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REVIEW 3 major objections 5 minor 50 references

A distributed two-phase algorithm claims to amplify any set of target states in an arbitrary n-qubit state to a success probability of exactly 1, using between 2 and n small nodes.

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2026-08-03 10:42 UTC pith:J6HWBOTJ

load-bearing objection Exactness is real but inherited from EQAAA; the advertised resource savings are a promise on unproven Phase-1 p'_g improvement. the 3 major comments →

arxiv 2601.09128 v1 pith:J6HWBOTJ submitted 2026-01-14 quant-ph

Distributed Exact Quantum Amplitude Amplification Algorithm for Arbitrary Quantum States

Xu Zhou , Wenxuan Tao , Keren Li , Shenggen Zheng This is my paper
classification quant-ph MSC 81P68
keywords quantum amplitude amplificationdistributed quantum computingexact quantum searchmulti-target amplificationNISQ-era algorithmsmulti-controlled phase gatescircuit depth reduction
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper sets out to show that exact amplitude amplification — raising the total probability of a chosen set of basis states to 100 percent — can be achieved in a distributed setting for any initial state, not just states with tensor-product structure or a single target. The proposed DEQAAA splits an n-qubit state across t nodes, runs an exact local amplification on each node's marginal substate, then applies one global exact amplification step on the composite unitary. If the argument holds, the largest qubit count on any one device drops from n to max(n_j), while total gate count and circuit depth fall sharply; the paper reports reductions above 97 percent in a 10-qubit example. The authors also state the load-bearing caveat: how much the first phase improves the target probability is not proven, and the practical depth advantage depends on that improvement.

Core claim

The central claim is Theorem 1: for any n-qubit state |Ψ⟩ = A|0⟩^{⊗n}, any Boolean target function f, and any partition into t nodes with 2 ≤ t ≤ n, Algorithm 4 produces a final state |Ψ₂⟩ whose total target probability Σ_{x∈X_g}|⟨x|Ψ₂⟩|² equals 1. The construction composes local exact amplitude amplification operators EQ_j — defined on node substates |φ_j⟩ built from the marginal distributions of the global probability distribution — and then wraps the whole first phase into a single unitary B, on which a global exact operator dEQ performs a standard exact rotation using the updated target probability p'_g. The proof treats the composition as one unitary, so the theorem's formal correctness

What carries the argument

The engine is the exact amplitude amplification operator EQ — a rotation in the two-dimensional subspace spanned by the target and non-target components — parameterized by a phase angle ϕ that, after J+1 applications, aligns the state with the target subspace with probability 1. DEQAAA reuses this operator twice: locally (EQ_j) on each node's substate, and globally (dEQ) on the composite B. The distributed advantage comes from replacing one large multi-controlled phase gate C^{n−1}PS with smaller ones on fewer qubits, thanks to a decomposition lemma that expresses C^{n−1}PS as a repeated pattern of single-qubit phase gates and CNOT gates.

Load-bearing premise

The load-bearing premise is that the first phase — built from exact local rotations on substates that match each node's marginal probability distribution — substantially raises the target probability p'_g for arbitrary, including entangled, global states; the paper provides no bound on that improvement and explicitly flags it as an open question.

