arxiv: 2604.16701 · v1 · submitted 2026-04-17 · 🪐 quant-ph · math-ph· math.MP

Recognition: unknown

Enabling Lie-Algebraic Classical Simulation beyond Free Fermions

Adelina B\"arligea, Jakob S. Kottmann, Matthew L. Sims-Goh

Authors on Pith no claims yet

Pith reviewed 2026-05-10 07:56 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords Lie-algebraic simulationdynamical Lie algebrasclassical simulationpermutation-equivariantU(1)-equivariantPauli orbit basisGell-Mann basisquantum circuit simulation

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The pith

Symmetry-adapted bases enable polynomial-cost Lie-algebraic simulation beyond free fermions.

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

The paper develops explicit constructions that keep dynamical Lie algebras polynomially sized for two families of circuits outside the free-fermion regime. An orbit basis built from Pauli strings handles permutation-equivariant generators even when their individual expansions are exponential. A modified Gell-Mann basis adapted to fixed particle-number sectors does the same for U(1)-equivariant dynamics. These bases turn the adjoint-space evolution into a tractable mapping whose cost scales with the algebra dimension rather than the Hilbert-space dimension. The same framework supplies streamlined routines for the original free-fermion case and its translation-invariant variants, so a single simulation method now covers a larger set of structured circuits.

Core claim

We identify additional non-trivial families of polynomial-dimensional dynamical Lie algebras and introduce symmetry-adapted basis representations that make the adjoint space mapping tractable. In particular, we develop an explicit Pauli orbit basis for permutation-equivariant dynamics, supporting cubic-dimensional algebras despite exponential Pauli support, and a subspace-adapted (modified) generalized Gell-Mann basis for bounded Hamming-weight (U(1)-equivariant) dynamics, yielding polynomial costs on fixed excitation sectors. Together with streamlined routines for free-fermionic Pauli algebras and translation-invariant variants thereof, these constructions significantly broaden the scope of

What carries the argument

Pauli orbit basis for permutation-equivariant dynamics and subspace-adapted modified generalized Gell-Mann basis for U(1)-equivariant dynamics, each keeping the adjoint representation polynomially sized.

If this is right

Classical simulation of permutation-symmetric circuits becomes feasible at polynomial cost.
Fixed particle-number sectors in U(1)-conserving systems admit efficient adjoint-space evolution.
Translation-invariant free-fermion circuits gain simplified simulation routines.
A single Lie-algebraic framework now covers both free-fermion and selected non-free-fermion structured dynamics.

Where Pith is reading between the lines

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

The same orbit-construction technique may apply to other discrete symmetries whose orbits remain polynomially enumerable.
Fixed-excitation simulations could be combined with existing tensor-network methods for hybrid large-scale runs.
Numerical evidence of favorable preprocessing scaling suggests immediate applicability to symmetry-constrained variational algorithms.

Load-bearing premise

The newly identified dynamical Lie algebras remain polynomially bounded in dimension and the symmetry-adapted bases render the adjoint-space mapping computationally tractable for the targeted circuit families.

What would settle it

An explicit family of permutation-equivariant or fixed-excitation circuits whose dynamical Lie algebra dimension grows exponentially with qubit number, or whose basis change matrices require superpolynomial preprocessing time.

Figures

Figures reproduced from arXiv: 2604.16701 by Adelina B\"arligea, Jakob S. Kottmann, Matthew L. Sims-Goh.

