Pith. sign in

REVIEW 2 major objections 5 minor 2 cited by

A deterministic, ancilla-free Gray-code circuit prepares any U(1)-eigenstate of a spin-s chain, and yields the first exact higher-spin Bethe-state preparations.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-03 09:12 UTC pith:PFFO26YB

load-bearing objection A legitimate qudit extension of the fixed-Hamming-weight preparation scheme, with the first s>1/2 Bethe-state prep as the real novelty; the math checks out, but the control-pruning rule is asserted without proof and needs a referee's attention. the 2 major comments →

arxiv 2601.14513 v2 pith:PFFO26YB submitted 2026-01-20 quant-ph

Spin-s U(1)-eigenstate preparation

Nabi Zare Harofteh , Rafael I. Nepomechie This is my paper
classification quant-ph
keywords U(1) eigenstate preparationspin-s chainsGray codebounded integer compositionsqudit circuitsBethe statesDicke statesAKLT state
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper describes a deterministic algorithm that prepares, with no ancilla qudits, any normalized superposition of computational basis states of an n-site spin-s chain that has a fixed sum of digits—equivalently, any U(1)-eigenstate with a given Sz eigenvalue. The construction orders the basis states as a Gray code over bounded integer compositions, so consecutive ditstrings differ by moving one unit of spin between two sites. A circuit of two-qudit 'Gray gates,' each a controlled Givens rotation on the two differing sites with controls pruned by a simple rule, builds the target state one basis state at a time. The authors use it to prepare exact eigenstates of integrable spin-s XXX Hamiltonians for the first time for s > 1/2, and confirm that it correctly prepares AKLT ground states and spin-s Dicke states as special cases.

Core claim

The central claim is that any normalized amplitude vector a_m on the fixed-digit-sum subspace is exactly realized by the circuit (2.23). The circuit is formed by reading the Gray code left-to-right, preparing the initial basis state with X gates, and for each step applying the Gray gate with controls pruned according to the rule of Sec. 2.2.1, with rotation angles determined by the amplitudes via (2.20)–(2.21). The paper further claims that with the amplitudes (5.11), this circuit prepares exact eigenstates of the integrable spin-s XXX Hamiltonian for any s, including the first reported preparation of such Bethe states for spin s > 1/2. The complexity is O(D) gates, where D is the number of

What carries the argument

The key object is the Gray code for bounded integer compositions: a listing of all ditstrings with fixed digit sum such that consecutive entries differ only by moving one unit from position j to position i. This ordering guarantees that each basis state is generated exactly once. The carrying mechanism is the Gray gate—a two-qudit controlled Givens rotation that acts on the two positions where the ditstring changes, with controls on the unchanged nonzero positions; the pruning rule (2.12)–(2.15) removes redundant controls. The rotation angles are fixed so that the amplitudes telescope into the prescribed a_m. This reduces the problem of preparing any U(1)-eigenstate to a classical ordering p

Load-bearing premise

The load-bearing premise is the control-pruning rule: after removing the 'untouched' controls, every Gray gate fires on precisely one existing basis state and never on an already-finalized one; if that rule fails for some (n, k, s), the circuit produces a corrupted superposition.

What would settle it

A brute-force search over small instances (say n up to 6, all k and s) can settle the claim: simulate each step of circuit (2.23) and check that each controlled Gray gate's non-trivial action is confined to the two consecutive basis states and that the final state has fidelity 1 with the target (1.5). Any instance where a later gate acts nontrivially on an earlier basis state—e.g., through a pruned-away control position where the digits coincidentally match—would falsify the pruning rule.

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

If this is right

  • For any s and n, every state in a fixed-Sz sector of n qudits can be prepared with O(D) gates and zero ancillas, where D is the number of basis states in the sector.
  • Exact integrable spin-s Bethe states, previously out of reach for s > 1/2, become available as quantum circuits; these can serve as trial states in variational eigensolvers with Bethe roots as variational parameters.
  • The same circuit, with the appropriate amplitudes, prepares AKLT ground states and su(2) symmetric Dicke states, providing a unified preparation routine for three families of physically important states.
  • Because the Gray code and the gate angles depend only on the subspace, not on the specific state, the classical preprocessing can be reused for any target amplitudes in that sector.

