Virtual Qudits for Simon's Problem: Dimension-Lifted Algorithms on Qubit Hardware

TL;DR

This paper presents a method to simulate qudit-based Simon's algorithm on qubit hardware by constructing virtual qudits, enabling higher-dimensional quantum algorithms to run on existing qubit-based quantum computers.

Contribution

It introduces a general construction for simulating qudit versions of Simon's algorithm on qubit hardware using controlled permutations and phase operations.

Findings

Verified that the lifted circuit reproduces correct measurement statistics.

Analyzed depth overhead tradeoffs as a function of qudit dimension d.

Provided numerical simulations demonstrating the approach.

Abstract

Simon's problem admits an exponential quantum speedup, but current quantum devices support only qubits. This work introduces a general construction for simulating qudit versions of Simon's algorithm on qubit hardware by defining virtual qudits implemented through controlled permutations and qudit phase operations. We build a dimension lifted oracle that encodes the hidden shift in dimension d and show how to realize its action using only qubit gates. We mathematically verify that the lifted circuit reproduces the correct measurement statistics, analyze the depth overhead tradeoffs as a function of d, and provide numerical simulations in QuTiP for example values. Our approach demonstrates how higher-dimensional structures can be embedded into qubit devices and provides a general method for extending qudit algorithms to current hardware.