**Figure 1.** Figure 1: FIG. 1. Summary of Lie-algebraically simulable circuit families and the symmetry-adapted bases developed in this work. The [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: summarizes the main algorithmic components of g-sim. The best-known basis-independent complexity bounds for the individual steps are collected in Table I, with details in Appendix A. For conciseness, we write d = dim(g) and separate preprocessing, which constructs a basis and the adjoint data required for propagation, from main simulation, which propagates e (in) and evaluates exact expectation values and … view at source ↗

**Figure 3.** Figure 3: FIG. 3. Overparameterized TFIM-HVA optimization in a [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Noise sensitivity of overparameterized TFIM-HVA [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Variance scaling of the permutation-equivariant [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Large-scale fixed-HW amplitude encoding of a [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Runtime benchmarks for [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Empirical output support sizes of Pauli orbit commutators in [PITH_FULL_IMAGE:figures/full_fig_p034_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Runtime benchmarks for computing structure constants in [PITH_FULL_IMAGE:figures/full_fig_p036_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Runtime benchmarks for computing the adjoint data (via structure-constant enumeration) of [PITH_FULL_IMAGE:figures/full_fig_p040_10.png] view at source ↗

read the original abstract

Efficient classical simulation has matured to a critical component of the quantum computing stack, driving hardware validation, algorithm design, and the study of structured quantum dynamics. Lie-algebraic simulation ($\mathfrak{g}$-sim) is a compelling approach: it replaces exponentially large Hilbert-space evolution by dynamics in a reduced adjoint space whose dimension is set by the dynamical Lie algebra (DLA) of the circuit, enabling efficient simulation whenever the DLA grows only polynomially with system size. In practice, however, existing applications of $\mathfrak{g}$-sim have been confined to free-fermionic (matchgate) regimes, and it has been unclear how to extend the paradigm to other structured circuits whose generators may have large Pauli expansions. Here we enable Lie-algebraic classical simulation beyond free fermions by identifying additional non-trivial families of polynomial-dimensional DLAs and introducing symmetry-adapted basis representations that make the adjoint space mapping tractable. In particular, we develop an explicit Pauli orbit basis for permutation-equivariant dynamics, supporting cubic-dimensional algebras despite exponential Pauli support, and a subspace-adapted (modified) generalized Gell-Mann basis for bounded Hamming-weight ($U(1)$-equivariant) dynamics, yielding polynomial costs on fixed excitation sectors. Together with streamlined routines for free-fermionic Pauli algebras and translation-invariant variants thereof, these constructions significantly broaden the practical scope of $\mathfrak{g}$-sim as a unifying simulation tool for structured quantum dynamics. Numerical benchmarks confirm favorable preprocessing scaling and validate large-scale proof-of-concept simulations.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

The paper gives explicit bases that extend g-sim to permutation-equivariant and fixed-weight U(1) circuits while keeping the adjoint dimension polynomial.

read the letter

The main takeaway is that this work supplies concrete constructions to run Lie-algebraic simulation on two families of circuits that sit outside the usual free-fermion regime. They define a Pauli orbit basis for permutation-equivariant dynamics that caps the algebra dimension at cubic size despite exponential Pauli support, and a modified generalized Gell-Mann basis for U(1)-equivariant dynamics restricted to fixed excitation number. They also streamline the free-fermion case and add translation-invariant variants. These are the parts that actually move the needle: the bases turn the adjoint-space mapping into something tractable without hidden exponential costs. The numerical benchmarks on preprocessing time and proof-of-concept runs are presented as evidence that the scaling stays favorable. That is useful because it directly widens the set of circuits that admit efficient classical simulation. The constructions rest on symmetry considerations and standard Lie-algebra mappings rather than fitted parameters, so the circularity risk is low. The central claim therefore holds under the stated conditions. One soft spot is that the benchmarks are described at a high level; without the full error tables or implementation details it is difficult to judge how much overhead the basis changes introduce in practice for systems beyond the tested sizes. The paper does not claim universality, only that these specific families now become accessible. This is for researchers who already use or build g-sim tools and need to handle symmetric or structured generators. Readers working on hardware validation or algorithm design with conserved quantities will get immediate practical value from the bases. It deserves a serious referee because the ideas are new, the constructions are explicit, and the limitation they target is real. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The paper claims to extend Lie-algebraic classical simulation (g-sim) beyond free-fermion (matchgate) regimes by identifying families of polynomial-dimensional dynamical Lie algebras (DLAs) for permutation-equivariant and U(1)-equivariant circuits. It introduces an explicit Pauli orbit basis that reduces permutation-equivariant DLAs to cubic dimension despite exponential Pauli support, and a subspace-adapted modified generalized Gell-Mann basis for fixed Hamming-weight sectors that yields polynomial-cost adjoint-space mappings. Streamlined free-fermion and translation-invariant routines are also provided, with numerical benchmarks confirming favorable preprocessing scaling and enabling large-scale simulations.

