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

Abed Semre, Steven Frankel

TL;DR

The paper develops a dimension-lifted framework for Simon’s problem by introducing virtual qudits and a lifted d-to-one oracle implementable on qubit hardware via packing/unpacking of qubits. It formalizes the problem over Z_d^n, derives the uniform measurement distribution over S^⊥, and analyzes how increasing the local dimension d reduces the required repetition budget while preserving the same O(n) oracle-call scaling. The approach is validated through QuTiP simulations for dimensions {2,3,4}, and a concrete qubit-to-qudit lift demonstrates how to realize qudit algorithms using only a binary Simon oracle Uf. This work provides a practical blueprint for exploring qudit-style advantages on existing hardware, enabling dimension-based performance studies without native multilevel devices.

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.

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

TL;DR

The paper develops a dimension-lifted framework for Simon’s problem by introducing virtual qudits and a lifted d-to-one oracle implementable on qubit hardware via packing/unpacking of qubits. It formalizes the problem over Z_d^n, derives the uniform measurement distribution over S^⊥, and analyzes how increasing the local dimension d reduces the required repetition budget while preserving the same O(n) oracle-call scaling. The approach is validated through QuTiP simulations for dimensions {2,3,4}, and a concrete qubit-to-qudit lift demonstrates how to realize qudit algorithms using only a binary Simon oracle Uf. This work provides a practical blueprint for exploring qudit-style advantages on existing hardware, enabling dimension-based performance studies without native multilevel devices.

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.

Paper Structure

This paper contains 36 sections, 1 theorem, 50 equations, 4 figures, 1 table.

Table of Contents

  1. Background and Motivation
  2. From Qubits to Qudits
  3. Notation and assumptions.
  4. Why Generalize Simon's Algorithm?
  5. Goals and Contributions of This Work
  6. Quantum Gates and Operations for Qudit Based Algorithms
  7. States
  8. Gates
  9. X gate
  10. H gate
  11. Generalized Simon’s Problem over Qudits
  12. Complexity
  13. Classical complexity
  14. Coset decomposition
  15. Quantum algorithm
  16. ...and 21 more sections

Key Result

Lemma 6.1

For all $\eta\in\mathbb{Z}_d^n$ and all $(\delta_0,\dots,\delta_{\ell-1})\in\{0,1\}^{\ell}$, Consequently, $f_{(d)}$ is $d$-to-$1$, with each fiber equal to the orbit $\{\eta \;\oplus_{\mathrm{layers}}\; \delta\!\cdot\! s:\; \delta\in\{0,1\}^{\ell}\}$.

Figures (4)

  • Figure 1: Asymptotic repetitions $k$ required for $P_{\mathrm{fail}}\le 1/3$ as a function of $d$: $f(d)=\frac{\log 3}{2\log d - \log(d+1)}$.
  • Figure 2: Effect of decreasing $\epsilon$ on the surface $f(d,n,\epsilon)$. Lower values of $\epsilon$ increase the required repetition count, while larger dimensions $d$ reduce it, shifting the feasible region toward smaller $k$ as $d$ grows.
  • Figure 3: Empirical measurement distribution for $d=4$, $n=4$: samples are uniformly distributed over the orthogonal subspace $S^{\perp}$ containing $4^3 = 64$ outcomes.
  • Figure 4: Schematic representation of the lifted oracle construction. Each horizontal wire corresponds to a physical qubit $\ket{\psi}^{(k)}_{j}$, where $k \in \{0,\ldots,\ell-1\}$ indexes the layer within a logical qudit and $j \in \{0,\ldots,n-1\}$ indexes the logical qudit position. For each fixed $j$, the $\ell$ wires detour through a common oracle block $U_f$, indicating that all $\ell$ physical qubits jointly encode a single $d = 2^\ell$-dimensional qudit.

Theorems & Definitions (2)

  • Lemma 6.1: Oracle invariance for the lifted construction
  • proof