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Distributed Quantum Amplitude Amplification

Ximing Hua, Daowen Qiu

TL;DR

The paper addresses scalable search on NISQ devices by introducing distributed quantum amplitude amplification (DQAA), which partitions a global amplitude amplification into sub-functions via a tensor-product circuit $\mathcal{A}=\mathcal{A}_1\otimes\mathcal{A}_2$. Each sub-function $f_k$ has its own initial probability $a_k$, and at least one satisfies $a_k\ge a$, enabling the use of fixed-point quantum amplitude amplification to reach an overall success probability of at least $1-\varepsilon^2$ with $l\ge\left\lceil\frac{\ln(2/\varepsilon)}{2\sqrt{a}}\right\rceil$ iterations; the total query complexity is $O\left(\frac{\log(2/\varepsilon)}{2\sqrt{a}}\right)$. The authors define local operators $Q_k(\phi,\varphi)$ for each sub-function and prove correctness via a distributed framework, supplemented by Qiskit simulations demonstrating qubit-reduction advantages over prior approaches. They compare their method to distributed Grover and standard QAA, highlighting that no quantum communication is needed between processors in their scheme. The work advances practical quantum search on near-term devices and extends distributed quantum computing by exploiting sub-function partitioning to balance resources and success probability.

Abstract

Quantum amplitude amplification algorithm is an important and basic technique in quantum computing. In this paper, our goal is to study distributed quantum amplitude amplification algorithms, and the main contributions are: (1) A distributed quantum amplitude amplification algorithm is proposed. (2) We simulate the proposed algorithm in a particular situation by Qiskit. (3) Compared to other related works, our algorithm has certain advantages concerning the number of qubits.

Distributed Quantum Amplitude Amplification

TL;DR

The paper addresses scalable search on NISQ devices by introducing distributed quantum amplitude amplification (DQAA), which partitions a global amplitude amplification into sub-functions via a tensor-product circuit . Each sub-function has its own initial probability , and at least one satisfies , enabling the use of fixed-point quantum amplitude amplification to reach an overall success probability of at least with iterations; the total query complexity is . The authors define local operators for each sub-function and prove correctness via a distributed framework, supplemented by Qiskit simulations demonstrating qubit-reduction advantages over prior approaches. They compare their method to distributed Grover and standard QAA, highlighting that no quantum communication is needed between processors in their scheme. The work advances practical quantum search on near-term devices and extends distributed quantum computing by exploiting sub-function partitioning to balance resources and success probability.

Abstract

Quantum amplitude amplification algorithm is an important and basic technique in quantum computing. In this paper, our goal is to study distributed quantum amplitude amplification algorithms, and the main contributions are: (1) A distributed quantum amplitude amplification algorithm is proposed. (2) We simulate the proposed algorithm in a particular situation by Qiskit. (3) Compared to other related works, our algorithm has certain advantages concerning the number of qubits.
Paper Structure (11 sections, 4 theorems, 29 equations, 4 figures)

This paper contains 11 sections, 4 theorems, 29 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum Amplitude Amplification
  4. Fixed-point Quantum Amplitude Amplification
  5. Distributed Quantum Amplitude Amplification
  6. Sub-functions
  7. Distributed Quantum Amplitude Amplification algorithm
  8. Simulation and Comparison
  9. Simulation
  10. Comparison
  11. Conclusions

Key Result

Theorem 1

Let $\mathcal{A}$ be any quantum algorithm that uses no measurements, and let a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$. Suppose $a>0$, and set $m=\lfloor\frac{\pi}{4\arcsin(\sqrt{a})}\rfloor$, where $0<\arcsin(\sqrt{a})<\pi/2$. If we measure the quantum register in step 3 of Algorithm QAA

Figures (4)

  • Figure 1: Quantum amplitude amplification.
  • Figure 2: Distributed quantum amplitude amplification algorithm.
  • Figure 3: The circuit of an iteration in Algorithm \ref{['DQAFPQS']}.
  • Figure 4: Result of distributed Grover's algorithm in each quantum computer with $\varepsilon=0.3$ and $\mathcal{A}_1=H^{\otimes 2}$. The target states are $\{110110,111111,011001\}$

Theorems & Definitions (12)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Remark 4
  • Remark 5
  • Theorem 3
  • proof
  • Remark 6
  • ...and 2 more