Virtual Qudits for Simon's Problem: Dimension-Lifted Algorithms on Qubit Hardware
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.
Cite
@article{arxiv.2512.06756,
title = {Virtual Qudits for Simon's Problem: Dimension-Lifted Algorithms on Qubit Hardware},
author = {Abed Semre and Steven Frankel},
journal= {arXiv preprint arXiv:2512.06756},
year = {2025}
}