Zassenhaus Expansion in Solving the Schr\"odinger Equation

Molena Nguyen; Naihuan Jing; Yali Du; Yiling Wang

REVIEW 4 major objections 5 minor 1 cited by

Zassenhaus Expansion in Solving the Schr\"odinger Equation

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Folding second-order Zassenhaus commutator corrections into a Cartan/KAK template gives $U(t)=K_c^\dagger e^{-ih_0 t}K_c$ with constant circuit depth and local error $\mathcal{O}(t^3)$, beating first-order product formulas on gate count…

desk verdict A plausible blend of Zassenhaus and KAK simulation that is let down by an unproven diagonalization claim, but worth refereeing for the idea. read the letter →

arxiv 2505.09441 v2 pith:XXXB3XFN submitted 2025-05-14 quant-ph

Yali Du , Naihuan Jing , Molena Nguyen , Yiling Wang This is my paper · ORCID

classification quant-ph

keywords HamiltoniansimulationZassenhausexpansionCartandecompositionKAKfixed-depthquantumcircuitsNISQhardwaredynamicalLiealgebraproductformulas

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 proposes a fixed-depth strategy for simulating $U(t)=e^{-iHt}$ when $H$ is a bounded sum of Pauli operators. It combines the Cartan/KAK decomposition of the dynamical Lie algebra with the second-order Zassenhaus expansion, so that the time evolution is written as $K_c^\dagger e^{-ih_0 t}K_c$ with $h_0$ in a Cartan subalgebra. The claim is that this representation has circuit depth independent of $t$ and local error $\mathcal{O}(t^3)$ in operator norm, while using fewer gates than first-order Trotterization when time is large and depth is constrained. A sympathetic reader would care because, if the diagonalization step works, lattice and chemistry Hamiltonians with locality structure could be compiled into near-term, depth-limited circuits without explicit matrix exponentiation.

What carries the argument

The central object is the truncated multivariable Zassenhaus expansion used as the Cartan factor: $K(\theta)=\prod_i e^{i\theta_i k_i}\prod_{i<j} e^{-\frac{1}{2}\theta_i\theta_j[k_i,k_j]}\cdots$, a product of exponentials of generators and nested commutators that approximates the rotation moving $H$ into $\mathfrak{h}$. The KAK theorem justifies the target form $K^\dagger e^{-ih_0t}K$, and closure of Pauli commutators inside the Pauli-generated Lie algebra lets every correction term be computed symbolically. The companion mechanism is the optimization of $f(\theta)=\operatorname{tr}\big(K^\dagger(\theta)vK(\theta)H\big)$ for a fixed diagonal $v\in\mathfrak{h}$ with irrational coefficients, minimized by a quasi-Newton local optimizer; convergence of that optimization is what is asserted to yield $h_0\in\mathfrak{h}$.

What would settle it

Take a benchmark model such as the transverse-field Ising chain, run the same optimization to get $\theta^*$, and directly compute $h_0=K_cHK_c^\dagger$. If any Pauli component of $h_0$ lies outside the chosen Cartan subalgebra, or if $[h_0,D]\neq 0$ for a basis $D$ of $\mathfrak{h}$, then $e^{-ih_0t}$ is not the claimed commuting evolution and Eq. (17) does not equal $e^{-iHt}$; a log-log plot of the operator-norm error versus $t$ would also show whether the promised $\mathcal{O}(t^3)$ local scaling actually holds.

Watch

Extended reading notes

Core claim

For a Hamiltonian built from Pauli strings, the paper's central claim is that an optimized unitary rotation $K_c$, approximated by a truncated multivariable Zassenhaus expansion, conjugates $H$ into a Cartan subalgebra $\mathfrak{h}$: $h_0=K_cHK_c^\dagger\in\mathfrak{h}$. Then $e^{-iHt}=K_c^\dagger e^{-ih_0t}K_c$, and because $h_0$ is diagonal, evolution reduces to commuting gates flanked by a fixed-depth pair $K_c,K_c^\dagger$. Including the second-order Zassenhaus correction $e^{-\frac{1}{2}\theta_i\theta_j[k_i,k_j]}$ suppresses the leading non-commutativity error, giving local error $\mathcal{O}(t^3)$ in operator norm under boundedness assumptions. On six benchmark spin models, the optimized second- and higher-order expansions match or beat the first-order fixed-depth baseline in operator-norm error.

