Semiclassical Formulation of Gottesman-Knill and Universal Quantum Computation

TL;DR

This paper develops a semiclassical path integral framework for qudit quantum systems, revealing how Clifford operations are classical and universal gates introduce quantum complexity.

Contribution

It introduces a path integral formulation for qudit evolution, showing Clifford operations are classical and universal gates require a semiclassical expansion with exponentially scaling complexity.

Findings

Clifford operations are fully described at order , indicating classicality.

Universal gates introduce  order contributions, reflecting quantum complexity.

The number of semiclassical terms scales exponentially with qudit number.

Abstract

We give a path integral formulation of the time evolution of qudits of odd dimension. This allows us to consider semiclassical evolution of discrete systems in terms of an expansion of the propagator in powers of $\hbar$. The largest power of $\hbar$ required to describe the evolution is a traditional measure of classicality. We show that the action of the Clifford operators on stabilizer states can be fully described by a single contribution of a path integral truncated at order $\hbar^0$ and so are "classical," just like propagation of Gaussians under harmonic Hamiltonians in the continuous case. Such operations have no dependence on phase or quantum interference. Conversely, we show that supplementing the Clifford group with gates necessary for universal quantum computation results in a propagator consisting of a finite number of semiclassical path integral contributions truncated at order $\hbar^1$ , a number that nevertheless scales exponentially with the number of qudits. The same sum in continuous systems has an infinite number of terms at order $\hbar^1$.