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.

翻译：西蒙问题展现出指数级的量子加速优势，但当前量子设备仅支持量子比特。本研究提出了一种通用构造方法，通过定义由受控置换和量子比特相位操作实现的虚拟量子比特，在量子比特硬件上模拟西蒙算法的量子比特版本。我们构建了一个维度提升的预言机，用于在维度d中编码隐藏位移，并展示了如何仅使用量子比特门实现其作用。我们通过数学验证了提升后的电路能够复现正确的测量统计特性，分析了深度开销随d变化的权衡关系，并利用QuTiP对示例值进行了数值模拟。我们的方法展示了如何将高维结构嵌入量子比特设备，并为将量子比特算法扩展至当前硬件提供了一种通用途径。