What would settle it

Take two n-qubit states with identical one-node marginals, one a tensor product and one entangled (for n=2: |+⟩⊗|+⟩ and (|00⟩+|11⟩)/√2), and run Phase 1 of the algorithm with the same target set and the same node substates. If the entangled case yields a different p'_g than the product case — or if p'_g becomes zero — then the substate construction does not capture the global evolution and the algorithm's claimed applicability to arbitrary amplitude distributions fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Any device with at least max(n_j) qubits per node and any node count t between 2 and n can in principle realize multi-target quantum search with probability 1, without auxiliary qubits.
  • The paper's circuit-depth formula (Theorem 4) gives a concrete criterion for choosing the node partition that minimizes the maximum per-node depth for a fixed p_g and target set.
  • After decomposing multi-controlled phase gates into elementary gates, the distributed scheme's advantage in gate count and depth grows with n; the 10-qubit simulations show a reduction of over 97% compared with centralized exact and non-exact amplification.
  • The algorithm removes the restrictive assumption A = A₁ ⊗ A₂ that earlier distributed amplitude amplification required, so it applies to arbitrary local state-preparation unitaries.
  • Practical use requires the exact probability distribution of the intermediate state (or a good estimate from measurement), because the second phase's iteration count and phase angle are computed from p'_g.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The two-phase structure suggests a general template: run a cheap parallel local preprocessing pass, then correct with one global exact step whose success probability is measured. Whether this template is useful for entangled states depends on the local preprocessing actually concentrating target amplitude; the paper leaves that question open.
  • A concrete test would compare two n-qubit states with identical single-node marginals but different global entanglement — e.g., |+⟩^{⊗2} versus (|00⟩+|11⟩)/√2 — and run Phase 1 with the same substates; the resulting p'_g values differ, which would show that substate marginals alone do not determine the evolution.
  • The reported resource advantage is demonstrated only for a two-target example at 4–10 qubits; whether it persists for target sets whose size scales with n (where |X_j| grows per node) remains untested.
  • If Phase 1 fails to raise p'_g, the depth formula reduces to that of a global EQAAA with a composite B instead of A; then the only guaranteed benefit is the lower per-node qubit count, not the gate/depth reduction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes DEQAAA, a two-phase distributed exact amplitude amplification algorithm for an n-qubit state prepared by a unitary A and an arbitrary target set X_g. Phase 1 runs local EQAAA operations on t nodes, where each node's substate |φ_j> is built from the marginal probability distribution P_j; Phase 2 runs a global EQAAA whose state-preparation operator B is exactly the unitary implemented in Phase 1. The paper proves exactness (Theorem 1) by reducing the whole procedure to EQAAA with preparation B, derives circuit-depth formulas (Theorems 2-4), and reports MindSpore Quantum simulations for 4-, 6-, 8-, and 10-qubit examples with a claimed >97% reduction in gate count and depth at 10 qubits after decomposing multi-controlled phase gates.

Significance. The exactness claim is sound under the stated assumptions: if the probability distributions P and P' are known exactly and all phase angles are computed exactly, Phase 2 is exactly EQAAA with preparation B, so the final success probability is 1. The distributed formulation is potentially useful, the depth formulas are explicit, and the paper provides a reproducibility URL for its simulations. However, the paper's headline resource advantage is not established for arbitrary amplitude distributions. The Phase-1 local operations are built from marginal distributions only, and the authors explicitly leave the crucial question of whether p'_g improves after Phase 1 to future work. Since the depth saving depends on p'_g being large, the general resource claim is a load-bearing open gap rather than a proven theorem.

major comments (3)
  1. [§4.2, Theorem 4, Eqs. (48)-(53)] The depth advantage rests on p'_g being large after Phase 1. From Eq. (36)/(53), \hat J ≈ π/(4 arcsin√p'_g) − 1/2, so the second phase costs O(1/√p'_g) iterations of a circuit that contains the entire Phase 1 as B and B†. The authors explicitly state in §4.2 that whether p'_g is sufficiently improved after Phase 1 is an open question. No lower bound on p'_g is proved for arbitrary amplitude distributions. Without such a bound, DEQAAA can be deeper than EQAAA in the worst case, and the >97% reductions in Tables 3-4 cannot be claimed for general states. This is load-bearing for the abstract's resource claim.
  2. [§3.1, Algorithm 3, Eq. (18); Eq. (25)] The substate |φ_j> is chosen from only the diagonal marginals P_j(x), discarding all inter-node coherences. For a genuinely entangled global state, the actual reduced state ρ_j at node j is mixed and has rank up to 2^{n_j}. A local unitary EQ_j acts on ρ_j, not on the pure substate |φ_j>, so it cannot in general map ρ_j into the local target subspace when rank(ρ_j) > |X_j|. Consequently Phase 1 is not guaranteed to increase p'_g; it can even decrease it. The manuscript needs either a concrete bound on p'_g for a stated class of states/partitions, explicit counterexamples, or a restriction of the resource claims to the cases actually demonstrated.
  3. [Appendix G, Lemma G1; Table 4] The C^3PS(φ) decomposition is stated with only 'It is straightforward to confirm' as its proof, and the extension to n=6,8,10 is asserted without derivation. Tables 3-4 and the headline >97% reduction are computed from this decomposition. Since the resource comparison is a central contribution, the decomposition must be fully verified — either by a complete algebraic proof or by machine-checked unitary equivalence for each n used in the experiments. As written, the resource numbers depend on an unproved lemma.
minor comments (5)
  1. [§5.4, Eq. (66)] The amplitudes in Eq. (55) are real, while the post-Phase-1 state in Eq. (66) is complex. Clarify that the chosen A and B introduce complex phases; otherwise the notation is confusing.
  2. [Algorithm 4, step 12] The algorithm says to 'obtain the exact probability distribution P′' after Phase 1. In a physical experiment this requires full state tomography or equivalent. The assumption is stated earlier, but Step 12 should explicitly say this is a computational/theoretical step under that assumption, not a measurement-free procedure.
  3. [Table 3, row 4] The row 'Repetition number of amplification operators' is ambiguous: for DEQAAA it should specify whether the number counts local EQ_j applications, global dEQ applications, or the total. The 4-qubit text reports J0+1=J1+1=1 and \hat J+1=1, but the table entry is simply 1.
  4. [Appendix F, Eq. (F2)] The text says the depths of R^{φ_j}_{f_j} and R^{φ_j}_{|0>} are 'both set to 1' for the local node, while Eq. (50) uses the factor 3|X_j|+3. Align the wording with the formula: each single-target rotation has depth 3, hence |X_j| rotations contribute 3|X_j|.
  5. [Figures 17-18] Several figure captions contain garbled placeholder character sequences (e.g., '/uni00000025/...'). These should be replaced with readable text describing the plotted quantities.

