Pith. sign in

REVIEW 3 major objections 5 minor 72 references

Spectral amplification for ground-state energy estimation of electronic structure in first quantization

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read This paper establishes that first-quantized ground-state energy estimation can be made asymptotically cheaper by rewriting the plane-wave Hamiltonian as a charge-density sum of squares, reducing the effective block-encoding normalization fr

desk verdict Real algorithmic improvement for first-quantized plane-wave simulation, but the molecular asymptotic speedup leans on a scaling assumption the authors themselves flag as not guaranteed. read the letter →

arxiv 2607.15358 v1 pith:T57SWUXN submitted 2026-07-16 quant-ph physics.chem-ph

Alicja Dutkiewicz , Alec F. White , Guang Hao Low , A. Eugene DePrince III , Matthew P. Harrigan , Marika Kieferova , Ryan Babbush , Dominic W. Berry
show 1 more author
Nicholas C. Rubin
This is my paper
classification quant-phphysics.chem-ph MSC 81P6881V55
keywords spectralgapamplificationsumofsquaresfirstquantizationquantumphaseestimationblockencodingplane-wavebasiselectronicstructureToffolicomplexity
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

This paper claims that the ground-state energy estimation cost for first-quantized electronic structure in a plane-wave basis can be reduced by rewriting the Hamiltonian as a sum of squares generated by the total charge density operator. The rewrite makes the Hamiltonian non-negative up to a shift and enables spectral gap amplification, lowering the effective block-encoding normalization from O(ηΔ^{-2}+η^2Δ^{-1}) to O(ηΔ^{-1.5}+η^{1.5}Δ^{-1}). Because the block-encoding circuitry stays almost the same as the standard linear-combination-of-unitaries method, the asymptotic gain translates into end-to-end Toffoli reductions of 2–44× across molecular, metallic, and fusion-target benchmarks. A reader should care because this directly lowers the estimated cost of classically intractable materials and chemistry simulations without adding physical approximations.

What carries the argument

The central object is the sum-of-squares decomposition of H+βI built from three families of generators: kinetic generators T_{j,w}, potential generators Vν = sqrt(Cν)(Σ_j U_shift,j(ν) − Σ_ℓ ζ_ℓ e^{-ikν·Rℓ} I), and self-interaction generators X_{j,ν} that detect out-of-bounds momentum shifts. Squaring Vν reproduces the electron-electron and electron-nuclear Coulomb terms; the diagonal self-interaction left over is absorbed into the identity shift β via the X terms. Block encoding is implemented as a nested LCU using the Chebyshev/oblivious-amplitude-amplification structure of SOSSA, with branch-selection registers, nested-boxes preparation for ν, overflow-flag reflection, and QROM-based data

What would settle it

Compute β_nuc from Eq. (33) for a sequence of molecular supercells with a fixed fraction of vacuum and increasing η, holding density and momentum cutoff fixed; if β_nuc grows faster than linearly in η, the asymptotic O(η^{0.5}) speedup in the large-system limit disappears.

Watch

Extended reading notes

Core claim

The central discovery is an analytical sum-of-squares (SOS) representation of the first-quantized electronic structure Hamiltonian, H+βI = Σ_α O_α†O_α, whose block encoding has essentially the same per-query cost as the previous LCU implementation. The key generator Vν is the Fourier component of the total charge density operator: the electron translation operator minus the classical nuclear density. Its square produces both the electron-electron and electron-nuclear Coulomb terms, while the spurious diagonal self-interaction is converted into an identity shift by companion operators X_{j,ν} that flag out-of-bounds momentum shifts. The resulting SOS norm λ_SOS has the same asymptotic form as

Load-bearing premise

For the claimed O(η^{1.5}Δ^{-1}) term to hold, the nuclear self-interaction energy must grow only linearly with system size; in clustered-ion or small-molecule supercells it could instead grow quadratically, which would degrade the bound λeff ≤ sqrt(λ_SOS(β+E0)).

Editorial extensions

If this is right

  • In the continuum limit (Δ^{-1}≫η), the kinetic-energy-dominated cost drops from O(ηΔ^{-2}) to O(ηΔ^{-1.5}), an O(Δ^{-0.5}) speedup.
  • In the large-system limit (η≫Δ^{-1}), the Coulomb-dominated cost drops from O(η^2Δ^{-1}) to O(η^{1.5}Δ^{-1}), an O(η^{0.5}) speedup.
  • End-to-end logical Toffoli counts for the benchmark systems fall by 2.5–44×, with the largest reduction (44×) for the deuterium fusion-target system; the uniform electron gas improves by up to 10.6×.
  • The block-encoding walk-operator cost stays comparable to the LCU method, so the query-count reduction is realized as a full end-to-end speedup.
  • Because the asymptotic scaling is better, the speedup grows with system size, which is the regime relevant to materials simulations approaching the thermodynamic limit.