Load-bearing premise

The method's fixed-depth claim and error bound rest on the assumption that the optimizer's $K_c$ really conjugates the full Hamiltonian into the chosen Cartan subalgebra, even though the cost function only measures overlap with one diagonal reference matrix and no proof of diagonalization is given.

Editorial extensions

If this is right

Circuit depth for the simulation becomes independent of the evolution time $t$; longer times only change the angles of the commuting gates inside $e^{-ih_0 t}$.
The second-order Zassenhaus correction removes the leading commutator error of first-order product formulas, improving local error from $\mathcal{O}(t^2)$ to $\mathcal{O}(t^3)$ without the depth growth of higher-order Trotter–Suzuki splitting.
For the six benchmark spin models, the optimized second- through fourth-order expansions match or beat the first-order fixed-depth method, with the XY and Kitaev models unchanged because their commutator corrections vanish.
Pauli commutator closure allows all correction terms to be evaluated symbolically, eliminating explicit matrix exponentiation from classical preprocessing.

Reading between the lines

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

The paper leaves diagonalization certification open; an editorially natural extension is to compute the norm of the off-Cartan part of $h_0=K_cHK_c^\dagger$ and to accept the ansatz only when that norm is below the claimed error.
If the fixed-depth claim survives certification, the protocol should generalize beyond spin models to any Hamiltonian whose dynamical Lie algebra has a small Cartan subalgebra, such as quadratic fermionic systems, as long as the commutators close symbolically.
The optimization landscape is a testable risk: at large qubit number, the scalar overlap $f(\theta)$ may have stationary points that do not diagonalize $H$, so random-initialization or small-$n$ plateau experiments would quantify how generic the reported convergence is.
The error bound is local in $t$, so long-time behavior is not automatically controlled; a concrete extension is to bound the global error of $K_c^\dagger e^{-ih_0t}K_c$ over an interval $[0,T]$ and to compare it with the reported $t=20$ errors.

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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

The paper extends the fixed-depth Cartan/KAK simulation framework by adding Zassenhaus commutator corrections to the ansatz for the diagonalizing rotation K. That is a natural and sensible thing to try, and the symbolic use of Pauli-closure to evaluate commutators is a useful detail. The multivariable Zassenhaus formulas are cited correctly, and the authors run benchmarks on six spin models. So the combination is new and the intent is clear.

The problem is that the central mathematical claim is unsupported. The paper asserts that after BFGS optimization of f(theta)=tr(K†vKH) the optimized Kc conjugates H into the Cartan subalgebra, h0=KcHKc† in h. That conclusion does not follow. When K is restricted to the finite-dimensional Zassenhaus product, the scalar overlap with a fixed diagonal v cannot guarantee exact diagonalization of a generic H; the parameter count is far too small. No argument is given that the constrained optimum reaches the Cartan bound. If KcHKc† is not in h, then U(t)=Kc†e^{-ih0t}Kc is just a circuit, not the time evolution under H, and no bound is supplied for the difference.

The claimed O(t^3) local error also lacks a derivation. Equation (18) bounds the Zassenhaus truncation error for e^{A+B}, not the error from replacing H by Kc†h0Kc. The gate-count advantage over first-order product formulas mentioned in the abstract is never measured. The numerical errors at t=20 are all around 1e-14, which suggests the optimizer is finding essentially exact diagonalizations for these small models; that is interesting, but it does not demonstrate a scaling advantage in t or system size. There are also smaller slips: the abstract calls the approximation non-unitary even though the circuit is unitary, and the text says the cost function is minimized when alignment would presumably maximize it.

