Quantifying nonstabilizerness of matrix product states

Tobias Haug and Lorenzo Piroli
Phys. Rev. B 107, 035148 – Published 27 January 2023

Abstract

Nonstabilizerness, also known as magic, quantifies the number of non-Clifford operations needed to prepare a quantum state. As typical measures either involve minimization procedures or a computational cost exponential in the number of qubits N, it is notoriously hard to characterize for many-body states. In this paper, we show that nonstabilizerness, as quantified by the recently introduced stabilizer Rényi entropies (SREs), can be computed efficiently for matrix product states (MPSs). Specifically, given an MPS of bond dimension χ and integer Rényi index n>1, we show that the SRE can be expressed in terms of the norm of an MPS with bond dimension χ2n. For translation-invariant states, this allows us to extract it from a single tensor, the transfer matrix, while for generic MPSs this construction yields a computational cost linear in N and polynomial in χ. We exploit this observation to revisit the study of ground-state nonstabilizerness in the quantum Ising chain, providing accurate numerical results up to large system sizes. We analyze the SRE near criticality and investigate its dependence on the local computational basis, showing that it is, in general, not maximal at the critical point.

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  • Received 2 August 2022
  • Revised 10 January 2023
  • Accepted 17 January 2023

DOI:https://doi.org/10.1103/PhysRevB.107.035148

©2023 American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Tobias Haug1,* and Lorenzo Piroli2

  • 1QOLS, Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom
  • 2Philippe Meyer Institute, Physics Department, École Normale Supérieure (ENS), Université PSL, 24 rue Lhomond, F-75231 Paris, France

  • *tobias.haug@u.nus.edu

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Vol. 107, Iss. 3 — 15 January 2023

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