Trotter error compensation with polylogarithmic precision and nested-commutator scaling without ancillas

Jinzhao Sun; Pei Zeng; Qi Zhao; Shuo Zhou; Tom Gur; Tongyang Li; Xinzhao Wang; Ziruo Wang

REVIEW 2 major objections 4 minor 38 references

Trotter error compensation with polylogarithmic precision and nested-commutator scaling without ancillas

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A channel-level Trotter-error compensation method achieves polylogarithmic precision dependence in circuit size while keeping the standard O(ε⁻²) repetition cost and no ancillary qubits.

desk verdict Genuinely new algorithm for Trotter error compensation, but the central precision bound rests on unproved lemmas from the authors' companion preprint; deserving of serious review with a request to make that dependence self-contained. read the letter →

arxiv 2607.11856 v2 pith:QDQNX376 submitted 2026-07-13 quant-ph

Xinzhao Wang , Shuo Zhou , Ziruo Wang , Pei Zeng , Jinzhao Sun , Qi Zhao , Tom Gur , Tongyang Li This is my paper

classification quant-ph MSC 81P68

keywords TrottererrorHamiltoniansimulationBaker-Campbell-Hausdorffexpansionproductformulaslinearcombinationofquantumchannelsobservableestimationcommutatorscalingpolylogarithmicprecision

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 introduces high-order nested-commutator compensation (HNCC), a Hamiltonian simulation method that keeps the practical advantages of Trotter product formulas while improving the precision dependence of circuit size from polynomial to polylogarithmic. For a fixed K-th order product formula on a k-local Hamiltonian, HNCC estimates tr[O e^{-itH} ρ e^{itH}] to additive precision ε||O|| with the standard O(ε⁻²) independent circuit repetitions and no ancillary qubits. The central claim is that by writing the Trotter remainder through a truncated Baker–Campbell–Hausdorff expansion and compensating it at the level of quantum channels via randomly sampled Pauli rotations, the maximum gate count per circuit grows only polylogarithmically in 1/ε, effectively matching the scaling of a product formula of order 2K+1 while preserving commutator scaling. A sympathetic reader would care because HNCC is the first product-formula-based scheme that simultaneously offers commutator scaling, effective higher-order time scaling, polylogarithmic precision dependence, O(ε⁻²) sampling, and an ancilla-free implementation.

What carries the argument

The key identities are the parameter-shift rule ad_{iP}(ρ) = U_{e^{iπ/4}P}(ρ) − U_{e^{−iπ/4}P}(ρ), which turns each nested commutator of Pauli operators into a difference of two Pauli-rotation channels, and the companion identity-pairing identity I + c·ad_{iP} = (1+c²)U_{e^{iθP}} − c²U_{iP} with θ = tan⁻¹(c), which lowers the 1-norm from 1+O(c) to 1+O(c²). These are combined with a truncated BCH expansion of the Trotter remainder, a light-cone sampler that draws nested commutators from overlapping supports, and a cutoff on the total BCH order per circuit that keeps the accumulated bias at O(ε).

What would settle it

Take the 1D periodic Heisenberg chain with N=6, g0=1, k=2 and compute the doubly right-nested commutator sum in Eq. (25) for a specific sequence such as q1=q2=2, checking whether the bound (1/(2k q_d))·(∏_{r=1}^d P_{r+1} q_r! (2k g̃)^{q_r})·N is respected for every term ordering; a violation would disprove Lemma 4 and hence Lemma 5. Alternatively, numerically evaluate ‖V_K(x) − exp(∑_{q=K+1}^{q0} Φ_q(x))‖ for the Heisenberg chain at q0=8, N=10, t=1, and x = t/ν with the paper's chosen ν, and compare it against the claimed bound 2(4e q0 k g̃)^{q0+1} N; if the actual deviation exceeds that bound

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the multiplicative Trotter remainder V_K(x) = U(x) S_K(x)† can be approximated, to arbitrarily high order and with nested-commutator structure intact, by a linear combination of quantum channels that are compositions of π/4 Pauli rotations (Clifford gates) plus one additional Pauli rotation. The BCH expansion of V_K(x) is truncated at order q₀ = Θ(log(1/ε)); the paper proves that with a light-cone sampler the LCQC 1-norm stays bounded, and that the truncation plus a cutoff on the total BCH order contribute only O(ε) bias. Pairing the terms linear in the BCH generator with the identity channel reduces the 1-norm from 1 + O(c) to 1 + O(c²), which

Load-bearing premise

The bound that controls how much error comes from cutting off the Baker-Campbell-Hausdorff expansion is proved using lemmas from a companion preprint that are not proved inside this paper. If those lemmas are wrong, the whole precision guarantee fails.

