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REVIEW 3 major objections 4 minor 21 references

Better product formulas for quantum phase estimation

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For a gapped eigenstate, the energy-estimation error of an order-$p$ product formula equals the average of a nested-commutator expectation plus $O(s^{2p})$, and a custom 9-term formula attains fourth-order energy accuracy.

desk verdict The Magnus-based QPE error analysis is worth a look, but the custom product formula construction fails as printed: the supplementary condition does not enforce the claimed error scaling. read the letter →

arxiv 2412.16811 v1 pith:S4LRX27L submitted 2024-12-22 quant-ph

Kasra Hejazi , Jay Soni , Modjtaba Shokrian Zini , Juan Miguel Arrazola This is my paper · ORCID
classification quant-ph
keywords quantumphaseestimationproductformulasTrotter-SuzukiformulaMagnusexpansionnestedcommutatorsenergyerrorlow-energyboundsHamiltoniansimulation
verification ladder T0 review T1 audit T2 compute T3 formal

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 when product formulas are used inside quantum phase estimation, the error in the estimated energy can be much smaller than the worst-case spectral error of the time-evolution operator. For an order-$p$ product formula on a gapped eigenstate, it derives a closed-form expansion: $\delta E = \frac{1}{s}\int_0^s \langle\psi|E(\sigma)|\psi\rangle\,d\sigma + O(s^{2p})$, where $E(\sigma)$ is a sum of nested commutators of Hamiltonian terms. This formula turns the design of product formulas into a cancellation problem: any nested commutator with the full Hamiltonian on its outermost slot contributes nothing to the energy error. The paper exploits this to construct a 9-term second-order formula whose energy error is $O(s^4)$, matching a fourth-order Trotter-Suzuki formula that uses 11 terms.

What carries the argument

The central object is the effective Hamiltonian $\tilde{H}(s)$ defined by the single-step exponential identity $e^{-i\tilde{H}(s)s}=W(s)$. The product formula is first written as a time-ordered exponential $W(s)=T\exp(-i\int_0^s F(\sigma)\,d\sigma)$ with $F(\sigma)=H+E(\sigma)$, where $E(\sigma)$ is an exponent operator built from nested commutators of the Hamiltonian terms, each weighted by product-formula coefficients. A Magnus expansion then converts the time-ordered exponential into the single exponential $e^{-i\tilde{H}(s)s}$, giving $\tilde{H}-H = \frac{1}{s}\int_0^s E(\sigma)\,d\sigma$ plus higher nested-commutator terms. The identity $\delta E=\frac{1}{s}\int\langle\psi|E(\sigma)|\psi\rangle\,d\sigma+O(s^{2p})$ follows because terms with $H$ in the outermost commutator have zero expectation on an eigenstate. The custom product formula $W_2(s)=e^{-iAa_1s}e^{-iBa_2s}e^{-iAa_3s}e^{-iBa_4s}e^{-iAa_5s}$ is constructed so that the leading surviving term in $E(\sigma)$ is proportional to $[H,[H,B]]$.

What would settle it

Compute the exact QPE energy error $\delta E(s)$ for a two-level Hamiltonian $H=A+B$ whose effective Hamiltonian $\tilde{H}(s)$ has a gap that closes at some step size inside $[0,s]$. If the measured error deviates from $\frac{1}{s}\int_0^s \langle\psi|E(\sigma)|\psi\rangle\,d\sigma$ by a term that is not $O(s^{2p})$, the perturbation expansion fails. Alternatively, on a random multi-qubit Hamiltonian, if the 9-term formula $W_2(s/2)W_2'(s/2)$ shows energy-error scaling $s^2$ or $s^3$ rather than $s^4$ at small step sizes, the central design claim is wrong.