Where Pith is reading between the lines

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

  • The paper asserts the control-pruning rule by analogy with a known qubit subroutine and verifies it numerically; a rigorous combinatorial proof of this lemma would be needed to firmly establish the claim for all (n, k, s).
  • If the pruning lemma holds, the construction immediately generalizes the fixed-Hamming-weight encoder to arbitrary qudit subspaces, so the algorithm may be adaptable as a generic sparse state-preparation tool for qudits.
  • The Bethe-root-to-angle map in Sec. 5 suggests a natural testable extension: use the same circuit as a variational ansatz for spin-s Heisenberg ground states and check whether optimizing the roots recovers the Bethe equation solutions.
  • Because the classical computation of Bethe amplitudes sums over all permutations, the gate count is not the only cost; for large systems the classical bottleneck may exceed the quantum circuit cost—a point the authors explicitly flag.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper presents a deterministic quantum circuit that prepares any normalized state in a fixed S^z sector of n spin-s qudits, i.e., any linear combination of computational basis states whose ditstrings have a fixed digit sum k. The circuit is built from a Gray code over bounded integer compositions, controlled two-qudit "Gray gates," and classically computed rotation angles. The authors then use the construction to prepare AKLT ground states, spin-s Dicke states, and—for the first time for s>1/2—exact Bethe eigenstates of integrable spin-s XXX chains. The central algorithm is a qudit generalization of the fixed-Hamming-weight encoder of [18].

Significance. The main result is plausible and, if correct, is a useful contribution: it extends deterministic U(1)-eigenstate preparation from qubits to qudits and provides the first concrete preparation scheme for higher-spin Bethe states. The basis-state count (2.16) is correct, the angle recurrences (2.17)-(2.21) reproduce the target amplitudes by direct substitution for nonzero amplitudes, and the applications are well chosen. The paper is not circular: the target amplitudes come from externally defined MPS, Dicke, and Bethe-ansatz inputs, and the authors correctly acknowledge the high circuit complexity for Bethe states. However, the proof of the control-pruning rule is missing, and the complex-phase formula has an edge-case gap, so the central claim is not yet fully established.

major comments (2)
  1. [§2.2.1 and Eq. (2.23)] The correctness of the circuit (2.23) depends on the claim that the controls defined by (2.12)-(2.15) make each G[l] act nontrivially only on |m[l]> and leave all previously generated states invariant. This is asserted without proof; the reference to 'Subroutine 2 of [18]' and the cirq verification do not establish it. If a control is missing, a later gate can rotate an earlier finalized amplitude and corrupt every subsequent amplitude. Please add an inductive proof: show that U[l] is exactly the set of positions not touched by gates 0,...,l-1, and that any p<l with the same (m_i[l],m_j[l]) and equal to m[l] on all remaining C[l] must equal m[l].
  2. [§2.3, Eq. (2.21)] The complex-phase recurrence divides by a_l and by tanθ_l. Thus it is undefined whenever some a_l=0, and the preliminary rescaling 'a_l ↦ a_l |a_0|/a_0' fails when a_0=0. Since the stated target (1.5) is 'any normalized amplitudes', the general claim is not covered. Please provide a well-defined limiting or recursive phase formula for the case of zero amplitudes, or explicitly restrict the theorem to amplitudes with a_l≠0.
minor comments (5)
  1. [§5] Typo: 'Hamiltonain' should be 'Hamiltonian'.
  2. [References] Reference [42] cites 'qsim', but the text says implementations are in 'cirq'; please cite the correct software and clarify which package was used for the numerical checks.
  3. [GitHub [43]] For reproducibility, include a commit hash or versioned archive of the repository; the verification claims are otherwise not independently checkable.
  4. [Eq. (2.18)] Specify the convention for atan2(0,0), which is needed when trailing amplitudes vanish.
  5. [§2.3] The statement that circuit size and depth are O(D) refers to the abstract Gray gates. The actual cost of implementing multi-controlled qudit gates is not analyzed; please clarify to avoid an overstrong resource claim.

