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Clifford synthesis via generalized S and CZ gates

Vadym Kliuchnikov, Marcus P. da Silva

Abstract

We show that any $n$-qubit Clifford unitary can be implemented using at most $2n$ multi-qubit joint measurements. All the multi-qubit joint measurements used for implementing the Clifford unitary can be chosen to form at most two sets of independent mutually-commuting measurements. Each of these sets is of size at most $n$. This enables very flexible space-time trade-offs when implementing Clifford unitaries. We also discuss a version of the result that relies on multi-target CNOTs and is more relevant for targeting fault-tolerant hardware based on Quantum LDPC codes.

Clifford synthesis via generalized S and CZ gates

Abstract

We show that any -qubit Clifford unitary can be implemented using at most multi-qubit joint measurements. All the multi-qubit joint measurements used for implementing the Clifford unitary can be chosen to form at most two sets of independent mutually-commuting measurements. Each of these sets is of size at most . This enables very flexible space-time trade-offs when implementing Clifford unitaries. We also discuss a version of the result that relies on multi-target CNOTs and is more relevant for targeting fault-tolerant hardware based on Quantum LDPC codes.

Paper Structure

This paper contains 9 sections, 2 theorems, 18 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pauli unitaries
  4. Clifford unitaries
  5. Symplectic matrices
  6. Remote execution of diagonal gates and circuits for generalized S and CZ gates
  7. Clifford synthesis algorithm
  8. Clifford synthesis using A* search
  9. Conclusions

Key Result

Theorem 2.1

Each symplectic matrix is a product of two symplectic involutions.

Figures (4)

  • Figure 2.1: Quantum circuit notation
  • Figure 2.2: Remote execution of S gate. The circuit on the right is described in Appendix D in EDP2022; the circuit on the left is obtained from the circuit on the right by replacing the CNOT-based fanout gate with a measurement-based one, similar to B2022.
  • Figure 2.3: Remote execution of CZ gate. The circuit on the right is described in Appendix D in EDP2022; the circuit on the left is obtained from the circuit on the right by replacing the CNOT-based fanout gate with a measurement-based one, similar to B2022.
  • Figure 2.4: Circuits for executing generalized S and CZ gates using one and two multi-qubit Pauli measurements. Similar circuits are in Figure 11(b) and Figure 33(b) in GSC2019.

Theorems & Definitions (2)

  • Theorem 2.1: The main theorem in GOW1981
  • Theorem 2.2: Theorem 2.1.16 in O1978