Reading between the lines

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

  • The same charge-density SOS construction should transfer to other translation-based interactions, such as dipole or higher multipole terms, wherever a classical background density can be subtracted; the paper only sketches broader applicability.
  • A concrete stress test is to compute β_nuc for large molecular supercells with fixed vacuum fraction; if β_nuc grows faster than linearly in η, the molecular-case speedup would not hold, even though periodic and random-ion cases are proven.
  • The compiled costs show that data movement (controlled swaps of electron registers) dominates the Toffoli count, so hardware-aware optimization of that routing would multiply the practical benefit.
  • The non-Born-Oppenheimer generalization has the same λ_SOS and β scaling, suggesting a direct extension to electron-nuclear dynamics with the same asymptotic improvement.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper constructs an analytical sum-of-squares (SOS) representation of the first-quantized electronic structure Hamiltonian in a plane-wave basis (Eq. 26), based on the total charge density operator. It shows that the SOS representation has a block-encoding normalization λ_SOS with the same asymptotic scaling as the prior LCU method, while spectral gap amplification lowers the effective normalization to λ_eff = O(ηΔ^{-1.5} + η^{1.5}Δ^{-1}) (Eq. 58) under assumptions β_nuc = O(ηΔ^{-1}) and E_0 = O(η). The authors provide a full logical-level compilation of the SOSSA walk operator, detailed Toffoli counts, and resource comparisons for molecular, periodic, and uniform-electron-gas benchmarks, reporting speedups between 2× and 44× relative to the LCU method of Ref. [16]. The central mathematical identity is exact and the compilation is unusually concrete, but the asymptotic claim for molecules rests on a scaling assumption that the paper itself concedes is not guaranteed.

Significance. If the asymptotic claim holds, the paper gives the first first-quantized, plane-wave ground-state energy estimation algorithm whose effective block-encoding normalization improves on the standard LCU approach in both the continuum limit and the large-system limit. The SOS construction is parameter-free and algebraically checkable from Eqs. (26)–(38); this is a genuine structural insight rather than a numerical curve fit. The paper also ships a detailed logical compilation with explicit Toffoli counts, reflection costs, and state-preparation tables, making the resource estimates reproducible in principle. The resource improvements for periodic lithium cells and the deuterium fusion target are significant even as finite-size numbers. However, the universal asymptotic statement is conditional on an assumption about nuclear configurational scaling that is proven for periodic supercells, argued for disordered and random systems, but explicitly left open for the molecular case; this limits the scope of the central claim.

major comments (3)
  1. [Section III, Appendix A, Eqs. (33), (47), (58)] The molecular asymptotic speedup is not established. The bound λ_eff ≤ sqrt(λ_SOS(β+E_0)) yields Eq. (58) only if β+E_0 = O(ηΔ^{-1}), which in turn requires β_nuc = O(ηΔ^{-1}). Appendix A proves this exactly for periodic supercells and gives numerical evidence for randomly distributed ions, but explicitly concedes that for molecular systems the linear regime may not appear until the molecule dwarfs the vacuum region. For a compact non-periodic nuclear configuration, the small-|ν| Fourier amplitude in Eq. (33) can be as large as Z_tot = η without phase cancellation, so β_nuc can be superlinear; the triangle-inequality η^2 bound is not the only superlinear scenario. If β_nuc is superlinear, the claimed O(η^{0.5}) advantage in the large-system limit is weakened or lost. Table V's molecular speedups are finite-size resource data and do not certify Eq. (58) for molecules.
  2. [Section III, Eqs. (47)–(58)] The derivation of the second term in Eq. (58) implicitly assumes that the gap Egap = E0 + β is dominated by β = O(ηΔ^{-1}). The stated assumption E0 = O(η) only gives Egap = O(η + ηΔ^{-1}). In the large-system regime, if E0 dominates, the same algebra gives λ_eff = O(η^{1.5}Δ^{-0.5}) rather than O(η^{1.5}Δ^{-1}). The η-scaling speedup survives, but the precise Δ dependence in Eq. (58) and the corresponding slopes in Fig. 2 need to be justified with an explicit statement about the relative size of β and E0, or corrected.
  3. [Section IV.D, Table V] The resource comparison mixes two different tasks: it reports speedups for molecular benchmarks whose asymptotic protection is not established, and it also reports large periodic systems where the proof in Appendix A does apply. The abstract and introduction present a universal speedup, but the table's molecular rows should be separated from the periodic-crystal rows, with a caveat that the molecular speedups are finite-size results. This would not change the numerical values but would align the presentation with the actual proof coverage.
minor comments (5)
  1. [Abstract and Conclusion] The phrase 'lowest cost estimates for ab initio materials simulation' overclaims: the benchmarks compare only with a limited set of prior algorithms (mainly Ref. [16], some Ref. [23] and Ref. [19]), not with all existing pseudopotential or compact-basis first-quantized approaches.
  2. [Eq. (38)] The symbol λ_T is used both for the LCU normalization (Eq. 18) and as a shorthand inside λ_SOS. This may confuse readers; a separate symbol such as λ_T^SOS would clarify that the kinetic contribution enters squared.
  3. [Appendix D.2, Eq. (D43)] The alternative T+U SOS is shown to scale as Θ(η^4Δ^{-3}), which is asymptotically worse. It would be helpful to mention this rejected alternative in the main text so that readers do not wonder why only one SOS representation is pursued.
  4. [Figure 2] The fitted slopes for SOSSA are 1.33 and 1.47, close but not equal to the predicted 1.5. The scatter is attributed to the Hartree-Fock upper bound, but the text should state whether finite-Δ corrections or the Egap dominance issue in major comment 2 contribute to the systematic deviation.
  5. [Table V] Several column headers are difficult to parse ('LCU SOS', 'LCU SOS', repeated labels). The table would be more readable with explicit units and a legend explaining the 'AA'/'No AA' choice.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central asymptotic claim follows from exact SOS algebra plus an explicitly stated scaling assumption, not from a fitted input or a self-citation chain.

