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The sorted-list fermion encoding matches first-quantized Trotter scaling in the plane-wave basis while using only O(N log M) qubits, and becomes the cheaper option than Jordan-Wigner for compact orbital bases at low electron filling.

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2026-08-04 12:49 UTC pith:IKTKB5AR

load-bearing objection Useful resource counts for sorted-list encoding, but the plane-wave Trotter parity claim leans on an unverified same-author preprint. the 3 major comments →

arxiv 2510.01710 v2 pith:IKTKB5AR submitted 2025-10-02 quant-ph

Benchmarking Quantum Simulation of Chemical Hamiltonians using the Sorted-List Encoding

Calvin Ku , Yu-Cheng Chen , Alice Hu , Min-Hsiu Hsieh This is my paper
classification quant-ph PACS 03.67.Ac
keywords sorted-list encodingfermion-to-qubit encodingquantum phase estimationtrotterizationqubitizationplane-wave basismolecular orbital basisresource estimation
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.

Quantum phase estimation for chemistry needs a concrete answer to which fermion encoding to build a circuit with. This paper argues that the sorted-list encoding, which stores only the occupied orbitals as a sorted list, can deliver near-first-quantized Trotter performance in the plane-wave basis while using only O(N log M) qubits, and that in the molecular-orbital basis it becomes the cheaper choice than Jordan-Wigner at low electron-filling ratios, with a numerical break-even near N/M ≈ 0.1. The cost is a 2–4 order-of-magnitude increase in T gates in the MO regime. For qubitization, the paper finds the encoding choice matters less in the MO basis but that first-quantization keeps a clear lead in the plane-wave basis because its Hamiltonian has a smaller 1-norm. If right, this gives resource planners a concrete crossover rule instead of a qualitative preference.

Core claim

The paper claims three things. In the plane-wave basis, Trotterized robust phase estimation with the sorted-list encoding reaches the same asymptotic gate scaling as the first-quantized encoding, roughly O~((M^{2/3}N^{4/3}+M^{1/3}N^{8/3}) 5^{p/2-1}(M^{1/3}N^{2/3})^{1/p}/eps^{1+1/p}), with only O(N log M) qubits, while Jordan-Wigner carries an extra M^2 factor. In the molecular-orbital basis, each sorted-list fermionic term costs O(N log M) Clifford and T gates, and the total becomes cheaper than Jordan-Wigner in the N/M → 0 limit, with a numerical break-even at N/M ≈ 0.1 despite 2–4 more orders of T gates. For qubitization, the sorted-list plane-wave circuit costs O~((N^{1/3}M^{5/3}+N^{5/3}M

What carries the argument

The central object is the sorted-list encoding, which writes a Slater determinant as the sorted concatenation of binary orbital indices of occupied orbitals, padding with an 'infinity' sentinel. Fermionic operators are implemented through compare/bubble circuits (=p, <p, U_p) that sort registers and accumulate parity; the workhorse identity is that on any sorted-list Slater determinant, e^{-i T^(2)} = e^{-i T^(1)} for the kinetic term (and similarly for potential terms), so the first-quantized Trotter circuit can be reused. The paper bridges the two encodings by a hybrid quantization scheme that converts sorted-list to first-quantized, applies the QFT there, and converts back, claimed to cos

Load-bearing premise

Theorem 2's parity between sorted-list and first-quantized plane-wave scaling rests on an O(N)-gate conversion between the two encodings, asserted in the proof of Theorem 2 by reference to a prior work rather than derived or verified here.

What would settle it

Implement the conversion used in the proof of Theorem 2 on a small plane-wave system and count the gates. If converting between sorted-list and first-quantized registers needs Ω(N log M) gates rather than O(N), the claimed asymptotic parity fails. A complementary check is to verify the identities e^{-i T^(2)} = e^{-i T^(1)} and the analogous potential-term identities on generic sorted-list basis states; they hold on single Slater determinants, so the check should use a superposition.

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

If this is right

  • Plane-wave Trotterized phase estimation can be performed with O(N log M) qubits at the same asymptotic gate cost as first-quantized, making the sorted-list encoding a low-qubit choice in that regime.
  • In the MO basis, sorted-list total resource advantage is confined to N/M below roughly 0.1; above that, Jordan-Wigner's small T-gate cost dominates despite more qubits.
  • For qubitization in the MO basis, PREPARE dominates, so sorted-list and Jordan-Wigner have similar T counts, and sorted-list saves qubits without hurting gates.
  • For qubitization in the plane-wave basis, first-quantized retains a clear lead (5–10 orders in benchmark T counts), so sorted-list is not recommended there.
  • The break-even filling ratio provides a quantitative rule for choosing between the two encodings.

