Distributed Quantum Error Correction with Permutation-Invariant Approximate Codes

TL;DR

This work tackles the challenge of error correction in modular, multi-processor quantum computers by proposing Distributed Approximate Quantum Error Correction (DAQEC) that leverages permutation-invariant, approximate codes. The core idea is to distribute code blocks across processors so that most quantum gates act locally within a processor, reducing processor-nonlocal gates and enabling universal transversal operations via codes that evade the Eastin–Knill constraint, exemplified by the W-state code. The authors provide concrete constructions: an explicit W-state encoding/decoding scheme, improved W-state preparation circuits, and a framework for collective fault tolerance when concatenating heterogeneous inner codes, together with a formal bound and numerical evidence showing advantages under noise asymmetry and spatially correlated errors. These results suggest a practical path toward scalable, fault-tolerant, distributed quantum computation without the overhead typically associated with magic-state distillation or code switching, and they motivate exploring more advanced approximate codes for near-term hardware implementations.

Abstract

Modular quantum computing architectures require error correction schemes that remain effective in the presense of noisy inter-processor operations. We introduce a distributed quantum error correction framework based on approximate codes to address this challenge. Our approach enables concatenation of distinct local codes across modules while allowing logical operations composed primarily of processor-local gates. We derive a lower bound and present corresponding simulations which indicate that this nontraditional approach can provide marked advantage over existing approaches in the highly non-uniform error landscape of a distributed quantum computer. As a concrete realization, we present encoding and decoding circuits for the permutation-invariant W- state code and propose efficient methods for its preparation. These results highlight the potential of approximate distributed error correction strategies for scalable, modular, fault-tolerant quantum computation.

Figures (14)

• Figure 1: Comparison of interactions required for local QEC and distributed QEC in a 4-QPU system. Each QPU holds four qubits and qubits of the same color belong to a single code block. Distributed QEC enables logical 2-qubit gates comprised entirely of local physical gates (assuming transversality), at the expense of a more complex distributed decoding procedure.
• Figure 2: Relative performance of distributed and local code block allocation schemes as a function of circuit depth between decoding stages. Note that these results apply to the 7-qubit Steane code only for circuits composed of transversal gates; a method applicable to general circuits is presented in Section \ref{['adqec']}.
• Figure 3: Logical error rates of distributed QEC (red dots) vs local QEC (blue dots) under spatially correlated error conditions. We simulate seven $[[7,1,3]]$ Steane codes across seven processors, comparing the cases where code blocks are each contained within their own processor (local) and each spread among all processors (fully distributed). The processor error rates are sampled from normal distributions, with the standard deviation set to 0.5 times the mean error rate. Distributed code blocks consistently outperform local code blocks, resulting in 13-20% lower logical error rates (green curve).
• Figure 4: W state preparation procedure. We construct a $2^k$-qudit W state by first (a) constructing the 2-qudit W state and then (b) recursively doubling its size. In the $d=2$ case we can replace the circuit in (a) with a simple Bell state preparation circuit, which is composed only of Clifford gates. Note that the ancilla (${|{+}\rangle}$) in each circuit is a qubit, while other subsystems are qudits. The open-circle CNOT gates are controlled-on-${|{0}\rangle}$-NOT gates; a decomposition is given in yeh-qudit-w-state.
• Figure 5: Encoding circuit for the W-state code. The encoding procedure consists of a W state preparation followed by analog rotations. (a) Encoding circuit for $n=3, d_L=2$, involving a Y-rotation about $\phi=2\cos^{-1}(1/\sqrt{3})$. (b) General encoding circuit. The encoding simplifies significantly for W states with $n=2^k$, as the input W state can be prepared according to Figure \ref{['fig:w-prep']}.

Theorems & Definitions (7)

• Definition 1: Fully Distributed QEC
• Lemma 1: Advantage of the Distributed W-state Code
• proof
• Lemma 2
• proof
• Theorem 3: Lower Bound on Distributed W-state Code Advantage
• proof