full rationale

The derivation of Eq. (58) is self-contained. The SOS representation in Eqs. (26)-(30) is verified by direct expansion: squaring T_j,w, V_nu, and X_j,nu reproduces H plus a shift, with beta given in Eq. (31) and beta_nuc in Eq. (33). Substituting the asymptotics lambda_T = O(eta Delta^-2), lambda_out = O(Delta^-1), and Z_tot = O(eta) into Eq. (38) gives lambda_SOS = O(eta Delta^-2 + eta^2 Delta^-1); with E0 = O(eta) and beta = O(eta Delta^-1), Eq. (47) directly yields lambda_eff = O(eta Delta^-1.5 + eta^1.5 Delta^-1). No parameter is fitted to the claimed improvement; the compiled Toffoli numbers in Table V are backed by explicit per-subroutine arithmetic accounting and external Hartree-Fock energies. The paper's reliance on the SOSSA framework [14,26] and first-quantized LCU compilation [16] is a citation of published, parameter-free protocols, not a reduction of the present claim to those citations. The one genuinely load-bearing assumption is beta_nuc = O(eta), proven exactly for periodic supercells in Appendix A and numerically supported for random ion configurations; the paper explicitly concedes for molecular systems that 'the asymptotic behavior may not become apparent until the molecule is so large that it dwarfs the vacuum region.' That is a limitation on the scope of Eq. (58), not a circular step.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central asymptotic claim has no fitted free parameters: λ_SOS, β, and λeff are defined by exact algebraic quantities. The free parameters listed are compilation precision settings that affect the resource tables but not the asymptotic scaling. The key assumptions are physical scaling laws (E0 extensive, β_nuc extensive) that are stated and partially justified, not hidden fitted values.

free parameters (2)
  • nested-boxes scaling κ = system-dependent, not specified
    Appendix B 6: κ adjusts the accuracy of the nested-boxes momentum-transfer state preparation and enters the hardware-inflated 1-norms Λ^κ_V and Λ^κ_X. It affects only compiled resource estimates, not the asymptotic claim.
  • state-preparation precision parameters n_T, n_M, n_R = system-dependent, not specified
    Appendix B: finite-precision rotation synthesis and nested-boxes precision M=2^{n_M} inflate λeff. These are standard compilation parameters, not fitted to data.
assumptions (6)
  • domain assumption Born-Oppenheimer approximation and plane-wave Hamiltonian model, Eqs. (5)–(7)
    The entire derivation is set in the plane-wave first-quantized Coulomb Hamiltonian, including the Born-Oppenheimer separation. This is the standard model for the target application.
  • domain assumption Charge neutrality Z_tot = O(η)
    Used in Section III to obtain λ_SOS = O(ηΔ^{-2}+η^2Δ^{-1}). The resource estimates assume neutral or near-neutral systems.
  • domain assumption Extensivity of the ground-state energy, E0 = O(η), and basis-set convergence of Hartree-Fock, Eqs. (51)–(53)
    The λeff scaling requires β+E0 = O(ηΔ^{-1}). The paper invokes the linked-cluster theorem and HF basis-set-limit convergence; for strongly correlated or non-extensive systems this could fail.
  • domain assumption β_nuc = O(η) for physically relevant systems
    Appendix A is the load-bearing scaling assumption. It is exact for periodic supercells, numerically supported for random ion configurations, but conceded to fail for worst-case molecular geometries and possibly not reached for practical molecules with large vacuum regions.
  • domain assumption SOSSA framework and the phase-estimation error formula from Refs. [14,26]
    The walk-operator construction and query-count formula λeff = sqrt(E_gap(λ_SOS-E_gap)) are taken from prior work rather than re-derived.
  • domain assumption LCU compilation subroutines from Ref. [16], including nested-boxes state preparation and phase-gradient arithmetic
    The gate-cost tables rely on these prior implementations; they are not re-verified in this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Spectral amplification for ground-state energy estimation of electronic structure in first quantization." pith.science (2026). https://pith.science/paper/T57SWUXN