Who is this for? Researchers working on fixed-depth Lie-theoretic simulation might find the Zassenhaus-KAK ansatz worth exploring, but they should not rely on the claimed guarantees until the diagonalization step is either proven or replaced by a rigorous variational bound. The core idea is not obviously wrong; it is under-analyzed.

My recommendation: send this to peer review. A serious referee can demand a real error bound and actual gate-count measurements, and the paper deserves that attention despite the current gaps.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a fixed-depth Hamiltonian simulation method that combines a Cartan/KAK decomposition with a truncated Zassenhaus expansion. The authors define a parameterized unitary K(θ) as a product of exponentials of Lie algebra generators and their commutators, optimize the cost f(θ)=tr(K†(θ)vK(θ)H) in Eq. (15) using BFGS, and then claim that the optimizer Kc conjugates the Hamiltonian into a Cartan subalgebra, h0=KcHKc†∈h, yielding the fixed-depth decomposition U(t)=Kc†e^{-ih0t}Kc. The paper claims a local error O(t^3), reduced gate counts relative to first-order product formulas, and reports numerical results on six spin models.

Significance. If the central diagonalization claim were established, the proposed fixed-depth simulation protocol would be a notable contribution to Lie-theoretic quantum simulation. The paper usefully recalls the multivariate Zassenhaus formulas of Ref. [41] and the idea of symbolic commutator evaluation within Pauli-generated Lie algebras. However, the load-bearing mathematical step—that the optimized Kc actually maps H into the Cartan subalgebra—is unproved and, as written, the numerical experiments measure a mathematical identity rather than an approximation error. The gate-count reductions claimed in the abstract are not quantified anywhere. At present the manuscript does not provide a sound basis for its main claims.

major comments (4)

[Algorithm of a fixed-depth Hamiltonian simulation using the Zassenhaus expansion, after Eq. (15)] The assertion that BFGS convergence yields h0=KcHKc†∈h is unsupported. Maximizing the scalar f(θ)=tr(K†(θ)vK(θ)H) over the restricted d-parameter family in Eq. (14) does not imply that KcHKc† is diagonal or that it commutes with the chosen diagonal v. Even if K ranged over the full unitary group, a global maximum with nondegenerate v would only force KHK† to commute with v; for the constrained Zassenhaus ansatz, no argument shows that the optimum lies in the Cartan subalgebra, and for a generic H whose eigenvector data has dimension much larger than d, the finite-parameter family cannot contain an exact diagonalizing unitary. Since the fixed-depth and O(t^3) claims rest entirely on h0∈h, this missing proof is a central gap.
[Experiments, Fig. 1(b) and Table I] Equation (17) is an exact identity for every unitary Kc when h0 is defined as KcHKc†. The reported 2-norm errors, on the order of 1e-14 across all models and orders, therefore measure only floating-point precision in evaluating the identity, not the quality of any approximation. The numerical results provide no evidence that the Zassenhaus expansion improves simulation, and the identical values for second, third, and fourth orders in Table I are consistent with this interpretation. Moreover, the same Hamiltonian H is used both to fit the parameters of Kc and to compute the reported error, so the demonstration is circular in an operational sense.
[Avoiding Trotter error, Eq. (18)] The claimed local error O(t^3) is not derived for the proposed method. Equation (18) bounds the truncation error of the Zassenhaus product formula e^{A+B}≈e^Ae^Be^{-[A,B]/2}, which is not the error of replacing e^{-iHt} by Kc†e^{-ih0t}Kc. Since Eq. (17) is exact for any Kc, the quantity that actually needs a bound is the distance of h0 from the Cartan subalgebra, or alternatively the error incurred by approximating e^{-ih0t} with a diagonal circuit when h0∉h; no such bound is supplied. The paper therefore does not substantiate the central O(t^3) local-error statement.
[Abstract and Performance and outlook] The abstract and discussion claim that the method 'substantially reduces gate counts relative to first-order product formulas' and 'reduces circuit depth', but no gate counts, circuit widths, or depth comparisons are reported anywhere in the manuscript. The only quantitative results are normalized cost values and the meaningless identity errors described above. Without resource estimates, the central practical claim of the paper is unmeasured.

