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The Heisenberg Representation of Quantum Computers

Daniel Gottesman

TL;DR

The paper presents a Heisenberg-picture framework for quantum computation that tracks the evolution of Pauli operators under Clifford gates and Pauli measurements, enabling efficient analysis of a large class of quantum networks through stabilizer formalism. It defines the Clifford group as the set of gates preserving the Pauli group, introduces stabilizers to describe a wide range of quantum states (stabilizer states) and codes, and demonstrates how error-detecting and error-correcting codes arise naturally within this formalism. By applying these ideas to teleportation, remote operations, and gate synthesis via measurements, the work shows how many quantum communication tasks and fault-tolerant primitives can be analyzed without full state tomography. A central result (Knill's theorem) states that Clifford-group circuits with Pauli measurements and classical conditioning can be efficiently simulated on a classical computer, clarifying the boundary between classically simulable and universal quantum computation and highlighting the practical relevance of stabilizer codes and Clifford-based subroutines.

Abstract

Since Shor's discovery of an algorithm to factor numbers on a quantum computer in polynomial time, quantum computation has become a subject of immense interest. Unfortunately, one of the key features of quantum computers - the difficulty of describing them on classical computers - also makes it difficult to describe and understand precisely what can be done with them. A formalism describing the evolution of operators rather than states has proven extremely fruitful in understanding an important class of quantum operations. States used in error correction and certain communication protocols can be described by their stabilizer, a group of tensor products of Pauli matrices. Even this simple group structure is sufficient to allow a rich range of quantum effects, although it falls short of the full power of quantum computation.

The Heisenberg Representation of Quantum Computers

TL;DR

The paper presents a Heisenberg-picture framework for quantum computation that tracks the evolution of Pauli operators under Clifford gates and Pauli measurements, enabling efficient analysis of a large class of quantum networks through stabilizer formalism. It defines the Clifford group as the set of gates preserving the Pauli group, introduces stabilizers to describe a wide range of quantum states (stabilizer states) and codes, and demonstrates how error-detecting and error-correcting codes arise naturally within this formalism. By applying these ideas to teleportation, remote operations, and gate synthesis via measurements, the work shows how many quantum communication tasks and fault-tolerant primitives can be analyzed without full state tomography. A central result (Knill's theorem) states that Clifford-group circuits with Pauli measurements and classical conditioning can be efficiently simulated on a classical computer, clarifying the boundary between classically simulable and universal quantum computation and highlighting the practical relevance of stabilizer codes and Clifford-based subroutines.

Abstract

Since Shor's discovery of an algorithm to factor numbers on a quantum computer in polynomial time, quantum computation has become a subject of immense interest. Unfortunately, one of the key features of quantum computers - the difficulty of describing them on classical computers - also makes it difficult to describe and understand precisely what can be done with them. A formalism describing the evolution of operators rather than states has proven extremely fruitful in understanding an important class of quantum operations. States used in error correction and certain communication protocols can be described by their stabilizer, a group of tensor products of Pauli matrices. Even this simple group structure is sufficient to allow a rich range of quantum effects, although it falls short of the full power of quantum computation.

Paper Structure

This paper contains 8 sections, 1 theorem, 31 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Basics of the Heisenberg Representation
  3. The Clifford Group
  4. Applying the Heisenberg Representation
  5. Stabilizers
  6. Quantum Error-Correcting Codes
  7. Measurements
  8. Discussion and Summary

Key Result

Theorem 1

Any quantum computer performing only: a) Clifford group gates, b) measurements of Pauli group operators, and c) Clifford group operations conditioned on classical bits, which may be the results of earlier measurements, can be perfectly simulated in polynomial time on a probabilistic classical comput

Figures (9)

  • Figure 1: Alice's quantum computer: a) network, b) analysis.
  • Figure 2: Bob's ${\rm CNOT}(2 \rightarrow 1)$: a) network, b) analysis.
  • Figure 3: Ancient Hittite gate network: a) network, b) analysis
  • Figure 4: Alice's improved quantum computer: a) network, b) analysis.
  • Figure 5: Making a Bell state: a) network, b) analysis.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Example 9
  • Example 10: Quantum Teleportation
  • ...and 2 more