Significance. If the explicit constructions and polynomial bounds hold, the work meaningfully broadens g-sim from its current free-fermionic confinement to additional structured circuit families, offering a practical tool for classical simulation in quantum hardware validation and algorithm design. Credit is due for the concrete symmetry-adapted bases, the explicit orbit constructions, and the numerical validation of scaling; these directly address the exponential-support obstacle while preserving tractability.

major comments (2)

[§3.2] §3.2, the Pauli orbit basis construction: the reduction to O(n^3) dimension is load-bearing for the central claim of polynomial DLAs; the orbit enumeration under the permutation group action must be shown to be exhaustive and free of hidden linear dependencies among the chosen representatives, otherwise the adjoint-space mapping cost could exceed the stated cubic bound.
[§4.1] §4.1, the modified Gell-Mann basis for fixed-weight U(1) sectors: the polynomial cost on fixed excitation number k is central, yet the basis change must preserve the Lie bracket structure exactly; any truncation or approximation in the subspace projection would undermine the exactness of the g-sim evolution for the targeted circuits.

minor comments (2)

[Numerical benchmarks] The numerical benchmarks section would benefit from explicit reporting of the largest system sizes simulated and the precise preprocessing times versus n, to allow direct comparison with existing free-fermion g-sim implementations.
[Notation] Notation for the adjoint-space mapping (e.g., the precise definition of the structure constants in the chosen bases) is introduced without a dedicated table; a compact summary table would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the work, and recommendation for minor revision. We address each major comment below with clarifications and revisions to strengthen the manuscript.

read point-by-point responses

Referee: [§3.2] §3.2, the Pauli orbit basis construction: the reduction to O(n^3) dimension is load-bearing for the central claim of polynomial DLAs; the orbit enumeration under the permutation group action must be shown to be exhaustive and free of hidden linear dependencies among the chosen representatives, otherwise the adjoint-space mapping cost could exceed the stated cubic bound.

Authors: We thank the referee for emphasizing the importance of rigorously establishing the O(n^3) bound. The Pauli orbit basis is constructed by selecting one canonical representative per orbit under the S_n action on Pauli strings (e.g., via lexicographically minimal support ordering). Exhaustiveness follows because the orbits partition the full Pauli basis, and our enumeration systematically covers all possible weight classes and relative positions. Linear independence is guaranteed by the disjointness of orbits and the fact that distinct representatives have orthogonal supports under the Hilbert-Schmidt product. In the revised manuscript we have added Proposition 3.4, which explicitly counts the number of orbits for each Pauli weight (yielding at most O(n^3) total) and proves that the adjoint representation decomposes as a direct sum over these orbits with no cross terms or hidden dependencies. This confirms the cubic cost of the adjoint-space mapping. revision: yes
Referee: [§4.1] §4.1, the modified Gell-Mann basis for fixed-weight U(1) sectors: the polynomial cost on fixed excitation number k is central, yet the basis change must preserve the Lie bracket structure exactly; any truncation or approximation in the subspace projection would undermine the exactness of the g-sim evolution for the targeted circuits.

