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The paper constructs the first quantum algorithm that estimates fidelity susceptibility with Heisenberg-limited precision, via a resolvent identity and quantum singular value transformation.

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A quantum algorithm estimates fidelity susceptibility in O~(1/epsilon) queries using a resolvent reformulation, achieving Heisenberg-limited precision.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection A genuine QSVT+QAE algorithm with clean error analysis, but the exact-E0 assumption is load-bearing and uncosted. the 1 major comments →

arxiv 2509.01359 v1 pith:6SWHRBUG submitted 2025-09-01 quant-ph

Heisenberg limited quantum algorithm for estimating the fidelity susceptibility

Yukun Zhang , Xiao Yuan This is my paper
classification quant-ph
keywords fidelity susceptibilityHeisenberg limitquantum singular value transformationquantum amplitude estimationfrustration-free Hamiltoniansquantum Fisher informationblock encodingquantum phase transitions
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.

The reading

The paper claims that fidelity susceptibility—the standard order-parameter-free probe for quantum phase transitions—can be estimated on a quantum computer at the Heisenberg precision limit, meaning the query count grows as 1/ε for additive error ε rather than 1/ε². Its key move is a resolvent identity that rewrites the susceptibility as the squared norm of the Moore-Penrose pseudoinverse of the shifted Hamiltonian applied to the ground state, collapsing an exponentially long excited-state sum. The paper backs this with a constructive algorithm: block-encode the shifted Hamiltonian and the perturbation, use QSVT to approximate the inverse, multiply the encodings, and read out the norm with quantum amplitude estimation. It proves explicit query counts and shows the same pipeline covers static susceptibilities and quantum Fisher information. For frustration-free Hamiltonians, a spectral-amplified block encoding yields a further quadratic speedup in the gap dependence. If the construction is correct, this is the first quantum algorithm with guaranteed efficiency for the quantity, enabling finite-size scaling studies of critical points on fault-tolerant hardware.

Core claim

Fidelity susceptibility χ_F, the second-order response of the ground-state fidelity to a perturbation H_I, is usually an exponential sum over excited states. The paper's central identity is χ_F = ||(H−E0)^+ H_I|Ψ0⟩||², where (H−E0)^+ is the Moore-Penrose pseudoinverse: it reciprocates excited-state energy differences and zeroes the ground component. Spectral decomposition proves the identity. The algorithm block-encodes H−E0 and H_I, uses QSVT to approximate 1/x over the spectrum, multiplies the encodings to form G, and uses quantum amplitude estimation to read ||G|Ψ0⟩||². Theorem 4 guarantees an ε-additive estimate with probability ≥ 8/π² using O(α_H α_I²/(Δ³ ε) log(α_I/(Δε))) queries to th

What carries the argument

The load-bearing object is the resolvent identity χ_F = ||(H−E0)^+ H_I|Ψ0⟩||². The Moore-Penrose pseudoinverse leaves the ground component at zero and inverts every excited-state energy difference, collapsing the spectral sum into one norm. Two tools carry the argument: QSVT, which applies a polynomial approximation of 1/x to a block-encoded matrix (a unitary holding the normalized matrix in its top-left block) and yields the pseudoinverse encoding; and quantum amplitude estimation, which reads the squared norm with a quadratic precision boost. For frustration-free Hamiltonians, the spectral-amplified operator H_SA = Σ_j |j⟩⟨j|⊗Π_j, together with a reflection, gives a normalization-one block

Load-bearing premise

The load-bearing premise is that the ground-state energy is known exactly: if it must be estimated from scratch, the estimation error must be much smaller than the energy gap or the inversion step breaks, and that cost is left out of the stated query complexity.

What would settle it

Take a small transverse-field Ising chain at fixed size, where χ_F can be computed by exact diagonalization. Implement the QSVT pseudoinverse block encoding with a certified gap lower bound Δ and count queries to U_H for a sequence of decreasing target errors ε and for two different Δ values. The theorem predicts query counts scaling as ε^{-1} and Δ^{-3} (or Δ^{-2.5} in the frustration-free case); observing ε^{-2} scaling, or a weaker Δ dependence, would falsify the central complexity claim.

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

If this is right

  • An additive-ε estimate of χ_F costs roughly 1/ε queries to the Hamiltonian block encoding, reaching the optimal precision scaling allowed by quantum mechanics.
  • The same resolvent-plus-amplitude-estimation pipeline gives Heisenberg-limited estimates of linear static susceptibilities and of quantum Fisher information for pure states, since both share the same spectral-sum form.
  • For frustration-free Hamiltonians, the inverse-query count improves from 1/Δ³ to √r/Δ^{2.5}, keeping local projector Hamiltonians with small gaps tractable.
  • Because the underlying linear-system inversion is BQP-complete, a general classical polynomial-time algorithm for the same task would collapse BQP to BPP; the paper's reduction therefore gives a concrete form of quantum advantage.

