Classical Simulability of Quantum Circuits with Shallow Magic Depth

TL;DR

This paper explores how the distribution and depth of quantum magic in circuits affect their classical simulability, revealing sharp complexity transitions and providing new simulation techniques for certain shallow circuits.

Contribution

It demonstrates the impact of magic distribution on classical simulation complexity and introduces algorithms for simulating shallow, decomposable circuits with limited magic depth.

Findings

Amplitude estimation and sampling are classically hard with all T gates in one layer.

Pauli observable evaluation becomes complex with a single T layer but is easy with T^{1/2} gates.

Polynomial-time algorithms exist for certain shallow circuits with decomposable structures.

Abstract

Quantum magic is a necessary resource for quantum computers to be not efficiently simulable by classical computers. Previous results have linked the amount of quantum magic, characterized by the number of $T$ gates or stabilizer rank, to classical simulability. However, the effect of the distribution of quantum magic on the hardness of simulating a quantum circuit remains open. In this work, we investigate the classical simulability of quantum circuits with alternating Clifford and $T$ layers across three tasks: amplitude estimation, sampling, and evaluating Pauli observables. In the case where all $T$ gates are distributed in a single layer, performing amplitude estimation and sampling to multiplicative error are already classically intractable under reasonable assumptions, but Pauli observables are easy to evaluate. Surprisingly, with the addition of just one $T$ gate layer or merely replacing all $T$ gates with $T^{\frac{1}{2}}$, the Pauli evaluation task reveals a sharp complexity transition from P to GapP-complete. Nevertheless, when the precision requirement is relaxed to 1/poly($n$) additive error, we are able to give a polynomial time classical algorithm to compute amplitudes, Pauli observable, and sampling from $\log(n)$ sized marginal distribution for any magic-depth-one circuit that is decomposable into a product of diagonal gates. Our research provides new techniques to simulate highly magical circuits while shedding light on their complexity and their significant dependence on the magic depth.