@misc{pith2026260715358,
  author       = {Pith},
  title        = {Pith review of: Spectral amplification for ground-state energy estimation of electronic structure in first quantization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T57SWUXN}},
  note         = {Machine review of arXiv:2607.15358}
}
abstract

We demonstrate an asymptotic gate complexity improvement in first-quantized ground-state energy estimation of electronic structure Hamiltonians in a plane wave basis by employing the sum-of-squares spectral gap amplification protocol. The improvement relies on identifying a sum-of-squares representation of the Hamiltonian which provides a lower bound certificate and low cost block encoding that leads to a provably lower quantum phase estimation gate cost. This is achieved by using a sum-of-squares operator generated by the total charge density operator resulting in a block encoding normalization improvement of $\lambda = \mathcal{O}\left(\eta\Delta^{-1.5}+\eta^{1.5}\Delta^{-1} \right)$ compared to prior work $\lambda = \mathcal{O}(\eta\Delta^{-2}+\eta^2\Delta^{-1})$ where $\eta$ is the number of electrons and $\Delta$ is the simulation grid spacing. The asymptotic reduction in block encoding normalization and similar block encoding costs to prior work is demonstrated to reduce resource estimates for materials and chemical systems by a factor of $2 - 44\times$ corresponding to the lowest cost estimates for ab initio materials simulation.

Figures

Figures reproduced from arXiv: 2607.15358 by the authors.

Figure 1
Figure 1. FIG. 1. Overall structure of the circuit for the block-encoding. For clarity we have grouped the registers as follows: “branch” [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Λ [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The top five subroutines and summed remainder costs for the Hydrogen system in Table [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Structure of the mixed outer/inner prepare [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Structure of inner prepare [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The structure of the select oracle ( [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

72 extracted references · 3 linked inside Pith

  1. [16]

    Performing the inequality test with a quantum register instead of a classically chosen number increases the cost by 1 Toffoli

    The inequality test needs to be with a number output into a quantum register. Performing the inequality test with a quantum register instead of a classically chosen number increases the cost by 1 Toffoli. The inequality test is performed twice to make it two extra Toffolis. (The inversion of the inequality test is performed with Cliffords.)

  2. [1]

    Preparation oracles Utilizing the direct sum decomposition we can now construct the states and unitaries needed for the block encoding described in Eq. (60). The states we need to prepare are |Ψprep⟩=P out/in|0. . .0⟩anc = 2M s=0 s Λ(s) λSOS |s⟩S ⊗ |ϕ(s) out⟩ ⊗ |ϕ(s) in ⟩,(86) where |ϕ(0) out⟩=   1√η ηX j=1 |j⟩idx   ⊗ 1√ 3 2X w=0 |w⟩ ! ,(87) |ϕ(0) in ...

  3. [2]

    The unary iteration step requires max[η, Ztot]−2 Toffoli gates

    Selection oracles TheSELoracle proceeds in three distinct phases: (A) unary iteration over indexj=m∈[0,max[η, Z tot]−1] where for each value we swap electronjinto a designated workspace register and writeR ℓ(m) to an ancilla register, (B) the conditional Hamiltonian arithmetic controlled by the execution flags (Exec T ,Exec V ,Exec X ), and (C) uncomputat...

  4. [3]

    Here we provide the total costs in Table IV

    Reflection costs The qubits we reflect on are described in Appendix B 5. Here we provide the total costs in Table IV. TABLE IV. Comprehensive Toffoli costs for the walk operator reflections and control for phase estimation. For the global reflection, in each step of phase estimation there is one more Toffoli to make this reflection controlled, as well as ...

    arXiv

  5. [4]

    Modification forν-state prep failure Highlighted in Refs [15, 16], the nested boxes state preparation for the outer state over the momentum difference registerνhas a failure rate because it uses rejection sampling to prepare a discrete probability distribution. The failure rate is approximately 1/4, so a single round of amplitude amplification can boost t...

  6. [5]

    In other words, the circuit produces more trash to be recycled than is needed (in terms of Λ T )

    ΛT ΛV +ΛX < (1−pν ) pν corresponds to a case where the failure during theνstate preparation is sufficient to cover all the weight forTand then some. In other words, the circuit produces more trash to be recycled than is needed (in terms of Λ T ). This is handled by throttling the application ofTby imposing that not only should we have a failure in state p...

