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Distributed quantum approximate counting algorithm

Huaijing Huang, Daowen Qiu

TL;DR

The paper tackles quantum counting under qubit-constrained, noisy conditions by introducing a distributed DIQC framework that combines Grover-based amplitude estimation with classical post-processing. Each node runs a localized MIQAE-inspired procedure to estimate partial counts, and a classical aggregator sums these to yield a global count with provable accuracy and confidence bounds. The method is applied to inner product and Hamming distance tasks, with Qiskit-based simulations demonstrating reduced circuit depth and robust performance compared to MIQAE and traditional quantum counting. Overall, the work offers a scalable, parallelizable approach suitable for NISQ-era devices and provides a path toward tighter theoretical bounds and adaptive amplification strategies.

Abstract

In this article, we propose a distributed quantum algorithm for solving counting problem using Grover operator and a classical post-processing procedure. We apply the proposed algorithm to estimate inner products and Hamming distances. Simulations are conducted on the Qisikit platform, further demonstrating the effectiveness of our algorithm and its suitability for the NISQ era. Compared to existing counting algorithms, the proposed algorithm has advantages in terms of the number of qubits, circuit depth, and the number of quantum gates.

Distributed quantum approximate counting algorithm

TL;DR

The paper tackles quantum counting under qubit-constrained, noisy conditions by introducing a distributed DIQC framework that combines Grover-based amplitude estimation with classical post-processing. Each node runs a localized MIQAE-inspired procedure to estimate partial counts, and a classical aggregator sums these to yield a global count with provable accuracy and confidence bounds. The method is applied to inner product and Hamming distance tasks, with Qiskit-based simulations demonstrating reduced circuit depth and robust performance compared to MIQAE and traditional quantum counting. Overall, the work offers a scalable, parallelizable approach suitable for NISQ-era devices and provides a path toward tighter theoretical bounds and adaptive amplification strategies.

Abstract

In this article, we propose a distributed quantum algorithm for solving counting problem using Grover operator and a classical post-processing procedure. We apply the proposed algorithm to estimate inner products and Hamming distances. Simulations are conducted on the Qisikit platform, further demonstrating the effectiveness of our algorithm and its suitability for the NISQ era. Compared to existing counting algorithms, the proposed algorithm has advantages in terms of the number of qubits, circuit depth, and the number of quantum gates.

Paper Structure

This paper contains 9 sections, 8 theorems, 91 equations, 9 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The modified iterative quantum amplitude estimation algorithm
  4. The distributed quantum approximate counting algorithm
  5. Correctness analysis
  6. Comparison to related works
  7. Application
  8. Numerical experiment
  9. Conclusion

Key Result

Theorem 1

Given a confidence level $1-\alpha \in (0, 1)$, a target accuracy $\epsilon > 0$, and an $(n + 1)$-qubit unitary $\mathcal{A}$ satisfying where $\ket{\psi_0}$ and $\ket{\psi_1}$ are $n$-qubit states and $a \in [0, 1]$, MIQAE algorithm outputs a confidence interval for $a$ that satisfies where $a_u - a_l < 2\epsilon$, leading to an estimate $\hat{a}=\frac{a_l+a_u}{2}$ for $a$ such that $|a - \hat

Figures (9)

  • Figure 1: The circuit for $Q_j$ of DIQC algorithm.
  • Figure 2: The circuit of quantum counting algorithm in Ref.brassard2000quantum.
  • Figure 3: The circuit for $\mathcal{A}_0$ in DIQC algorithm.
  • Figure 4: The circuit for $\mathcal{A}_1$ in DIQC algorithm.
  • Figure 5: The circuit for $Q_0$ in DIQC algorithm.
  • ...and 4 more figures

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • proof
  • proof
  • ...and 7 more