Where Pith is reading between the lines

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

  • If the O(N)-gate conversion underlying Theorem 2 holds, sorted-list Trotter simulations in the plane-wave basis would be asymptotically as gate-efficient as first-quantized while using fewer qubits than Jordan-Wigner; that would make them attractive for early fault-tolerant chemistry without full first-quantization overhead.
  • The same conversion trick could be used to import other first-quantized circuits into second-quantized simulations, not just Trotter steps; nothing in the paper rules this out, but it is not claimed.
  • The MO crossover at N/M ≈ 0.1 suggests sorted-list merits reconsideration whenever basis sets are enlarged for high precision, e.g., for dispersion-bound or hyperfine calculations; this is a natural place to test the paper's cost model.
  • A concrete test of the paper's model would be to run the same RPE cost formulas with higher-order Trotter formulas (p > 2) and compare the crossover location, since the paper's numerical work uses p = 2.

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

3 major / 5 minor

Summary. This paper presents a resource-cost analysis of quantum phase estimation (QPE) for chemical Hamiltonians using the sorted-list encoding, comparing it with the Jordan-Wigner and first-quantized encodings. The comparison is carried out for both Trotterization-based robust phase estimation (RPE) and qubitization-based QPE, in both molecular-orbital and plane-wave bases. The main formal claims are: (Theorem 1) in the MO basis, a fermionic term costs O(N log M) gates/qubits with the sorted-list encoding, giving an advantage in the N/M → 0 limit; (Theorem 2) in the plane-wave basis, the sorted-list encoding has the same asymptotic gate scaling as the first-quantized encoding for Trotterization, namely O~((M^{2/3}N^{4/3}+M^{1/3}N^{8/3}) 5^{p/2-1}(M^{1/3}N^{2/3})^{1/p}/ε^{1+1/p}); (Theorem 3) in the plane-wave basis, qubitization with the sorted-list encoding costs O~((N^{1/3}M^{5/3}+N^{5/3}M^{4/3}+N^{2/3}M^{7/3})/ε), which is worse than first-quantized qubitization. The paper also includes numerical T-gate and qubit estimates for H2O, CO2, EC, LiPF6, and FeMoCo model systems.

Significance. If the asymptotic claims are correct, the paper gives a useful quantitative map of the regime in which the qubit savings of the sorted-list encoding outweigh its T-gate overhead. The supporting information contains detailed, original circuit constructions and exact gate-count tables for the sorted-list encoding in both Trotterization and qubitization settings, which are a valuable reference even independently of the numerical benchmarks. The paper is also honest about the regimes where the sorted-list encoding loses, especially in plane-wave qubitization, where the first-quantized encoding's superior 1-norm scaling dominates. However, the key plane-wave Trotterization parity claim in Theorem 2 depends on an unproven, same-author companion result [32] for an O~(N) conversion between the sorted-list and first-quantized encodings. Since this conversion is the mechanism by which the fermionic Fourier transform is implemented, the central claim is not yet self-contained.