minor comments (5)

[Introduction] The text describes the approximation as 'controlled, non-unitary' and as 'relaxing strict unitarity constraints', yet Kc is unitary and e^{-ih0t} is unitary, so the proposed circuit is manifestly unitary. This inconsistency should be corrected.
[Experiments] No parameters are given for the six spin models (number of sites, coefficients, boundary conditions, or time-step details), so the numerical section is not reproducible. The figure captions also omit axis labels and units, and no data or code availability statement is provided.
[Eq. (19)] The expression e^{-iHt}≈(e^{-iAt/m}e^{-iBt/m}e^{it^2[A,B]/2m^2})^m is stated without derivation, and the sign of the commutator exponent should be checked against Eq. (6); a brief derivation or reference would help the reader.
[References] Reference [8], titled 'Distributionally Robust Receive Beamforming', appears unrelated to unitary synthesis or quantum simulation and should be replaced or removed.
[Eq. (14) and notation] The notation in Eq. (14) is ambiguous: the imaginary unit i is used alongside the index i in θ_i, and the product ordering of the second-order correction is not fully specified. Clarifying the notation would improve readability.

Circularity Check

1 steps flagged · score 4.0 of 10

Numerical validation is a fitting benchmark; the core Zassenhaus/Cartan identity is independent.

fitted input called prediction [Eqs. (15)-(17); Experiments, Fig. 1(b), Table I]
"The optimization objective is to find a parameter vector θ ∈ Rd such that the conjugated operator K†(θ)vK(θ) closely aligns with the system Hamiltonian H. ... Upon convergence, the optimizer yields an optimal unitary transformation Kc := K(θ∗), that conjugates the original Hamiltonian into the Cartan subalgebra: h0 = KcHK†c ∈ h."

The parameters θ are fitted to H through the BFGS minimization of f(θ)=tr(K†(θ)vK(θ)H), i.e., the same H is the training data. The reported validation quantity, ||e^{-iHt}−K†ce^{-ih0t}Kc||_2 in Fig. 1(b) and Table I, is then evaluated on that same H. The small errors therefore measure how well the optimized Kc diagonalizes the training Hamiltonian, not an independent prediction. In addition, the asserted membership h0∈h is not a mathematical consequence of maximizing a scalar overlap with a fixed diagonal v; it is assumed upon convergence. Thus the numerical 'predictions' reduce by construction to fitting benchmarks, while the exactness of Eq. (17) remains an unproved assumption.

full rationale

The core mathematical ingredient—the second-order Zassenhaus identity e^{A+B}=e^Ae^Be^{-[A,B]/2}+O(t^3) and the Cartan/KAK decomposition—is an external, parameter-free result and is not defined in terms of the paper's own target. The citation of Ref. [41] (by a co-author) for the multivariable Zassenhaus recursion is a self-citation, but the formula is independently established in the operator-splitting literature, so it is not load-bearing. The main proof gap is that Eq. (16)/(17) require h0=KcHKc†∈h, which is asserted after BFGS convergence but not implied by the scalar cost f(θ)=tr(K†vKH); this is an unproved correctness assumption rather than a circular reduction. The one genuinely circular element is the numerical demonstration: the same Hamiltonian H is used both to fit θ and to report the error ||e^{-iHt}−K†ce^{-ih0t}Kc||_2 (Fig. 1(b), Table I). Those errors are fitting benchmarks and cannot independently validate the claimed fixed-depth simulation protocol. Score 4 reflects this single fitted-input-called-prediction step while the underlying Zassenhaus/Cartan derivation retains independent content.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entity. Its free parameters are the optimized circuit parameters theta, the hand-chosen diagonal target v, and the expansion truncation order. The key axioms are the existence of a Cartan-compatible decomposition for the chosen Hamiltonians, the ability of the truncated Zassenhaus parameterization to realize an exact or near-exact diagonalizing K, the standard convergence of Zassenhaus for bounded operators, the Pauli-closure property, and unproven global convergence of BFGS.

