Almost Optimal Synthesis of Reversible Function in Qudit Model

TL;DR

This paper presents an asymptotically optimal method for synthesizing reversible functions in the qudit model, leveraging multi-level quantum systems to improve efficiency over previous qubit-focused approaches.

Contribution

It introduces a novel synthesis technique for reversible functions in the qudit model, achieving asymptotic optimality in sub-circuit count and gate complexity.

Findings

Synthesizes even permutations in $A_{d^{n}}$ with $ heta(d)$ sub-circuits.

Uses $O(n d^{n})$ gates with only one ancilla, asymptotically tight.

Achieves asymptotic optimality in the count of sub-circuits.

Abstract

Quantum oracles are widely adopted in problems, like query oracle in Grover's algorithm, cipher in quantum cryptanalytic and data encoder in quantum machine learning. Notably, the bit-flip oracle, capable of flipping the state based on a given classical function, emerges as a fundamental component in the design and construction of quantum algorithms. Devising methods to optimally implement the bit-flip oracle essentially translates to the efficient synthesis of reversible functions. Prior research has primarily focused on the qubit model, leaving the higher dimensional systems, i.e. qudit model, largely unexplored. By allowing more than two computational bases, qudit model can fully utilize the multi-level nature of the underlying physical mechanism. We propose a method to synthesize even permutations in $A_{d^{n}}$ using $\Theta(d)$ $(n - 1)$-qudit sub-circuits, which achieve asymptotic optimality in the count of sub-circuits. Moreover, we introduce a technique for synthesizing reversible functions employing $O\left( n d^{n} \right)$ gates and only a single ancilla. This is asymptotically tight in terms of $d$ and asymptotically almost tight in terms of $n$.