Circularity Check

0 steps flagged

No significant circularity: the angle construction is an exact inverse of the target amplitudes; only minor, non-load-bearing self-citations appear.

full rationale

The central derivation is a direct constructive synthesis, not a fit or a self-reference. Given target amplitudes a_l in (1.5), the angles (2.20)-(2.21) are derived by inverting the amplitude recurrence (2.17)/(2.19), which itself is the algebraic statement of what the gate sequence (2.3) does. The circuit (2.23) implements exactly that recurrence, so the output state is equal to the input amplitudes by construction, not by circular reasoning. The Gray code is supplied by an external result (Walsh [44], reviewed in Appendix A), and the Gray gate (2.5) is a defined unitary. The applications are benchmarked against externally defined amplitudes: AKLT MPS (3.2) from [34,35], Dicke amplitudes (4.1) from the standard definition [24,25], and spin-s Bethe amplitudes (5.5)-(5.11) from the coordinate Bethe ansatz [41]. Self-citations occur ([17], [24,25], [49], [51,52]) but they provide inputs, prior work, or numerical Bethe roots; none is invoked as the proof that the circuit prepares the state. The one significant gap is the control-pruning rule (2.12)-(2.15): the paper states it as a 'generalization of Subroutine 2 in [18]' without proof, relying on cirq checks ([43]). This is an unproven combinatorial lemma and a correctness risk, but it is not circular: the rule is not an input recycled as the conclusion, and the paper does not define the target state in terms of the controls. Accordingly, the circularity score is low (1), reflecting minor self-citation that is not load-bearing.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No free parameters are fitted: the angles (θ_l, φ_l) are deterministic inverses of the specified amplitudes (a constructive map, not a fit), and the Bethe roots u_j are external solutions of the Bethe equations (5.7) taken from [49]. No invented entities: the Gray gate is explicitly decomposed into standard X and R gates (Fig. 2). Six external or asserted premises carry the construction: Walsh's composition Gray code, the unproved pruning lemma, the (sign-corrected) coordinate Bethe ansatz, the Babujian Hamiltonians, the AKLT MPS amplitudes, and the standard combinatorial count (2.16).

axioms (6)
  • domain assumption Bounded integer compositions admit a Gray code with property (1.4) for every (n,k,s); Walsh's O(n) successor algorithm (Appendix A) generates one.
    Invoked in §2.1 and needed for the whole construction (the circuit index order m[0],...,m[D-1] is the Gray list); the paper reviews but does not prove Walsh's theorem, and correctness of the successor rules is load-bearing for the algorithm.
  • ad hoc to paper The pruning rule C[l] := C[l] \ U[l] (2.12)-(2.15) removes exactly the redundant controls, so that every later Gray gate acts only on the intended basis state.
    Asserted in §2.2.1 as a generalization of Subroutine 2 of [18], with no proof in the preprint; the entire claim that (2.23) equals (1.5) depends on this lemma. Only numerical cirq verification is cited.
  • domain assumption The coordinate Bethe ansatz of Crampé–Ragoucy–Alonzi [41], with the sign correction to A_P in (5.5), gives the exact eigenstates of H(s).
    §5 builds the Bethe-state amplitudes entirely on [41]; the sign correction is asserted (footnote 7) without derivation, so the application inherits a non-independent local assumption — if the corrected sign were wrong, the prepared state would not be an eigenstate.
  • domain assumption The Hamiltonians H(s) = Σ h(x_i, s) (5.1)-(5.2), with h from Table 2, are the Babujian integrable spin-s XXX Hamiltonians.
    Imported from [39] (§5); required so that the prepared Bethe states are eigenstates with energies (5.8).
  • domain assumption The MPS amplitudes (3.2) are the exact AKLT ground state of (3.1).
    Standard AKLT result [34,35] used for the §3 application; the algorithm itself does not depend on it.
  • standard math The dimension formula (2.16) (inclusion–exclusion) and the claimed scalings D ~ n^k/k! and D ~ (2s+1)^n hold.
    Standard combinatorics; verified by hand for the s=1/2 reduction and small cases; used in the complexity claims in §2.3.

pith-pipeline@v1.3.0-alltime-deepseek · 14626 in / 33216 out tokens · 330315 ms · 2026-08-03T09:12:26.456639+00:00 · methodology

0 comments
read the original abstract

We formulate a deterministic algorithm for preparing a general $U(1)$-eigenstate of a spin-$s$ chain of length $n$. These states consist of linear combinations of computational basis states $|\vec{m}\rangle$ of $n$ qudits, each with $(2s+1)$ levels and $s= 1/2, 1, 3/2, \ldots$, whose ditstrings $\vec{m}$ have a fixed digit sum. Exploiting a Gray code for bounded integer compositions, whose consecutive ditstrings obey the Gray property, the quantum state is prepared by applying corresponding ``Gray gates.'' We use this algorithm to prepare exact eigenstates of integrable spin-$s$ XXX Hamiltonians. We also consider the preparation of AKLT states and spin-$s$ Dicke states.

