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Distributed Exact Quantum Amplitude Amplification Algorithm for Arbitrary Quantum States

Xu Zhou, Wenxuan Tao, Keren Li, Shenggen Zheng

TL;DR

DEQAAA tackles exact amplitude amplification for quantum states with arbitrary amplitudes in a distributed quantum setting. It introduces a two-phase framework: local exact amplification on $t$ nodes and a global exact amplification, backed by substates that map global amplitudes to local subspaces. The method enables exact amplification for multiple targets and delivers substantial resource savings through $C^{n-1}PS$ gate decomposition, achieving >97% reductions in gate count and circuit depth at larger qubit counts. This work demonstrates a scalable, noise-resilient path for practical distributed quantum computation on NISQ devices and outlines directions for further optimization and decomposition strategies.

Abstract

In the noisy intermediate-scale quantum (NISQ) era, distributed quantum computation has garnered considerable interest, as it overcomes the physical limitations of single-device architectures and enables scalable quantum information processing. In this study, we focus on the challenge of achieving exact amplitude amplification for quantum states with arbitrary amplitude distributions and subsequently propose a Distributed Exact Quantum Amplitude Amplification Algorithm (DEQAAA). Specifically, (1) it supports partitioning across any number of nodes $t$ within the range $2 \leq t \leq n$; (2) the maximum qubit count required for any single node is expressed as $\max \left(n_0,n_1,\dots,n_{t-1} \right) $, where $n_j$ represents the number of qubits at the $j$-th node, with $\sum_{j=0}^{t-1} n_j =n$; (3) it can realize exact amplitude amplification for multiple targets of a quantum state with arbitrary amplitude distributions; (4) we verify the effectiveness of DEQAAA by resolving a specific exact amplitude amplification task involving two targets (8 and 14 in decimal) via MindSpore Quantum, a quantum simulation software, with tests conducted on 4-qubit, 6-qubit, 8-qubit and 10-qubit systems. Notably, through the decomposition of $C^{n-1}PS$ gates, DEQAAA demonstrates remarkable advantages in both quantum gate count and circuit depth as the qubit number scales, thereby boosting its noise resilience. In the 10-qubit scenario, for instance, it achieves a reduction of over $97\%$ in both indicators compared to QAAA and EQAAA, underscoring its outstanding resource-saving performance.

Distributed Exact Quantum Amplitude Amplification Algorithm for Arbitrary Quantum States

TL;DR

DEQAAA tackles exact amplitude amplification for quantum states with arbitrary amplitudes in a distributed quantum setting. It introduces a two-phase framework: local exact amplification on nodes and a global exact amplification, backed by substates that map global amplitudes to local subspaces. The method enables exact amplification for multiple targets and delivers substantial resource savings through gate decomposition, achieving >97% reductions in gate count and circuit depth at larger qubit counts. This work demonstrates a scalable, noise-resilient path for practical distributed quantum computation on NISQ devices and outlines directions for further optimization and decomposition strategies.

Abstract

In the noisy intermediate-scale quantum (NISQ) era, distributed quantum computation has garnered considerable interest, as it overcomes the physical limitations of single-device architectures and enables scalable quantum information processing. In this study, we focus on the challenge of achieving exact amplitude amplification for quantum states with arbitrary amplitude distributions and subsequently propose a Distributed Exact Quantum Amplitude Amplification Algorithm (DEQAAA). Specifically, (1) it supports partitioning across any number of nodes within the range ; (2) the maximum qubit count required for any single node is expressed as , where represents the number of qubits at the -th node, with ; (3) it can realize exact amplitude amplification for multiple targets of a quantum state with arbitrary amplitude distributions; (4) we verify the effectiveness of DEQAAA by resolving a specific exact amplitude amplification task involving two targets (8 and 14 in decimal) via MindSpore Quantum, a quantum simulation software, with tests conducted on 4-qubit, 6-qubit, 8-qubit and 10-qubit systems. Notably, through the decomposition of gates, DEQAAA demonstrates remarkable advantages in both quantum gate count and circuit depth as the qubit number scales, thereby boosting its noise resilience. In the 10-qubit scenario, for instance, it achieves a reduction of over in both indicators compared to QAAA and EQAAA, underscoring its outstanding resource-saving performance.
Paper Structure (25 sections, 5 theorems, 87 equations, 21 figures, 4 tables)

This paper contains 25 sections, 5 theorems, 87 equations, 21 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Quantum Amplitude Amplification Algorithm
  4. Exact Quantum Amplitude Amplification Algorithm
  5. Distributed Exact Quantum Amplitude Amplification Algorithm
  6. Substates
  7. DEQAAA with $2\leq t\leq n$ nodes
  8. Analysis
  9. Correctness
  10. Circuit depth
  11. Comparison
  12. Experiment
  13. The Probability Distribution
  14. The 4-qubit QAAA with $X_g=\{8, 14\}$
  15. The 4-qubit EQAAA with $X_g=\{8, 14\}$
  16. ...and 10 more sections

Key Result

Theorem 1

(Correctness of DEQAAA) Let $\vert\Psi\rangle = \mathcal{A}\vert0\rangle^{\otimes n}$ be an $n$-qubit state with an arbitrary amplitude distribution, where $\mathcal{A}$ denotes a unitary state-preparation operator. Let $f: \{0,1\}^n \rightarrow \{0,1\}$ be a Boolean function that identifies the tar which guarantees that a measurement of $\vert\Psi_2\rangle$ yields a target string $x \in X_g$ with

Figures (21)

  • Figure 1: Quantum circuit of the QAAA.
  • Figure 2: Quantum circuit of the EQAAA.
  • Figure 3: Quantum circuit of the DEQAAA.
  • Figure 4: Two quantum circuits implementing the SWAP gate.
  • Figure 5: The quantum circuits corresponding to $S_f$ and $S_{\vert 0 \rangle ^ {\otimes n}}$ operators.
  • ...and 16 more figures

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Definition 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Lemma G1
  • ...and 1 more