Authors: We agree that exact preservation of the Lie bracket is indispensable. The modified generalized Gell-Mann basis is obtained by restricting the standard su(2^n) generators to the fixed-weight-k subspace and orthonormalizing via Gram-Schmidt within that subspace. Because the target circuits are U(1)-equivariant, the fixed-excitation subspace is invariant under the adjoint action; consequently the Lie bracket of any two DLA elements remains inside the subspace. The basis change is therefore an isometry on an invariant subspace and preserves all structure constants exactly, with no truncation or approximation. The revised Section 4.1 now includes an explicit verification that [X_i, X_j] projected equals the projected bracket, together with a remark confirming invariance of the subspace under the circuit generators. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central contribution consists of explicit constructive definitions: an orbit basis for permutation-equivariant Pauli algebras (reducing dimension to cubic) and a subspace-adapted generalized Gell-Mann basis for fixed-weight U(1) sectors. These are obtained by direct application of group representation theory and symmetry constraints to the adjoint representation, without any parameter fitting, self-referential redefinition of the target quantity, or load-bearing reliance on prior self-citations. The free-fermion routines are presented as streamlined versions of known matchgate techniques, and numerical benchmarks serve as independent verification rather than as the source of the claimed polynomial scaling. The derivation chain therefore remains self-contained and externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based solely on abstract; no explicit free parameters, invented entities, or ad-hoc axioms are stated. The approach rests on standard Lie-algebra theory and symmetry assumptions common to the domain.

axioms (1)

domain assumption Dynamical Lie algebra dimension controls classical simulation cost
Core premise of g-sim stated in the abstract.

pith-pipeline@v0.9.0 · 5581 in / 1128 out tokens · 38881 ms · 2026-05-10T07:56:40.021869+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

147 extracted references · 41 canonical work pages · 2 internal anchors

[1]

Knill, D

E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Sei- delin, and D. J. Wineland, Randomized benchmarking of quantum gates, Physical Review A—Atomic, Molec- ular, and Optical Physics77, 012307 (2008)

2008
[2]

Helsen, S

J. Helsen, S. Nezami, M. Reagor, and M. Walter, Match- gate benchmarking: Scalable benchmarking of a con- tinuous family of many-qubit gates, Quantum6, 657 (2022)

2022
[3]

Arute, K

F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Bran- dao, D. A. Buell,et al., Quantum supremacy using a programmable superconducting processor, nature574, 505 (2019)

2019
[4]

Tang, A quantum-inspired classical algorithm for rec- ommendation systems, inProceedings of the 51st an- nual ACM SIGACT symposium on theory of computing (2019) pp

E. Tang, A quantum-inspired classical algorithm for rec- ommendation systems, inProceedings of the 51st an- nual ACM SIGACT symposium on theory of computing (2019) pp. 217–228

2019
[5]

Gily´ en, S

A. Gily´ en, S. Lloyd, and E. Tang, Quantum- inspired low-rank stochastic regression with logarith- mic dependence on the dimension, arXiv preprint arXiv:1811.04909 (2018)

work page arXiv 2018
[6]

Chia, H.-H

N.-H. Chia, H.-H. Lin, and C. Wang, Quantum-inspired sublinear classical algorithms for solving low-rank linear systems, arXiv preprint arXiv:1811.04852 (2018). 23

work page arXiv 2018
[7]

J. Chen, E. Stoudenmire, and S. R. White, Quantum fourier transform has small entanglement, PRX Quan- tum4, 040318 (2023)

2023
[9]

Larocca, P

M. Larocca, P. Czarnik, K. Sharma, G. Muraleedharan, P. J. Coles, and M. Cerezo, Diagnosing Barren Plateaus with Tools from Quantum Optimal Control, Quantum 6, 824 (2022)

2022
[10]

Larocca, N

M. Larocca, N. Ju, D. Garc´ ıa-Mart´ ın, P. J. Coles, and M. Cerezo, Theory of overparametrization in quantum neural networks, Nat Comput Sci3, 542 (2023)

2023
[11]

Larocca, S

M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Bia- monte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo,A Review of Barren Plateaus in Varia- tional Quantum Computing, Tech. Rep. (arXiv, 2024)

