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REVIEW 4 major objections 7 minor 74 references

A Time-Symmetric Quantum Algorithm for Direct Eigenstate Determination

T0 review · 4 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Forward-backward time evolution about a preset energy is a spectral filter: it amplifies the eigenstate nearest that energy in one subspace and the farthest in the other, so any eigenstate appears without computing lower states first.

desk verdict Correct spectral-filter identity, but the 'direct' eigenstate claim is overstated: the method needs a target energy estimate and non-negligible overlap, and the numerical demos seed those inputs from known spectra. read the letter →

arxiv 2506.10283 v1 pith:MSROQGCI submitted 2025-06-12 quant-ph

Shijie Wei , Jingwei Wen , Xiaogang Li , Peijie Chang , Bozhi Wang , Franco Nori , Guilu Long This is my paper
classification quant-ph
keywords time-symmetricquantumeigensolverdirecteigenstatedeterminationforward-backwardtimeevolutionlinearcombinationofunitariesMonteCarloenergybandwidthtopologicalphaseserroraccumulation
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 a Hamiltonian's eigenstates can be obtained directly by evolving a trial state forward and backward in time at once, without variational optimization and without first solving for lower states. After $k$ rounds, each eigenstate component in the $\lvert 0\rangle$ subspace is multiplied by $[2\cos((E_i-e_s)t)]^k$, so the eigenstate whose energy is closest to the preset shift $e_s$ dominates; the sine version amplifies the eigenstate farthest from $e_s$ instead. Because both appear in the same circuit, the ground state and the highest excited state, and hence the energy bandwidth, can be read out in a single run, and any chosen excited state can be targeted directly. The authors argue this direct mechanism eliminates the error accumulation of sequential excited-state solvers and the barren-plateau problem of variational ones, and they give two implementations: linear combination of unitaries and quantum Monte Carlo sampling, with the latter avoiding exponential post-selection decay. Numerical demonstrations cover hydrogen and LiH molecular spectra, the Kane-Mele topological insulator, the SSH chain with Hubbard interaction, and twisted bilayer graphene flat bands.

What carries the argument

The engine is the energy-shifted, time-symmetric evolution pair $U_f^{e_s}=e^{-i(H-e_s)t}$ and $U_b^{e_s}=e^{+i(H-e_s)t}$, whose sum and difference convert the Hamiltonian spectrum into a filter: $(U_f^{e_s}+U_b^{e_s})^k=[2\cos((H-e_s)t)]^k$ acts in the $\lvert 0\rangle$ ancilla subspace and $(U_f^{e_s}-U_b^{e_s})^k=[-2i\sin((H-e_s)t)]^k$ acts in the $\lvert 1\rangle$ subspace. This turns eigenvalue selection into a spectral filtering problem: after $k$ rounds the target component is amplified relative to every other component by the $k$-th power of a cosine (or sine) ratio, giving convergence exponential in $k$ and an iteration count $O(\log(1/\epsilon))$ for accuracy $\epsilon$. The non-unitary filter is realized either by expanding the Hilbert space with a control ancilla (the linear-combination-of-unitaries route, with an iteration-free variant that trades $k-1$ extra ancillas against repeated measurements) or by a quantum Monte Carlo estimator of the same cosine and sine powers, which keeps the sampling overhead polynomial rather than exponential.

What would settle it

Take a two-level system $H=\sigma_z$ with the shift exactly halfway between the levels, $e_s=0$: both eigenstates have equal factors $\cos((\pm 1)t)$, the cosine filter amplifies them identically, and the output stays the equal superposition for every $k$, so the claimed convergence to the nearest eigenstate fails whenever the nearest state is not unique. Similarly, if the initial state has zero overlap with the target, $\langle E_s|\psi_0\rangle=0$, that component remains identically zero under the filter, so overlap-blind extraction is impossible; both cases are checkable on a two-qubit device.

