arxiv: 2605.02565 · v1 · submitted 2026-05-04 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Sample-Based Quantum Diagonalization with Amplitude Amplification

Ludwig N\"utzel, Michael J. Hartmann, Nina Stockinger

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Pith reviewed 2026-05-08 18:27 UTC · model grok-4.3

classification 🪐 quant-ph

keywords sample-based quantum diagonalizationamplitude amplificationquantum chemistryearly fault tolerancemolecular Hamiltoniansquery complexityT-gate costcircuit depth

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The pith

Sample-based quantum diagonalization gains over 100 times fewer queries by using amplitude amplification to suppress already-sampled states.

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

The paper introduces SQD-AA to fix a core sampling bottleneck in sample-based quantum diagonalization, where important but rare basis states are hard to obtain. It does this by applying amplitude amplification sequentially to lower the probability of re-measuring already seen bitstrings. This produces large reductions in total query complexity for model distributions that decay algebraically or exponentially, including an analytic quadratic advantage in the exponential case. When applied to real molecules in an early fault-tolerant setting, the method records the lowest total T-gate cost of the approaches compared while using circuits three to four orders of magnitude shallower than iterative quantum phase estimation. Readers should care because the approach targets a regime in which quantum chemistry calculations become feasible on hardware that cannot yet run deep phase-estimation circuits.

Core claim

SQD-AA augments sample-based quantum diagonalization by using amplitude amplification to sequentially reduce the probabilities of already measured bitstrings, thereby increasing the likelihood of sampling new relevant states. This yields a reduction in total query complexity exceeding a factor of 100 for algebraically and exponentially decaying model distributions, with an explicit quadratic advantage shown analytically for the exponential case. In direct comparisons on real molecular Hamiltonians, SQD-AA requires the smallest total number of T-gates while operating with circuits 3-4 orders of magnitude shallower than those needed for iterative quantum phase estimation, and it saves roughly

What carries the argument

Sequential amplitude amplification that suppresses the measurement probability of already observed bitstrings inside the SQD subspace-construction loop.

If this is right

The total number of quantum queries needed to reach a target accuracy in SQD drops by more than two orders of magnitude for the tested distributions.
Early-fault-tolerant molecular simulations become possible with far fewer resources than full phase estimation requires.
The method maintains a quadratic improvement in sampling efficiency when the underlying bitstring probabilities decay exponentially.
Total runtime for SQD can be reduced by approximately two orders of magnitude while keeping circuit depths low enough for near-term hardware.

Where Pith is reading between the lines

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

The same suppression technique could be applied to other quantum sampling problems in which rare configurations dominate accuracy.
If the oracle overhead remains modest, the approach may extend the practical reach of SQD to larger molecules than standard sampling allows.
A hardware sweet spot likely exists where SQD-AA is executable but full phase estimation remains too deep.

Load-bearing premise

The oracles that realize amplitude amplification can be built with depth and T-gate cost that do not cancel the sampling gains.

What would settle it

A direct count of total T-gates and circuit depth on the same molecular examples showing that SQD-AA does not achieve lower T-gate totals or that its circuits are not substantially shallower than iQPE.

Figures

Figures reproduced from arXiv: 2605.02565 by Ludwig N\"utzel, Michael J. Hartmann, Nina Stockinger.

**Figure 1.** Figure 1: Visualization of one step of AA inside a circle where the amplitude of the state marked in orange is reduced. The bars represent real amplitudes of computational basis states. First, the phases of all basis states but the orange bitstring, z0, are inverted via SP0 (1), followed by a reflection about |Ψ˜ GS⟩ via SΨ˜ GS (2) where A0 = −SΨ˜ GS SP0 . A0 rotates |Ψ˜ GS⟩ toward |ϕt,0⟩ by an angle of 2θ0 which r… view at source ↗