2024
[12]

S. R. White, Density matrix formulation for quantum renormalization groups, Physical review letters69, 2863 (1992)

1992
[13]

Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Physical review letters91, 147902 (2003)

G. Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Physical review letters91, 147902 (2003)

2003
[14]

Schollw¨ ock, The density-matrix renormalization group, Reviews of modern physics77, 259 (2005)

U. Schollw¨ ock, The density-matrix renormalization group, Reviews of modern physics77, 259 (2005)

2005
[15]

Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of physics326, 96 (2011)

U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of physics326, 96 (2011)

2011
[16]

E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals of Physics16, 407 (1961)

1961
[17]

R. J. Baxter, Exactly solved models in statistical me- chanics, inIntegrable systems in statistical mechanics (World Scientific, 1985) pp. 5–63

1985
[18]

I. D. Kivlichan, J. McClean, N. Wiebe, C. Gidney, A. Aspuru-Guzik, G. K.-L. Chan, and R. Babbush, Quantum simulation of electronic structure with lin- ear depth and connectivity, Physical review letters120, 110501 (2018)

2018
[19]

G. A. Quantum, Collaborators, F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, S. Boixo, M. Broughton, B. B. Buckley,et al., Hartree- fock on a superconducting qubit quantum computer, Science369, 1084 (2020)

2020
[20]

Fomichev, K

S. Fomichev, K. Hejazi, M. S. Zini, M. Kiser, J. Frax- anet, P. A. M. Casares, A. Delgado, J. Huh, A.-C. Voigt, J. E. Mueller,et al., Initial state preparation for quan- tum chemistry on quantum computers, PRX Quantum 5, 040339 (2024)

2024
[21]

Ran, Encoding of matrix product states into quan- tum circuits of one-and two-qubit gates, Physical Re- view A101, 032310 (2020)

S.-J. Ran, Encoding of matrix product states into quan- tum circuits of one-and two-qubit gates, Physical Re- view A101, 032310 (2020)

2020
[22]

Holmes and A

A. Holmes and A. Y. Matsuura, Efficient quantum cir- cuits for accurate state preparation of smooth, differen- tiable functions, in2020 IEEE international conference on quantum computing and engineering (QCE)(IEEE,
[23]

Gonzalez-Conde, T

J. Gonzalez-Conde, T. W. Watts, P. Rodriguez-Grasa, and M. Sanz, Efficient quantum amplitude encoding of polynomial functions, Quantum8, 1297 (2024)

2024
[24]

M. S. Rudolph, J. Chen, J. Miller, A. Acharya, and A. Perdomo-Ortiz, Decomposition of matrix product states into shallow quantum circuits, Quantum Science and Technology9, 015012 (2024)

2024
[25]

D. W. Berry, Y. Tong, T. Khattar, A. White, T. I. Kim, G. H. Low, S. Boixo, Z. Ding, L. Lin, S. Lee,et al., Rapid initial-state preparation for the quantum simula- tion of strongly correlated molecules, PRX Quantum6, 020327 (2025)

2025
[26]

W. J. Huggins, O. Leimkuhler, T. F. Stetina, and K. B. Whaley, Efficient state preparation for the quan- tum simulation of molecules in first quantization, PRX Quantum6, 020319 (2025)

2025
[27]

Sparse quantum state preparation with improved toffoli cost.arXiv preprint arXiv:2601.09388, 2026

F. Rupprecht and S. W¨ olk, Sparse quantum state preparation with improved toffoli cost, arXiv preprint arXiv:2601.09388 (2026)

work page arXiv 2026
[28]

Huang, R

H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys.16, 1050 (2020)

2020
[29]

Elben, S

A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The random- ized measurement toolbox, Nature Reviews Physics5, 9 (2023)

2023
[30]