Circularity Check

0 steps flagged

No circular derivation: the final exactness step is a direct application of the EQAAA construction to the composite unitary B, not an identification of input with output.

full rationale

Theorem 1 is self-contained rather than circular. The paper explicitly reduces the two-phase procedure to the standard EQAAA by defining B = (⊗ EQ_j^{J_j+1}) A and then applying dEQ = B R^φ̂_{|0>} B† R^φ̂_f with Ĵ and φ̂ computed from p'_g. Appendix C states: “The ‘Phase 1 + Phase 2’ process is equivalent to the standard EQAAA process with B as the state preparation operator,” and the EQAAA construction is itself derived in Appendix B from the rotation-angle condition, not imported by citation. No parameter is fitted to data: p_j, J_j, φ_j, p'_g, Ĵ, and φ̂ are all closed-form functions of computed probabilities or of the actual state after Phase 1. The choice of local substates |φ_j> from the marginal probabilities P_j is an ansatz, but the correctness of the final exact amplification does not depend on Phase 1 achieving p'_g = 1; Phase 2 corrects any residual probability. The paper itself flags the unproven improvement of p'_g as an open question in §4.2 (“whether p′_g is sufficiently improved compared to p_g after the first phase”), which is a support gap in the resource-advantage claim, not a circularity in the exactness derivation. Self-citations such as Refs. [22,23] for exact Grover and for C^{n-1}PS decomposition are used as context, and the relevant operators and decompositions are derived within the paper (Appendices B and G), so they are not load-bearing. No prediction reduces to its input by construction; the final success probability 1 is a theorem consequence of the EQAAA rotation analysis applied to B.|0>^n.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The central claim depends on exact knowledge of the initial (and post-Phase-1) probability distributions, on an implicit assumption that local EQ_j acts on node reduced states as it would on the constructed pure substates, and on an unproved generalization of the C^(n-1)PS decomposition.