Editorial extensions

If this is right

For any fixed Trotter order K, the maximum gate count per circuit scales polylogarithmically in 1/ε, replacing the polynomial precision dependence of standard Trotter formulas.
The repetition cost stays at the standard O(ε⁻²), in contrast to Richardson extrapolation approaches that incur extra log factors from coefficient amplification.
HNCC is ancilla-free: the compensation is applied at the channel level, so no Hadamard tests and no controlled compensation operations are needed, which can improve circuit depth on limited-connectivity devices.
An unpaired variant achieves polylog precision while using only O(k log(1/ε)) extra Clifford gates beyond the original product formula, at the cost of keeping the original system-size and time scaling.
Finite-size resource estimates for the periodic Heisenberg chain show up to 25.9× reduction in estimated T-gate count per circuit relative to the uncompensated second-order Trotter formula.

Reading between the lines

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

The identity-pairing trick that converts a 1+O(c) 1-norm into 1+O(c²) may be applicable beyond this specific Trotter-compensation setting, potentially reducing sampling overhead in other linear-combination-of-channels constructions.
The paper explicitly leaves open whether channel-level compensation extends to Lindbladian simulation and quantum singular value transformation settings that use product-formula approximations; a natural testable extension is to apply the same truncated-BCH channel compensation to those algorithms.
The connected-cluster preprocessing that merges identical Pauli terms may give a practical classical preprocessing advantage for any local Hamiltonian with bounded overlap degree, but the paper does not claim an efficient reduction for general k-local Hamiltonians; that is an open gap a reader could investigate.
Because the polylog guarantee rests on companion-paper lemmas not proved here, the most direct way to gain confidence is to verify those lemmas independently or to numerically test the BCH truncation bound on small local Hamiltonians.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Let me give you the short version: this is a real contribution with a load-bearing external dependency. HNCC is a genuine new combination—nested-commutator scaling, polylog(1/eps) circuit size, O(eps^-2) repetition count, no ancillas—that none of the previous methods achieved. The internal machinery is coherent: channel-level LCQC with identity pairing to cut the 1-norm from 1+O(c) to 1+O(c^2), the light-cone sampler, the cutoff scheme. The unpaired variant is a nice bonus: polylog precision with only extra Clifford gates.

The problem is the foundation. The main theorem's polylog precision depends on the BCH truncation bound in Lemma 5, and that bound is built on Lemmas 3 and 4, which are not proved in this manuscript—they are quoted from the authors' companion preprint [26]. It doesn't stop there: even Lemma 2, the norm bound on individual BCH terms, uses [26, Cor. 2] for the final nested-commutator sum. So a large part of the technical stack is outsourced. The paper is explicit about this, and the lemmas look plausible, but they are not independently verified here. If any of those bounds has an extra N-dependent prefactor or a norm factor that isn't 1, the q0 = Θ(log 1/eps) choice fails and the polylog scaling and O(eps^-2) repetition claim both break. That's not a speculative issue; it's the single point the whole theorem rests on.

Numerics are also a bit soft: no code or data shipped, and the m=5 parameter is picked at one operating point (t=1, eps=1e-3) and then held fixed across the scans. That's a minor issue—the asymptotic results are what matter—but it limits the weight you can put on the 'lowest T-gate count' claim.

None of this makes me think the authors are wrong. The rest of the proof is careful and the algorithm design is smart. It just means a referee can't verify the central theorem without also verifying [26]. I'd send this to review, but I'd ask the authors to include full proofs of the imported lemmas in the paper or a supplementary appendix. A conditional acceptance would be reasonable.

