Positive Wigner functions render classical simulation of quantum computation efficient (1208.3660v2)

Abstract: 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 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 simulatable 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.

• The paper generalizes the Gottesman-Knill theorem, showing that circuits with positive Wigner functions can be simulated efficiently using a novel sampling algorithm.
• It employs a unified phase space framework applicable to both continuous- and discrete-variable systems, thereby broadening classical simulation methods.
• The study reveals that limited Wigner negativity constrains quantum advantage, offering insights for practical, error-tolerant quantum computation.

Efficient Classical Simulation of Quantum Circuits via Positive Wigner Functions

The paper "Positive Wigner functions render classical simulation of quantum computation efficient" by A. Mari and J. Eisert addresses a critical challenge in the field of quantum computation: delineating the boundary between quantum circuits that can be efficiently simulated classically and those that cannot. The authors extend these boundaries using the concept of the Wigner function, a well-known quasi-probability distribution in quantum mechanics.

Summary of the Findings

The central result of the paper is the demonstration that quantum circuits, which can be entirely described using positive Wigner functions, are amenable to efficient classical simulation. This finding holds for both continuous-variable systems and discrete-variable systems with a local dimension that is an odd prime. The approach hinges on deriving an explicit sampling algorithm that respects the positivity of Wigner functions, thereby generalizing the Gottesman-Knill theorem.

Key Contributions:

• Generalization of the Gottesman-Knill Theorem: The paper broadens the Gottesman-Knill theorem from stabilizer circuits with Clifford gates to a broader class that includes continuous-variable systems and discrete-variable systems with positive Wigner functions. This encompasses systems that might be only partially described by stabilizer states or Gaussian states, thanks to the algorithm's robustness to certain positive Wigner function representations.
• Phase Space Approach: By utilizing phase space representations for both discrete and continuous systems, the authors construct an elegant theoretical framework. The same mathematical expressions apply, highlighting the universality of their approach across different quantum systems.
• Efficient Algorithm: A sampling algorithm is developed based on classical stochastic processes, capable of drawing samples from the probability distributions defined by the quantum circuits. The procedure is polynomial in the number of constituents and the number of gates, showcasing the practical efficiency in classical simulation for quantum circuits with positive Wigner functions.
• Robustness to Errors: The paper includes a discussion on the robustness of their procedures to errors in sampling due to positive Wigner function approximations. It reveals that small deviations from the ideal Wigner functions accumulate only linearly with the depth of the circuit, allowing for practical applications even with slight imperfections.

Implications for Quantum Computing

The implications of this work are substantial for both the theoretical understanding of quantum computational complexity and practical quantum algorithm development:

• Resource Theory of Quantum Negativity: The negativity of the Wigner function is underscored as a resource necessary for quantum computational advantage. This parallels the use of entanglement and other quantum resources, motivating further analysis within a resource-theoretic framework.
• Boundary Identification: By determining the conditions under which quantum operations remain classically simulatable, this research provides clearer boundaries between quantum systems that offer true quantum advantage and those effectively classical in nature.
• Universal Computing Considerations: The research asserts that although negative Wigner functions are vital for surpassing classical capabilities, these can be diluted in various components without fully forfeiting universal quantum computational power. This poses interesting avenues for constructing practically realizable quantum protocols with limited non-classical resources.

Future Prospects

The paper opens several directions for future research, including:

• Extending these findings to systems with more complex configurations and expanding beyond initial product states.
• Exploring the quantification of negativity as a computational resource, akin to measures of entanglement.
• Investigating potential applications in error-correction and fault-tolerant quantum computing where Wigner function considerations could aid in resource-efficient problem-solving.

This paper represents a pivotal step in understanding the interplay between quantum characteristics and classical simulations, paving the way for more nuanced quantum mechanical and computational insights.