  7. [6]

    In other words, there istoo little trashgenerated by nested-boxes state preparation to be recycled intoTto cover all ofT

    ΛT ΛV +ΛX ≥ (1−pν ) pν corresponds to the case where failure during theνstate preparation is insufficient to cover all the weight needed for applying theToperator. In other words, there istoo little trashgenerated by nested-boxes state preparation to be recycled intoTto cover all ofT. In this case we consider applyingTif nested-boxes state preparation fai...

  8. [7]

    The choice of which part of the Hamiltonian to block encode is governed by the flags Exec V , ExecX , and ExecT

    State Preparation for Selection betweenT, V, andX The logical routing depends entirely on whether the target Hamiltonian falls into the AND or OR recycling regime. The choice of which part of the Hamiltonian to block encode is governed by the flags Exec V , ExecX , and ExecT . The Boolean logic used to compute these flags is determined classically during ...

Show all 72 references
  1. [8]

    Exec T is computed asf ν = 1 ANDq T = 0

  2. [9]

    Exec V is computed asf ν = 0 ANDq V X= 0

  3. [10]

    Regime 2: The OR Strategy The execution flags are computed as:

    Exec X is computed asf ν = 0 ANDq V X= 1 Only two Toffolis are needed, because it is possible to copy the NOT of the value off ν and useq V X to control the swap of the qubits storing Exec V and ExecX . Regime 2: The OR Strategy The execution flags are computed as:

  4. [11]

    Exec T is computed asf ν = 1 ORq T = 0

  5. [12]

    Exec V is computed as Exec T = 0 ANDq V X= 0

  6. [13]

    One Toffoli is used to compute the OR for Exec T

    Exec X is computed as Exec T = 0 ANDq V X= 1 Again only two Toffolis are needed. One Toffoli is used to compute the OR for Exec T . The NOT of that is copied into the one of the qubits for Exec V and ExecX with a CNOT, thenq V Xis used to control the swap of the 1 into the cor...

  7. [14]

    Thew-bit preparation takes 3n+2b r −9 withb r = 8 andn= 2 yielding 13 Toffolis

    Structure of PREP out/in The outer state preparation includes the signal state preparation for theν-register,{q T , qV X}control registers, execution flag preparation, and unconditional preparation over indexwandr. Thew-bit preparation takes 3n+2b r −9 withb r = 8 andn= 2 yiel...

  8. [15]

    These Hadamards need to be made controlled 3 times, though one is an inversion

    A number of Hadamards corresponding to the difference in the number of bits for the two numbersn η bits for ηandn Ztot bits forZ tot. These Hadamards need to be made controlled 3 times, though one is an inversion. Because the controlled Hadamards are performed on|0⟩states, the...

  9. [17]

    The rotation is performed twice, increasing the Toffoli count by 2

    The rotation on the ancilla qubit also needs to be by an angle stored in a quantum register, increasing the Toffoli cost by 1. The rotation is performed twice, increasing the Toffoli count by 2. Overall, the Toffoli cost to control between preparing an equal superposition of t...

  10. [18]

    Kinetic Energy This is very similar to Ref

    Select implementations a. Kinetic Energy This is very similar to Ref. [16] in that we expand the absolute value in a sum over the binary representation of the magnitude part of the signed integer to construct the LCU Tj,w = 1 2 X b∈{0,1} np−2X r=0 √ 2π Ω1/3 2r X p∈G (−1)b(1+pw...

  11. [19]

    From the 3n p-bit workspace register, controlled by the spatial index|w⟩ ∈ {0,1,2}, we copy the corresponding np −1 magnitude bits into a temporary 1D register. In Ref. [16] its cost was given as 3(n p −1) Toffoli gates, but it may be reduced to 2(n p −1). The reason is that w...

  12. [20]

    From the temporary register, controlled by the bit-index|r⟩, we copy ther-th bitp w,r into a dedicated ancilla qubitq r. In Ref. [16] the cost was given asn p −1, but it may be reduced ton p −2 using the same method as before (copy the first without control)

  13. [21]

    The|b⟩register is prepared in the|+⟩ b state

    Finally, we must apply the LCU phase (−1) b(1+pw,r ). The|b⟩register is prepared in the|+⟩ b state. This phase dictates that a global−1 phase should be applied if and only ifb= 1 andp w,r = 0. We achieve this by applying anXgate toq r, followed by a CCZ gate controlled by|b⟩,q...

  14. [22]

    [16] Total Cost:The kinetic arithmetic requires 3(n p −1) Toffolis

    The ancilla qubits may be erased with measurements and Clifford gates, as in Ref. [16] Total Cost:The kinetic arithmetic requires 3(n p −1) Toffolis. 27 qT R(θ) EXEC branch qV X R(φ) ExecT ExecV ExecX ν NestedBoxes outer µ m fν w PREPw fw Idx PREPIdx mix flag-nuc PREPflag-nuc ...