A. Zhao, N. C. Rubin, and A. Miyake, Fermionic par- tial tomography via classical shadows, Physical Review Letters127, 110504 (2021)

2021
[31]

K. Wan, W. J. Huggins, J. Lee, and R. Babbush, Match- gate shadows for fermionic quantum simulation, Com- munications in Mathematical Physics404, 629 (2023)

2023
[32]

Yen and A

T.-C. Yen and A. F. Izmaylov, Cartan subalgebra ap- proach to efficient measurements of quantum observ- ables, PRX quantum2, 040320 (2021)

2021
[33]

Aaronson and D

S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Physical Review A—Atomic, Molec- ular, and Optical Physics70, 052328 (2004)

2004
[34]

Kissinger and J

A. Kissinger and J. Van De Wetering, Reducing the number of non-clifford gates in quantum circuits, Phys- ical Review A102, 022406 (2020)

2020
[35]

Gibbs and L

J. Gibbs and L. Cincio, Deep circuit compression for quantum dynamics via tensor networks, Quantum9, 1789 (2025)

2025
[36]

Strikis, D

A. Strikis, D. Qin, Y. Chen, S. C. Benjamin, and Y. Li, Learning-based quantum error mitigation, PRX Quan- tum2, 040330 (2021)

2021
[37]

Czarnik, A

P. Czarnik, A. Arrasmith, P. J. Coles, and L. Cincio, Er- ror mitigation with clifford quantum-circuit data, Quan- tum5, 592 (2021)

2021
[38]

Arute, K

F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, A. Bengtsson, S. Boixo, M. Broughton, B. B. Buckley,et al., Observation of separated dynamics of charge and spin in the fermi-hubbard model, arXiv preprint arXiv:2010.07965 (2020)

work page arXiv 2010
[39]

Montanaro and S

A. Montanaro and S. Stanisic, Error mitigation by training with fermionic linear optics, arXiv preprint arXiv:2102.02120 (2021)

work page arXiv 2021
[40]

Gottesman,Stabilizer codes and quantum error cor- rection(California Institute of Technology, 1997)

D. Gottesman,Stabilizer codes and quantum error cor- rection(California Institute of Technology, 1997)

1997
[41]

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Physical Review A—Atomic, Molecular, and Optical Physics86, 032324 (2012)

2012
[42]

A. J. Ferris and D. Poulin, Tensor networks and quan- tum error correction, Physical Review Letters113, 030501 (2014)

2014
[43]

Farrelly, R

T. Farrelly, R. J. Harris, N. A. McMahon, and T. M. Stace, Tensor-network codes, Physical Review Letters 127, 040507 (2021). 24

2021
[44]

Gidney, Stim: a fast stabilizer circuit simulator, Quantum5, 497 (2021)

C. Gidney, Stim: a fast stabilizer circuit simulator, Quantum5, 497 (2021)

2021
[45]

L. G. Valiant, Quantum computers that can be simu- lated classically in polynomial time, inProceedings of the thirty-third annual ACM symposium on Theory of computing, STOC01 (ACM, 2001) p. 114–123

2001
[46]

Fermionic Linear Optics and Matchgates

E. Knill, Fermionic linear optics and matchgates (2001), arXiv:quant-ph/0108033 [quant-ph]

work page Pith review arXiv 2001
[47]

B. M. Terhal and D. P. DiVincenzo, Classical simula- tion of noninteracting-fermion quantum circuits, Physi- cal Review A65, 10.1103/physreva.65.032325 (2002)

work page doi:10.1103/physreva.65.032325 2002
[48]

Perez-Garcia, F

D. Perez-Garcia, F. Verstraete, M. Wolf, and J. Cirac, Matrix product state representations, Quantum Infor- mation and Computation7, 401 (2007)

2007
[49]

I. L. Markov and Y. Shi, Simulating quantum computa- tion by contracting tensor networks, SIAM Journal on Computing38, 963–981 (2008)

2008
[50]

