# Computing quopit Clifford circuit amplitudes by the sum-over-paths technique

## TL;DR

This paper presents an alternative proof that Clifford circuits on odd prime-dimensional systems can be efficiently simulated by expressing their amplitudes as products of Weil sums, leveraging the sum-over-paths technique.

## Contribution

It introduces a novel proof for the efficient classical simulation of quopit Clifford circuits using the sum-over-paths approach and Weil sums, extending the Gottesman-Knill theorem.

## Key findings

- Amplitudes of quopit Clifford circuits can be computed efficiently.
- Sum-over-paths for these circuits reduces to Weil sums with quadratic polynomials.
- Provides a new method for simulating quantum circuits using polynomial and sum-over-paths techniques.

## Abstract

By the Gottesman-Knill Theorem, the outcome probabilities of Clifford circuits can be computed efficiently. We present an alternative proof of this result for quopit Clifford circuits (i.e., Clifford circuits on collections of $p$-level systems, where $p$ is an odd prime) using Feynman's sum-over-paths technique, which allows the amplitudes of arbitrary quantum circuits to be expressed in terms of a weighted sum over computational paths. For a general quantum circuit, the sum over paths contains an exponential number of terms, and no efficient classical algorithm is known that can compute the sum. For quopit Clifford circuits, however, we show that the sum over paths takes a special form: it can be expressed as a product of Weil sums with quadratic polynomials, which can be computed efficiently. This provides a method for computing the outcome probabilities and amplitudes of such circuits efficiently, and is an application of the circuit-polynomial correspondence which relates quantum circuits to low-degree polynomials.

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Source: https://tomesphere.com/paper/1702.03316