  15. [23]

    This physically implementsU shift,j(ν)

    Iff nuc-flag = 0 we apply the momentum shift for the electron branch. This physically implementsU shift,j(ν). Note we will use the overflow bits to flag (p−ν)/∈G. (a) Convertq w to two’s complement, which costsn p −2 Toffolis. (b) Perform a controlled swap ofν w (controlled by...

  16. [24]

    This is because the control of the phase is implemented by this controlled swap

    Iff nuc-flag = 1: As each component ofνis swapped in 1(b) above, we also apply the phasing for the corresponding component for thef nuc-flag = 1 branch. This is because the control of the phase is implemented by this controlled swap. The value of ˜Rℓ,w is output from the itera...

  17. [25]

    This physically implements−U phase,ℓ(ν)

    Apply aZgate to|f nuc-flag⟩to kick back the−1 phase. This physically implements−U phase,ℓ(ν). The SOSSA walk operator requires executing the block encoding unitary twice per step (U R forward, andU † R in reverse). This meansS in must apply the phasee −ikν ·Rℓ , whileS † in mu...

  18. [26]

    The execution bits, which are one-hot encoded Exec T,V,X , are in the outer preparation and can be used to control the reflection application

    Reflection costs Recall that, because of our flag qubits, we have a direct-sum definition of the inner Hilbert space Πin = 2X s=0 |s⟩⟨s|S ⊗I (s) out ⊗ |0⟩⟨0|(s) in .(B26) The inner Hilbert space qubitsH (s) in must be reflected upon, and thus we must use multi-controlledZgates...

  19. [27]

    (b)f ν: The nested boxes failure flag (1 qubit, outer)

    Branch & Logic Qubits: (a)q T , qV X: The branch selection qubits (2 qubits, outer). (b)f ν: The nested boxes failure flag (1 qubit, outer). (c) Total: 3 qubits

  20. [28]

    (b)r: The bit-index (needsn p −1 qubits for 0 ton p −2, inner)

    Spatial (w) & Bit-Index (r, b): (a)w: The spatial dimension (2 qubits, sincen= 2 for 3 states, outer). (b)r: The bit-index (needsn p −1 qubits for 0 ton p −2, inner). (c)b: As part of block encodingT(1 qubit, inner). (d) Total:n p + 2 qubits

  21. [29]

    (b)µ: The box index (n p qubits in unary, outer)

    Nested Boxes (|ν⟩): (a)ν: The 3D momentum transfer vector (3n p + 3 qubits, accounting for sign bits, outer). (b)µ: The box index (n p qubits in unary, outer). (c)m nb: The inequality test uniform state (n M qubits, outer). (d) Total: We eliminate the inner boxes, unlike in Re...

  22. [30]

    30 (b)f nuc-flag: The nuclear flag (1 qubit, inner)

    Shared Index (idx, fnuc-flag): (a)idx: The shared index register (⌈log 2 max(η, Ztot)⌉qubits, outer and inner). 30 (b)f nuc-flag: The nuclear flag (1 qubit, inner). (c) Total:⌈log 2 max(η, Ztot)⌉+ 1 qubits

  23. [31]

    The cost of a reflection on this number of qubits is 2 Toffolis less than this, though there are 2 extra needed for the phase estimation

    Overflow bits forXandVand the flag (a) two ancilla bits for each directionw(6 qubits, outer) (b) one qubit flagged when the addition overflows inVorX(inner) Overall, the outer projection qubits are qT , qV X, fν, w, ν, µ, m,idx,overflow-w.(B27) As a result, the total number of...

  24. [32]

    This inflation accounts for the finite precision of the state prepa- ration subroutines and the inherent failure probabilities of the quantum arithmetic

    Hardware-Inflated 1-Norms and the Effectiveλ eff To physically execute the SOSSA block encoding on quantum hardware, the ideal analytical 1-norms (Λ T ,Λ V ,Λ X ) defined previously must be systematically inflated. This inflation accounts for the finite precision of the state ...

  25. [33]

    It is thenλ ˜Vν =η, without theZ tot term

    We can redefine our inner LCU 1-norm for the potential term as it no longer involves the nuclear interaction. It is thenλ ˜Vν =η, without theZ tot term. The SOS Jellium block encoding normalization is thus λ(jellium) SOS = ηX j=1 X w∈{x,y,z} λ2 Tj,w + X ν∈G0 Cνλ2 ˜Vν + ηX j=1 ...

  26. [34]

    For the kinetic termT, we use the same decomposition as in Eq

    Separate SOS decompositions ofT,U, andV The most direct approach is to construct independent SOS representations of the kinetic, interaction, and potential terms. For the kinetic termT, we use the same decomposition as in Eq. (27) in the main text: T= ηX j=1 X w∈{x,y,z} T † j,...