Figures

Figures reproduced from arXiv: 2601.14513 by Nabi Zare Harofteh, Rafael I. Nepomechie.

Figure 1
Figure 1. Figure 1: The convex polytope for bounded compositions of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Circuit diagram for the Gray gate G mi,mj i,j (θ, ϕ) (2.5). In its symbolic form shown on the right, we use sharp (♯) and flat (♭) to designate qudits i and j, respectively. The horizontal wires represent the two d-level qudits. A circle µ denotes a control on the value µ. The 1-qudit gate X and its inverse are defined as (see e.g. [48]) X∣µ⟩ = ∣µ + 1⟩, X † ∣µ⟩ = ∣µ − 1⟩, (2.7) where the sums are defined m… view at source ↗
Figure 3
Figure 3. Figure 3: The quantum circuit (2.23) for preparing a state with [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

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

  1. Preparing multi-qudit states in a definite-weight subspace

    quant-ph 2026-06 unverdicted novelty 6.0

    A deterministic algorithm prepares arbitrary multi-qudit states in definite-weight subspaces via Gray codes for multiset permutations and applies it to SU(3) Bethe states and SU(d) Dicke states.

  2. Preparing multi-qudit states in a definite-weight subspace

    quant-ph 2026-06 unverdicted novelty 6.0

    A deterministic algorithm prepares arbitrary multi-qudit states in a definite-weight subspace via Gray-code ordering of multiset permutations, reducing preparation to controlled 2-qudit Gray rotations, and is demonstr...

Reference graph

Works this paper leans on

53 extracted references · 31 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Deterministic preparation of Dicke states,

    A. B¨ artschi and S. Eidenbenz, “Deterministic preparation of Dicke states,”Lecture Notes in Computer Science(2019) 126–139,arXiv:1904.07358 [quant-ph]

    Pith/arXiv arXiv 2019

  2. [2]

    Preparing Dicke states in a spin ensemble using phase estimation,

    Y. Wang and B. M. Terhal, “Preparing Dicke states in a spin ensemble using phase estimation,”Phys. Rev. A104no. 3, (2021) ,arXiv:2104.14310 [quant-ph]

    Pith/arXiv arXiv 2021

  3. [3]

    State preparation by shallow circuits using feed forward,

    H. Buhrman, M. Folkertsma, B. Loff, and N. M. P. Neumann, “State preparation by shallow circuits using feed forward,”Quantum8(2024) 1552,arXiv:2307.14840 [quant-ph]

    Pith/arXiv arXiv 2024

  4. [4]

    Global variational quantum circuits for arbitrary symmetric state preparation,

    L. J. Bond, M. J. Davis, J. Min´ aˇ r, R. Gerritsma, G. K. Brennen, and A. Safavi-Naini, “Global variational quantum circuits for arbitrary symmetric state preparation,”Phys. Rev. Res.7no. 2, (2025) L022072,arXiv:2312.05060 [quant-ph]

    Pith/arXiv arXiv 2025

  5. [5]

    Approximating Many-Body Quantum States with Quantum Circuits and Measurements,

    L. Piroli, G. Styliaris, and J. I. Cirac, “Approximating Many-Body Quantum States with Quantum Circuits and Measurements,”Phys. Rev. Lett.133no. 23, (2024) 230401,arXiv:2403.07604 [quant-ph]

    Pith/arXiv arXiv 2024

  6. [6]

    Efficient preparation of Dicke states,

    J. Yu, S. R. Muleady, Y.-X. Wang, N. Schine, A. V. Gorshkov, and A. M. Childs, “Efficient preparation of Dicke states,”arXiv:2411.03428 [quant-ph]

    arXiv

  7. [7]

    Low-depth quantum symmetrization,

    Z. Liu, A. M. Childs, and D. Gottesman, “Low-depth quantum symmetrization,”arXiv:2411.04019 [quant-ph]

    Pith/arXiv arXiv

  8. [8]

    Constant-Depth Unitary Preparation of Dicke States,

    F. Vasconcelos and M. R. Joshi, “Constant-Depth Unitary Preparation of Dicke States,” arXiv:2601.10693 [quant-ph]

    arXiv

  9. [9]

