Reducing the Depth of Linear Reversible Quantum Circuits

Timothee Goubault De Brugiere,Cyril Allouche,Benoit Valiron,Marc Baboulin,Simon Martiel

doi:10.1109/tqe.2021.3091648

Abstract

In quantum computing the decoherence time of the qubits determines the computation time available and this time is very limited when using current hardware. In this paper we minimize the execution time (the depth) for a class of circuits referred to as linear reversible circuits, which has many applications in quantum computing (e.g., stabilizer circuits, CNOT+T circuits, etc.). We propose a practical formulation of a divide and conquer algorithm that produces quantum circuits that are twice as shallow as those produced by existing algorithms. We improve the theoretical upper bound of the depth in the worst case for some range of qubits. We also propose greedy algorithms based on cost minimization to find more optimal circuits for small or simple operators. Overall, we manage to consistently reduce the total depth of a class of reversible functions, with up to 92% savings in an ancilla-free case and up to 99% when ancillary qubits are available.

Highlights

Quantum computing is getting closer to the moment when it will be able to solve problems insoluble using current computers
If a hardware improvement is possible, it is possible to compress the set of instructions so that their execution takes less time. These instructions are usually represented by a quantum circuit and, assuming that two non-overlapping gates can be executed in parallel, the execution time of the circuit is strongly related to its depth
In this paper we focus on improving the depth of linear reversible circuits with a full qubit connectivity, meaning that any Controlled-Not gate (CNOT) gate between any pair of qubits can be done — equivalently this means that any row operation is available

Summary

Introduction

Quantum computing is getting closer to the moment when it will be able to solve problems insoluble using current computers. If a hardware improvement is possible, it is possible to compress the set of instructions so that their execution takes less time These instructions are usually represented by a quantum circuit and, assuming that two non-overlapping gates can be executed in parallel, the execution time of the circuit is strongly related to its depth. K k and the function f can be represented with a column vector α = [f (e1), ..., f (en)]T such that f (x) = α · x, where · stands for the scalar product on F2n and (−)T is the matrixtranspose operation.

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