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An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation

Daniel Gottesman

TL;DR

This paper surveys quantum error correction and fault-tolerant quantum computation, emphasizing stabilizer codes, the stabilizer formalism, and their connections to GF$(4)$ and CSS constructions. It explains how error models and the threshold theorem enable scalable quantum computation with concatenated codes and fault-tolerant gadgets. Key contributions include the discussion of transversal gates, gate teleportation, Steane and Knill error correction, and the apparatus for universal fault-tolerant computation. The work highlights the stabilizer/Clifford framework for efficient classical simulation and the practical implications of thresholds, overhead, and error models.

Abstract

Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements. To build a quantum computer which behaves correctly in the presence of errors, we also need a theory of fault-tolerant quantum computation, instructing us how to perform quantum gates on qubits which are encoded in a quantum error-correcting code. The threshold theorem states that it is possible to create a quantum computer to perform an arbitrary quantum computation provided the error rate per physical gate or time step is below some constant threshold value.

An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation

TL;DR

This paper surveys quantum error correction and fault-tolerant quantum computation, emphasizing stabilizer codes, the stabilizer formalism, and their connections to GF and CSS constructions. It explains how error models and the threshold theorem enable scalable quantum computation with concatenated codes and fault-tolerant gadgets. Key contributions include the discussion of transversal gates, gate teleportation, Steane and Knill error correction, and the apparatus for universal fault-tolerant computation. The work highlights the stabilizer/Clifford framework for efficient classical simulation and the practical implications of thresholds, overhead, and error models.

Abstract

Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements. To build a quantum computer which behaves correctly in the presence of errors, we also need a theory of fault-tolerant quantum computation, instructing us how to perform quantum gates on qubits which are encoded in a quantum error-correcting code. The threshold theorem states that it is possible to create a quantum computer to perform an arbitrary quantum computation provided the error rate per physical gate or time step is below some constant threshold value.

Paper Structure

This paper contains 21 sections, 13 theorems, 69 equations, 7 figures, 5 tables.

Table of Contents

  1. Background: the need for error correction
  2. Basic properties and structure of quantum error correction
  3. The nine-qubit code
  4. General properties of quantum error-correcting codes
  5. Stabilizer codes
  6. More quantum error-correcting codes and their structure
  7. Some other important codes
  8. Codes over ${\rm GF}(4)$
  9. Even more quantum error-correcting codes
  10. The logical Pauli group
  11. The Clifford group
  12. Fault-tolerant gates
  13. The need for fault tolerance
  14. Definition of fault tolerance
  15. Transversal gates
  16. ...and 6 more sections

Key Result

Theorem 1

There is no quantum operation that takes a state $|\psi\rangle$ to $|\psi\rangle \otimes |\psi\rangle$ for all states $|\psi\rangle$.

Figures (7)

  • Figure 1: A non-fault-tolerant implementation of the measurement of $U$, which has eigenvalues $\pm 1$.
  • Figure 2: A component of a fault-tolerant implementation of the measurement of $U = \bigotimes U_i$, which has eigenvalues $\pm 1$.
  • Figure 3: Steane Error Correction. Each horizontal line represents a full $n$-qubit block of the code, and each gate or measurement represents a transversal implementation of that operation.
  • Figure 4: Knill EC. Each horizontal line represents a full $n$-qubit block of the code, and each gate or measurement represents a transversal implementation of that operation. $Q$ is a Pauli operator and contains a correction for both the teleportation outcome and error correction.
  • Figure 5: Gate teleportation of $\overline{U}$. The process of teleporting the state followed by $\overline{U}$ is the same as teleporting the state through a special ancilla with an appropriately modified correction operation.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Theorem 1: No-Cloning
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • proof
  • Definition 1
  • ...and 25 more