    Preparing Bethe Ansatz Eigenstates on a Quantum Computer,

    J. S. Van Dyke, G. S. Barron, N. J. Mayhall, E. Barnes, and S. E. Economou, “Preparing Bethe Ansatz Eigenstates on a Quantum Computer,”PRX Quantum2(2021) 040329,arXiv:2103.13388 [quant-ph]

    Pith/arXiv arXiv 2021

  10. [10]

    Preparing exact eigenstates of the open XXZ chain on a quantum computer,

    J. S. Van Dyke, E. Barnes, S. E. Economou, and R. I. Nepomechie, “Preparing exact eigenstates of the open XXZ chain on a quantum computer,”J. Phys. A55no. 5, (2022) 055301,arXiv:2109.05607 [quant-ph]

    Pith/arXiv arXiv 2022

  11. [11]

    Bethe states on a quantum computer: success probability and correlation functions,

    W. Li, M. Okyay, and R. I. Nepomechie, “Bethe states on a quantum computer: success probability and correlation functions,”J. Phys. A55no. 35, (2022) 355305,arXiv:2201.03021 [quant-ph]

    Pith/arXiv arXiv 2022

  12. [12]

    Algebraic Bethe Circuits,

    A. Sopena, M. H. Gordon, D. Garc ´ ıa-Mart ´ ın, G. Sierra, and E. L´ opez, “Algebraic Bethe Circuits,” Quantum6(2022) 796,arXiv:2202.04673 [quant-ph]

    Pith/arXiv arXiv 2022

  13. [13]

    The Bethe Ansatz as a Quantum Circuit,

    R. Ruiz, A. Sopena, M. H. Gordon, G. Sierra, and E. L´ opez, “The Bethe Ansatz as a Quantum Circuit,”arXiv:2309.14430 [quant-ph]

    Pith/arXiv arXiv

  14. [14]

    Bethe Ansatz, quantum circuits, and the F-basis,

    R. Ruiz, A. Sopena, E. L´ opez, G. Sierra, and B. Pozsgay, “Bethe Ansatz, quantum circuits, and the F-basis,”SciPost Phys.18no. 6, (2025) 187,arXiv:2411.02519 [quant-ph]

    Pith/arXiv arXiv 2025

  15. [15]

    Fractal decompositions and tensor network representations of Bethe wavefunctions,

    S. Sahu and G. Vidal, “Fractal decompositions and tensor network representations of Bethe wavefunctions,”SciPost Phys. Core8(2025) 067,arXiv:2412.00923 [quant-ph]

    arXiv 2025

  16. [16]

    Reducing circuit depth in quantum state preparation for quantum simulation using measurements and feedforward,

    H. Yeo, H. E. Kim, I. Sohn, and K. Jeong, “Reducing circuit depth in quantum state preparation for quantum simulation using measurements and feedforward,”Phys. Rev. Applied23no. 5, (2025) 054066,arXiv:2501.02929 [quant-ph]

    Pith/arXiv arXiv 2025

  17. [17]

    Deterministic Bethe state preparation,

    D. Raveh and R. I. Nepomechie, “Deterministic Bethe state preparation,”Quantum8(2024) 1510, arXiv:2403.03283 [quant-ph]

    Pith/arXiv arXiv 2024

  18. [18]

    Quantum encoder for fixed-Hamming-weight subspaces,

    R. M. S. Farias, T. O. Maciel, G. Camilo, R. Lin, S. Ramos-Calderer, and L. Aolita, “Quantum encoder for fixed-Hamming-weight subspaces,”Phys. Rev. Applied23no. 4, (2025) 044014, arXiv:2405.20408 [quant-ph]. 11

    Pith/arXiv arXiv 2025

  19. [19]

    Toward optimal circuit size for sparse quantum state preparation,

    R. Mao, G. Tian, and X. Sun, “Toward optimal circuit size for sparse quantum state preparation,” Phys. Rev. A110no. 3, (2024) 032439,arXiv:2404.05147 [quant-ph]

    Pith/arXiv arXiv 2024

  20. [20]

    Preparation of Hamming-Weight-Preserving Quantum States with Log-Depth Quantum Circuits,

    Y. Li, G. Tian, X. He, and X. Sun, “Preparation of Hamming-Weight-Preserving Quantum States with Log-Depth Quantum Circuits,”arXiv:2508.14470 [quant-ph]

    Pith/arXiv arXiv

  21. [21]