Watch

Extended reading notes

Core claim

The central discovery is a perturbation-theoretic identity for the error in energy estimates obtained from product formulas. For any product formula $W(s)$ of order $p$, define the effective Hamiltonian $\tilde{H}(s)$ by $e^{-i\tilde{H}(s)s}=W(s)$. Under a non-vanishing spectral gap, the shift in the target eigenvalue is, up to $O(s^{2p})$, the time-average of the expectation value of the exponent operator $E(\sigma)$: $\delta E = \frac{1}{s}\int_0^s \langle\psi|E(\sigma)|\psi\rangle\,d\sigma + O(s^{2p})$ (Eq. (9)). Because $H$ appears as a factor in many of the nested commutators inside $E(\sigma)$, and $\langle\psi|[H,\cdot]|\psi\rangle=0$ for an eigenstate, many leading contributions vanish. For $H=A+B$, the paper exhibits a five-exponential second-order formula $W_2(s)$ with $E(\sigma)=3\alpha_3\sigma^2[H,[H,B]]+O(\sigma^3)$, making the energy error $O(s^3)$, and its symmetrized combination $W_2(s/2)W_2'(s/2)$ yields an error $O(s^4)$. For low-lying eigenstates of a class of $k$-local positive Hamiltonians, the coefficients in the expansion grow only polylogarithmically in system size up to order $2p-1$, which leads to a step-size scaling $s=O(\epsilon^{1/p}/\log(N)^{1+1/p}+\epsilon^{1/(2p)}\operatorname{poly}(N))$ and an up-to-quadratic asymptotic speedup in the target error.

Load-bearing premise

The analysis assumes the target energy level stays separated from all other levels of the simulated effective Hamiltonian at every step size up to the chosen one; if that gap closes, the predicted error formula no longer holds.

Editorial extensions

If this is right

  • For a gapped eigenstate, an order-$p$ product formula with exponent operator $E(\sigma)$ has energy-estimation error $\delta E = \frac{1}{s}\int_0^s \langle\psi|E(\sigma)|\psi\rangle\,d\sigma + O(s^{2p})$, so any nested-commutator term with $H$ on the outermost slot contributes nothing at leading order.
  • For $H=A+B$, the five-exponential formula $W_2(s)$ achieves $O(s^3)$ energy error, improving the required step size from $s=O(\epsilon^{1/2})$ for second-order Trotter-Suzuki to $s=O(\epsilon^{1/3})$.
  • The symmetrized 9-term formula $W_2(s/2)W_2'(s/2)$ achieves $O(s^4)$ energy error, matching a fourth-order Trotter-Suzuki formula while using 9 exponentials per step instead of 11.
  • For low-energy eigenstates of $k$-local positive-semidefinite fast-forwardable Hamiltonians, the step size can scale as $s=O(\epsilon^{1/p}/\log(N)^{1+1/p}+\epsilon^{1/(2p)}\operatorname{poly}(N))$, giving up to a quadratic asymptotic speedup in the target error $\epsilon$.
  • The results apply to any energy-estimation routine that consumes simulated time evolution, not only to quantum phase estimation's internal protocol.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same cancellation criterion is generic: any order-$p$ formula whose nested commutators of orders $p$ through $2p-1$ all have $H$ on the outermost slot should inherit an $O(s^{2p})$ energy error; constructing such formulas at higher orders is left for future work.
  • Because the analysis depends only on the input/output structure of the energy-estimation routine, the step-size savings should carry over to other energy-estimation schemes that query simulated time evolution, such as amplitude-estimation-based approaches.
  • The observed $s^4$ scaling for $W_2$ on the XY model, instead of the generic $O(s^3)$, suggests that model-specific commutator structure can further improve the energy error beyond the universal family; this could be tested systematically on other lattice models.
  • If the paper's conjecture that the positivity assumption in the low-energy bound can be relaxed is correct, the polylogarithmic improvement would extend to fermionic Hamiltonians used in electronic structure simulation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a Magnus-based perturbative analysis of the error in energy estimation (e.g., QPE) when the Hamiltonian evolution is implemented with product formulas. The main theoretical result is Eq. (9), which expresses the leading energy error as an expectation value of the exponent operator of the product formula, and generalizes previous first-order results to arbitrary order. Based on this, the authors propose a family of five-exponential second-order product formulas W2 for H=A+B, and a nine-exponential symmetrized combination W2(s/2)W2'(s/2), claiming energy errors O(s^3) and O(s^4), respectively. They also connect their analysis to low-energy bounds from prior work [12] to argue for improved cost scaling for low-energy states of certain k-local Hamiltonians.