Who's the reader? Anyone working on product formulas, Trotter error mitigation, or early fault-tolerant simulation. It's worth reading for the construction and the open problem it leaves.

Referee Report

2 major / 4 minor

Summary. The paper introduces HNCC, an ancilla-free product-formula-based Hamiltonian simulation algorithm that compensates Trotter error at the channel level. The Trotter remainder is expanded in a truncated BCH series; each BCH term is sampled as a nested commutator using a light-cone sampler, and each adjoint-Pauli factor is represented by a difference of Pauli-rotation channels. An identity-pairing trick reduces the LCQC 1-norm from 1+O(λ) to 1+O(λ^2), giving an effective product-formula order 2K+1. The main theorem (Theorem 1/Theorem 4) claims O(ε^{-2}) repetitions and a maximum gate count per circuit that is polylogarithmic in 1/ε and exhibits nested-commutator scaling in N and Γ. An unpaired variant (Theorem 5) achieves polylog precision with the system-size/time scaling of the original formula and only Clifford compensation gates. Finite-size estimates for the periodic Heisenberg chain are presented, showing the lowest estimated T-gate count among the methods compared.

Significance. If the main theorem holds, this is a substantial contribution: it is the first product-formula-based method to combine polylogarithmic precision dependence in circuit size, standard O(ε^{-2}) sampling cost, nested-commutator scaling, and an ancilla-free implementation in a single construction. The technical machinery is rich and mostly coherent: the channel-level LCQC sampling, the light-cone sampler, the identity-pairing reduction, and the cutoff analysis are internally consistent and the proofs of Theorems 2–5 are detailed and checkable. The paper is also honest about the companion-preprint dependency. However, the central polylog-precision guarantee is conditional on importable bounds that are not proved here. The construction is attractive and the claimed result is plausible, but the manuscript as submitted is not self-contained at its load-bearing point.

major comments (2)

[Section IV.B, Lemmas 3–5 and Section IV.D, Theorem 4] The claim that q0 = Θ(log(1/ε)) and hence the polylog gate count is load-bearing rests on unproved external results. Lemma 3 is stated as the 'Hamiltonian specialization of [26, Theorem 2]' and its proof is essentially a citation; Lemma 4 is quoted verbatim from [26, Theorem 9]; Lemma 5 is proved only from these two. In the proof of Theorem 4, the accumulated BCH bias is bounded by 2νN(4eq0k g̃)^{q0+1}, and this exact exponential tail is what justifies q0 = Θ(log(1/ε)). These imported lemmas supply both the norm factor 1 in Eq. (24) and the prefactor in Eq. (25). If either bound has an extra polynomial factor in N or q0, the bias budget would force q0 = Θ(log N + log(1/ε)), changing the stated gate-count scaling. The same dependency appears already in Lemma 2, whose proof invokes [26, Corollary 2] for the commutator sum defining λ_q; Eq. (34) and the entire 1-norm analysis rely on that b
[Section IV.D, Theorem 4 proof, discarded-sample bias] The proof that the s0 cutoff introduces only O(2^{-s0}) bias uses the weighted identity Σ_j 2^{d_j}|β_j| = (1 + λ_single(1)^2/2 + Σ_{r≥2} λ_single(2)^r/r!)^{mν}. This is correct, but it is not fully derived in the text. In particular, for the paired sampler the linear part is implemented by the identity-pairing formula, not by coefficients λ_q, and the reader must reconstruct the exact coefficients of the one-step LCQC to verify the equality. I checked the construction and it works, but the proof should spell out this step: state the one-step weighted LCQC norm, then take the mν-fold product. Since the whole cutoff argument depends on this inequality, it should be a formal lemma rather than a displayed equation in the proof of Theorem 4.

minor comments (4)

