Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits

Abstract

One of the lowest-order corrections to Gaussian quantum mechanics in infinite-dimensional Hilbert spaces are Airy functions: a uniformization of the stationary phase method applied in the path integral perspective. We introduce a "periodized stationary phase method" to discrete Wigner functions of systems with odd prime dimension and show that the $\frac{\pi }{8}$ gate is the discrete analog of the Airy function. We then establish a relationship between the stabilizer rank of states and the number of quadratic Gauss sums necessary in the periodized stationary phase method. This allows us to develop a classical strong simulation of a single qutrit marginal on $t$ qutrit $\frac{\pi }{8}$ gates that are followed by Clifford evolution, and show that this only requires $3^{\frac{t}{2}+1}$ quadratic Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as $\sim 3^{0.8t}$ for $10^{-2}$ precision) for any number of $\frac{\pi }{8}$ gates to full precision.

arXiv commentQuantum Journal version

References in corpus (29)

• Quantum Resource Theories
• Positive Wigner functions render classical simulation of quantum computation efficient
• Application of a resource theory for magic states to fault-tolerant quantum computing
• Negative Quasi-Probability as a Resource for Quantum Computation
• Hudson's Theorem for finite-dimensional quantum systems
• Improved classical simulation of quantum circuits dominated by Clifford gates
• Negativity and contextuality are equivalent notions of nonclassicality
• Trading classical and quantum computational resources
• Simulation of quantum circuits by low-rank stabilizer decompositions
• Resource theory of quantum non-Gaussianity and Wigner negativity
• Estimating outcome probabilities of quantum circuits using quasiprobabilities
• Convex resource theory of non-Gaussianity
• Magic state distillation in all prime dimensions using quantum Reed-Muller codes
• The contextual fraction as a measure of contextuality
• Wigner function negativity and contextuality in quantum computation on rebits
• Qudit versions of the qubit "pi-over-eight" gate
• Framed Hilbert space: hanging the quasi-probability pictures of quantum theory
• Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation
• Resource theory of non-Gaussian operations
• The Weyl representation on the torus
• Anticoncentration theorems for schemes showing a quantum speedup
• Quantum Commuting Circuits and Complexity of Ising Partition Functions
• Resource theory of contextuality for arbitrary prepare-and-measure experiments
• Discrete Wigner Formalism for Qubits and Non-Contextuality of Clifford Gates on Qubit Stabilizer States
• Approximate stabilizer rank and improved weak simulation of Clifford-dominated circuits for qudits
• Discrete Wigner Function Derivation of the Aaronson-Gottesman Tableau Algorithm
• Semiclassical Formulation of Gottesman-Knill and Universal Quantum Computation
• Measurement Contextuality and Planck's Constant
• The Non-Disjoint Ontic States of the Grassmann Ontological Model, Transformation Contextuality, and the Single Qubit Stabilizer Subtheory

Cited by in corpus (5)

• LIMDD: A Decision Diagram for Simulation of Quantum Computing Including Stabilizer States
• Classical simulation of quantum circuits by half Gauss sums
• Improved Simulation of Quantum Circuits by Fewer Gaussian Eliminations
• Improved Strong Simulation of Universal Quantum Circuits
• Noncontextual Pauli Hamiltonians