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Simulating Clifford Circuits with Gaussian Elimination

Yuchen Pang, Edgar Solomonik

TL;DR

This work reframes Clifford circuit simulation as a linear-algebra problem over GF(2) by representing circuits as phased graph states and linking their amplitudes to LDL decompositions. Through ZX-calculus and graph-like ZX-diagrams, circuits are mapped to graphs whose LDL factors drive both strong and weak simulations, with the treewidth guiding the runtime via a fast LDL algorithm. The approach matches or surpasses state-of-the-art graph-state methods for several regimes, and extends to Clifford+T circuits by leveraging stabilizer decompositions. Beyond simulation, the amplitudes formula yields new insights into locally Clifford-equivalent graph states and enables efficient learning of low-rank graph structures. Overall, the paper provides a unifying, algebraic framework that broadens the toolkit for classical simulation of quantum circuits using fast linear algebra techniques.

Abstract

Quantum circuits are considered more powerful than classical circuits and require exponential resources to simulate classically. Clifford circuits are a special class of quantum circuits that can be simulated in polynomial time but still show important quantum effects such as entanglement. In this work, we present an algorithm that simulates Clifford circuits by performing Gaussian elimination on a modified adjacency matrix derived from the circuit structure. Our work builds on an ZX-calculus tensor network representation of Clifford circuits that reduces to quantum graph states. We give a concise formula of amplitudes of graph states based on the LDL decomposition of matrices over GF(2), and use it to get efficient algorithms for strong and weak simulation of Clifford circuits using tree-decomposition-based fast LDL algorithm. The complexity of our algorithm matches the state of art for weak graph state simulation and improves the state of art for strong graph state simulation by taking advantage of Strassen-like fast matrix multiplication. Our algorithm is also efficient when computing many amplitudes or samples of a Clifford circuit. Further, our amplitudes formula provides a new characterization of locally Clifford equivalent graph states as well as an efficient protocol to learn graph states with low-rank adjacency matrices.

Simulating Clifford Circuits with Gaussian Elimination

TL;DR

This work reframes Clifford circuit simulation as a linear-algebra problem over GF(2) by representing circuits as phased graph states and linking their amplitudes to LDL decompositions. Through ZX-calculus and graph-like ZX-diagrams, circuits are mapped to graphs whose LDL factors drive both strong and weak simulations, with the treewidth guiding the runtime via a fast LDL algorithm. The approach matches or surpasses state-of-the-art graph-state methods for several regimes, and extends to Clifford+T circuits by leveraging stabilizer decompositions. Beyond simulation, the amplitudes formula yields new insights into locally Clifford-equivalent graph states and enables efficient learning of low-rank graph structures. Overall, the paper provides a unifying, algebraic framework that broadens the toolkit for classical simulation of quantum circuits using fast linear algebra techniques.

Abstract

Quantum circuits are considered more powerful than classical circuits and require exponential resources to simulate classically. Clifford circuits are a special class of quantum circuits that can be simulated in polynomial time but still show important quantum effects such as entanglement. In this work, we present an algorithm that simulates Clifford circuits by performing Gaussian elimination on a modified adjacency matrix derived from the circuit structure. Our work builds on an ZX-calculus tensor network representation of Clifford circuits that reduces to quantum graph states. We give a concise formula of amplitudes of graph states based on the LDL decomposition of matrices over GF(2), and use it to get efficient algorithms for strong and weak simulation of Clifford circuits using tree-decomposition-based fast LDL algorithm. The complexity of our algorithm matches the state of art for weak graph state simulation and improves the state of art for strong graph state simulation by taking advantage of Strassen-like fast matrix multiplication. Our algorithm is also efficient when computing many amplitudes or samples of a Clifford circuit. Further, our amplitudes formula provides a new characterization of locally Clifford equivalent graph states as well as an efficient protocol to learn graph states with low-rank adjacency matrices.

Paper Structure

This paper contains 32 sections, 32 theorems, 107 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Preliminaries and Notations
  4. Matrix-related notations
  5. Matrices over ${\mathbb{Z}}_4$ and ${\mathbb{F}}_2$
  6. Tree decomposition and treewidth
  7. Quantum circuits
  8. Graph states
  9. Clifford circuits and stabilizer states
  10. Other Clifford circuit simulation methods
  11. Tensor networks
  12. ZX-diagrams
  13. Graph states as ZX-diagrams
  14. Graph-like ZX-diagrams
  15. Clifford circuit simulation as phased graph state simulation
  16. ...and 17 more sections

Key Result

Theorem 2

Let ${A}\in{\mathbb{J}}_n$. Then we have where $\omega_1({A})={A}\bmod2$, $k$ is rank of $\omega_1({A})$ over ${\mathbb{F}}_2$, and ${B}\in{\mathbb{J}}_n$, ${w}\in{\mathbb{F}}_2^n$, ${v}\in{\mathbb{F}}_2^n$ could be calculated along with an LDL factorization of $\omega_1({A})$. In particular, if the leading block of $A$, $A_{11}$, is full where $A_{11}^{\#}$ is the inverse of $A_{11}$ over $\mat

Figures (2)

  • Figure 1: A tensor diagram representing \ref{['eq:tn-example']}
  • Figure 1: A graph-like ZX-diagram viewed as an inner product between graph states and a tensor product of one-qubit states. The Z-spiders with phases $p_j$ represent a tensor product of one-qubit states, while the remaining ZX-diagram represents a graph state.

Theorems & Definitions (65)

  • Definition 1: Clifford circuit simulation
  • Theorem 2: Informal form of \ref{['thm:amp-formula']}
  • Definition 3: Phased graph state simulation
  • Theorem 4: Summary of \ref{['thm:weak-sim-pgs']}, \ref{['thm:strong-sim-pgs']} and \ref{['thm:strong-sim-pgs-fixed-bits']}
  • Theorem 5: Restatement of \ref{['thm:gs-sim-complexity']}
  • Theorem 6: Restatement of \ref{['thm:clifford-complexity']}
  • Theorem 7: Restatement of \ref{['thm:lc-eq']}
  • Definition 1: LDL decomposition over ${\mathbb{F}}_2$
  • Definition 2: Reduced LDL decomposition over ${\mathbb{F}}_2$
  • Definition 3: Tree decomposition
  • ...and 55 more