  27. [35]

    SOS decomposition ofT+U A second natural attempt is motivated by factorization techniques for the Coulomb Hamiltonian originating in early quantum mechanics [36]. For a single electron interacting with a single nucleus of chargeZ, the Hamiltonian H=− 1 2 ∇2 − Z r (D14) can be ...

  28. [36]

    For example, approximating the electron-electron interaction as a sum of separable real-space functions naturally leads to an SOS form

    Other SOS decompositions Many other SOS representations are possible. For example, approximating the electron-electron interaction as a sum of separable real-space functions naturally leads to an SOS form. The various terms can also be split up further. Taking the electron-nuc...

  29. [37]

    Aspuru-Guzik, A

    A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Simulated quantum computation of molecular energies, Science309, 1704 (2005)

    2005

  30. [38]

    J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik, Simulation of electronic structure hamiltonians using quantum com- puters, Molecular Physics109, 735 (2011)

    2011

  31. [39]

    Babbush, D

    R. Babbush, D. W. Berry, I. D. Kivlichan, A. Y. Wei, P. J. Love, and A. Aspuru-Guzik, Exponentially more precise quantum simulation of fermions in second quantization, New Journal of Physics18, 033032 (2016)

    2016

  32. [40]

    Reiher, N

    M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, Elucidating reaction mechanisms on quantum computers, Proceedings of the national academy of sciences114, 7555 (2017)

    2017

  33. [41]

    Babbush, C

    R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Encoding electronic spectra in quantum circuits with linear t complexity, Physical Review X8, 041015 (2018)

    2018

  34. [42]

    D. W. Berry, C. Gidney, M. Motta, J. R. McClean, and R. Babbush, Qubitization of arbitrary basis quantum chemistry leveraging sparsity and low rank factorization, Quantum3, 208 (2019)

    2019

  35. [43]

    von Burg, G

    V. von Burg, G. H. Low, T. H¨ aner, D. S. Steiger, M. Reiher, M. Roetteler, and M. Troyer, Quantum computing enhanced computational catalysis, Physical Review Research3, 033055 (2021)

    2021

  36. [44]

    J. Lee, D. W. Berry, C. Gidney, W. J. Huggins, J. R. McClean, N. Wiebe, and R. Babbush, Even more efficient quantum computations of chemistry through tensor hypercontraction, PRX quantum2, 030305 (2021)

    2021

  37. [45]

    J. J. Goings, A. White, J. Lee, C. S. Tautermann, M. Degroote, C. Gidney, T. Shiozaki, R. Babbush, and N. C. Rubin, Reliably assessing the electronic structure of cytochrome P450 on today’s classical computers and tomorrow’s quantum computers, Proceedings of the National Acade...

    2022

  38. [46]

    N. C. Rubin, D. W. Berry, F. D. Malone, A. F. White, T. Khattar, A. E. DePrince III, S. Sicolo, M. K¨ uehn, M. Kaicher, J. Lee, and R. Babbush, Fault-tolerant quantum simulation of materials using bloch orbitals, PRX Quantum4, 040303 (2023)

    2023

  39. [47]

    A. V. Ivanov, C. S¨ underhauf, N. Holzmann, T. Ellaby, R. N. Kerber, G. Jones, and J. Camps, Quantum computation for periodic solids in second quantization, Physical Review Research5, 013200 (2023)

    2023

  40. [48]

    Oumarou, M

    O. Oumarou, M. Scheurer, R. M. Parrish, E. G. Hohenstein, and C. Gogolin, Accelerating quantum computations of chemistry through regularized compressed double factorization, Quantum8, 1371 (2024)

    2024

  41. [49]

    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 simulation of strongly correlated molecules, PRX Quantum6, 020327 (2025)

    2025

  42. [50]

    G. H. Low, R. King, D. W. Berry, Q. Han, A. E. DePrince, A. F. White, R. Babbush, R. D. Somma, and N. C. Rubin, Fast quantum simulation of electronic structure by spectral amplification, Phys. Rev. X15, 041016 (2025)

    2025

  43. [51]

    Babbush, D

    R. Babbush, D. W. Berry, J. R. McClean, and H. Neven, Quantum simulation of chemistry with sublinear scaling in basis size, npj Quantum Information5, 92 (2019)

    2019

  44. [52]

    Y. Su, D. W. Berry, N. Wiebe, N. Rubin, and R. Babbush, Fault-tolerant quantum simulations of chemistry in first quantization, PRX Quantum2, 040332 (2021)

    2021

  45. [53]

    Delgado, P

    A. Delgado, P. A. Casares, R. Dos Reis, M. S. Zini, R. Campos, N. Cruz-Hern´ andez, A.-C. Voigt, A. Lowe, S. Jahangiri, M. A. Martin-Delgado,et al., Simulating key properties of lithium-ion batteries with a fault-tolerant quantum computer, Physical Review A106, 032428 (2022)