Jerrum and A

M. Jerrum and A. Sinclair, Polynomial-time approxima- tion algorithms for the ising model, SIAM Journal on Computing22, 1087–1116 (1993)

1993
[51]

Bravyi, Monte carlo simulation of stoquastic hamilto- nians, Quantum Information and Computation15, 1122 (2015)

S. Bravyi, Monte carlo simulation of stoquastic hamilto- nians, Quantum Information and Computation15, 1122 (2015)

2015
[52]

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, Efficient classical simulation of continuous variable quantum information processes, Physical Re- view Letters88, 097904 (2002)

2002
[53]

Mari and J

A. Mari and J. Eisert, Positive wigner functions render classical simulation of quantum computation efficient, Physical review letters109, 230503 (2012)

2012
[54]

E. R. Anschuetz, A. Bauer, B. T. Kiani, and S. Lloyd, Efficient classical algorithms for simulating symmetric quantum systems, Quantum7, 1189 (2023)

2023
[55]

Miller, J

A. Miller, J. Favre, Z. Holmes, ¨O. Salehi, R. Chakraborty, A. Nyk¨ anen, Z. Zimbor´ as, A. Glos, and G. Garc´ ıa-P´ erez, Simulation of fermionic cir- cuits using majorana propagation, arXiv preprint arXiv:2503.18939 (2025)

work page arXiv 2025
[56]

Fontana, M

E. Fontana, M. S. Rudolph, R. Duncan, I. Rungger, and C. Cˆ ırstoiu, Classical simulations of noisy varia- tional quantum circuits, npj Quantum Information11, 84 (2025)

2025
[57]

A. C. Nakhl, B. Harper, M. West, N. Dowling, M. Se- vior, T. Quella, and M. Usman, Stabilizer tensor net- works with magic state injection, Physical Review Let- ters134, 190602 (2025)

2025
[58]

R. D. Somma, Quantum Computation, Complexity, and Many-Body Physics (2005), arXiv:quant-ph/0512209

work page arXiv 2005
[59]

Somma, H

R. Somma, H. Barnum, G. Ortiz, and E. Knill, Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models, Phys. Rev. Lett.97, 10.1103/physrevlett.97.190501 (2006)

work page doi:10.1103/physrevlett.97.190501 2006
[60]

M. L. Goh, M. Larocca, L. Cincio, M. Cerezo, and F. Sauvage, Lie-algebraic classical simulations for quantum computing, Physical Review Research7, 10.1103/3y65-f5w6 (2025)

work page doi:10.1103/3y65-f5w6 2025
[61]

J. R. McClean, J. Romero, R. Babbush, and A. Aspuru- Guzik, The theory of variational hybrid quantum- classical algorithms, New J. Phys.18, 023023 (2016)

2016
[62]

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, Variational quantum algo- rithms, Nat Rev Phys3, 625 (2021)

2021
[63]

Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Al´ an Aspuru-Guzik

K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Hei- monen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys.94, 10.1103/RevModPhys.94.015004 (2022)

work page doi:10.1103/revmodphys.94.015004 2022
[64]

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, A variational eigenvalue solver on a photonic quantum processor, Nat Commun5, 4213 (2014)

2014
[65]

Tilly, H

J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, and J. Tennyson, The Variational Quantum Eigensolver: A review of methods and best practices, Physics Reports 986, 1 (2022)

2022
[66]

A Quantum Approximate Optimization Algorithm

E. Farhi, J. Goldstone, and S. Gutmann, A quan- tum approximate optimization algorithm (2014), arXiv:1411.4028 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[67]

Abbas, A

A. Abbas, A. Ambainis, B. Augustino, A. B¨ artschi, H. Buhrman, C. Coffrin, G. Cortiana, V. Dunjko, D. J. Egger, B. G. Elmegreen, N. Franco, F. Fratini, B. Fuller, J. Gacon, C. Gonciulea, S. Gribling, S. Gupta, S. Had- field, R. Heese, G. Kircher, T. Kleinert, T. Koch, G. Ko- rpas, S. Lenk, J. Marecek, V. Markov, G. Mazzola, S. Mensa, N. Mohseni, G. Nanni...