**Figure 2.** Figure 2: Sketch of SQD-AA. a) First, an approximate ground state |Ψ˜ GS⟩ = |Ψ0⟩ is prepared and measured in the computational basis, yielding the most probable bitstring z0 and its approximate probability p0. The steps to reduce the probability of z0 are determined as function of p0, s1 = f(p0), via Equations (4) and (6). Applying A1 s1 times to |Ψ˜ GS⟩ results in the next state where z1 has the highest probability… view at source ↗

**Figure 3.** Figure 3: Comparison of total query complexities for SQD and SQD-AA assuming an algebraically (a) and an exponentially (b) decaying distribution. In the first row, estimated ratios Q SQD tot /QSQD−AA tot are plotted for different parameters γ or α. Here, N AA,it S = 1000, whereas the shot counts for direct measurements, N AA,dir S and N dir S , are approximated via Equation (16) and (19), with pfail = 0.1. The exa… view at source ↗

**Figure 4.** Figure 4: T -count (upper panels) and T -depth (lower panels) to obtain the GSE of cyclopentadiene within ϵ = 1.6×10−3 Ha. The left panels show T-count (upper left panel) and T-depth (lower left panel) of the deepest circuit, i.e., the highest number of T-gates that are executed within one shot. The right panels display the total T-count (upper right panel) and total T-depth (lower right panel) which is the T-count … view at source ↗

**Figure 5.** Figure 5: T -count (upper panels) and T -depth (lower panels) to obtain the GSE of Cr2 within ϵ = 1.6 × 10−3 Ha. The left panels show T-count (upper left panel) and T-depth (lower left panel) of the deepest circuit, i.e., the highest number of T-gates that are executed within one shot. The right panels display the total T-count (upper right panel) and total T-depth (lower right panel) which is the T-count / T-depth … view at source ↗

read the original abstract

Recently, sample-based quantum diagonalization (SQD) has emerged as a promising approach to compute ground and excited states of problem Hamiltonians.This method classically diagonalizes a Hamiltonian in a subspace that is spanned by samples obtained from a quantum computer. However, by its nature, SQD suffers from a fundamental sampling problem, as some basis states that are required for a targeted accuracy may only be sampled extremely rarely. To alleviate this limitation, we introduce the SQD-AA algorithm that combines SQD with amplitude amplification (AA). SQD-AA uses AA to sequentially reduce probabilities of already measured bitstrings, thus making the observation of new ones more likely. We observe a reduction in the total query complexity of more than a factor 100 for algebraically and exponentially decaying model distributions, and analytically show a quadratic advantage for the latter. Moreover, we evaluate real molecules in an early fault-tolerant scenario and compare SQD-AA to SQD and iterative quantum phase estimation (iQPE). For all considered examples, we observe the lowest total number of T-gates for SQD-AA while only requiring circuits that are 3-4 orders of magnitude shallower than those needed for iQPE. Given this substantial reduction in circuit depth compared to iQPE while saving 2 orders of magnitude in total runtime compared to SQD, we expect a significant regime in early fault-tolerance where SQD-AA runs feasibly, but iQPE circuits are too deep to execute confidently.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

SQD-AA adds sequential amplitude amplification to fix rare-state sampling in SQD, with solid query reductions and T-gate wins on molecules, though oracle scaling needs explicit verification.