    2022

  46. [54]

    M. S. Zini, A. Delgado, R. dos Reis, P. A. M. Casares, J. E. Mueller, A.-C. Voigt, and J. M. Arrazola, Quantum simulation of battery materials using ionic pseudopotentials, Quantum7, 1049 (2023)

    2023

  47. [55]

    N. C. Rubin, D. W. Berry, A. Kononov, F. D. Malone, T. Khattar, A. White, J. Lee, H. Neven, R. Babbush, and A. D. Baczewski, Quantum computation of stopping power for inertial fusion target design, Proceedings of the National Academy of Sciences121, e2317772121 (2024). 38

    2024

  48. [56]

    D. W. Berry, N. C. Rubin, A. O. Elnabawy, G. Ahlers, A. E. DePrince III, J. Lee, C. Gogolin, and R. Babbush, Quantum simulation of realistic materials in first quantization using non-local pseudopotentials, npj Quantum Information10, 130 (2024)

    2024

  49. [57]

    Pocrnic, I

    M. Pocrnic, I. Loaiza, J. M. Arrazola, N. Wiebe, and D. Motlagh, Efficient simulation of pre-born-oppenheimer dynamics on a quantum computer, arXiv preprint arXiv:2602.11272 (2026)

    2026

  50. [58]

    F. H. da Jornada, M. Lostaglio, S. Pallister, B. S ¸ahino˘ glu, and K. I. Seetharam, A comprehensive framework to simulate real-time chemical dynamics on a fault-tolerant quantum computer, arXiv preprint arXiv:2504.06348 (2025)

    2025 arXiv

  51. [59]

    T. N. Georges, M. Bothe, C. S¨ underhauf, B. K. Berntson, R. Izs´ ak, and A. V. Ivanov, Quantum simulations of chemistry in first quantization with any basis set, npj Quantum Information11, 55 (2025)

    2025

  52. [60]

    E. C. Eklund, A. Tikku, P. Sinnott, W. J. Huggins, G. H. Low, D. W. Berry, and I. Kassal, End-to-end simulation of chemical dynamics on a quantum computer, arXiv preprint arXiv:2603.19007 (2026)

    2026

  53. [61]

    A. V. Ivanov, A. Patterson, M. Bothe, C. S¨ underhauf, B. K. Berntson, J. J. Mortensen, M. Kuisma, E. Campbell, and R. Izs´ ak, Quantum computation of electronic structure with projector augmented-wave method and plane wave basis set, Journal of Chemical Theory and Computation...

    2025

  54. [62]

    R. King, G. H. Low, R. Babbush, R. D. Somma, and N. C. Rubin, Quantum simulation with sum-of-squares spectral amplification, Physical Review Letters136, 110601 (2026)

    2026

  55. [63]

    J. Leng, J. Jiang, and L. Lin, Accelerating quantum gibbs sampling without quantum walks, arXiv preprint arXiv:2604.22996 (2026)

    2026 arXiv

  56. [64]

    R. D. Somma and S. Boixo, Spectral gap amplification, SIAM Journal on Computing42, 593 (2013)

    2013

  57. [65]

    G. H. Low, W. J. Huggins, D. W. Berry, T. Khattar, A. F. White, N. C. Rubin, and R. Babbush, A denser planar surface code, arXiv preprint arXiv:2605.30455 (2026)

    2026 arXiv

  58. [66]

    N. C. Rubin, G. H. Low, and A. E. DePrince III, Near-frustration-free electronic structure hamiltonian representations and lower bound certificates, arXiv preprint arXiv:2602.05069 (2026)

    2026

  59. [67]

    A. L. Fetter and J. D. Walecka,Quantum theory of many-particle systems(Dover, 2003)

    2003

  60. [68]

    R. D. Mattuck,A guide to Feynman diagrams in the many-body problem(Dover, 1992)

    1992

  61. [69]

    Giannozzi, S

    P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo,et al., Quantum espresso: a modular and open-source software project for quantum simulations of materials, Journal of physics: Condensed matter21, 395502 (2009)

    2009

  62. [70]

    Giannozzi, O

    P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni,et al., Advanced capabilities for materials modelling with quantum espresso, Journal of physics: Condensed matter29, 465901 (2017)

    2017

  63. [71]

    Y. R. Sanders, D. W. Berry, P. C. Costa, L. W. Tessler, N. Wiebe, C. Gidney, H. Neven, and R. Babbush, Compilation of fault-tolerant quantum heuristics for combinatorial optimization, PRX Quantum1, 020312 (2020)

    2020

  64. [72]

    Infeld and T

    L. Infeld and T. Hull, The factorization method, Reviews of modern Physics23, 21 (1951)

    1951

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.