free parameters (2)
  • node qubit partition {n_j} = e.g., {2,2} for 4-qubit; multi-node variants for 6/8/10-qubit
    The algorithm allows any 2≤t≤n, but resource counts depend on the chosen partition, and no optimal selection rule is provided.
  • substate |φ_j> amplitudes = arbitrary pure states matching marginal probabilities (positive square roots used in Eq. 18)
    For an entangled global state, the actual reduced density matrix has different coherences; the arbitrary choice changes B, p'_g, and all gate/depth numbers.
axioms (4)
  • domain assumption Exact knowledge of the probability distributions P and P' of |Ψ> and |Ψ_1>
    Section 3.1 and Algorithm 4 step 12 require exact P and P' to set phase angles and iteration counts; the paper admits this is approximated statistically in practice (§5.1).
  • ad hoc to paper Each node's reduced state behaves like the pure substate |φ_j> under local EQ_j
    Algorithm 3 Eq. (18) builds a pure substate from marginals only, and EQ_j (Eq. 25) is designed for that substate. No proof covers mixed reduced states of entangled global states.
  • standard math Long/EQAAA exact-amplification formulas are correct
    The phase and iteration formulas (Eqs. 13–14) are adopted from prior exact Grover/EQAAA results and not rederived.
  • ad hoc to paper The generalized C^(n-1)PS decomposition is valid for n=6,8,10
    Lemma G1 is stated for n=4 only; the text says the generalization is not included, yet Table 4 resource counts rely on it.

pith-pipeline@v1.3.0-alltime-deepseek · 35188 in / 22385 out tokens · 239776 ms · 2026-08-03T10:42:14.583504+00:00 · methodology

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read the original 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.

Figures

Figures reproduced from arXiv: 2601.09128 by Keren Li, Shenggen Zheng, Wenxuan Tao, Xu Zhou.

Figure 1
Figure 1. Figure 1: A detailed analysis of the QAAA is provided in Appendix A. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Quantum circuit of the QAAA. 2.2. Exact Quantum Amplitude Amplification Algorithm The QAAA can output x ∈ Xg with a high probability of success. However, except in special cases, it does not achieve exact precision, which would require a 100% probability of outputting x ∈ Xg. Inspired by the exact Grover’s algorithm by Long [45, 22, 23], the key concept involves extending the Grover operator G to the exact… view at source ↗
Figure 2
Figure 2. Figure 2: Quantum circuit of the EQAAA. 3. Distributed Exact Quantum Amplitude Amplification Algorithm We present the Distributed Exact Quantum Amplitude Amplification Algorithm (DEQAAA) in this section. It operates on a distributed quantum system composed of t nodes, with 2 ≤ t ≤ n. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Quantum circuit of the DEQAAA. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two quantum circuits implementing the SWAP gate. [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The quantum circuits corresponding to Sf and S|0⟩⊗n operators. The phase rotation operator R ϕ f in the EQAAA shares a similar construction principle with the Sf in QAAA. Its 15 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The quantum circuits corresponding to R ϕ f and R ϕ |0⟩⊗n operators. Here, the core phase operation is enacted by the multi-controlled phase shift gate C n−1P S(ϕ), where the single￾qubit phase gate is P S(ϕ) =   1 0 0 e iϕ   . (42) The surrounding X gates perform the same basis transformation role. Specifically, for the all-zero state |0⟩ ⊗n, the construction (see Figure 6b) is R ϕ |0⟩⊗n = [PITH_FULL… view at source ↗
Figure 7
Figure 7. Figure 7: Quantum circuit of the operator A to prepare the 4-qubit state |Ψ⟩. Suppose the global target set is defined as Xg = {8, 14} = {1000, 1110}. Based on the amplitude distribution of |Ψ⟩, the target measurement probability (success probability) is: pg = 0.31642 + 0.30462 = 0.1929. (56) It is well-established that the square of the amplitude of a quantum state corresponds to the measurement probabil￾ity, so if… view at source ↗
Figure 8
Figure 8. Figure 8: The distribution plots of measurement outcomes of [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Quantum circuit of the 4-qubit QAAA with the set of target strings [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Measurement distribution of |Ψ⟩ after QAAA (10,000 measurements). 5.3. The 4-qubit EQAAA with Xg = {8, 14} We then employ EQAAA to enhance the success probability to 1. Similarly, the complete quantum circuit for this case is constructed via Algorithm 2, as shown in [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Quantum circuit of the 4-qubit EQAAA with the set of target strings [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Measurement distribution of |Ψ⟩ after EQAAA (10,000 measurements). 5.4. The 4-qubit DEQAAA with t = 2 nodes and Xg = {8, 14} Lastly, we extend our exploration to the application of DEQAAA for this example. Specifically, we assume a small-scale distributed system consisting of t = 2 quantum computing nodes, where each node is equipped with n0 = n1 = 2 qubits, resulting in a total of n = 4 computing qubits.… view at source ↗
Figure 13
Figure 13. Figure 13: Quantum circuit of 4-qubit DEQAAA (first phase) with [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Measurement distribution of |Ψ⟩ after the first phase of DEQAAA (10,000 measurements). 24 [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Complete quantum circuit of the 4-qubit DEQAAA (first [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Measurement distribution of |Ψ⟩ after DEQAAA (10,000 measurements). 5.5. Data comparison For a comprehensive performance comparison of the three algorithms, we present a summary of the key ex￾perimental results across multiple dimensions in [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Comparison of the computational complexity of different QAAAs before and after decomposition. [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Scalability trends of quantum gates and circuit depth of 6-qubit, 8-qubit and 10-qubit examples after decomposition. [PITH_FULL_IMAGE:figures/full_fig_p029_18.png] view at source ↗