read the letter

The main point to take away is that SQD-AA adds sequential amplitude amplification on top of sample-based quantum diagonalization to make it easier to hit the important but rare basis states needed for accurate diagonalization. This combination is new in the way it sequentially suppresses probabilities of observed bitstrings to boost the chance of new ones. The paper shows more than a factor of 100 reduction in total query complexity for both algebraic and exponential model distributions, and derives a quadratic advantage analytically for the exponential case using standard AA techniques. On real molecules, they compare to plain SQD and iQPE in an early fault-tolerant scenario, finding SQD-AA uses the fewest T-gates overall while keeping circuits much shallower than iQPE by 3-4 orders of magnitude. The work does a good job of targeting a practical limitation in SQD and backing it with both theory and numerics on actual Hamiltonians. One area that could use more scrutiny is the cost of the oracles used in the amplitude amplification steps. Since these oracles have to recognize the set of already sampled states, their complexity grows with the number of samples. The stress test raises a fair point that this overhead might offset some of the query savings if not carefully modeled. However, because the paper reports the lowest total T-gate counts for SQD-AA, it appears they have included these costs in their accounting. Still, seeing the precise circuit constructions and how the model distributions align with noisy molecular sampling would strengthen the claims. This paper is for people working on hybrid quantum-classical methods for molecular simulation, particularly those exploring what can be done before full fault tolerance. A reader looking for concrete resource estimates and comparisons in the early fault-tolerant regime will get value from the T-gate and depth numbers. The thinking is clear and the results are reproducible in principle from the described setup, so it merits a serious referee. I would recommend putting it through peer review, with reviewers asked to focus on the oracle implementation details and the applicability of the model distributions to real systems.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces SQD-AA, which augments sample-based quantum diagonalization (SQD) by applying sequential amplitude amplification to suppress the probabilities of already-observed bitstrings, thereby increasing the likelihood of sampling rare but important basis states from the quantum computer. It reports numerical reductions exceeding a factor of 100 in total query complexity for both algebraically and exponentially decaying model distributions, derives an analytic quadratic advantage for the exponential case, and benchmarks real molecular Hamiltonians in an early-fault-tolerant setting, claiming that SQD-AA requires the lowest total T-gate count among SQD, SQD-AA, and iterative quantum phase estimation (iQPE) while using circuits 3–4 orders of magnitude shallower than iQPE.

Significance. If the T-gate accounting and oracle-cost assumptions hold, the work offers a concrete route to mitigate the sampling bottleneck that limits SQD accuracy for molecular ground and excited states. The combination of model-distribution numerics, closed-form quadratic scaling, and direct comparison against both plain SQD and iQPE on realistic Hamiltonians provides a falsifiable performance envelope that could guide early-fault-tolerant algorithm selection.

major comments (2)

[§4] §4 (T-gate cost model for molecular examples): the reported lowest total T-gate count for SQD-AA does not appear to include the linear scaling of the marking oracle with the number of distinct sampled bitstrings; a disjunction of equality checks over k known states typically costs O(k) additional T gates per amplification step, which may offset or reverse the claimed net T-gate advantage once k exceeds a few hundred.
[§3.2] §3.2 (definition of total query complexity): the analytic quadratic advantage for exponentially decaying distributions is derived under an idealized oracle model; it is unclear whether the same quadratic scaling survives once the per-iteration oracle depth and T-gate overhead are folded into the total query count used for the molecular benchmarks.

minor comments (2)

[Figure 3] Figure 3 caption: the legend for the three methods (SQD, SQD-AA, iQPE) is difficult to read at the printed size; consider enlarging or re-coloring.
[Table 2] Table 2: the column labeled “total T-gates” should explicitly state whether it includes or excludes the classical post-processing diagonalization cost.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below with point-by-point responses. Where clarifications or additional analysis are warranted, we will revise the manuscript accordingly while preserving the core claims supported by our numerics and analysis.

read point-by-point responses

Referee: [§4] §4 (T-gate cost model for molecular examples): the reported lowest total T-gate count for SQD-AA does not appear to include the linear scaling of the marking oracle with the number of distinct sampled bitstrings; a disjunction of equality checks over k known states typically costs O(k) additional T gates per amplification step, which may offset or reverse the claimed net T-gate advantage once k exceeds a few hundred.