major comments (3)
  1. [Theorem 2 / Proof of Theorem 2 (Methods)] The parity claim for plane-wave Trotterization rests on the hybrid quantization scheme of [32]. The proof says: 'This can be done efficiently using the hybrid quantization scheme introduced in a previous work [32], where we convert the wavefunction to the first-quantized encoding, performed the QFT in the first-quantization, and converting back to the sorted-list encoding with a gate cost of O~(N).' No derivation, circuit, or error analysis for this conversion is included. The issue is load-bearing: the conversion must be an isometry on the full (M choose N)-dimensional occupied subspace, must preserve fermionic parity signs through the Fourier transform, and must work for arbitrary superpositions, not just the single sorted basis state used in the proof. Without this, the claimed O~(N) FFFT cost fails and the sorted-list plane-wave Trotterization scaling could degrade to a Jordan-Wigner
  2. [Section A.3.3 (Simulation Methods for Trotterization cost)] The numerical T-gate costs in Table III are derived using the empirical constant C_gs = 3.470e-5 λ^2.081, fitted from Figure 8 of [2]. The paper gives no uncertainty estimate, no validation across molecules, and no comparison with the rigorous bound. Because the overall gate count scales as C_gs^{1/p} (p=2), a factor of two in C_gs changes costs by sqrt(2). The claim that the sorted-list encoding is a 'viable alternative' in the low-filling regime is partly based on these numbers; the fitted constant should be justified or the numerical conclusions should be qualified. At minimum, the authors should state the accuracy of the fit and test sensitivity.
  3. [Table IV (MO-basis qubitization) and surrounding text] The text says that for MO-basis qubitization the gate and qubit costs are similar between sorted-list and Jordan-Wigner because the PREPARE circuit dominates, yet Table IV appears to list different asymptotic T-gate scalings for the two encodings in methods such as Sparse and Single-Factorization (e.g., λM^2/ε for sorted-list versus λM^4/ε for Jordan-Wigner in some columns). This discrepancy is confusing: if PREPARE dominates, the leading-order total cost should be the same for both encodings. The table needs a clearer breakdown of which cost is PREPARE, which is SELECT, and how the total is obtained.
minor comments (5)
  1. [Table III] Several entries are typeset in a nonstandard way, e.g., '885×10^17' and '996×10^19', which should be written as 8.85×10^19 and 9.96×10^21. The reader cannot immediately compare magnitudes.
  2. [Tables III and VI] The definition of electron-filling ratio N/M is inconsistent: N is the number of electrons while M sometimes refers to spatial orbitals and sometimes to spin orbitals. Please state this explicitly in each table caption, because the break-even point N/M ≈ 0.1 is a headline result.
  3. [Proof of Theorem 2] The Trotter error bound is quoted from [38] without stating whether it applies to the first-quantized, second-quantized, or both representations. Since the paper claims the same bound for Jordan-Wigner and sorted-list encodings, a one-sentence justification would help.
  4. [Section A.3.1 / Figure A.16] The T-gate counts in Table A.6 assume that an Rz rotation can be executed with constant T count. No circuit synthesis for arbitrary-angle rotations is given; if each Rz is synthesized to precision δ, the T count should include a factor ~log(1/δ). The authors should specify whether T counts are idealized.
  5. [Abstract and Discussion] The phrase 'best known scaling to date' for first-quantized qubitization should be accompanied by a direct comparison with the cited work [16] and, if possible, with recent subsequent works, to avoid an unsupported superlative.

Circularity Check

1 steps flagged

Theorem 2's plane-wave parity is imported from the authors' own prior hybrid-quantization preprint [32] without proof or verification.

specific steps
  1. self citation load bearing [Proof of Theorem 2 (Methods), final paragraph; also Discussion: '...we are able to leverage the hybrid quantization scheme [32]...']
    "We are left with the implementation of the fermionic fast fourier transform circuits required to switch between the plane-wave for T^(2) and its dual basis for U^(2)+V^(2). This can be done efficiently using the hybrid quantization scheme introduced in a previous work [32], where we convert the wavefunction to the first-quantized encoding, performed the QFT in the first-quantization, and converting back to the sorted-list encoding with a gate cost of O~(N). This results in identical gate cost scaling for the entire RPE algorithm, as shown in the main text."

    The central new claim of Theorem 2 -- that sorted-list and first-quantized encodings have identical plane-wave Trotter scaling -- depends on an O~(N) conversion between the two encodings that is not derived, bounded, or verified anywhere in this paper. It is imported from the authors' own prior preprint [32] (same four authors). No circuit diagram, isometry proof, or complexity analysis is supplied, and the paper provides no external check of [32]. If the conversion is not O~(N) or introduces approximation error, the FFFT cost enters every Trotter step and the claimed parity with the first-quantized encoding collapses toward the Jordan-Wigner-like scaling. Thus the theorem's load-bearing step is a self-citation rather than a self-contained derivation.

full rationale

Most of the paper's cost analysis is grounded in external, independent references: the RPE repetition counts come from [2], the plane-wave Trotter error bound from [38], the Jordan-Wigner plane-wave circuit from [39], the first-quantized circuit costs from [14,16], and the sorted-list fermionic-operation costs from [17]. The MO-basis Theorem 1 and the qubitization Theorem 3 are derived from those external pieces plus the SI circuit decompositions, with numerical tables computed from the stated formulas; I find no fitted-input-called-prediction or definitional circularity in those parts. The single problematic link is the plane-wave Trotter result: the proof's final step, which is essential for the sorted-list/first-quantized parity, delegates the fermionic Fourier transform to the authors' own prior hybrid-quantization preprint [32]. That is a same-author citation carrying the central new scaling claim, with no in-paper verification. Since the remaining results are independent, the overall circularity is partial rather than total; score 4 reflects a load-bearing unverified self-citation without making the whole derivation equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