free parameters (3)

theta (K-circuit parameters) = not reported
BFGS optimizes theta in K(theta)=prod e^{i theta_i k_i} prod e^{-1/2 theta_i theta_j [k_i,k_j]} to align K^dag v K with H; these parameters are fitted to the Hamiltonian itself and directly determine the approximation.
v coefficients gamma_i = irrational, e.g. pi, unspecified per model
The target diagonal element v=sum gamma_i h_i is chosen by hand from transcendental numbers to break degeneracy; since v's spectrum is not matched to H, this choice is a free design input affecting the cost function and any approximate K.
Zassenhaus expansion order = 2, 3, 4 (discrete)
The authors choose the truncation order of the expansion; performance, parameter count, and circuit depth depend on this choice, but no systematic ablation or selection criterion is reported.

assumptions (5)

domain assumption The Cartan involution Theta(g)=-g^T gives a Cartan decomposition g=k direct sum m with H in m for the Hamiltonians considered.
Invoked when choosing Theta such that Theta(H)=-H; only true for Hamiltonians with the required symmetry, and the paper does not characterize which spin models satisfy it.
ad hoc to paper The truncated Zassenhaus parameterization of K can realize or sufficiently approximate the exact Cartan-conjugating unitary.
The paper asserts after BFGS convergence that K_c H K_c^dag is in h, but this is not proven and is the load-bearing premise of the fixed-depth ansatz. Located around Eq. (16).
standard math The Zassenhaus expansion converges for the bounded operators under consideration.
Used to justify truncation at second order; standard for bounded operators with small norms, but no norm bound is given for the specific Hamiltonians.
standard math The set of Pauli strings P_n forms a closed algebra under commutation up to phases.
The paper relies on closure of Pauli commutators to evaluate commutators symbolically; this is a standard property of the Pauli group.
domain assumption BFGS converges to a global optimum of f(theta).
The paper claims irrational gamma_i promote a unique optimizer, but no convergence or global-optimality proof is given for the nonconvex cost function.

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Pith. "Pith review of Zassenhaus Expansion in Solving the Schr\"odinger Equation." pith.science (2026). https://pith.science/paper/XXXB3XFN

@misc{pith2026250509441,
  author       = {Pith},
  title        = {Pith review of: Zassenhaus Expansion in Solving the Schr\"odinger Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XXXB3XFN}},
  note         = {Machine review of arXiv:2505.09441}
}

abstract

A fundamental challenge in quantum simulation is approximating the time-evolution operator $U(t)=e^{-i\mathcal{H}t}$ generated by a large sum of typically non-commuting Hamiltonians using resource-efficient circuits compatible with near-term devices. We present a refinement of fixed-depth Lie-theoretic simulation that incorporates second-order Zassenhaus commutator corrections into a Cartan/KAK decomposition template. The resulting approximation retains constant circuit depth while achieving local error $\mathcal{O}(t^3)$ in operator norm under standard boundedness assumptions, and it substantially reduces gate counts relative to first-order product formulas when time is large and depth is constrained. The method leverages closure of Pauli commutators inside Pauli-generated Lie algebras, enabling symbolic commutator evaluation and avoiding explicit matrix exponentiation in classical preprocessing. This yields a structured pathway to compile lattice and chemistry-inspired Hamiltonians with locality constraints into fixed-depth circuits suitable for noisy intermediate-scale quantum hardware.

Figures

Figures reproduced from arXiv: 2505.09441 by the authors.

**Figure 1.** (a) demonstrates that all higher-order Zassenhaus expansions consistently outperform the baseline protocol of Ref. [67], achieving significantly lower cost values. Given that the objective is minimizing f(θ), these results correspond directly to enhanced diagonalization accuracy and a more precise approximation of the target unitary evolution. Notably, the second- and third-order expansions achieve convergence with… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Error bounds for the truncated Baker--Campbell--Hausdorff and Zassenhaus formulas in unitary problems
math-ph 2026-07 accept novelty 7.0 of 10

Explicit commutator-scaling error bounds are derived for truncated BCH and Zassenhaus formulas in the skew-adjoint (unitary) setting, generalizing Lie–Trotter and Strang splitting bounds.

