Stabilizer extent is not multiplicative

TL;DR

This paper proves that the stabilizer extent, a measure of complexity for quantum states in classical simulation, is not multiplicative under tensor products, impacting how we understand quantum resource measures.

Contribution

The paper demonstrates that the stabilizer extent is not multiplicative, resolving an open problem and influencing classical simulation strategies for quantum circuits.

Findings

Stabilizer extent is not multiplicative under tensor products.

The result relies on properties of the set of stabilizer states.

The approach can be adapted to other resource theories.

Abstract

The Gottesman-Knill theorem states that a Clifford circuit acting on stabilizer states can be simulated efficiently on a classical computer. Recently, this result has been generalized to cover inputs that are close to a coherent superposition of logarithmically many stabilizer states. The runtime of the classical simulation is governed by the stabilizer extent, which roughly measures how many stabilizer states are needed to approximate the state. An important open problem is to decide whether the extent is multiplicative under tensor products. An affirmative answer would yield an efficient algorithm for computing the extent of product inputs, while a negative result implies the existence of more efficient classical algorithms for simulating largescale quantum circuits. Here, we answer this question in the negative. Our result follows from very general properties of the set of stabilizer states, such as having a size that scales subexponentially in the dimension, and can thus be readily adapted to similar constructions for other resource theories.