    Optimal Circuit Size for Fixed-Hamming-Weight Quantum States Preparation,

    J. Luo and L. Li, “Optimal Circuit Size for Fixed-Hamming-Weight Quantum States Preparation,” arXiv:2508.17197 [quant-ph]

    Pith/arXiv arXiv

  22. [22]

    Deterministic Constant-Depth Preparation of the AKLT State on a Quantum Processor Using Fusion Measurements,

    K. C. Smith, E. Crane, N. Wiebe, and S. M. Girvin, “Deterministic Constant-Depth Preparation of the AKLT State on a Quantum Processor Using Fusion Measurements,”PRX Quantum4no. 2, (2023) 020315,arXiv:2210.17548 [quant-ph]

    Pith/arXiv arXiv 2023

  23. [23]

    Transmon qutrit-based simulation of spin-1 AKLT systems,

    K. Kumaran, F. Alam, N. Eassa, K. Ferris, X. Xiao, L. Cincio, N. Bronn, and A. Banerjee, “Transmon qutrit-based simulation of spin-1 AKLT systems,”arXiv:2412.19786 [quant-ph]

    arXiv

  24. [24]

    Spin-s Dicke states and their preparation,

    R. I. Nepomechie, F. Ravanini, and D. Raveh, “Spin-s Dicke states and their preparation,”Adv. Quantum Technol.7no. 12, (2024) 2400057,arXiv:2402.03233 [quant-ph]

    Pith/arXiv arXiv 2024

  25. [25]

    Dicke states as matrix product states,

    D. Raveh and R. I. Nepomechie, “Dicke states as matrix product states,”Phys. Rev. A110no. 5, (2024) 052438,arXiv:2408.04729 [quant-ph]

    Pith/arXiv arXiv 2024

  26. [26]

    Simple ways of preparing qudit Dicke states,

    N. B. Kerzner, F. Galeazzi, and R. I. Nepomechie, “Simple ways of preparing qudit Dicke states,” arXiv:2507.13308 [quant-ph]

    arXiv

  27. [27]

    On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain,

    H. Bethe, “On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain,”Z. Phys.71(1931) 205–226

    1931

  28. [28]

    Gaudin,La fonction d’onde de Bethe

    M. Gaudin,La fonction d’onde de Bethe. Masson, 1983. English translation by J.-S. Caux,The Bethe wavefunction, CUP, 2014

    1983

  29. [29]

    How algebraic Bethe ansatz works for integrable models,

    L. D. Faddeev, “How algebraic Bethe ansatz works for integrable models,” inSym´ etries Quantiques (Les Houches Summer School Proceedings vol 64), A. Connes, K. Gawedzki, and J. Zinn-Justin, eds., pp. 149–219. North Holland, 1998.arXiv:hep-th/9605187 [hep-th]

    Pith/arXiv arXiv 1998

  30. [30]

    H. S. Wilf,Combinatorial algorithms:an update. SIAM, 1989

    1989

  31. [31]

    A survey of combinatorial Gray codes,

    C. Savage, “A survey of combinatorial Gray codes,”SIAM Rev.39(1997) 605–629

    1997

  32. [32]

    Combinatorial Gray codes—an updated survey,

    T. M¨ utze, “Combinatorial Gray codes—an updated survey,”Elect. J. Combinatorics(2012) DS26–Sep,arXiv:2202.01280 [math.CO]

    Pith/arXiv arXiv 2012

  33. [33]

    Coherence in Spontaneous Radiation Processes,

    R. H. Dicke, “Coherence in Spontaneous Radiation Processes,”Phys. Rev.93(1954) 99–110

    1954

  34. [34]

    Rigorous Results on Valence Bond Ground States in Antiferromagnets,

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Rigorous Results on Valence Bond Ground States in Antiferromagnets,”Phys. Rev. Lett.59(1987) 799

    1987

  35. [35]

    Valence Bond Ground States in Isotropic Quantum Antiferromagnets,

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Valence Bond Ground States in Isotropic Quantum Antiferromagnets,”Commun. Math. Phys.115(1988) 477

    1988

  36. [36]

    Model factorized S matrix and an integrable Heisenberg chain with spin 1,

    A. B. Zamolodchikov and V. A. Fateev, “Model factorized S matrix and an integrable Heisenberg chain with spin 1,”Sov. J. Nucl. Phys.32(1980) 298–303

    1980

  37. [37]

