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

Abstract

Determining the quantum-classical boundary between quantum circuits which can be efficiently simulated classically and those which 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 multi-qubit gates of the form $\alpha I+\beta P$. We give a simple algebraic criterion which characterizes the individual quantum circuits for which OFD generates an efficient CAMPS - the bond-dimension is exponential in the null space of a 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 poly-logarithmic 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 R\'enyi 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.