Linear-Time Encodable and Decodable Quantum Error-Correcting Codes

TL;DR

This work constructs explicit and asymptotically good quantum codes whose encoding, unencoding and decoding all use a linear number of gates, and additionally whose encoding and unencoding may be run in logarithmic depth.

Abstract

Recent years have seen rapid development in the subject of quantum coding theory, with breakthroughs on many exciting classes of codes, including quantum LDPC codes, quantum locally testable codes, and quantum codes with interesting transversal gates. However, a natural class of quantum codes, which has been well-studied classically, has not yet been treated: those which can be quickly encoded and decoded. This problem concerns the channel capacity setting, where a noise channel sits between perfect encoding and unencoding/decoding operations; this is the setting that is relevant for communication between fault-tolerant quantum computers. In this work, we construct asymptotically good quantum codes that can be encoded and unencoded by quantum circuits of logarithmic depth and consisting of a linear total number of gates. The classical decoding algorithms also run in logarithmic depth and use $\mathcal{O}(n \log n)$ gates, or alternatively a linear number of gates but with higher depth. We further construct explicit and asymptotically good quantum codes whose encoding, unencoding and decoding all use a linear number of gates, and additionally whose encoding and unencoding may be run in logarithmic depth.

Figures (5)

• Figure 1: High-level construction of the classical error-correcting code $\mathcal{Q}_k$ from the classical error-correcting code $\mathcal{Q}_{k-1}$ and the classical error-reduction code $\mathcal{R}$. We will use the same structure to form quantum error-correcting codes from quantum error-reduction codes.
• Figure 2: A first attempt at constructing and operating a quantum error-reduction code. This figure depicts the process of encoding, noise, unencoding, and stabiliser measurement.
• Figure 3: The encoding, noise, unencoding, and syndrome measurement of our quantum error-reduction codes. The Pauli noise on each group of qubits is labelled. For example, the $X$ support of the noise on the message qubits may be labelled by a $X$-type Pauli $X_q$, which may equivalently be thought of as a bit string in $\mathbb{F}_2^n$.
• Figure 4: Definition of the matrices $A, B$ and $D$ which define our quantum error-reduction code, from the lossless $Z$-graph. Vertices in $L_1$ and $R_1$ all have degree $\Delta_1$. Vertices in $R_2$ and $L_2$ all have $\Delta_1' = \frac{n}{m}\Delta_1$ edges joining them to vertices in $L_1$ and $R_1$, respectively, and $\Delta_2$ edges joining them to each other.
• Figure 7: Depiction of a simple encoding circuit for a quantum CSS code of the form of Equations \ref{['eq:error_reduction_X_parity']} and \ref{['eq:error_reduction_Z_parity']}. This is intended for illustrative purposes only.

Theorems & Definitions (47)

• Theorem 1
• Theorem 2
• Definition 4.1: Quantum Error-Reduction Code
• Remark 4.1
• Proposition 4.1
• Lemma 4.1
• proof
• Remark 4.2
• Lemma 4.2
• Remark 4.3