Where Pith is reading between the lines

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

  • Editorial inference: the advertised query counts treat the ground-state energy E0 as given. A fully self-contained protocol must also estimate E0 to precision well below Δ and prepare the ground state; that extra cost is not in Theorems 4 and 5, so the end-to-end advantage over classical methods depends on those subroutines staying polynomial.
  • Editorial inference: the resolvent formulation should extend to other zero-frequency response functions beyond static susceptibility and QFI—for example imaginary-time correlation integrals at zero frequency—whenever the target admits a spectral representation separated from the inverted eigenvalue.
  • Editorial inference: a simulator-based test can separate the two regimes by counting queries for a frustration-free and a matched non-frustration-free Hamiltonian with equal r and Δ; the theory predicts a half-power gap in the Δ dependence between the two cases.
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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

1 major / 4 minor

Summary. The paper proposes a quantum algorithm for estimating the fidelity susceptibility chi_F of a nondegenerate ground state. The central idea is a resolvent reformulation chi_F = ||(H - E0)^+ H_I |Psi0>||^2, where the Moore-Penrose pseudoinverse of the shifted Hamiltonian is block-encoded using quantum singular value transformation (QSVT), and the resulting norm is estimated by quantum amplitude estimation (QAE). The formal results state query complexities O(alpha_H alpha_I^2 / (Delta^3 epsilon) log(alpha_I/(Delta epsilon))) for U_H and O(alpha_I^2/(Delta^2 epsilon)) for U_I, with success probability at least 8/pi^2. A second result for frustration-free Hamiltonians improves the gap dependence to O(sqrt(r) alpha_I^2/(Delta^{2.5} epsilon) log(r alpha_I/(Delta epsilon))). Applications to static susceptibilities and quantum Fisher information are also discussed.

Significance. If the technical gaps are repaired, this is a solid and useful contribution. The resolvent reformulation is elegant and is explicitly shown to be equivalent to the standard perturbative sum in Eq. (3). The QSVT polynomial degree, block-encoding error propagation, and QAE scaling are internally consistent, and the complexity statements are tied to explicit block-encoding input assumptions. There are no fitted parameters or circular arguments. The frustration-free speedup is a genuine extra result. The main weakness is that the algorithm assumes exact knowledge of E0, and the cost of obtaining E0 is not included in the stated query complexity.

major comments (1)
  1. [Supplementary, 'Main results' (after Eq. (9)) and Theorem 4 (Eqs. (23)-(24))] The assumption that E0 is known exactly is load-bearing, not a convenience. Lemma 4 and Theorem 3 block-encode the pseudoinverse (H-E0)^+ using an odd polynomial p_d whose approximation guarantee (Lemma 3) holds only outside the interval [-delta, delta] with delta = Delta/alpha_H. If E0 is replaced by an estimate E0 + delta0, the shifted operator H - E0 - delta0 I has eigenvalue -delta0 on |Psi0>. For any reasonable estimation error delta0 << Delta, this eigenvalue lies inside the excluded interval, so p_d(-delta0/alpha_H) is uncontrolled. Oddness gives only p_d(0)=0, not smallness at -delta0/alpha_H. The block encoding then contains a spurious ground-state term p_d(-delta0/alpha_H)|Psi0><Psi0|H_I|Psi0>, which can induce an error in chi_F of order |p_d(-delta0/alpha_H)| |<Psi0|H_I|Psi0>|, potentially exceeding epsilon. Moreover, the query counts in Theorem 4 do not include the cost of es
minor comments (4)
  1. [Abstract and Theorems 1-2 (Informal)] The abstract and informal theorems say 'with high probability', but the formal Theorems 4 and 5 guarantee only probability at least 8/pi^2. Since 8/pi^2 is a constant, the success probability can be boosted to 1-delta by O(log(1/delta)) repetitions, but this overhead should be stated or the wording changed.
  2. [Proof of Theorem 4, after Eq. (26)] 'the final estimation is given by alpha_Q E' should read alpha_Q^2 E; the subsequent display correctly uses alpha_Q^2 E.
  3. [FF section, Lemma 6 and definition of frustration-free Hamiltonians] Minor typos: 'PREPARA' should be 'PREPARE'; 'intersection_i Pi != empty' should be 'intersection_j ker(Pi_j) != empty'. Also in Eq. (34), P^perp is used before being defined; please define it before the display.
  4. [Supplementary, 'Main results'] The sentence 'we can get an estimation of E0 to an arbitrary desirable accuracy' is not a substitute for the exact-knowledge assumption. This sentence should be reconciled with the exact-E0 assumption, or removed.