[Theorem 1 / Algorithm 1] The 'maximum gate count per circuit' guarantee applies to circuits that pass the cutoff in Line 7; circuits that fail the cutoff are not executed and output X=0. This should be stated in the theorem so the reader does not interpret the maximum over all sampled indices as including discarded terms.
[Section VI, Figures 3 and 4] The axis labels appear to read '10 4' and '10 3' instead of 10^{-4} and 10^{-3}. Please fix the formatting; also specify in the captions that the target precision decreases from left to right.
[Figure 1 and Algorithm 1] The notation is inconsistent between the figure and the algorithm: the figure uses ω_{ℓ,a}, ω_tot, and λ^{mν}_{paired}, while the algorithm uses η_{ℓ,a}, η, and λ^{mν}_{paired}. Unify the symbols.
[Section V.E, Proposition 1] The connected-cluster preprocessing is analyzed only for the classical cost of building the BCH generator table. This cost is not part of either Theorem 1's gate count or the numerical resource estimates. It would help to state explicitly that the theorem's circuit-size bounds exclude classical preprocessing, and that the numerical results include this preprocessing only indirectly through the merged Pauli table.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is a constructed parameter choice; the load-bearing [26] commutator bounds are general external lemmas, so reliance on them is a verifiability gap, not a circular reduction.

full rationale

The derivation chain for Theorem 1 is: Lemma 2 bounds individual BCH terms via [29, Prop. 5] and [26, Cor. 2]; Lemma 5 bounds the BCH truncation error using Lemma 3 (specialization of [26, Thm. 2]) and Lemma 4 (quoted from [26, Thm. 9]); Theorem 4 then chooses q0 = Theta(log 1/eps), nu, and s0 so that the truncation, cutoff, and sampling errors each fall below eps/4 and the LCQC 1-norm lambda_paired^{m nu} = O(1), giving O(eps^-2) repetitions and the stated gate count. This is proof-by-construction: the free parameters are selected to satisfy the proved inequalities, not fitted to reproduce a known simulation outcome. The only debatable point is that Lemmas 3 and 4 are not proved in this manuscript; they are imported from the authors' companion preprint [26], whose author list overlaps. Under the rubric, however, self-citation is circular only when the cited result is itself the target, or is an unverified ansatz that forbids alternatives. Here [26, Thm. 2] and [26, Thm. 9] are general, parameter-free commutator bounds for k-local, g-extensive operator sets; they do not state or assume HNCC's polylogarithmic-precision claim. Their validity is a correctness/verifiability concern, not a case of the conclusion being equivalent to its assumptions. No step fits a parameter to the target expectation, renames a known result, or invokes a uniqueness theorem to force the construction. Hence the circularity score is 0.

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

The algorithm's guarantee rests on the standard mathematical machinery of BCH expansions, k-locality and g0-extensiveness, plus a critical external bound from the authors' companion preprint. The main free choices are the algorithmic parameters q0, s0, m, and ν, together with unspecified constant prefactors used in the proofs.

free parameters (5)

q0 (BCH truncation order) = Θ(log(1/ε))
Chosen in the proof to keep accumulated BCH truncation error below ε/4; contributes the log(1/ε) factor in the gate count.
s0 (total BCH-order cutoff) = Θ(log(1/ε))
Cutoff for discarding high-order sampled compensation terms; chosen to keep the resulting bias at most ε/4.
m (number of fractional Trotter-remainder factors) = Θ(min{N, Γ/log(1/ε)}); in numerics m=5
Optimized to minimize the gate-count bound; in the numerical section m=5 is selected at t=1, ε=10^-3 and reused for all scans.
ν (number of Trotter steps) = Θ((N^2/m)^{1/(2K+1)} (kg0t log(1/ε))^{1+1/(2K+1)}) with unspecified constant
Chosen by the proof to keep the LCQC 1-norm bounded and the truncation condition 8e q0 k g~ ≤ 1 satisfied; the constant prefactor is not given explicitly.
Implicit constants in q0, s0, ν = large enough constants
The proofs invoke 'sufficiently large constant prefactors' to make error contributions ε/4 and to ensure λ_paired^{mν}=O(1); these constants are not computed.

