Predicting Magic from Very Few Measurements

TL;DR

This paper introduces a method to estimate the nonstabilizerness (magic) of quantum states using a small set of Pauli measurements, making the quantification more practical despite inherent computational complexity.

Contribution

It presents a framework for witnessing and quantifying quantum magic with limited measurements and analyzes the computational hardness of related problems.

Findings

Magic can be estimated from any set of Pauli measurements containing anti-commuting pairs.

Deciding membership in the reduced stabilizer polytope is NP-hard.

The method effectively computes nonstabilizerness in complex quantum states beyond previous techniques.

Abstract

The nonstabilizerness of quantum states is a necessary resource for universal quantum computation, yet its characterization is notoriously demanding. Quantifying nonstabilizerness typically requires an exponential number of measurements and a doubly exponential classical post-processing cost to evaluate its standard monotones. In this work, we show that nonstabilizerness is, to a large extent, in the eyes of the beholder: it can be witnessed and quantified using any set of $m$ $n$-qubit Pauli measurements, provided the set contains anti-commuting pairs. We introduce a general framework that projects the stabilizer polytope onto the subspace defined by these observables and provide an algorithm that estimates magic from Pauli expectation values with runtime exponential in the number of measurements $m$ and polynomial in the number of qubits $n$. By relating the problem to a stabilizer-restricted variant of the quantum marginal problem, we also prove that deciding membership in the corresponding reduced stabilizer polytope is NP-hard. In particular, unless $\mathrm{P} = \mathrm{NP}$, no algorithm polynomial in $m$ can solve the problem in full generality, thus establishing fundamental complexity-theoretic limitations. Finally, we employ our framework to compute nonstabilizerness in different Hamiltonian ground states, demonstrating the practical performance of our method in regimes beyond the reach of existing techniques.