Reference graph

Works this paper leans on

72 extracted references · 43 canonical work pages · cited by 1 Pith paper

[41]

Vatan and C

F. Vatan and C. Williams, Phys. Rev. A 69, 032315 (2004)

work page 2004
[1]

The search direction is computed as pk =−Hk∇f(θk),

work page
[2]

A line search determines a step size αk satisfying Wolfe or Armijo conditions [47],

work page
[3]

The parameters are updated: θk+1 =θk +αkpk,

work page
[4]

The algorithm achieves superlinear convergence near a local minimum, making it well-suited for smooth cost functions such as Eq

The inverse Hessian estimate Hk+1 is updated us- ing gradient and step differences. The algorithm achieves superlinear convergence near a local minimum, making it well-suited for smooth cost functions such as Eq. (15), which inherit analyticity from the matrix exponential and trace operations. Upon convergence, the optimizer yields an optimal uni- tary tr...

work page
[5]

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

work page 2010
[6]

Barenco, C

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Phys. Rev. A 52, 3457 (1995)

1995
[7]

J. J. Vartiainen, M. M¨ ott¨ onen, and M. M. Salomaa, Phys. Rev. Lett. 92, 177902 (2004)

work page 2004

Show all 72 references

[8]

V. V. Shende, S. S. Bullock, and I. L. Markov, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 25, 1000 (2006)

2006
[9]

Kamakari, S.-N

H. Kamakari, S.-N. Sun, M. Motta, and A. J. Minnich, PRX Quantum 3, 010309 (2022)

2022
[10]

Babbush, N

R. Babbush, N. Wiebe, J. McClean, J. McClain, H. Neven, and G. K.-L. Chan, Phys. Rev. X 8, 011044 (2018)

2018
[11]

G. H. Low, Y. Su, Y. Tong, and M. C. Tran, PRX Quan- tum 4, 020323 (2023)

2023
[12]

S. Wang, W. Dai, and G. Y. Li, Distributionally Robust Receive Beamforming, arXiv:2401.12345 (2024)

2024 arXiv
[13]

Khatri, R

S. Khatri, R. LaRose, A. Poremba, L. Cincio, A. T. Sorn- borger, and P. J. Coles, Quantum 3, 140 (2019)

2019
[14]

Y. Nam, N. J. Ross, Y. Su, A. M. Childs, and D. Maslov, npj Quantum Inf. 4, 23 (2018)

2018
[15]

K. N. Smith, G. S. Ravi, P. Murali, J. M. Baker, N. Earnest, A. Javadi-Abhari, and F. T. Chong, arXiv:2105.01760 [quant-ph] (2021)

2021 arXiv
[16]

M. Amy, D. Maslov, M. Mosca, and M. Roetteler, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.32, 818 (2013)

2013
[17]

Arute, K

F. Arute, K. Arya et al. , Nature 574, 505 (2019)

2019
[18]

I. D. Kivlichan, J. McClean, N. Wiebe, C. Gidney, ´A. Aspuru-Guzik, G. K.-L. Chan, and R. Babbush, Phys. Rev. Lett. 120, 110501 (2018)

2018
[19]

Lloyd, Science 273, 1073 (1996)

S. Lloyd, Science 273, 1073 (1996)

1996
[20]

D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Phys. Rev. Lett. 114, 090502 (2015)

2015
[21]

A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Phys. Rev. X 11, 011020 (2021)

2021
[22]

G. H. Low and I. L. Chuang, Phys. Rev. Lett. 118, 010501 (2017)

2017
[23]

Peruzzo, J

A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, ´A. Aspuru-Guzik, and J. L. O’Brien, Nat. Commun. 5, 4213 (2014)