assumptions (5)

standard math BCH expansion Eq. (14) holds as a formal power-series identity for the Trotter remainder.
Used throughout Section IV to represent the Trotter remainder as exp(Σ Φ_q).
domain assumption The Hamiltonian satisfies the g0-extensiveness condition Eq. (11) and admits a Pauli decomposition with each term supported on at most k sites.
Definition in Section IV; all commutator bounds and light-cone sampling probabilities depend on it.
ad hoc to paper Doubly right-nested commutator bounds from [26, Theorem 2 and Theorem 9] are correct.
Imported as Lemmas 3 and 4 to bound the BCH truncation error; not proven in this manuscript and sourced from an overlapping-author preprint.
standard math Imaginary Pauli operators close under commutation, with each nonzero nesting yielding a factor of 2.
Used to derive the parameter-shift LCQC and light-cone sampler weights.
standard math Hoeffding's inequality for bounded measurement outcomes.
Used in Lemma 1 to convert LCQC 1-norm into O(λ^2/ε^2) sample complexity.

invented entities (1)

None
purpose: No new physical entities are introduced.
HNCC channels, fractional Trotter remainders, and the light-cone sampler are algorithmic constructions; they do not postulate new particles, forces, dimensions, or conserved quantities.

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Cite this review

Pith. "Pith review of Trotter error compensation with polylogarithmic precision and nested-commutator scaling without ancillas." pith.science (2026). https://pith.science/paper/QDQNX376

@misc{pith2026260711856,
  author       = {Pith},
  title        = {Pith review of: Trotter error compensation with polylogarithmic precision and nested-commutator scaling without ancillas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QDQNX376}},
  note         = {Machine review of arXiv:2607.11856}
}

abstract

Product formulas are among the most practical approaches to Hamiltonian simulation, requiring no ancillary qubits and exhibiting error bounds governed by nested commutators rather than only by Hamiltonian norms. Their circuit size, however, scales polynomially with the inverse precision. We develop a high-order nested-commutator compensation (HNCC) algorithm that preserves the main advantages of product formulas while achieving polylogarithmic precision dependence in the circuit size and the standard $\mathcal{O}(\varepsilon^{-2})$ sampling cost. HNCC uses a truncated Baker--Campbell--Hausdorff expansion to represent high-order Trotter errors by products of nested commutators and compensates these errors at the channel level through randomly sampled Pauli-rotation channels, avoiding Hadamard tests and ancillary qubits. For a fixed $K$-th order product formula applied to a $k$-local Hamiltonian on $N$ qubits with $\Gamma$ Pauli terms and local interaction strength $g_0$, HNCC estimates $\operatorname{tr}[Oe^{-i tH}\rho e^{i tH}]$ to additive precision $\varepsilon\|O\|$ using $\mathcal{O}(\varepsilon^{-2})$ repetitions. Its maximum gate count per circuit is $\mathcal{O}\bigl( kN^{\frac{1}{2K+1}} \Gamma^{1-\frac{1}{2K+1}} \max\{\Gamma,N\log(1/\varepsilon)\}^{\frac{1}{2K+1}} (kg_0t\log(1/\varepsilon))^{1+\frac{1}{2K+1}} \bigr)$. Finite-size resource estimates for the periodic Heisenberg chain indicate that HNCC has the lowest estimated $T$-gate count per circuit among the product-formula-based methods considered.

Figures

Figures reproduced from arXiv: 2607.11856 by the authors.

**Figure 2.** FIG. 2. Calling relation among the routines used for one [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** FIG. 3. Finite-size resource comparison for the periodic Heisenberg chain with [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Finite-size resource comparison as a function of the simulation time [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

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Works this paper leans on

38 extracted references · 6 linked inside Pith

[26]

1, 010359 (2025)

Pei Zeng, Jinzhao Sun, Liang Jiang, and Qi Zhao,Simple and high-precision Hamiltonian simulation by compen- sating Trotter error with linear combination of unitary operations, PRX Quantum6, no. 1, 010359 (2025)

2025
[1]

vgZKQ0lCPN0dKZ3epZBplO5N720=

Unpaired HNCC Unpaired HNCC represents the linear part using the parameter-shift rule, so one fractional Trotter remain- 7 der has LCQC 1-normλ unpaired = 1 +O(λ single). Al- thoughλ single scales as 1/m, the fractional remainder is sampledmtimes after each product-formula step. In- creasingmtherefore does not reduce the leading con- tribution to the full...
[2]

