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Good Quantum Error-Correcting Codes Exist

A. R. Calderbank, Peter W. Shor

TL;DR

Addresses the challenge of decoherence in quantum information processing. Proposes a construction of quantum error-correcting codes from pairs of classical codes with C2 subset of C1 and analyzes decoding via two complementary bases, enabling recovery of the original k qubits after t errors. Proves the existence of t-error-correcting codes with asymptotic rate $1-2H_2(2t/n)$ and develops a Gilbert-Varshamov-type argument via weakly self-dual codes. Provides capacity bounds for quantum channels using Levitin-Holevo and entanglement bounds, illustrating fundamental limits and practical performance.

Abstract

A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n = 1 - 2H(2t/n) where H(p) is the binary entropy function -p log p - (1-p) log (1-p). Upper bounds on this asymptotic rate are given.

Good Quantum Error-Correcting Codes Exist

TL;DR

Addresses the challenge of decoherence in quantum information processing. Proposes a construction of quantum error-correcting codes from pairs of classical codes with C2 subset of C1 and analyzes decoding via two complementary bases, enabling recovery of the original k qubits after t errors. Proves the existence of t-error-correcting codes with asymptotic rate and develops a Gilbert-Varshamov-type argument via weakly self-dual codes. Provides capacity bounds for quantum channels using Levitin-Holevo and entanglement bounds, illustrating fundamental limits and practical performance.

Abstract

A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n = 1 - 2H(2t/n) where H(p) is the binary entropy function -p log p - (1-p) log (1-p). Upper bounds on this asymptotic rate are given.

Paper Structure

This paper contains 6 sections, 4 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions
  3. Quantum Codes
  4. Decoding Quantum Codes
  5. Weakly Self-Dual Codes
  6. Quantum Channels

Key Result

Lemma 1

Suppose that ${\cal C}$ is a binary linear code of length $n$. Let $e$, $E$$\in$${\sf F}_2^n$, with $e \preceq E$ and ${\rm wt}(E) < d({\cal C}^\perp)$. Then there exists a vector $v_e \in {\cal C}$ such that $v_e|_{{\rm supp}(E)} = e$.

Figures (1)

  • Figure 1: The solid line shows the asymptotic rate $R$ of our quantum codes versus the error rate of the channel $t/n$. Two upper bounds for this quantity are also plotted: the Levitin--Holevo upper bound with a dashed line and the entanglement upper bound with a dotted line.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Lemma 3
  • proof