Stabilizer extent is not multiplicative

Arne Heimendahl,David Gross,Felipe Montealegre-Mora,Frank Vallentin

doi:10.22331/q-2021-02-24-400

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 thestabilizer 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.

Highlights

Introduction and Summary of resultsIn the model of quantum computation with magic states [5], stabilizer circuits, whose computational power is limited by the Gottesmann-Knill theorem [1, 13], are promoted to universality by implementing non-Clifford gates via the injection of magic states
We will continue with the proof of Lemma 3, which is a consequence of complementary slackness
Ifs∈D is optimal for the primal and (y,s∈D) optimal for the dual, complementary slackness [2] enforces but as we have · zs = 0, s∈D

Summary

Introduction and Summary of results

In the model of quantum computation with magic states [5], stabilizer circuits, whose computational power is limited by the Gottesmann-Knill theorem [1, 13], are promoted to universality by implementing non-Clifford gates via the injection of magic states. No efficient methods are known for computing the stabilizer rank analytically or numerically To address this issue, Bravyi et al [8] introduced a computationally better-behaved convex relaxation: the stabilizer extent (see Definition 1). The central sparsification lemma of [8] states that a stabilizer decomposition with small extent can be transformed into a sparse decomposition that is close to the original state. Since the set of stabilizer states is closed under taking tensor products, one can see that the stabilizer extent is submultiplicative, that is ξ(⊗jψj) ≤ j ξ(ψj) for any input state ⊗jψj. The properties used — prime among them that the size of the dictionaries scales subexponentially with the Hilbert space dimension — are listed as Properties (i) to (v) in the following theorem.

Proof strategy

Proof of the main theorem

An optimality condition for the stabilizer extent

Summary and outlook

A Formulating the extent as a second order cone program