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Improved Simulation of Stabilizer Circuits

Scott Aaronson, Daniel Gottesman

TL;DR

This paper refines the Gottesman–Knill theorem by introducing a faster tableau-based simulation that eliminates Gaussian elimination, enabling efficient classical simulation of large stabilizer circuits with a memory- and time-efficient representation. It implements CHP (CNOT-Hadamard-Phase) to demonstrate practical scalability to thousands of qubits, and proves ⊕L-completeness for stabilizer circuit simulation, indicating limits of classical computation within this model. A canonical form with only $O(n^2/\, ext{log} )$ gate equivalents is established, supporting circuit minimization and hardware synthesis considerations. The authors also extend the framework beyond pure stabilizer circuits to mixed states, circuits with a few non-stabilizer gates, and circuits with restricted measurements, broadening the scope of efficient classical simulation and highlighting rich avenues for further research and engineering implications.

Abstract

The Gottesman-Knill theorem says that a stabilizer circuit -- that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates -- can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. First, by removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor-2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freely-available program called CHP (CNOT-Hadamard-Phase), which can handle thousands of qubits easily. Second, we show that the problem of simulating stabilizer circuits is complete for the classical complexity class ParityL, which means that stabilizer circuits are probably not even universal for classical computation. Third, we give efficient algorithms for computing the inner product between two stabilizer states, putting any n-qubit stabilizer circuit into a "canonical form" that requires at most O(n^2/log n) gates, and other useful tasks. Fourth, we extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of non-stabilizer gates, and circuits acting on general tensor-product initial states but containing only a limited number of measurements.

Improved Simulation of Stabilizer Circuits

TL;DR

This paper refines the Gottesman–Knill theorem by introducing a faster tableau-based simulation that eliminates Gaussian elimination, enabling efficient classical simulation of large stabilizer circuits with a memory- and time-efficient representation. It implements CHP (CNOT-Hadamard-Phase) to demonstrate practical scalability to thousands of qubits, and proves ⊕L-completeness for stabilizer circuit simulation, indicating limits of classical computation within this model. A canonical form with only gate equivalents is established, supporting circuit minimization and hardware synthesis considerations. The authors also extend the framework beyond pure stabilizer circuits to mixed states, circuits with a few non-stabilizer gates, and circuits with restricted measurements, broadening the scope of efficient classical simulation and highlighting rich avenues for further research and engineering implications.

Abstract

The Gottesman-Knill theorem says that a stabilizer circuit -- that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates -- can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. First, by removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor-2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freely-available program called CHP (CNOT-Hadamard-Phase), which can handle thousands of qubits easily. Second, we show that the problem of simulating stabilizer circuits is complete for the classical complexity class ParityL, which means that stabilizer circuits are probably not even universal for classical computation. Third, we give efficient algorithms for computing the inner product between two stabilizer states, putting any n-qubit stabilizer circuit into a "canonical form" that requires at most O(n^2/log n) gates, and other useful tasks. Fourth, we extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of non-stabilizer gates, and circuits acting on general tensor-product initial states but containing only a limited number of measurements.

Paper Structure

This paper contains 12 sections, 9 theorems, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Efficient Simulation of Stabilizer Circuits
  4. Implementation and Experiments
  5. Complexity of Simulating Stabilizer Circuits
  6. Canonical Form
  7. Beyond Stabilizer Circuits
  8. Mixed States
  9. Non-Stabilizer Initial States
  10. Non-Stabilizer Gates
  11. Open Problems
  12. Acknowledgments

Key Result

Theorem 1

Given an $n$-qubit state $\left| \psi\right\rangle$, the following are equivalent:

Figures (2)

  • Figure 1: The four types of gate allowed in the stabilizer formalism
  • Figure 2: Average time needed to simulate a measurement after applying $\beta n\log_{2}n$ unitary gates to $n$ qubits, on a 650MHz Pentium III with 256MB RAM.

Theorems & Definitions (9)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Theorem 8
  • Corollary 9