(7) a fractional Trotter remainder

Sampling for the Trotter remainder For a positive integerm, we factorize the truncated Trotter remainder as ˜V (q0) K (x) = exp 1 m q0X q=K+1 Φq(x) m .(7) We call the factor repeatedmtimes on the right-hand side of Eq. (7) a fractional Trotter remainder. We con- struct an LCQC for this map and applymindependently sampled copies after each product-formula ...
[3]

Each nonzero sampled channel is a Pauli rotation by the common angle described above

Gate complexity of a sampled channel We first consider the part linear in the BCH terms. Each nonzero sampled channel is a Pauli rotation by the common angle described above. Its generator arises from aq-fold right-nested commutator of Pauli operators, each supported on at mostkqubits. Its support therefore contains at mostkq≤kq 0 qubits, so the channel u...
[4]

Feynman,Simulating physics with computers, International Journal of Theoretical Physics21, no

Richard P. Feynman,Simulating physics with computers, International Journal of Theoretical Physics21, no. 6/7, 467–488 (1982)

1982
[5]

5278, 1073–1078 (1996)

Seth Lloyd,Universal quantum simulators, Science273, no. 5278, 1073–1078 (1996)

1996
[6]

1, 015003 (2020)

Sam McArdle, Suguru Endo, Al´ an Aspuru-Guzik, Si- mon C Benjamin, and Xiao Yuan,Quantum computa- tional chemistry, Reviews of Modern Physics92, no. 1, 015003 (2020)

2020
[7]

Childs, Dmitri Maslov, Yunseong Nam, Neil J

Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, and Yuan Su,Toward the first quantum simulation with quantum speedup, Proceedings of the National Academy of Sciences115, no. 38, 9456–9461 (2018)

2018

Show all 38 references

[8]

Yulong Dong, Lin Lin, and Yu Tong,Ground-state prepa- ration and energy estimation on early fault-tolerant quan- tum computers via quantum eigenvalue transformation of unitary matrices, PRX Quantum3, 040305 (2022)

2022
[9]

3, eaeb1622 (2026)

Jinzhao Sun, Pei Zeng, Tom Gur, and MS Kim, High-precision and low-depth quantum algorithm design for eigenstate problems, Science Advances12, no. 3, eaeb1622 (2026)

2026
[10]

Harrow, Avinatan Hassidim, and Seth Lloyd, Quantum algorithm for linear systems of equations, Phys- ical Review Letters103, 150502 (2009)

Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, Quantum algorithm for linear systems of equations, Phys- ical Review Letters103, 150502 (2009)

2009
[11]

Childs, Robin Kothari, and Rolando D

Andrew M. Childs, Robin Kothari, and Rolando D. Somma,Quantum algorithm for systems of linear equa- tions with exponentially improved dependence on preci- sion, SIAM Journal on Computing46, no. 6, 1920–1950 (2017)

1920
[12]

Dong An, Jin-Peng Liu, and Lin Lin,Linear combination of Hamiltonian simulation for nonunitary dynamics with optimal state preparation cost, Physical Review Letters 131, 150603 (2023)

2023
[13]

Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, A quantum approximate optimization algorithm, 2014, arXiv:1411.4028

2014 arXiv
[14]

2, 400– 23 407 (1991)

Masuo Suzuki,General theory of fractal path integrals with applications to many-body theories and statistical physics, Journal of Mathematical Physics32, no. 2, 400– 23 407 (1991)

1991
[15]

Childs, Yuan Su, Minh C

Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu,Theory of Trotter error with commutator scaling, Physical Review X11, no. 1, 011020 (2021)

2021
[16]

Berry, Andrew M

Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma,Exponential im- provement in precision for simulating sparse Hamiltoni- ans, Proceedings of the Forty-Sixth Annual ACM Sym- posium on Theory of Computing, STOC ’14, p. 283–292, 2014

2014
[17]

Berry, Andrew M

Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma,Simulating Hamiltonian dynamics with a truncated Taylor series, Physical Review Letters114, no. 9, 090502 (2015)

