Positive Wigner functions render classical simulation of quantum computation efficient.

By Andrea Mari, Jens Eisert. Physical Review Letters, 2012

421 citations recorded by the source corpus.

We show that quantum circuits where the initial state and all the following quantum operations can be represented by positive Wigner functions can be classically efficiently simulated. This is true both for continuous-variable as well as discrete variable systems in odd prime dimensions, two cases which will be treated on entirely the same footing. Noting the fact that Clifford and Gaussian operations preserve the positivity of the Wigner function, our result generalizes the Gottesman-Knill theorem. Our algorithm provides a way of sampling from the output distribution of a computation or a simulation, including the efficient sampling from an approximate output distribution in the case of sampling imperfections for initial states, gates, or measurements. In this sense, this work highlights the role of the positive Wigner function as separating classically efficiently simulable systems from those that are potentially universal for quantum computing and simulation, and it emphasizes the role of negativity of the Wigner function as a computational resource.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/b8242871dfe9de5aa9ae6b1950b8f40ba96e5af6

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Citations in the corpus

Contextuality supplies the ‘magic’ for quantum computation (2014, neutral): “The independence number of the exclusivity graph associated with Σ~r is α(Γ~r) = p3 for all A~r ∈ ASIM so that, relative to our construction, exactly the states ρ /∈ PSIM are those that exhibit contextuality.”
Negative quasi-probability as a resource for quantum computation (2012, methodology): “After the completion of this work we became aware of a complementary work by Eisert and Mari (2012) which develops an alternative efficient simulation for systems with positively represented Wigner function in odd and infinite dimensions.”
Modes and states in quantum optics (2019, background): “, 2002), a property which can be extended to any positive Wigner function state (Mari and Eisert, 2012).”
The resource theory of stabilizer quantum computation (2013, methodology): “Some progress on this front has already been made: it has been shown that linear optics operations acting on states with positive Wigner function, which includes non-Gaussian states, is efficiently classically simulable[35, 46].”
Sufficient Conditions for Efficient Classical Simulation of Quantum Optics (2015, background): “In addition, for s 1⁄4 t 1⁄4 0, we have that when the transition function is non-negative and the input quantum state and output POVM elements have nonnegative Wigner functions, sampling from the output distribution can be simulated classically [2].”