Phase space simulation method for quantum computation with magic states on qubits

Robert Raussendorf (University of British Columbia), Juani Bermejo-Vega (Dahlem Center for Complex Quantum Systems – FU Berlin), Emily Tyhurst (University of Toronto), Cihan Okay (University of British Columbia), Michael Zurel (University of British Columbia)

We propose a method for classical simulation of finite-dimensional quantum systems, based on sampling from a quasiprobability distribution, i.e., a generalized Wigner function. Our construction applies to all finite dimensions, with the most interesting case being that of qubits. For multiple qubits, we find that quantum computation by Clifford gates and Pauli measurements on magic states can be efficiently classically simulated if the quasiprobability distribution of the magic states is non-negative. This provides the so far missing qubit counterpart of the corresponding result [V. Veitch, C. Ferrie, D. Gross, and J. Emerson, \emph{New J. Phys.} \textbf{14}, 113011 (2012)] applying only to odd dimension. Our approach is more general than previous ones based on mixtures of stabilizer states. Namely, all mixtures of stabilizer states can be efficiently simulated, but for any number of qubits there also exist efficiently simulable states outside the stabilizer polytope. Further, our simulation method extends to negative quasiprobability distributions, where it provides amplitude estimation. The simulation cost is then proportional to a robustness measure squared. For all quantum states, this robustness is smaller than or equal to stabilizer robustness.