2015
[18]

Berry, Andrew M

Dominic W. Berry, Andrew M. Childs, and Robin Kothari,Hamiltonian simulation with nearly optimal de- pendence on all parameters, Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Com- puter Science (FOCS) (USA), FOCS ’15, p. 792–809, IEEE Computer Society, 2015

2015
[19]

Chuang,Optimal Hamil- tonian simulation by quantum signal processing, Physical Review Letters118, 010501 (2017)

Guang Hao Low and Isaac L. Chuang,Optimal Hamil- tonian simulation by quantum signal processing, Physical Review Letters118, 010501 (2017)

2017
[20]

Chuang,Hamiltonian sim- ulation by qubitization, Quantum3, 163 (2019)

Guang Hao Low and Isaac L. Chuang,Hamiltonian sim- ulation by qubitization, Quantum3, 163 (2019)

2019
[21]

Suguru Endo, Qi Zhao, Ying Li, Simon Benjamin, and Xiao Yuan,Mitigating algorithmic errors in a Hamilto- nian simulation, Physical Review A99, 012334 (2019)

2019
[22]

Egger, David Ochsner, and Stefan Woerner,Well-conditioned multi- product formulas for hardware-friendly Hamiltonian sim- ulation, Quantum7, 1067 (2023)

Almudena Carrera Vazquez, Daniel J. Egger, David Ochsner, and Stefan Woerner,Well-conditioned multi- product formulas for hardware-friendly Hamiltonian sim- ulation, Quantum7, 1067 (2023)

2023
[23]

Gumaro Rendon, Jacob Watkins, and Nathan Wiebe, Improved Accuracy for Trotter Simulations Using Cheby- shev Interpolation, Quantum8, 1266 (2024)

2024
[24]

Watson and Jacob Watkins,Exponentially re- duced circuit depths using Trotter error mitigation, PRX Quantum6, no

James D. Watson and Jacob Watkins,Exponentially re- duced circuit depths using Trotter error mitigation, PRX Quantum6, no. 3, 030325 (2025)

2025
[25]

Berry, and Min-Hsiu Hsieh,Doubling the order of approximation via the ran- domized product formula, Physical Review A109, 062431 (2024)

Chien-Hung Cho, Dominic W. Berry, and Min-Hsiu Hsieh,Doubling the order of approximation via the ran- domized product formula, Physical Review A109, 062431 (2024)

2024
[27]

Santos,Im- proving time dynamics simulation by sampling the error unitary, 2025, arXiv:2508.17542

Lana Mineh, Adrian Chapman, and Raul A. Santos,Im- proving time dynamics simulation by sampling the error unitary, 2025, arXiv:2508.17542

2025 arXiv
[28]

Kaoru Mizuta,On the commutator scaling in Hamilto- nian simulation with multi-product formulas, Quantum 10, 1974 (2026)

1974
[29]

Xinzhao Wang, Shuo Zhou, Xiaoyang Wang, Yi- Cong Zheng, Shengyu Zhang, and Tongyang Li,Lind- bladian simulation with commutator bounds, 2026, arXiv:2603.28602

2026
[30]

Sanders,Higher order decompositions of ordered operator exponentials, Journal of Physics A: Mathemati- cal and Theoretical43, no

Nathan Wiebe, Dominic Berry, Peter Høyer, and Barry C. Sanders,Higher order decompositions of ordered operator exponentials, Journal of Physics A: Mathemati- cal and Theoretical43, no. 6, 065203 (2010)

2010
[31]

2, 53 (2021)

Ana Arnal, Fernando Casas, and Cristina Chiralt,A note on the baker–campbell–hausdorff series in terms of right- nested commutators, Mediterranean Journal of Mathe- matics18, no. 2, 53 (2021)

2021
[32]

Junaid Aftab, Dong An, and Konstantina Trivisa,Multi- product Hamiltonian simulation with explicit commutator scaling, 2024, arXiv:2403.08922

2024 arXiv
[33]

3, 557–564 (2013)