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Reference graph

Works this paper leans on

50 extracted references · 2 linked inside Pith

  1. [1]

    M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2010

    2010

  2. [2]

    Deutsch, Quantum theory, the Church–Turing principle and the universal quantum computer, Proceedings of the Royal Society of London

    D. Deutsch, Quantum theory, the Church–Turing principle and the universal quantum computer, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 400 (1818) (1985) 97–117

    1985

  3. [3]

    Deutsch, R

    D. Deutsch, R. Jozsa, Rapid solution of problems by quantum computation, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 439 (1907) (1992) 553–558

    1907

  4. [4]

    Bernstein, U

    E. Bernstein, U. Vazirani, Quantum complexity theory, in: Proceedings of the 25th Annual Symposium on Theory of Computing, ACM, 1993, pp. 11–20

    1993

  5. [5]

    D. R. Simon, On the power of quantum computation, SIAM Journal on Computing 26 (5) (1997) 1474–1483

    1997

  6. [6]

    P. W. Shor, Algorithms for quantum computation: discrete logarithms and factoring, in: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, IEEE, 1994, pp. 124–134

    1994

  7. [7]

    L. K. Grover, Quantum mechanics helps in searching for a needle in a haystack, Physical Review Letters 79 (2) (1997) 325

    1997

  8. [8]

    A. W. Harrow, A. Hassidim, S. Lloyd, Quantum algorithm for linear systems of equations, Physical Review Letters 103 (15) (2009) 150502

    2009

  9. [9]

    Peruzzo, J

    A. Peruzzo, J. McClean, P. Shadbolt, M. H. Yung, X. Q. Zhou, P. J. Love, A. Aspuru Guzik, J. L. O’brien, A variational eigenvalue solver on a photonic quantum processor, Nature Communications 5 (1) (2014) 4213

    2014

  10. [10]

    Farhi, J

    E. Farhi, J. Goldstone, S. Gutmann, A quantum approximate optimization algorithm, arXiv preprint arXiv:1411.4028 (2014)

    Pith/arXiv arXiv 2014

  11. [11]

    Arute, K

    F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell, et al., Quantum supremacy using a programmable superconducting processor, Nature 574 (7779) (2019) 505–510

    2019

  12. [12]

    Zhong, H

    H. Zhong, H. Wang, Y . Deng, M. Chen, L. Peng, Y . Luo, J. Qin, D. Wu, X. Ding, Y . Hu, et al., Quantum computational advantage using photons, Science 370 (6523) (2020) 1460–1463

    2020

  13. [13]

    G. Q. AI, Collaborators, Quantum error correction below the surface code threshold, Nature 638 (8052) (2025) 920–926. 39

    2025

  14. [14]

    D. Gao, D. Fan, C. Zha, J. Bei, G. Cai, J. Cai, S. Cao, F. Chen, J. Chen, K. Chen, et al., Establishing a new benchmark in quantum computational advantage with 105-qubit zuchongzhi 3.0 processor, Physical Review Letters 134 (9) (2025) 090601

    2025

  15. [15]

    Preskill, Quantum computing in the NISQ era and beyond, Quantum 2 (2018) 79

    J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2 (2018) 79

    2018

  16. [16]

    Barral, F

    D. Barral, F. J. Cardama, G. Díaz-Camacho, D. Faílde, I. F. Llovo, M. Mussa-Juane, J. Vázquez-Pérez, J. Vil- lasuso, C. Piñeiro, N. Costas, et al., Review of distributed quantum computing: from single qpu to high perfor- mance quantum computing, Computer Science Review 57 (2025) 100747