2014
[24]

Cerezo, A

M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Nat. Rev. Phys.3, 625 (2021)

2021
[25]

Gottesman, Ph.D

D. Gottesman, Ph.D. thesis, California Institute of Tech- nology (1997)

1997
[26]

B. M. Terhal, Rev. Mod. Phys. 87, 307 (2015)

2015
[27]

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Phys. Rev. A 86, 032324 (2012)

2012
[28]

D. E. Browne and T. Rudolph, Phys. Rev. Lett. 95, 010501 (2005)

2005
[29]

Raussendorf and H

R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001)

2001
[30]

H. J. Briegel, D. E. Browne, W. D¨ ur, R. Raussendorf, and M. Van den Nest, Nat. Phys. 5, 19 (2009)

2009
[31]

D’Alessandro, Introduction to Quantum Control and Dynamics (Chapman and Hall/CRC, New York, 2007)

D. D’Alessandro, Introduction to Quantum Control and Dynamics (Chapman and Hall/CRC, New York, 2007)

2007
[32]

D. W. Berry, A. M. Childs, Y. Su, X. Wang, and N. Wiebe, Quantum 4, 254 (2020)

2020
[33]

M. C. Tran, Y. Su, D. Carney, and J. M. Taylor, PRX Quantum 2, 010323 (2021)

2021
[34]

Campbell, Phys

E. Campbell, Phys. Rev. Lett. 123, 070503 (2019)

2019
[35]

Suzuki, J

M. Suzuki, J. Math. Phys. 32, 400 (1991)

1991
[36]

M. Koch, O. Schaudt, G. Mogk, T. Mrziglod, H. Berg, and M. E. Beck, ACS Omega 8, 22596 (2023)

2023
[37]

G. H. Low and I. L. Chuang, Quantum 3, 163 (2019)

2019
[38]

Preskill, Quantum 2, 79 (2018)

J. Preskill, Quantum 2, 79 (2018)

2018
[39]

A. M. Childs, A. Ostrander, and Y. Su, Quantum 3, 182 (2019)

2019
[40]

Wiersema, E

R. Wiersema, E. K¨ okc¨ u, A. F. Kemper, and B. N. Bakalov, npj Quantum Inf. 10, 110 (2024)

2024
[42]

Zassenhaus, Abh

H. Zassenhaus, Abh. Math. Semin. Univ. Hambg. 13, 1 (1939)

1939
[43]

Blanes, F

S. Blanes, F. Casas, J. A. Oteo, and J. Ros, Phys. Rep. 470, 151 (2009)

2009
[44]

Casas, A

F. Casas, A. Murua, and M. Nadinic, Comput. Phys. Commun. 183, 2386 (2012)

2012
[45]

L. Wang, Y. Gao, and N. Jing, Front. Math. China 14, 421-433, (2019)

2019
[46]

Khaneja and S

N. Khaneja and S. J. Glaser, Chem. Phys. 267, 11–23 (2001)

2001
[47]

Dirr and U

G. Dirr and U. Helmke, GAMM Mitt. 31, 59–93 (2008)

2008
[48]

Schulte-Herbr¨ uggen, A

T. Schulte-Herbr¨ uggen, A. Sp¨ orl, N. Khaneja, and S. J. Glaser, Phys. Rev. A 72, 042331 (2005)

2005
[49]

P. C. Moan and J. Niesen, Found. Comput. Math. 8, 291 (2008)

2008
[50]

S. G. Schirmer, H. Fu, and A. I. Solomon, Phys. Rev. A 63, 063410 (2001)

2001
[51]

Nocedal and S

J. Nocedal and S. J. Wright, Numerical Optimization , 2nd ed. (Springer, New York, 2006)

2006
[52]

R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982)

1982
[53]

Suzuki, Phys

M. Suzuki, Phys. Lett. A 146, 319–323 (1990)

1990
[54]

Hejazi, M

K. Hejazi, M. Motta, and G. K.-L. Chan, Phys. Rev. Res. 6, 033084 (2024)

2024
[55]