    Yang-Baxter Equation and Representation Theory. 1.,

    P. P. Kulish, N. Yu. Reshetikhin, and E. K. Sklyanin, “Yang-Baxter Equation and Representation Theory. 1.,”Lett. Math. Phys.5(1981) 393–403

    1981

  38. [38]

    Quantum spectral transform method. Recent developments,

    P. P. Kulish and E. K. Sklyanin, “Quantum spectral transform method. Recent developments,”Lect. Notes Phys.151(1982) 61–119

    1982

  39. [39]

    Exact solution of the isotropic Heisenberg chain with arbitary spins: thermodynamics of the model,

    H. M. Babujian, “Exact solution of the isotropic Heisenberg chain with arbitary spins: thermodynamics of the model,”Nucl. Phys. B215(1983) 317–336. 12

    1983

  40. [40]

    Bethe ans¨ atze for 19-vertex models,

    A. Lima-Santos, “Bethe ans¨ atze for 19-vertex models,”J. Phy. A32no. 10, (1999) 1819–1839, arXiv:hep-th/9807219 [hep-th]

    Pith/arXiv arXiv 1999

  41. [41]

    Coordinate Bethe Ansatz for Spin s XXX Model,

    N. Cramp´ e, E. Ragoucy, and L. Alonzi, “Coordinate Bethe Ansatz for Spin s XXX Model,”SIGMA7 (2011) 006,arXiv:1009.0408 [math-ph]

    Pith/arXiv arXiv 2011

  42. [42]

    https://doi.org/10.5281/zenodo.4023103

    Quantum AI team and collaborators, “qsim,” Sep, 2020. https://doi.org/10.5281/zenodo.4023103

    doi:10.5281/zenodo.4023103 2020

  43. [43]

    https://github.com/nepomechie/spin-s-U1-eigenstate-preparation

  44. [44]

    Loop-free sequencing of bounded integer compositions,

    T. R. Walsh, “Loop-free sequencing of bounded integer compositions,”J. Comb. Math. Comb. Comp. 33(2000) 323–345

    2000

  45. [45]

    A Gray code for compositions,

    P. Klingsberg, “A Gray code for compositions,”J. Algorithms3(1982) 41–44

    1982

  46. [46]

    Even,Algorithmic Combinatorics

    S. Even,Algorithmic Combinatorics. Macmillan, 1973

    1973

  47. [47]

    Gray codes and overlap cycles for restricted weight words,

    V. Horan and G. Hurlbert, “Gray codes and overlap cycles for restricted weight words,”Discrete Math. Alg. Appl.06(2014) 1450062,arXiv:1403.1818 [math.CO]

    Pith/arXiv arXiv 2014

  48. [48]

    Qudits and high-dimensional quantum computing,

    Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, “Qudits and high-dimensional quantum computing,” Front. Phys.8(2020) 479,arXiv:2008.00959 [quant-ph]

    Pith/arXiv arXiv 2020

  49. [49]

    Singular solutions, repeated roots and completeness for higher-spin chains,

    W. Hao, R. I. Nepomechie, and A. J. Sommese, “Singular solutions, repeated roots and completeness for higher-spin chains,”J. Stat. Mech.1403(2014) P03024,arXiv:1312.2982 [math-ph]

    Pith/arXiv arXiv 2014

  50. [50]

    The Variational Quantum Eigensolver: A review of methods and best practices,

    J. Tillyet al., “The Variational Quantum Eigensolver: A review of methods and best practices,”Phys. Rept.986(2022) 1–128,arXiv:2111.05176 [quant-ph]

    Pith/arXiv arXiv 2022

  51. [51]

    Bethe ansatz on a quantum computer?,

    R. I. Nepomechie, “Bethe ansatz on a quantum computer?,”Quantum Inf. Comp.21(2021) 255–265, arXiv:2010.01609 [quant-ph]

    Pith/arXiv arXiv 2021

  52. [52]

    Estimating Bethe roots with VQE,

    D. Raveh and R. I. Nepomechie, “Estimating Bethe roots with VQE,”J. Phys. A57no. 35, (2024) 355303,arXiv:2404.18244 [quant-ph]

    Pith/arXiv arXiv 2024

  53. [53]

    The inverse scattering method approach to the quantum Shabat-Mikhailov model,

    A. G. Izergin and V. E. Korepin, “The inverse scattering method approach to the quantum Shabat-Mikhailov model,”Commun. Math. Phys.79(1981) 303. 13

    1981