Circularity Check

0 steps flagged

No significant circularity: the resolvent reformulation is an exact identity, query bounds follow from QSVT/QAE and external results, and the exact-E0 assumption is a completeness gap rather than a circular step.

full rationale

Walking the derivation chain: (1) Eq. (5) introduces the resolvent expression chi_F = ||(H-E0)^+ HI |Psi0>||^2, and the paper explicitly verifies it: 'The formula is equivalent to the perturbative expansion given by Eq. (3). Our expression is verified by spectral decomposition of the resolvent...' This is an exact reformulation of the standard fidelity susceptibility, not a redefinition or a fit. (2) The complexity claims in Theorem 4 follow from Lemma 4 (QSVT polynomial approximation of the reciprocal of the shifted Hamiltonian) and Lemma 5 (quantum amplitude estimation), with errors propagated through Eqs. (26)-(29). The query counts depend on the normalization factors alpha_H, alpha_I, the gap lower bound Delta, and the target error epsilon, but not on the value of chi_F itself, so there is no fitted-input-called-prediction. (3) The frustration-free speedup in Theorem 5 invokes Lemmas 6-8 from Refs. [34] and [39], which are external prior works, not authored by Zhang/Yuan; the adaptation is explicit about its assumptions. There is no load-bearing self-citation chain. (4) The main caveat is the assumption of exact knowledge of E0: the Supplementary states 'For simplicity, we here assume that we know E0 exactly.' This is a real completeness gap — estimating E0 to precision much smaller than Delta would add cost not included in the theorem's query counts — but it is not circularity: E0 is an input oracle, and the derivation does not define E0 in terms of chi_F or vice versa. The algorithm is conditional on supplied ground-state access and E0; those assumptions are stated, and the derivation from them is self-contained. No circular step is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claim rests on standard QSVT/QAE machinery plus problem-specific oracles (ground state, energy, block encodings). No parameters are fitted and no new entities are postulated. The most fragile input is the exact-E0 assumption, which is outside the counted complexity.

axioms (6)
  • domain assumption Nondegenerate ground state and a known lower bound Delta on the spectral gap.
    Assumptions (1) and (2) in Main Results. Required for the inverse to be well-defined and for the QSVT polynomial approximation.
  • domain assumption Access to a ground-state preparation unitary and exact knowledge of E0.
    Assumption (3) and the exact-E0 sentence in Supplementary. The cost of obtaining these is outside the stated complexity.
  • domain assumption Exact block encodings of H - E0 and HI with known normalizations alpha_H and alpha_I.
    Theorems 1 and 3 assume (alpha, m, 0)-block encodings with zero error; in practice these are exact via LCU only when coefficients and E0 are known exactly.
  • standard math QSVT polynomial approximation lemma and product-of-block-encodings lemma from Gilyen et al. [17].
    Cornerstone tools, cited and used without proof.
  • standard math Quantum amplitude estimation lemma from Brassard et al. [19].
    Used for norm estimation; the paper quotes the lemma but does not prove it.
  • domain assumption Frustration-free block encoding lemmas from King et al. [34] (unpublished) and Proposition 12 of Orsucci and Dunjko [39].
    The FF speedup in Theorem 5 is derived by direct reference to these external results, not proved in the paper.

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Cite this review

Pith. "Pith review of Heisenberg limited quantum algorithm for estimating the fidelity susceptibility." pith.science (2026). https://pith.science/paper/6SWHRBUG

@misc{pith2026250901359,
  author       = {Pith},
  title        = {Pith review of: Heisenberg limited quantum algorithm for estimating the fidelity susceptibility},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6SWHRBUG}},
  note         = {Machine review of arXiv:2509.01359}
}
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read the original abstract

The fidelity susceptibility serves as a universal probe for quantum phase transitions, offering an order-parameter-free metric that captures ground-state sensitivity to Hamiltonian perturbations and exhibits critical scaling. Classical computation of this quantity, however, is limited by exponential Hilbert space growth and correlation divergence near criticality, restricting analyses to small or specialized systems. Here, we present a quantum algorithm that achieves efficient and Heisenberg-limited estimation of fidelity susceptibility through a novel resolvent reformulation, leveraging quantum singular value transformation for pseudoinverse block encoding with amplitude estimation for norm evaluation. This constitutes the first quantum algorithm for fidelity susceptibility with optimal precision scaling. Moreover, for frustration-free Hamiltonians, we show that the resolvent can be approximated with a further quadratic speedup. Our work bridges quantum many-body physics and algorithmic design, enabling scalable exploration of quantum criticality with applications in materials simulation, metrology, and beyond on fault-tolerant quantum platforms.

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.