Faster computation of nonstabilizerness

Abstract

The characterization of nonstabilizerness is fruitful due to its application in gate synthesis and classical simulation. In particular, the resource monotone called the stabilizer extent is a useful tool to estimate the simulation cost using rank-based simulators, one of the state-of-the-art simulators of Clifford+$T$ circuits. In this work, we propose faster numerical algorithms to compute the stabilizer extent. Our algorithm utilizes the Column Generation method, which iteratively updates the subset of pure stabilizer states used for calculation. This subset is selected based on the overlaps between all stabilizer states and a target state. In order to update the subset, we make use of a newly proposed subroutine for calculating the stabilizer fidelity that (i) achieves linear time complexity with respect to the number of stabilizer states, (ii) super-exponentially reduces the space complexity by in-place calculation, and (iii) prunes unnecessary states for the computation. As a result, our algorithm can compute the stabilizer fidelity and the stabilizer extent for Haar random pure states up to $n=9$ qubits, which naively requires a memory of 305 EiB. We further show that our algorithm runs faster when the target state vector is real. We prove that the problem size is reduced by $\mathcal{O}(2^n)$ compared to the general cases, which makes it computable for the case of $n=10$ qubits.