Clifford Circuit Optimization with Templates and Symbolic Pauli Gates

Sergey Bravyi,Dmitri Maslov,Shaohan Hu,Ruslan Shaydulin

doi:10.22331/q-2021-11-16-580

Abstract

The Clifford group is a finite subgroup of the unitary group generated by the Hadamard, the CNOT, and the Phase gates. This group plays a prominent role in quantum error correction, randomized benchmarking protocols, and the study of entanglement. Here we consider the problem of finding a short quantum circuit implementing a given Clifford group element. Our methods aim to minimize the entangling gate count assuming all-to-all qubit connectivity. First, we consider circuit optimization based on template matching and design Clifford-specific templates that leverage the ability to factor out Pauli and SWAP gates. Second, we introduce a symbolic peephole optimization method. It works by projecting the full circuit onto a small subset of qubits and optimally recompiling the projected subcircuit via dynamic programming. CNOT gates coupling the chosen subset of qubits with the remaining qubits are expressed using symbolic Pauli gates. Software implementation of these methods finds circuits that are only 0.2% away from optimal for 6 qubits and reduces the two-qubit gate count in circuits with up to 64 qubits by 64.7% on average, compared with the Aaronson-Gottesman canonical form.

Highlights

One of the central challenges in quantum computation is the problem of generating a short schedule of physically implementable quantum gates realizing a given unitary operation, otherwise known as the quantum circuit synthesis/optimization problem
We focus on the minimization of the cnot gate count, drawing motivation from physical layer realizations where entangling gates come at a higher cost than the single-qubit gates, and ignore the connectivity constraints
The methods we developed are evaluated on a collection of 2,264 circuits and shown to reduce the average cnot gate count by 64.7% compared with the methods proposed by Aaronson and Gottesman in [3]

Summary

Introduction

One of the central challenges in quantum computation is the problem of generating a short schedule of physically implementable quantum gates realizing a given unitary operation, otherwise known as the quantum circuit synthesis/optimization problem. Clifford group elements play a crucial role in quantum error correction [25], quantum state distillation [6, 20], randomized benchmarking [21, 23], study of entanglement [5, 25], and, more recently, shadow tomography [2, 16], to name some application areas. A special property of the Clifford group that plays the central role in many applications is being a unitary 2-design [11, 12]. It guarantees that a random uniformly distributed element of the Clifford group has exactly the same second order moments as the Haar random unitary operator. In contrast to Haar random unitaries, any Clifford operators admit an efficient implementation by a quantum circuit. We utilize the controlled-z (cz) gate, which can be constructed as a circuit with Hadamard and cnot gates as follows,

Objectives

Results

Conclusion