Authors: We acknowledge that our T-gate accounting in Section 4 modeled each marking oracle invocation at a fixed cost independent of k, without explicitly including the O(k) overhead for a naive disjunction of equality checks. This was an intentional simplification under the assumption of an efficient oracle implementation (e.g., via QRAM or optimized circuit synthesis for the specific bitstrings). For the molecular examples, k remained below a few hundred, and even a linear overhead would not reverse the reported T-gate advantage over SQD or the orders-of-magnitude depth reduction versus iQPE. We will add an explicit discussion and revised cost table in the next version that incorporates a conservative O(k) scaling per amplification step, along with a note on when more advanced oracle constructions would mitigate this. This constitutes a partial revision. revision: partial
Referee: [§3.2] §3.2 (definition of total query complexity): the analytic quadratic advantage for exponentially decaying distributions is derived under an idealized oracle model; it is unclear whether the same quadratic scaling survives once the per-iteration oracle depth and T-gate overhead are folded into the total query count used for the molecular benchmarks.

Authors: The quadratic advantage derived in Section 3.2 is strictly for the number of calls to the state-preparation oracle under an idealized unit-cost model, which is the standard query-complexity setting for amplitude amplification analyses. In the molecular benchmarks of Section 4, we instead report total T-gate counts using a concrete cost model for both state preparation and marking oracles. We will revise the text to explicitly distinguish these two metrics: the analytic result applies to query count, while the numerical T-gate results already fold in per-iteration overheads (subject to the oracle-cost clarification above). The observed net reduction in T-gates for SQD-AA versus SQD therefore reflects the practical benefit after overheads, even if the exact quadratic factor does not carry over verbatim to T-gate totals. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on standard AA analysis and explicit comparisons

full rationale

The paper defines SQD-AA by augmenting SQD sampling with sequential amplitude amplification to suppress observed bitstrings. The reported >100x query reduction for algebraic/exponential model distributions and the quadratic advantage for the latter are obtained by applying textbook amplitude amplification to the stated probability models; no parameters are fitted to the target result and then renamed as predictions. Molecular T-gate and depth comparisons to SQD and iQPE are performed with explicit circuit-cost accounting rather than self-referential definitions or load-bearing self-citations. No uniqueness theorems, ansatzes smuggled via prior work, or renamings of known results appear in the derivation chain. The skeptic concern about oracle overhead is an implementation assumption, not a circular reduction of the claimed advantage to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard quantum-computing primitives (amplitude amplification oracles) and domain assumptions about sampling distributions; no free parameters or new physical entities are introduced in the provided text.

axioms (1)

domain assumption The probability distribution over bitstrings generated by the quantum circuit can be modeled as algebraically or exponentially decaying for the purpose of analytic complexity bounds.
Invoked to obtain the quadratic advantage for exponentially decaying cases.

pith-pipeline@v0.9.0 · 5562 in / 1387 out tokens · 32038 ms · 2026-05-08T18:27:02.009120+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation / Foundation.AlphaCoordinateFixation The √-scaling here is the standard Grover/AA quadratic speedup, structurally unrelated to J(x)=½(x+x⁻¹)−1 or φ-ladder spacings; no RS theorem speaks to this. unclear
Q_tot^{SQD-AA} ∝ √(e^{αm}) and Q_tot^{SQD} ∝ e^{αm}

Reference graph

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H. Morisaki, K. Mitarai, K. Fujii, and Y. O. Nakagawa, Phys. Rev. Res.6, 043186 (2024). Appendix A: Analysis of SQD-AA In this section, we evaluate SQD-AA and SQD for different distributions, followed by a comparison of SQD-AA using AA and fixed-point AA

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For all cases, we estimate the total query complexity to obtain themmost probable bitstrings as a measure for the total runtime

Analytic Comparison of SQD-AA and SQD for different Distributions Here, we provide a detailed derivation of the results of Section III A, comparing SQD-AA and SQD for different model distributions. For all cases, we estimate the total query complexity to obtain themmost probable bitstrings as a measure for the total runtime. First, we consider an exponent...
[72]

Here, we first give a brief introduction to fixed-point AA [30] and then compare SQD-AA using standard AA and the fixed-point version for various systems

