Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions

Aleks Kissinger,John Van De Wetering

doi:10.1088/2058-9565/ac5d20

Aleks Kissinger, John Van De Wetering

Open Access

https://doi.org/10.1088/2058-9565/ac5d20

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Abstract

We introduce an enhanced technique for strong classical simulation of quantum circuits which combines the ‘sum-of-stabilisers’ method with an automated simplification strategy based on the ZX-calculus. Recently it was shown that quantum circuits can be classically simulated by expressing the non-stabiliser gates in a circuit as magic state injections and decomposing them in chunks of 2–6 states at a time, obtaining sums of (efficiently-simulable) stabiliser states with many fewer terms than the naive approach. We adapt these techniques from the original setting of Clifford circuits with magic state injection to generic ZX-diagrams and show that, by interleaving this ‘chunked’ decomposition with a ZX-calculus-based simplification strategy, we can obtain stabiliser decompositions that are many orders of magnitude smaller than existing approaches. We illustrate this technique to perform exact norm calculations (and hence strong simulation) on the outputs of random 50- and 100-qubit Clifford + T circuits with up to 70 T-gates as well as a family of hidden shift circuits previously considered by Bravyi and Gosset with over 1000 T-gates.

Highlights

Classical simulation of quantum circuits has a variety of applications, from verifying the correct behaviour of quantum hardware and software to general-purpose simulations of quantum manybody systems
The first step is that we reduce the diagram to graph-like form, so that all the spiders are Z-spiders, and the only connectivity is via Hadamard edges
Our method for classically simulating a quantum circuit consists of essentially combining the work done on stabiliser decompositions with that done on ZX-diagram simplification

Summary

Introduction

Classical simulation of quantum circuits has a variety of applications, from verifying the correct behaviour of quantum hardware and software to general-purpose simulations of quantum manybody systems. While Google’s quantum supremacy result [4] was originally believed to take 10,000 years to reproduce on a classical computer, improvements in simulation techniques reduced this to about 20 days [24] and later to 5 days [36]. Emulation, of quantum computations is widely believed to be a hard problem, requiring exponential classical resources. There are a variety of different approaches to classical simulation whose time and space requirements vary widely based on the size, shape, contents, or required fidelity of the quantum circuit to be simulated. Tensor-network based techniques scale exponentially not with the number of qubits, but with the treewidth of the underlying graph of the circuit [34]. Methods based on stabiliser rank [9] and stabiliser extent [8] have been used to simulate circuits which are suitably close to Clifford circuits

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