    2025

  17. [17]

    Buhrman, H

    H. Buhrman, H. Röhrig, Distributed quantum computing, in: Proceedings of Mathematical Foundations of Com- puter Science 2003: 28th International Symposium, Springer, 2003, pp. 1–20

    2003

  18. [18]

    Yimsiriwattana, S

    A. Yimsiriwattana, S. J. Lomonaco Jr, Distributed quantum computing: A distributed Shor algorithm, in: Quan- tum Information and Computation II, V ol. 5436, SPIE, 2004, pp. 360–372

    2004

  19. [19]

    Beals, S

    R. Beals, S. Brierley, O. Gray, A. W. Harrow, S. Kutin, N. Linden, D. Shepherd, M. Stather, Efficient distributed quantum computing, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469 (2153) (2013) 20120686

    2013

  20. [20]

    Avron, O

    J. Avron, O. Casper, I. Rozen, Quantum advantage and noise reduction in distributed quantum computing, Phys- ical Review A 104 (5) (2021) 052404

    2021

  21. [21]

    D. Qiu, L. Luo, L. Xiao, Distributed Grover’s algorithm, Theoretical Computer Science 993 (2024) 114461

    2024

  22. [22]

    X. Zhou, D. Qiu, L. Luo, Distributed exact Grover’s algorithm, Frontiers of Physics 18 (5) (2023) 51305

    2023

  23. [23]

    X. Zhou, X. Xu, S. Zheng, L. Luo, Distributed exact generalized Grover’s algorithm, Frontiers of Computer Science 20 (7) (2025) 2007905

    2025

  24. [24]

    H. Li, D. Qiu, L. Luo, Distributed Deutsch-Jozsa algorithm, The Journal of Supercomputing 81 (12) (2025) 1–44

    2025

  25. [25]

    X. Zhou, D. Qiu, L. Luo, Distributed Bernstein-Vazirani algorithm, Physica A: Statistical Mechanics and its Applications 629 (2023) 129209

    2023

  26. [26]

    J. Tan, L. Xiao, D. Qiu, L. Luo, P. Mateus, Distributed quantum algorithm for Simon’s problem, Physical Review A 106 (3) (2022) 032417

    2022

  27. [27]

    X. Zhou, Y . Wang, W. Tao, Z. Zhou, L. Luo, Distributed quantum algorithm for the NISQ era: A novel approach to solving Simon’s problem with reduced resources, Advanced Quantum Technologies 8 (5) (2025) 2500067. 40

    2025

  28. [28]

    L. Xiao, D. Qiu, L. Luo, P. Mateus, Distributed Shor’s algorithm, Quantum Information & Computation 23 (1-2) (2023) 0027–0044

    2023

  29. [29]

    L. Liu, Y . Zhang, Z. Li, R. Zhang, X. Yin, Y . Fei, L. Li, N. Liu, F. Xu, Y . Chen, et al., Distributed quantum phase estimation with entangled photons, Nature Photonics 15 (2) (2021) 137–142

    2021

  30. [30]

    C. Ying, B. Cheng, Y . Zhao, H. Huang, Y . Zhang, M. Gong, Y . Wu, S. Wang, F. Liang, J. Lin, et al., Experimental simulation of larger quantum circuits with fewer superconducting qubits, Physical Review Letters 130 (11) (2023) 110601

    2023

  31. [31]

    Akhtar, F

    M. Akhtar, F. Bonus, F. Lebrun Gallagher, N. Johnson, M. Siegele Brown, S. Hong, S. Hile, S. Kulmiya, S. Weidt, W. Hensinger, A high-fidelity quantum matter-link between ion-trap microchip modules, Nature Com- munications 14 (1) (2023) 531

    2023

  32. [32]

    X. Liu, X. Hu, T. Zhu, C. Zhang, Y . Xiao, J. Miao, Z. Ou, P. Li, B. Liu, Z. Zhou, et al., Nonlocal photonic quantum gates over 7.0 km, Nature Communications 15 (1) (2024) 8529

    2024

  33. [33]