Significance. If the custom product formula result were correct, it would be a significant advance: a task-specific (rather than worst-case) error analysis for product formulas, and a concrete formula that outperforms second-order Trotter-Suzuki in energy estimation at equal exponentials count. The general framework in Eq. (9) and the low-energy bound are useful and likely correct. The paper states its spectral-gap assumption explicitly and provides a supplementary derivation of the perturbation bound. However, the central custom-formula construction contains an algebraic error (see major comments), and the numerics are not reproducible as reported. The general contributions do not compensate for the collapse of the paper's headline claim.

major comments (3)
  1. [Supplementary material, Eq. (22)] Supplementary Eq. (22) does not enforce the equal-coefficient condition needed for the central claim in Eq. (16). The condition that the sigma^2 term in F(sigma) (Eq. (19)) be proportional to [H,[H,B]] is f_AA,B = f_BA,B. Expanding Eq. (15) for the five exponentials in Eq. (17) gives f_AA,B = (1/2)a2 a3^2 + a2 a3 a5 + (1/2)a2 a5^2 + (1/2)a4 a5^2 - (a1 a2 a3 + a1 a2 a5 + a1 a4 a5 + a3 a4 a5) and f_BA,B = a2 a3 a4 - ((1/2)a1 a2^2 + a1 a2 a4 + (1/2)a1 a4^2 + (1/2)a3 a4^2). The printed Eq. (22) is not algebraically equivalent to f_AA,B = f_BA,B after imposing Eqs. (20)-(21). For the paper's stated example a4=-0.3, solving Eqs. (20)-(22) gives a3 approx 0.021 or a3 approx 6.908, and in both cases f_AA,B - f_BA,B is nonzero (approx 0.137 and approx -23.8, respectively). Consequently, the effective Hamiltonian contains a nonzero component [A,[A,B]]-[B,[A,B]] at O(s^2), whose expectation in a generic eigenstate of H=A+B need not vanish, so deltaE is O(s^2), not O(s^3). This invalidates Eq. (16) and, through Eq. (13), the O(s^4) claim for W2(s/2)W2'(s/2) and Table I.
  2. [Section 'Customized product formulas'; Fig. 1] The numerical verification in Fig. 1 cannot be checked against the printed construction because the coefficient values a1, a2, a3, a5 used for a4=-0.3 are not reported. Given that the printed equations do not produce the claimed exponent operator, the observed scalings in Fig. 1 do not provide evidence for the validity of the construction as stated. The authors should provide the actual coefficient values and verify explicitly that they satisfy the correct equal-coefficient condition f_AA,B = f_BA,B, not Eq. (22).
  3. [Main text after Eq. (10); Fig. 1 (left)] The statement that 'one must choose the free variable outside of the range 0 < a4 < 1' is a property of the incorrect Eq. (22). The correct condition f_AA,B = f_BA,B leads to a different solution set; the existence of a real solution family and the admissible range of a4 must be re-established. Without a correct derivation, the claimed existence of the custom W2 family is unsupported.
minor comments (4)
  1. [Fig. 1 (middle) and text after Eq. (16)] The figure legend labels the W2 curve as '~ s4', while the text says 'We expect an s3 behavior for W2, but the s4 behavior appearing here is likely due to the structure of the model.' This is confusing; please clarify whether the plotted scaling is the observed numerical slope or the leading-order theoretical prediction, and reconcile the two statements.
  2. [Eq. (13)] The phrase 'by inspection' is too terse for the claimed expansion W2(s/2)W2'(s/2) - e^{-iHs} = alpha3 s^3 [H,[H,B]] + alpha4 s^4 [H,[H,[H,B]]] + O(s^5); provide a short derivation or at least specify how the coefficients alpha3 and alpha4 are obtained from the W2 solution.
  3. [Main text, Eq. (9) and surrounding text] The symbol E(sigma) is used in Eq. (9) and in the text as the 'exponent operator', but it is not defined in the main text; please define it there or refer explicitly to the supplementary definition, and avoid any confusion with the eigenvalue E.
  4. [After Eq. (4)] The nonvanishing spectral gap assumption is stated for all step sizes between 0 and s, but it is not checked for the numerical examples (XY model and random Hamiltonian). Please comment on whether this assumption is satisfied in the simulations, or how violation would affect the reported scalings.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: main QPE error derivation and custom product-formula construction are self-contained; only minor self-citation in low-energy section.

