Constant-space-overhead fault-tolerant quantum input/output and communication

TL;DR

This work develops constant-space-overhead fault-tolerant quantum input/output and communication by deploying interfaced circuits built from concatenated quantum Hamming codes. It introduces a level-conversion framework and threshold theorems tailored to circuits with quantum inputs/outputs, enabling faithful logical action across concatenated levels while maintaining a fixed space overhead. Applying these tools to fault-tolerant entanglement-assisted communication yields a new, tighter lower bound on capacity that approaches the noiseless rate and surpasses previous Steane-code-based results under practical noise levels. The approach has potential broader impact for distributed quantum computing and quantum repeater architectures, while leaving open the exploration of other constant-overhead codes and more general noise models.

Abstract

Fault-tolerant capacities quantify the ability of a quantum channel to reliably transmit information when every component of the encoding and decoding procedure is noisy. Earlier work analyzed achievable communication rates under such noise using fault-tolerant implementations based on concatenated codes with a single logical qubit. In this work, we develop an alternative approach using concatenations of quantum Hamming codes, which offer constant space overhead by encoding many logical qubits simultaneously. We introduce modular techniques for implementing fault-tolerant circuits with quantum input/output interfaces using the concatenated quantum Hamming code. These tools enable an analysis of fault-tolerant entanglement-assisted communication that is not only simpler, but also yields substantially higher achievable communication rates than previous methods, owing to the limited noise correlations in syndrome qubits of high-rate quantum Hamming codes.

Figures (7)

• Figure 1: A level-$2$ interfaced circuit for a unitary $U$, implemented in $\mathcal{Q}^{(2)}$ and concatenated with circuits $\mathop{\mathrm{Enc}}\nolimits_l$ and $\mathop{\mathrm{Dec}}\nolimits_l$ mapping between the levels for $l=1,2$. The original circuit $U$ is a 7-qubit unitary, and $U_{\mathcal{Q}^{(2)}}$ is its implementation in $\mathcal{Q}^{(2)}$ in terms of a total of 105 physical qubits. We emphasize that the wires in this diagram do not correspond to physical qubits, but to registers: $\mathop{\mathrm{Enc}}\nolimits_1$ takes a level-$0$ register (i.e. a physical qubit) and maps it to a level-$1$ register; $\mathop{\mathrm{Enc}}\nolimits_2$ maps $K_2=7$ level-$1$ registers to a level-$2$ register, $U_{\mathcal{Q}^{(2)}}$ takes a level-$2$ register to a a level-$2$ register.
• Figure 2: Compilation of a level-$2$ interfaced circuit $U$ ($L=2$), where we use the $[[7,1,3]]$ code at concentenation level $1$, and the $[[15,7,3]]$ code at level $2$. For each $l\in\{L,\ldots,1\}$, we recursively compile the level-$l$ circuit into the level-$(l-1)$ circuit by replacing every level-$l$ operation and interface location in the level-$l$ parts of the level-$l$ circuit with its corresponding level-$l$ gadget and by inserting the level-$l$ EC gadgets between all the adjacent pairs of two level-$l$ operations in $U^{Q^{(L)}}$. Then, the level-$1$ version of the circuit is obtained as the level-$2$ circuit circuit written in terms of level-$1$ gadgets and level-$1$ interfaces. Taking the interfaced circuit from Fig. \ref{['fig:level_l_circuit']}, $U_{\mathcal{Q}^{(2)}}$ in the first figure is understood as a unitary on level-$2$ registers (i.e. the wires going in and out of $U_{\mathcal{Q}^{(2)}}$ represent one level-$2$ register); in the second picture, it represents a unitary on $15$ level-$1$ registers; in the third picture, it represents the unitary on the total 105 physical qubits. These unitaries and interfaces at lower levels are obtained from the higher levels by the compilation procedure outlined below.
• Figure 3: The level-$l$ encoding gadget at the top and the level-$l$ decoding gadget at the bottom.
• Figure 4: Level conversion to identify the error rate of each of the level-$l$ operations and interfaces ($l\in\{0,1,\ldots,L\}$) in the intended level-$L$ circuit implemented by the fault-tolerant circuit. For each level $l\in\{1,\ldots,L\}$, the level conversion of the interfaced circuit for our protocol is performed by inserting a pair of the $*$-decoder and the $*$-encoder before every level-$l$ decoding gadget $\mathop{\mathrm{Dec}}\nolimits_l$ and every level-$l$ measurement gadget. The inserted $*$-encoders and $*$-decoders are moved through the level-$(l-1)$ circuit using the conditions on the correctness of the level-$l$ ExRecs until they are eliminated by the correctness conditions of the level-$l$ ExRecs or cancelled out. Then, the obtained level-$l$ circuit undergoes the local stochastic Pauli error model, and the error rate of each level-$l$ operation and interface location is upper bounded according to Theorem \ref{['thm-level-conversion']}.
• Figure 5: A schematic on the level-$L$ circuit for fault-tolerant channel coding for a quantum channel $\mathcal{N}$ on the top and the corresponding task for an effective channel ${\mathcal{N}}_{\text{eff},L}$ at the bottom.

Theorems & Definitions (16)

• Definition 1: Entanglement-assisted classical coding scheme
• Definition 2: Entanglement-assisted classical capacity
• Definition 3: Fault-tolerant entanglement-assisted classical coding scheme
• Definition 4: Fault-tolerant entanglement-assisted classical capacity
• Theorem 5: Threshold theorem, yamasaki2022timeefficient
• Theorem 6: Level conversion
• proof
• Corollary 7: Single-level interfaced circuit
• proof
• Corollary 8: Level conversion for interfaced circuits