Almost Optimal Synthesis of Reversible Function in Qudit Model

TL;DR

This work studies reversible function synthesis in the qudit model, addressing both high-level decomposition and low-level gate construction. It proves that any even permutation in $A_{d^{n}}$ can be implemented using $\Theta(d)$ $(n-1)$-qudit subcircuits, via a canonical decomposition that reduces to edge- and plane-based operations on a $d^n$-dimensional cube, with strong results for the $n=3$ base case and a recursive generalization to all $n$. It further develops a bottom-up approach using 2-qudit gates, introducing $dC^{m}X$ and $dX^{(m)}$ gadgets to realize $n$-dit reversible functions with $O(n d^{n})$ gates and a single clean ancilla, and establishes a matching lower bound up to logarithmic factors. Together, these results yield asymptotically tight synthesis bounds in $d$ and near-optimal bounds in $n$, advancing practical reversible circuit design for high-dimensional quantum architectures.

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 $Θ(d)$ $(n - 1)$-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 $n$.

Figures (13)

• Figure 1: Visualization of the state space of $3$-qudit system
• Figure 2: Overview for Section \ref{['section:result01:cube']}
• Figure 3: Operations to apply $\sigma^{-1} \pi^{-1} \sigma \pi$
• Figure 4: Operations to apply $\sigma_{i}^{-1} \sigma_{\pi(i)}$
• Figure 5: Visualization of the state space of multi-qudit system

Theorems & Definitions (27)

• Definition 1
• Example 1
• Theorem 1: sunGeneralizedShuffleexchangeProblem2022
• Theorem 2: informal version of Theorem \ref{['thm:thm01']}
• Definition 2: type-1 permutation
• Definition 3: type-2 permutation
• Lemma 3
• Theorem 4
• Lemma 5
• Lemma 6