Trading classical and quantum computational resources

TL;DR

This paper introduces hybrid quantum-classical simulation methods that enable efficient approximation of larger quantum systems using small quantum processors combined with classical computation, advancing quantum simulation capabilities.

Contribution

It presents novel algorithms for simulating sparse quantum circuits and Pauli-based computation with reduced quantum resources and improved classical simulation techniques.

Findings

Sparse quantum circuits can be simulated with classical processing in $2^{O(k)} poly(n)$ time.

Pauli-based computation can be simulated with classical processing in $2^{O(k)} poly(n)$ time.

A classical algorithm achieves $2^{0.94 n} poly(n)$ time simulation for PBC, surpassing brute-force methods.

Abstract

We propose examples of a hybrid quantum-classical simulation where a classical computer assisted by a small quantum processor can efficiently simulate a larger quantum system. First we consider sparse quantum circuits such that each qubit participates in O(1) two-qubit gates. It is shown that any sparse circuit on n+k qubits can be simulated by sparse circuits on n qubits and a classical processing that takes time $2^{O(k)} poly(n)$. Secondly, we study Pauli-based computation (PBC) where allowed operations are non-destructive eigenvalue measurements of n-qubit Pauli operators. The computation begins by initializing each qubit in the so-called magic state. This model is known to be equivalent to the universal quantum computer. We show that any PBC on n+k qubits can be simulated by PBCs on n qubits and a classical processing that takes time $2^{O(k)} poly(n)$. Finally, we propose a purely classical algorithm that can simulate a PBC on n qubits in a time $2^{c n} poly(n)$ where $c\approx 0.94$. This improves upon the brute-force simulation method which takes time $2^n poly(n)$. Our algorithm exploits the fact that n-fold tensor products of magic states admit a low-rank decomposition into n-qubit stabilizer states.