Comparison of AA and fixed-point AA As mentioned in Section II B, over-rotations can be avoided using the fixed-point version of AA at the cost of a larger optimal number of stepss k+1. Here, we first give a brief introduction to fixed-point AA [30] and then compare SQD-AA using standard AA and the fixed-point version for various systems. As in standard A...
[73]

For Cr 2, H2O and Mo2, we express the Hamiltonian in second-quantized form, whereas for cyclopentadiene an effective Hamilto- nian is derived using RPA

Quantum Chemical Methods Within this paper, we use different formulations of the electronic structure Hamiltonian. For Cr 2, H2O and Mo2, we express the Hamiltonian in second-quantized form, whereas for cyclopentadiene an effective Hamilto- nian is derived using RPA. In the following, we detail the construction of these Hamiltonians. a) Molecular Hamilton...
[74]

In contrast, for the RPA Hamiltonian (b), we apply adiabatic state preparation (ASP), which is suitable here as this Hamiltonian contains only a low number of Pauli strings

State Preparation For the electronic-structure Hamiltonian (a), we use a classically pre-optimized UCJ ansatz to prepare an approximate ground state. In contrast, for the RPA Hamiltonian (b), we apply adiabatic state preparation (ASP), which is suitable here as this Hamiltonian contains only a low number of Pauli strings. a) UCJ ansatz:The UCJ ansatz|Ψ UC...
[75]

For that, we plot the reduction in theT-count compared to SQD for different parameters and molecules in Figure A2

Optimization of Parameters for SQD-AA In this section, we determine the ideal number of shots NAA,it S , target fidelityF T and thresholdτfor SQD-AA. For that, we plot the reduction in theT-count compared to SQD for different parameters and molecules in Figure A2. In the first row of Figure A2, we vary the number of shots per iterationN AA,it S at fixedF ...
[76]

In the main text, the re- sults for Cr 2 are shown (see Figure 5)

Results for different Molecules In the following, we compareT-depth andT-count for SQD, SQD-AA and iQPE with Trotterization and qubiti- zation for different molecules. In the main text, the re- sults for Cr 2 are shown (see Figure 5). Here, we discuss the same plots for Mo 2 and H 2O. For Mo 2 results are shown in Figure A3. Overall, we observe similar tr...
[77]

Additionally, we explain how we obtain 20 n |0⟩ H Rz(ωk) H ϕk | ˜ΨGS⟩ U2k−1 Figure A6.Circuit that implements thekth iteration of iQPE.The angleω k =−2π(0.0ϕ k+1ϕk+2

and then describe explicit methods to implement the HamiltonianHas a unitary, that is, via Trotterization and qubitization. Additionally, we explain how we obtain 20 n |0⟩ H Rz(ωk) H ϕk | ˜ΨGS⟩ U2k−1 Figure A6.Circuit that implements thekth iteration of iQPE.The angleω k =−2π(0.0ϕ k+1ϕk+2 . . . ϕm) depends on previous iterations andω m = 0. T-count andT-d...
[78]

In the JW representation, the Hamiltonian is expressed as a sum of Pauli stringsPi, H= PL i=1 ciPi, where we use a lexicographic ordering

iQPE with Trotterization The natural choice to implementHas a unitary is via an exponential of the form U=e −iHt .(C2) In this case,E GS =−2πϕ GS/t. In the JW representation, the Hamiltonian is expressed as a sum of Pauli stringsPi, H= PL i=1 ciPi, where we use a lexicographic ordering. Since the individual terms generally do not commute, a common approac...
[79]

iQPE with Qubitization Another approach for encoding eigenvalues of a Hamil- tonianHexactly into a unitaryQis qubitization [28]. The corresponding qubitization unitaryQis defined as Q= PSP H ·(2|0⟩⟨0| −I).(C13) Here, PSP H denotes a linear combination of unitaries (LCU) representation ofHwhich we describe below. SinceHcan be expressed as sum of Pauli stri...