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

Abstract

We apply the periodized stationary phase method to discrete Wigner functions of systems with odd prime dimension using results from p-adic number theory. We derive the Wigner-Weyl-Moyal (WWM) formalism with higher order corrections representing contextual corrections to noncontextual Clifford operations. We apply this formalism to a subset of unitaries that include diagonal gates such as the π 8 gates. We characterize the stationary phase critical points as a quantum resource injecting contextuality and show that this resource allows for the replacement of the p 2t points that represent t magic state Wigner functions on p-dimensional qudits by ≤ p t points. We find that the π 8 gate introduces the smallest higher order correction possible, requiring the lowest number of additional critical points compared to the Clifford gates. We then establish a relationship between the stabilizer rank of states and the number of critical points necessary to treat them in the WWM formalism. This allows us to exploit the stabilizer rank decomposition of two qutrit π 8 gates to develop a classical strong simulation of a single qutrit marginal on t qutrit π 8 gates that are followed by Clifford evolution, and show that this only requires calculating 3 t 2 +1 critical points corresponding to Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as ∼3 0.8t for 10 −2 precision) for any number of π 8 gates to full precision.

Improved Simulation of Quantum Circuits by Fewer Gaussian Eliminations

“…In this direction, we previously showed that odd-primed dimensional qudit T gate magic states have the same stabilizer rank for t = 1 and t = 2 as has been found for qubits up to the exponential base factor-2 αt ↔ d αt , i.e. (χ 1 ) t = d t and (χ 2 ) t = d 0.5t [9]. In fact, we proved that stabilizer decompositions that achieve these stabilizer ranks for t = 1 and t = 2 have a one-to-one correspondence with the quadratic Gauss sums that decompose the T gate magic state's discrete Wigner function, which are operationally and cost-wise equivalent to the stabilizer rank.…”

“…see Eq. 5) [9]. Hence, ξ 1 = ξ 2 = 3, which leads to tensor bounds of ξ t ≤ (ξ 1 ) t = 3 t and ξ t ≤ (ξ 2 ) t/2 = t 0.5t , for even t.…”

“…for some h ∈ Z + and (x) i is the ith element of x. The Clifford sequence ÛC changes the restriction of the domain of the sum from x n+1 ≡ x q1 = 0 to D. We showed previously [9] that…”

“…sums require O(k 3 ) computations to evaluate for k qudits; their absolute value depends on the determinant of their covariance matrix and so they are governed by the cost of Gaussian elimination of a matrix of size k × k with entries in Z/dZ [9]. Importantly, in order to calculate stabilizer state inner products, a "tableau" matrix which has the same properties must also undergo Gaussian elimination [8].…”

“…More precisely, the overall cost of evaluating Eq. 2 scales as O(ξ k ) [9]. Since the product of two Wigner functions on separate qudits is also a Wigner function, ξ k satisfies the same trivial tensor bound property as χ k : ξ t ≤ (ξ k ) t/k for t a multiple of k. Therefore, calculating Eq.…”