No new physical entities are introduced. The dependency burden is heavy on external prior results and one same-author unpublished conversion scheme; numerical estimates also lean on an empirical fitted constant from [2]. The paper's own original circuit constructions are detailed in the SI and are a genuine contribution.

free parameters (1)
  • C_gs (empirical Trotter error constant) = 3.470e-5 * lambda^2.081
    Used in Eq. A.31 for deterministic Trotter cost estimates; fitted from Figure 8 of [2], not fitted in this paper, but the numerical T-gate tables depend on it.
axioms (5)
  • domain assumption RPE cost formulas of [2], including the empirical C_gs, correctly give the number of Trotter applications needed for ground-state energy to precision epsilon.
    Equations A.31-A.33 and Table A.2; this is external empirical methodology used without re-derivation.
  • domain assumption The Trotter error bound of [38] applies to the plane-wave Hamiltonian in second quantization with the stated scaling.
    Used in the proof of Theorem 2 to set the time step delta; if the bound is encoding-specific or inapplicable, Theorem 2 changes.
  • ad hoc to paper The hybrid quantization conversion [32] between sorted-list and first-quantized encodings costs O(N) gates and preserves the Trotter error.
    Same authors, unpublished preprint, no code or formal proof; load-bearing for the plane-wave Trotterization parity in Theorem 2.
  • domain assumption First-quantized plane-wave qubitization circuit and its Hamiltonian 1-norm scalings from [16] are correct.
    Used for Theorem 3 comparisons and Table V.
  • domain assumption Molecular-orbital Hamiltonian coefficients obey realness and 8-fold symmetry for the qubitization MO analysis.
    Assumed for single-factorization qubitization in the MO basis, Section A.2.2.1; typical for molecular systems.

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read the original abstract

Quantum Phase Estimation (QPE) is a cornerstone algorithm for fault-tolerant quantum computation, especially for electronic structure calculations of chemical systems. Optimal simulation relies on a complex trade-offs across many parameters including Hamiltonian simulation techniques, basis sets, and the fermion-to-qubit encodings. Here, we characterize the trade-offs and quantify the quantum resource costs of the sorted-list encoding as a particle-conserving, low-qubit alternative to the Jordan-Wigner encoding. We identify specific regimes, across different simulation techniques and basis sets, where the sorted-list encoding would be favorable compared to existing methods. Our findings are further supported through numerical benchmarks of real-world chemical systems. We found the sorted-list encoding to be a viable alternative to the Jordan-Wigner encoding for the compact molecular orbital basis when the electron-filling ratio is low, which typically occurs when high-precision results are required. In the plane-wave basis, we found similar asymptotic gate and qubit scaling between the sorted-list and the first-quantized encoding, although the first-quantized encoding still retains lower constant factors.

Figures

Figures reproduced from arXiv: 2510.01710 by Alice Hu, Calvin Ku, Min-Hsiu Hsieh, Yu-Cheng Chen.

Figure 1
Figure 1. Figure 1: Workflow for an energy estimation algorithm using quantum computers, using either trotterization or [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

discussion (0)

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Forward citations

Cited by 1 Pith paper

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

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    physics.chem-ph 2026-05 conditional novelty 5.0

    Chemical properties and symmetries, not variational energy, should guide UHF trial selection for ph-AFQMC on iron-sulfur clusters, yielding accurate energies despite suboptimal sampling and bias compensation.

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    However, the eigenvalue of (Ψ 𝑝 𝑞 + Ψ𝑞 𝑝)/ √ 2 is not necessarily +1. The sign of the eigenvalue can be flipped depending on the parity of the fermionic operators. Assuming that𝑝 > 𝑞, the parity of 𝑎† 𝑝𝑎𝑞 and𝑎† 𝑞𝑎𝑝 can be extracted from Equation A.26. PARITY{𝑎† 𝑝𝑎𝑞}= Θ 𝑝𝑞 ⎛ ⎝ 𝑝−1∏︁ 𝑖=𝑞 𝑍𝑖 ⎞ ⎠ =− 𝑝−1∏︁ 𝑖=𝑞 𝑍𝑖 = 𝑝−1∏︁ 𝑖=𝑞+1 𝑍𝑖, PARITY{𝑎† 𝑞𝑎𝑝}= Θ 𝑞𝑝 ⎛ ⎝ 𝑝−1∏...