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  • Open Access

Classical simulability of Clifford+T circuits with Clifford-augmented matrix product states

Zejun Liu and Bryan K. Clark*

  • The Anthony J. Leggett Institute for Condensed Matter Theory and IQUIST and NCSA Center for Artificial Intelligence Innovation and Department of Physics, University of Illinois at Urbana-Champaign, Illinois 61801, USA

  • *Contact author: bkclark@illinois.edu
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Phys. Rev. Research 8, 023116 – Published 4 May, 2026

DOI: https://doi.org/10.1103/ybnf-rjw8

Abstract

Determining the quantum-classical boundary between quantum circuits that can be efficiently simulated classically and those that cannot remains a fundamental question. One approach to classical simulation is to represent the output of a quantum circuit as a Clifford-augmented matrix product state (CAMPS), which, via a disentangling algorithm, decomposes the wave function into Clifford and MPS components, and from which Pauli expectation values can be computed in time polynomial in the MPS bond dimension. In this work, we develop an optimization-free disentangling (OFD) algorithm for Clifford circuits either doped with T gates or, equivalently, preceded by multiqubit gates of the form αI+βP. We give a simple and easily computed algebraic criterion that characterizes the individual quantum circuits for which OFD generates an efficient CAMPS—the bond dimension is exponential in the null space of a Galois binary field GF(2) matrix induced by a tableau of the twisted Pauli strings P. This significantly increases the number of circuits with rigorous polynomial-time classical simulations. We also give evidence that the typical N qubit random Clifford circuit doped with N uniformly distributed T gates of polylogarithmic depth or greater has a CAMPS with polynomial bond dimension. In addition, we compare OFD against disentangling by optimization. We further explore the representability of CAMPS for random Clifford circuits doped with more than N T gates. We also propose algorithms for sampling, probability, and amplitude estimation of bitstrings, and evaluation of entanglement Rényi entropy from CAMPS, which, though still having exponential complexity, are more efficient than standard MPS simulations. This work establishes a versatile framework for understanding classical simulatability of Clifford+T circuits and explores the interplay between quantum entanglement and quantum magic in quantum systems.

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Phys. Rev. Research 8, 023116– Published 4 May, 2026
  • Received 14 January 2025
  • Accepted 18 April 2026
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