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Distributed Quantum Discrete Logarithm Algorithm

Renjie Xu, Daowen Qiu, Ligang Xiao, Le Luo, Xu Zhou

Abstract

Solving the discrete logarithm problem (DLP) with quantum computers is a fundamental task with important implications. Beyond Shor's algorithm, many researchers have proposed alternative solutions in recent years. However, due to current hardware limitations, the scale of DLP instances that can be addressed by quantum computers remains insufficient. To overcome this limitation, we propose a distributed quantum discrete logarithm algorithm that reduces the required quantum register size for solving DLPs. Specifically, we design a distributed quantum algorithm to determine whether the solution is contained in a given set. Based on this procedure, our method solves DLPs by identifying the intersection of sets containing the solution. Compared with Shor's original algorithm, our approach reduces the register size and can improve the success probability, while requiring no quantum communication.

Distributed Quantum Discrete Logarithm Algorithm

Abstract

Solving the discrete logarithm problem (DLP) with quantum computers is a fundamental task with important implications. Beyond Shor's algorithm, many researchers have proposed alternative solutions in recent years. However, due to current hardware limitations, the scale of DLP instances that can be addressed by quantum computers remains insufficient. To overcome this limitation, we propose a distributed quantum discrete logarithm algorithm that reduces the required quantum register size for solving DLPs. Specifically, we design a distributed quantum algorithm to determine whether the solution is contained in a given set. Based on this procedure, our method solves DLPs by identifying the intersection of sets containing the solution. Compared with Shor's original algorithm, our approach reduces the register size and can improve the success probability, while requiring no quantum communication.

Paper Structure

This paper contains 17 sections, 12 theorems, 85 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Our Algorithm
  4. Quantum gates
  5. The superposition of eigenvectors
  6. Details of our algorithm
  7. Probability of observing $0^{m-1}1$ from the third register
  8. Strategy for searching solution
  9. Additional Analysis
  10. Algorithm
  11. Success probability
  12. The implementation of $\Gamma_{\tau}(M_{a})$
  13. Complexity
  14. Time and space complexity
  15. Communication complexity
  16. ...and 2 more sections

Key Result

Lemma 1

Given set $S_{n,\tau} = \{\tau+s \pmod{r}| s= 0,1,2,...,2^n-1 \}$ where $2^n<r$ and $\tau$ is integer. Then, if $t \in S_{n,\tau}$, Otherwise, if $t \notin S_{n,\tau}$, $\sum_{s \in S_{n,\tau}} \sum_{l=0}^{r-1} (-1)^s g((t-s)l) |\psi_l\rangle =0$.

Figures (5)

  • Figure 1: Original quantum discrete logarithm algorithm.
  • Figure 2: The total process of our Algorithm \ref{['DQDLA']}.
  • Figure 3: The frequency of the solutions ($t=23$) found by Algorithm \ref{['algorithm_strategy']} for $100$ times.
  • Figure 4: The circuit constructed via Pennylane.
  • Figure 5: The probability of observing different strings from the third register when $t \not\in S_{n,r}$ (upper figure) and $t\in S_{n,r}$ (lower figure).

Theorems & Definitions (25)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 15 more