) \circ (\otimes_{i = 1}^n {\cal N}_i)$ from Eq. (\ref{['eq:syndrome-noise-intro']}).
• Figure 2: Fault-tolerant realizations of quantum circuits with quantum systems only at the input or output. The channels $\overline{{\cal W}}$ and $\overline{{\cal N}}$ are either adversarial or local stochastic (depending on the circuit noise), with parameter $O(\delta)$.
• Figure 3: Schematic representation of the spaces in the derived code $\mathrm{Der}(\overline{L}, \mathscr{\mathscr{E}})$. The isometry $V: M \to N, \mathrm{Im}(V) = \overline{L}$ is the original one-to-one quantum code. The unitary $U: M \otimes F \to L$, where $L = \mathscr{E}{\overline{L}} \subseteq N$, defines the many-to-one code.
• Figure 4: Commutative diagram. ${\cal T}$ represents ${\cal P}$ in codes $(L, \mu)$ and $(L', \mu')$. Here, ${\cal J} : \mathbf{L}(L) \to \mathbf{L}(N), L \subseteq N$ is the natural embedding.
• Figure 5: The quantum channel ${\cal T}_g$ realized by the quantum gate $g \in \mathbf{A}$. Each double wire ($CC$ and $OC$) represents a set of classical bits and the single wire ($Q$) represents a set of qubits. The state of classical bits are diagonal in the computational basis (see Eq. \ref{['eq:hybrid_classical_quantum']}). Classical wires are divided into control and operational bits, labelled by $CC$ and $OC$, respectively. Depending on the state $\ket{\mathbf{u}}\bra{\mathbf{u}}_{CC}$ of the control wire, the quantum channel ${\cal T}_{\mathbf{u}}$ is applied on $OC$ and $Q$.

Theorems & Definitions (98)

• Theorem 1: Informal, see Corollary \ref{['cor:dec']} for details
• Theorem 2: Informal, see Corollary \ref{['cor:dec-stc']} for details
• Theorem 3: Informal, see Theorems \ref{['thm:ft-comm-code-general']} and \ref{['thm:ft-code-stc']} for details
• Remark 5
• Definition 6: Weight of an operator
• Definition 7: Weight of a superoperator
• Definition 8: Diamond norm
• Theorem 9
• Definition 10: one-to-one quantum code
• Definition 11: many-to-one quantum code