Watch

Extended reading notes

Core claim

The load-bearing claim is the filtering identity in Eqs. (4)--(5): writing the trial state as $\sum_i a_i \lvert E_i\rangle$, the $k$-fold application of the sum $e^{-i(H-e_s)t}+e^{+i(H-e_s)t}$ gives $\sum_i a_i [2\cos((E_i-e_s)t)]^k \lvert E_i\rangle$ in the $\lvert 0\rangle$ subspace, while the difference gives $\sum_i a_i [-2i\sin((E_i-e_s)t)]^k \lvert E_i\rangle$ in the $\lvert 1\rangle$ subspace. Provided every $(E_i-e_s)t$ lies in $[-\pi/2,\pi/2]$, the cosine factor is largest for the eigenstate nearest to $e_s$ and the sine factor is largest for the one farthest from $e_s$, so as $k$ grows those two states dominate their respective subspaces at a rate set by the ratio of the factors. The algorithm is therefore a direct eigensolver: set $e_s$ near the ground-state energy and the $\lvert 0\rangle$ subspace yields the ground state while the $\lvert 1\rangle$ subspace yields the highest excited state, giving the energy bandwidth in one run; set $e_s$ near any target eigenvalue and that eigenstate is amplified directly. Because the target is never found by first solving lower levels, the paper claims the method is free of error accumulation, and because no classical parameter optimization is involved, it cannot suffer barren plateaus.

Load-bearing premise

The scheme presupposes that the user already knows roughly where the target energy lies, so the shift $e_s$ can be set near it, and that the initial state already has a noticeable overlap with the target eigenstate; when the energy is unknown or the overlap $|a_s|^2$ is tiny, convergence slows and the Monte Carlo sampling cost, which the paper's Appendix A finds growing like $|a_s|^{-4}$, blows up.

Editorial extensions

If this is right

  • Choosing $e_s$ near the ground-state energy yields the ground state in the $\lvert 0\rangle$ subspace and the highest excited state in the $\lvert 1\rangle$ subspace, so the energy bandwidth comes out of a single run; the hydrogen-molecule simulation reaches chemical accuracy with error $1.5\times10^{-3}$ Hartree.
  • Any excited state can be targeted directly by setting $e_s$ near its energy: in the LiH study the per-level error stays flat as the level number increases, whereas the FQESS comparison shows errors growing with level number, demonstrating that the direct scheme avoids error accumulation.
  • Convergence is exponential in the iteration count $k$, with $O(\log(1/\epsilon))$ iterations needed for accuracy $\epsilon$, and the absence of classical optimization removes the barren-plateau problem of variational eigensolvers.
  • The quantum Monte Carlo implementation avoids the exponential post-selection decay of the LCU route, keeping sample counts polynomial with a leading dependence of order $|a_s|^{-4}$ and circuit depth low, which the authors argue suits near-term hardware.
  • The solver detects topological phase transitions from gap-edge data alone: it reproduces the Kane-Mele edge states for mass $M=0$ and their disappearance for $M=0.4$, and it locates the SSH+Hubbard transition where the ground-state degeneracy changes and the energy gap closes.

Reading between the lines

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

  • Editorial: because the convergence ratio depends on $t$ as well as on $e_s$, sweeping $e_s$ across the spectrum, one run per value, would turn the solver into a general spectral probe that maps band structure point by point, a use the paper only gestures at in its flat-band demonstrations.
  • Editorial: the filter amplifies equally any two eigenstates placed symmetrically around $e_s$, so combining two or more evolution times, for example a short $t$ to separate near-degenerate pairs, is a natural extension for resolving degeneracies that a single $t$ cannot split; the paper does not test this regime.
  • Editorial: the pipeline the paper itself adopts for the SSH+Hubbard case, seeding $e_s$ and the initial state from the non-interacting solution, suggests the algorithm's most realistic role is state refinement and gap detection inside a hybrid classical-quantum workflow rather than first-principles spectral discovery.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

Summary. The paper proposes TSQES, a nonvariational quantum algorithm for eigenstate determination that interferes forward and backward time evolution with an energy shift e_s. The central identity, Eq. (4), expands the k-fold filtered state in the |0> subspace as sum_i a_i [2 cos((E_i-e_s)t)]^k |E_i>, so the eigenstate closest to e_s dominates with increasing k; Eq. (5) gives the sine analogue for the eigenstate farthest from e_s. The authors present LCU and QMC implementations, an iteration-free version using additional ancillas, and numerical demonstrations for H2, LiH, the Kane-Mele model, the SSH model with Hubbard interaction, and twisted bilayer graphene. The paper claims simultaneous ground/highest-excited-state determination, direct extraction of arbitrary eigenstates without computing lower states, and avoidance of error accumulation.