Baoming Tang, Ehsan Khatami, and Marcos Rigol,A short introduction to numerical linked-cluster expansions, Computer Physics Communications184, no. 3, 557–564 (2013)

2013
[34]

Guang Hao Low, Vadym Kliuchnikov, and Nathan Wiebe,Well-conditioned multiproduct hamiltonian sim- ulation, 2019, 1907.11679

2019 arXiv
[35]

Svore, Efficient synthesis of universal repeat-until-success quan- tum circuits, Physical Review Letters114, no

Alex Bocharov, Martin Roetteler, and Krysta M. Svore, Efficient synthesis of universal repeat-until-success quan- tum circuits, Physical Review Letters114, no. 8, 080502 (2015)

2015
[36]

Wenjun Yu, Xiaogang Li, Qi Zhao, and Xiao Yuan, Lindbladian simulation with logarithmic precision scaling via two ancillas, Physical Review Letters135, 160602 (2025)

2025
[37]

Shantanav Chakraborty, Soumyabrata Hazra, Tongyang Li, Changpeng Shao, Xinzhao Wang, and Yuxin Zhang, Quantum singular value transformation without block en- codings: Near-optimal complexity with minimal ancilla, 2025, arXiv:2504.02385

2025 arXiv
[38]

Keisuke Murota, Yuta Kikuchi, Enrico Rinaldi, Fr´ ed´ eric Sauvage, and Synge Todo,Unbiased Hamiltonian sim- ulation by reversing Trotter error dynamics, 2026, arXiv:2606.29741

2026 arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [26]

1, 010359 (2025)

Pei Zeng, Jinzhao Sun, Liang Jiang, and Qi Zhao,Simple and high-precision Hamiltonian simulation by compen- sating Trotter error with linear combination of unitary operations, PRX Quantum6, no. 1, 010359 (2025)

2025

[2] [1]

vgZKQ0lCPN0dKZ3epZBplO5N720=

Unpaired HNCC Unpaired HNCC represents the linear part using the parameter-shift rule, so one fractional Trotter remain- 7 der has LCQC 1-normλ unpaired = 1 +O(λ single). Al- thoughλ single scales as 1/m, the fractional remainder is sampledmtimes after each product-formula step. In- creasingmtherefore does not reduce the leading con- tribution to the full...

[3] [2]

(7) a fractional Trotter remainder

Sampling for the Trotter remainder For a positive integerm, we factorize the truncated Trotter remainder as ˜V (q0) K (x) = exp 1 m q0X q=K+1 Φq(x) m .(7) We call the factor repeatedmtimes on the right-hand side of Eq. (7) a fractional Trotter remainder. We con- struct an LCQC for this map and applymindependently sampled copies after each product-formula ...

[4] [3]

Each nonzero sampled channel is a Pauli rotation by the common angle described above

Gate complexity of a sampled channel We first consider the part linear in the BCH terms. Each nonzero sampled channel is a Pauli rotation by the common angle described above. Its generator arises from aq-fold right-nested commutator of Pauli operators, each supported on at mostkqubits. Its support therefore contains at mostkq≤kq 0 qubits, so the channel u...

[5] [4]

Feynman,Simulating physics with computers, International Journal of Theoretical Physics21, no

Richard P. Feynman,Simulating physics with computers, International Journal of Theoretical Physics21, no. 6/7, 467–488 (1982)

1982

[6] [5]

5278, 1073–1078 (1996)

Seth Lloyd,Universal quantum simulators, Science273, no. 5278, 1073–1078 (1996)

1996

[7] [6]

1, 015003 (2020)

Sam McArdle, Suguru Endo, Al´ an Aspuru-Guzik, Si- mon C Benjamin, and Xiao Yuan,Quantum computa- tional chemistry, Reviews of Modern Physics92, no. 1, 015003 (2020)

2020

[8] [7]

Childs, Dmitri Maslov, Yunseong Nam, Neil J

Andrew M. Childs, Dmitri Maslov, Yunseong Nam, Neil J. Ross, and Yuan Su,Toward the first quantum simulation with quantum speedup, Proceedings of the National Academy of Sciences115, no. 38, 9456–9461 (2018)

2018