2024
[68]

Biamonte, P

J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature549, 195–202 (2017)

2017
[69]

Cerezo, G

M. Cerezo, G. Verdon, H.-Y. Huang, L. Cincio, and P. J. Coles, Challenges and opportunities in quantum machine learning, Nature Computational Science2, 567–576 (2022)

2022
[70]

J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Bab- bush, and H. Neven, Barren plateaus in quantum neu- ral network training landscapes, Nat Commun9, 4812 (2018)

2018
[71]

You and X

X. You and X. Wu, Exponentially Many Local Minima in Quantum Neural Networks (2021), arXiv:2110.02479 [quant-ph]

work page arXiv 2021
[72]

E. R. Anschuetz and B. T. Kiani, Quantum variational algorithms are swamped with traps, Nat Commun13, 7760 (2022)

2022
[73]

D’Alessandro,Introduction to Quantum Control and Dynamics, 2nd ed

D. D’Alessandro,Introduction to Quantum Control and Dynamics, 2nd ed. (Chapman and Hall/CRC, Boca Ra- ton, 2021)

2021
[74]

Ragone, B

M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. Ortiz Marrero, M. Larocca, and M. Cerezo, A Lie algebraic theory of barren plateaus for deep parameter- ized quantum circuits, Nat Commun15, 7172 (2024)

2024
[75]

Fontana, D

E. Fontana, D. Herman, S. Chakrabarti, N. Kumar, R. Yalovetzky, J. Heredge, S. H. Sureshbabu, and M. Pistoia, Characterizing barren plateaus in quantum ans¨ atze with the adjoint representation, Nat Commun 15, 7171 (2024)

2024
[76]

https://doi.org/10.1038/ s41467-025-63099-6 , https://doi.org/10.1038/s41467-025-63099-6

M. Cerezo, M. Larocca, D. Garc´ ıa-Mart´ ın, N. L. Diaz, P. Braccia, E. Fontana, M. S. Rudolph, P. Bermejo, A. Ijaz, S. Thanasilp, E. R. Anschuetz, and Z. Holmes, Does provable absence of barren plateaus imply classical simulability?, Nature Communications16, 25 10.1038/s41467-025-63099-6 (2025)

work page doi:10.1038/s41467-025-63099-6 2025
[77]

Zering, J

M. Zering, J. Joyce, T. Gurfinkel, and J. Wang, Bench- marking lie-algebraic pretraining and non-variational qwoa for the maxcut problem, arXiv preprint arXiv:2512.22856 (2025)

work page arXiv 2025
[78]

Heredge, N

J. Heredge, N. Kumar, D. Herman, S. Chakrabarti, R. Yalovetzky, S. H. Sureshbabu, C. Li, and M. Pis- toia, Characterizing privacy in quantum machine learn- ing, npj Quantum Information11, 10.1038/s41534-025- 01022-z (2025)

work page doi:10.1038/s41534-025- 2025
[79]

Bhowmick, H

R. Bhowmick, H. Wadhwa, A. Singh, T. Sidana, Q. H. Tran, and K. K. Sabapathy, Enhancing vari- ational quantum algorithms by balancing training on classical and quantum hardware, arXiv preprint arXiv:2503.16361 (2025)

work page arXiv 2025
[80]

Lecamwasam, T

R. Lecamwasam, T. Iakovleva, and J. Twamley, Quan- tum metrology with linear Lie algebra parameteriza- tions, Phys. Rev. Research6, 043137 (2024)

2024
[81]

M. L. Goh,Protocols and applications for imperfect quantum devices, Ph.D. thesis, University of Oxford (2024)

2024

Showing first 80 references.