Significance. If the claims were fully supported, the paper would offer a simple spectral filter that avoids the barren-plateau problem and error accumulation of variational and sequential excited-state methods. The mathematical core, Eq. (4), is a correct expansion, and the paper provides explicit circuits, a QMC sampling strategy, and several numerical studies. The main value is as a nonvariational eigenstate filter conditioned on an energy estimate and on initial-state overlap. However, the paper's stronger claims of 'direct eigenstate determination' and of mitigating exponential cost are not established: the algorithm must be supplied with e_s near the target energy, the numerical demonstrations often seed e_s and initial states from known spectra, and the QMC complexity depends polynomially on 1/|a_s|, which is exponential when overlap is exponentially small. The manuscript is therefore a potentially useful contribution whose significance is weakened by overstated framing and by an internal inconsistency in the QMC denominator estimator.

major comments (4)
  1. [Section II, Eq. (4) and Abstract] The central claim that TSQES enables 'direct identification of arbitrary eigenstates' overstates what the algorithm does. Eq. (4) is a spectral filter: after k iterations the amplitude of |E_i> is multiplied by [cos((E_i-e_s)t)]^k, so the output is the eigenstate nearest to the supplied energy shift e_s, not an eigenstate determined from the Hamiltonian alone. The paper itself acknowledges in Section V that 'the TSQES algorithm requires pre-knowledge of the target eigenenergy value, which guides the choice of e_s, and the initial state needs to have a limited overlap with the target eigenstate.' Because cos is even, if e_s is equidistant from two eigenvalues, the filter selects a superposition of both, so the nearest-eigenstate claim requires strict closeness to a nondegenerate target. The abstract and conclusion should be reworded to present TSQES as an eigenstate filter conditioned on an energy estimate and on initial overlap, not as a direct solver.
  2. [Section VI.B, Fig. 9] The SSH+Hubbard demonstration seeds the algorithm with the answer. The text states that 'the computable eigenstates and eigenvalues of the Hamiltonian without Hubbard interaction (U=0) [are taken] as the initial state and the energy-shift values e_s.' The initial state and energy shifts are therefore derived from the U=0 spectrum, and the algorithm's output tracks the strongly correlated eigenstates that connect to those U=0 states. This does not demonstrate ab initio determination of correlated eigenstates; it demonstrates continuation from a known noninteracting spectrum. The same limitation applies to the Bistritzer-MacDonald example in Appendix B, where e_s values are selected from the known spectrum. The numerical evidence should be presented as validation of the filter given controlled inputs, not as evidence for the direct-determination claim.
  3. [Appendix A, Eqs. (A6)-(A8) and Fig. 10] The QMC denominator estimator is internally inconsistent. In Eq. (15) the denominator is D = E_{k'_1,k'_2}[<psi0|(U^f)^{2k'_1-2k'_2}|psi0>], but the circuit in Appendix A implements U^f_{e_s}(k'_1,k'_2) = (U^f_{e_s})^{k'_1-k'_2}, so the measured estimator is E_{k'_1,k'_2}[<psi0|(U^f)^{k'_1-k'_2}|psi0>]. This is not equal to D; it corresponds to cos^{2k}((H-e_s)t/2) rather than cos^{2k}((H-e_s)t). The exponent in the circuit should be 2(k'_1-k'_2). As written, the QMC denominator estimation is incorrect, which calls into question the QMC implementation and the numerical results in Fig. 3(b) unless they were produced with the corrected circuit.
  4. [Appendix A, Eqs. (A19)-(A20)] The sampling complexity of the QMC estimator is O(1/|a_s|^4) (times lambda_s^{-4k}), so the cost grows as |a_s|^{-4}. If the initial state has exponentially small overlap with the target eigenstate, this is an exponential overhead. The paper's claim that QMC 'mitigates the issue of exponential decay in the success probability' is therefore conditional on the overlap |a_s|^2 being non-negligible, a condition that is not guaranteed and is not discussed in the complexity summary. The abstract's promise of an 'efficient solution for eigenvalue problems' should be qualified accordingly.
minor comments (7)
  1. [Section II, after Eqs. (4)-(5)] The condition (E_i-e_s)t in [-pi/2, pi/2] for all i is a global spectral condition that requires knowledge of the entire spectrum, not just the target energy; this assumption should be stated explicitly and its practical implications discussed.
  2. [Section V.A and Fig. 3] The parameters 'es1' and 'es2' are introduced without a clear definition of their roles in the two subspace filters; please define them consistently with the notation e_s.
  3. [Section V.B.2, Fig. 7] The relaxation of the constraint on evolution time is presented heuristically; the stated condition that |cos((E_i-e_s)t)| is maximal for the closest eigenstate is not sufficient when cos changes sign over the spectrum, and the empirical success at t=pi/2 should be framed as a case-specific observation rather than a general result.
  4. [Eq. (20)] The Hubbard interaction term contains an extra index s in hat n^s_{i up} hat n^s_{i down}; it should be sum_i U hat n_{i up} hat n_{i down}.
  5. [Appendix B] The model name is spelled 'Bistritizer-MacDonald' in the section title and text; the correct spelling is 'Bistritzer-MacDonald'.
  6. [Algorithms 1-2] The notation nD and nN should be typeset as n_D and n_N; more importantly, the Ensure line of Algorithm 1 says 'Estimation of the numerator D' but should read 'denominator D'.
  7. [Eqs. (A19)-(A20)] The parameters c and q in the exponent epsilon^{4c q} are not defined before they are used; define them or remove the intermediate expression.