full rationale

The central derivation is self-contained. Eq. (9) is obtained by combining the Magnus expansion with standard non-degenerate perturbation theory; the O(s^{2p}) remainder is an honest higher-order correction, and the argument that only the expectation value of E(σ) contributes up to that order is given explicitly in the observations preceding Eq. (9). No fitted parameter is renamed as a prediction: the custom W2 coefficients are solved algebraically from the conditions in Eqs. (20)-(22), and the numerical XY-lattice and random-Hamiltonian experiments are independent checks of the constructed formulas, not regression fits. The only self-citation is [12] in the low-energy section, where the paper imports a bound via 'an analysis identical to the one offered in the supplementary materials of [12]'. This is a self-citation and it is load-bearing for that section's speedup claim, but it is not a definitional reduction of the paper's main QPE error analysis, and it does not smuggle in the target result: [12] concerns low-energy dynamics, not energy-estimation error. The paper also flags the spectral-gap assumption limiting Eq. (9), which is a stated assumption rather than a hidden circular input. A possible algebraic inconsistency in Eq. (22), if confirmed, would be a correctness defect rather than a circularity, so it does not change this verdict.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central results rest on standard perturbation theory and Magnus expansion (standard math), plus domain assumptions about the Hamiltonian structure, spectral gap, and the validity of the low-energy bounds from the authors' prior work. No new entities are introduced. The only free parameter is the coefficient a4 in the custom product formula, which is a tunable degree of freedom, not a fit to data.

free parameters (1)
  • a4 = -0.3 (chosen for numerics)
    In the custom product formula W2(s), the coefficient a4 is left free; the equations (20)-(22) determine a1,a2,a3,a5 as functions of a4. The paper plots these solutions and picks a4=-0.3 for the numerical verification.
assumptions (5)
  • domain assumption The target eigenstate |ψ⟩ is separated from all other eigenstates of H̃(s) by a non-vanishing spectral gap for all s in [0,s].
    Stated in the main text after Eq. (4) ('a nonvanishing gap... assumed') and used in the supplementary perturbation theory (Eqs. (26)-(27)) to bound the projection difference f(s) via Lemma 8 of Ref. [17].
  • standard math The Magnus expansion for the effective Hamiltonian H̃(s) converges or is valid as an asymptotic series so that order-by-order terms can be computed.
    The expansion in Eq. (7) is taken from Ref. [15]; the paper uses it to derive Eq. (9) and assumes the perturbative order counting holds.
  • domain assumption For the custom product formula, the Hamiltonian decomposes as H=A+B with both A and B fast-forwardable.
    This is the setup for the custom formulas in Section 'Customized product formulas' and includes electronic structure Hamiltonians and spin chains.
  • domain assumption For the low-energy bound, H is a k-local Hamiltonian with M=O(1) fast-forwardable positive semidefinite terms, each a sum of O(N) k-local interactions with each qubit in at most d interactions, and q=O(1) in the product formula.
    These are the assumptions of Ref. [12] (arXiv:2402.10362), which the paper invokes for the low-energy analysis.
  • domain assumption The initial state has an appreciable overlap with the target eigenstate, which is required for QPE sampling efficiency.
    Mentioned in the main text: 'the only practical requirement is that it has an appreciable overlap with the target state'.

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Cite this review

Pith. "Pith review of Better product formulas for quantum phase estimation." pith.science (2026). https://pith.science/paper/S4LRX27L

@misc{pith2026241216811,
  author       = {Pith},
  title        = {Pith review of: Better product formulas for quantum phase estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S4LRX27L}},
  note         = {Machine review of arXiv:2412.16811}
}
read the original abstract

Quantum phase estimation requires simulating the evolution of the Hamiltonian, for which product formulas are attractive due to their smaller qubit cost and ease of implementation. However, the estimation of the error incurred by product formulas is usually pessimistic and task-agnostic, which poses problems for assessing their performance in practice for problems of interest. In this work, we study the error of product formulas for the specific task of quantum energy estimation. To this end, we employ the theory of Trotter error with a Magnus-based expansion of the effectively simulated Hamiltonian. The result is a generalization of previous energy estimation error analysis of gapped eigenstates to arbitrary order product formulas. As an application, we discover a 9-term second-order product formula with an energy estimation error that is quadratically better than Trotter-Suzuki. Furthermore, by leveraging recent work on low-energy dynamics of product formulas, we provide tighter bounds for energy estimation error in the low-energy subspace. We show that for Hamiltonians with some locality and positivity properties, the cost can achieve up to a quadratic asymptotic speedup in terms of the target error.

Figures

Figures reproduced from arXiv: 2412.16811 by the authors.

Figure 1
Figure 1. FIG. 1. Left: Solutions of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

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Reference graph

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