    D. Main, P. Drmota, D. Nadlinger, E. Ainley, A. Agrawal, B. Nichol, R. Srinivas, G. Araneda, D. Lucas, Dis- tributed quantum computing across an optical network link, Nature (2025) 1–6

    2025

  34. [34]

    Y . Wei, P. Stas, S. Aziza, B. Gefen, M. Francisco, H. Yanqi, K. Can, D. Sophie, M. Moritz, K. Erik, et al., Universal distributed blind quantum computing with solid-state qubits, Science 388 (6746) (2025) 509–513

    2025

  35. [35]

    Brassard, P

    G. Brassard, P. Höyer, M. Mosca, A. Tapp, Quantum amplitude amplification and estimation, Contemporary Mathematics 305 (2002) 53–74

    2002

  36. [36]

    Rajagopal, Q

    K. Rajagopal, Q. Zhang, S. Balakrishnan, P. Fakhari, J. Busemeyer, Quantum amplitude amplification for rein- forcement learning, Handbook of Reinforcement Learning and Control (2021) 819–833

    2021

  37. [37]

    Mandl, J

    A. Mandl, J. Barzen, M. Bechtold, F. Leymann, K. Wild, Amplitude amplification-inspired QAOA: improving the success probability for solving 3SAT, Quantum Science and Technology 9 (1) (2024) 015028

    2024

  38. [38]

    S. Perriello, Quantum circuit design for finding k-cliques via quantum amplitude amplification strategies, in: Proceedings of the 22nd ACM International Conference on Computing Frontiers, 2025, pp. 55–63

    2025

  39. [39]

    Diao, Exactness of the original Grover search algorithm, Physical Review A 82 (4) (2010) 044301

    Z. Diao, Exactness of the original Grover search algorithm, Physical Review A 82 (4) (2010) 044301

    2010

  40. [40]

    X. Hua, D. Qiu, Distributed quantum amplitude amplification, arXiv preprint arXiv:2510.16498 (2025)

    arXiv 2025

  41. [41]

    T. J. Yoder, G. H. Low, I. L. Chuang, Fixed-point quantum search with an optimal number of queries, Physical review letters 113 (21) (2014) 210501

    2014

  42. [42]

    L. K. Grover, Fixed-point quantum search, Physical Review Letters 95 (15) (2005) 150501. 41

    2005

  43. [43]

    URLhttps://gitee.com/mindspore/mindquantum

    MindQuantum, https://gitee.com/mindspore/mindquantum, version 0.9.0 (2021). URLhttps://gitee.com/mindspore/mindquantum

    2021

  44. [44]

    X. Xu, J. Cui, Z. Cui, R. He, Q. Li, X. Li, Y . Lin, J. Liu, W. Liu, J. Lu, et al., MindSpore Quantum: a user-friendly, high-performance, and AI-compatible quantum computing framework, arXiv preprint arXiv:2406.17248 (2024)

    Pith/arXiv arXiv 2024

  45. [45]

    Long, Grover algorithm with zero theoretical failure rate, Physical Review A 64 (2) (2001) 022307

    G. Long, Grover algorithm with zero theoretical failure rate, Physical Review A 64 (2) (2001) 022307

    2001

  46. [46]

    I. F. Araujo, D. K. Park, T. B. Ludermir, W. R. Oliveira, F. Petruccione, A. J. Da Silva, Configurable sublinear circuits for quantum state preparation, Quantum Information Processing 22 (2) (2023) 123

    2023

  47. [47]

    Möttönen, J

    M. Möttönen, J. J. Vartiainen, V . Bergholm, M. M. Salomaa, Transformation of quantum states using uniformly controlled rotations, Quantum Information and Computation 5 (6) (2005) 467–473

    2005

  48. [48]

    Zoufal, A

    C. Zoufal, A. Lucchi, S. Woerner, Quantum generative adversarial networks for learning and loading random distributions, npj Quantum Information 5 (1) (2019) 103

    2019

  49. [49]

    Kullback, R

    S. Kullback, R. A. Leibler, On information and sufficiency, Annals of Mathematical Statistics 22 (1951) 79–86

    1951

  50. [50]

    Barenco, C

    A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Wein- furter, Elementary gates for quantum computation, Physical Review A 52 (5) (1995) 3457. 42

    1995