Qvarfort and I

S. Qvarfort and I. Pikovski, PRX Quantum 6, 010201 (2025)

2025
[56]

Bertram, Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings, arXiv:math/0502168 (2005)

W. Bertram, Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings, arXiv:math/0502168 (2005)

2005 arXiv
[57]

A. W. Knapp, Lie Groups Beyond an Introduction , Progress in Mathematics, Vol. 140 (Birkh¨ auser, Boston, 1996)

1996
[58]

M. A. Nielsen, Quantum Inf. Comput. 6, 213 (2006)

2006
[59]

J. Haah, M. B. Hastings, R. Kothari, and G. H. Low, SIAM J. Comput. 52, FOCS18–250 (2021)

2021
[60]

F. G. S. L. Brand˜ ao, E. Crosson, M. B. S ¸ahino˘ glu, and J. Bowen, Phys. Rev. Lett. 123, 110502 (2019)

2019
[61]

Vidal, Phys

G. Vidal, Phys. Rev. Lett. 93, 040502 (2004)

2004
[62]

Zalka, Proc

C. Zalka, Proc. R. Soc. Lond. A 454, 313 (1998)

1998
[63]

D. S. Abrams and S. Lloyd, Phys. Rev. Lett. 79, 2586 (1997)

1997
[64]

S. P. Jordan, K. S. M. Lee, and J. Preskill, Science 336, 1130 (2012)

2012
[65]

Calaj´ o, G

G. Calaj´ o, G. Magnifico, C. Edmunds, M. Ringbauer, S. Montangero, and P. Silvi, PRX Quantum 5, 040309 (2024). 7

2024
[66]

Cˆ ırstoiu, Z

C. Cˆ ırstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, and A. Sornborger, npj Quantum Inf. 6, 82 (2020)

2020
[67]

Hatomura, Phys

T. Hatomura, Phys. Rev. A 105, L050601 (2022)

2022
[68]

Wiersema, C

R. Wiersema, C. Zhou, Y. de Sereville, J. F. Carrasquilla, Y. B. Kim, and H. Yuen, PRX Quantum 1, 020319 (2020)

2020
[69]

H. N. S´ a Earp and J. K. Pachos, J. Math. Phys. 46, 082108 (2005)

2005
[70]

Albertini and D

F. Albertini and D. D’Alessandro, Syst. Control Lett. 151, 104913 (2021)

2021
[71]

K¨ okc¨ u, T

E. K¨ okc¨ u, T. Steckmann, Y. Wang, J. K. Freericks, E. F. Dumitrescu, and A. F. Kemper, Phys. Rev. Lett. 129, 070501 (2022)

2022
[72]

Casas and A

F. Casas and A. Murua, J. Math. Phys. 50, 033513 (2009)

2009

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [41]

Vatan and C

F. Vatan and C. Williams, Phys. Rev. A 69, 032315 (2004)

work page 2004

[2] [1]

The search direction is computed as pk =−Hk∇f(θk),

work page

[3] [2]

A line search determines a step size αk satisfying Wolfe or Armijo conditions [47],

work page

[4] [3]

The parameters are updated: θk+1 =θk +αkpk,

work page

[5] [4]

The algorithm achieves superlinear convergence near a local minimum, making it well-suited for smooth cost functions such as Eq

The inverse Hessian estimate Hk+1 is updated us- ing gradient and step differences. The algorithm achieves superlinear convergence near a local minimum, making it well-suited for smooth cost functions such as Eq. (15), which inherit analyticity from the matrix exponential and trace operations. Upon convergence, the optimizer yields an optimal uni- tary tr...

work page

[6] [5]

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

work page 2010

[7] [6]

Barenco, C

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Phys. Rev. A 52, 3457 (1995)

1995

[8] [7]

J. J. Vartiainen, M. M¨ ott¨ onen, and M. M. Salomaa, Phys. Rev. Lett. 92, 177902 (2004)

work page 2004