Circularity Check

3 steps flagged · score 6.0 of 10

The filter identity in Eq. (4) is derived correctly, but the central 'direct eigenstate determination' claim reduces to inputting a close energy estimate e_s: every numerical demonstration seeds e_s (and often the initial state) from known spectra, so the converged eigenstates are selected by construction.

  1. self definitional [Section II, Eq. (4) and Section III (paragraph after Eq. (6))]
    "|ψk⟩0 = (U_f^{e_s}+U_b^{e_s})^k|ψ0⟩ = Σ_i a_i [2 cos((E_i-e_s)t)]^k |E_i⟩ ... If we set (E_i-e_s)t∈[−π/2, π/2] ∀i, then the m-th eigenstate |E_m⟩ with eigenvalue E_m closest to e_s in subspace |0⟩ will have the maximum amplitude ... It should be noted that the TSQES algorithm requires pre-knowledge of the target eigenenergy value, which guides the choice of e_s, and the initial state needs to have a limited overlap with the target eigenstate."

    The target eigenstate is defined by the input: Eq. (4) says the surviving component is precisely the eigenstate whose energy E_m is closest to the supplied e_s. Thus the 'direct determination' of any eigenstate is not an independent solve; it is a filter whose output is selected by the user-supplied energy shift. In the demonstrations e_s is placed at or near known eigenvalues (e.g., e_s1=-1.1 for the H2 ground energy -1.0458), so the convergent energy is read off from the input by construction. The paper itself acknowledges the pre-knowledge requirement in Section III, which undercuts the abstract claim of direct identification without prior spectral information.

  2. fitted input called prediction [Section VI.B, SSH+Hubbard simulation (Fig. 9)]
    "We take the computable eigenstates and eigenvalues of the Hamiltonian without Hubbard interaction (U=0) as the initial state and the energy-shift values e_s of the QDEDS algorithm, so that the algorithm results in the presence of strong many-body interactions can be well calculated."

    The U=10 interacting eigenstates are not independent targets here: their quantum numbers and approximate energies are inherited from the U=0 eigenstates used to seed both |ψ0⟩ and e_s. Eq. (4) then guarantees convergence to the eigenstate(s) with largest overlap with that seed and nearest to those energies. The SSH+Hubbard spectrum in Fig. 9 is therefore a continuation of the input U=0 spectrum, not a demonstration that the algorithm can find an excited state from a generic state with no prior eigenvalue information.

1 more flagged steps
  1. fitted input called prediction [Appendix B, Bistritzer-MacDonald simulation]
    "We select eight discontinuous energy levels {−1.098,−0.698,−0.502,−0.210,−0.002,0.698,1.016,1.099} as the simulation objects ... preset estimated values for the corresponding energy levels with energy-shifts e_s ∈ {−1.1,−0.7,−0.5,−0.2,0.0,0.7,1.0,1.1}"

    The energy shifts are chosen immediately next to the known target eigenvalues. Since Eq. (4) projects, after k iterations, onto the level closest to e_s, each converged output is exactly the input e_s rounded to the nearest level of the known spectrum. The claim that 'we no longer require prior determination of results for all preceding energy levels' only means the algorithm does not need the lower levels; it still requires a close estimate of each target level, and here those estimates are essentially the answers.

full rationale

The mathematical core, Eq. (4)-(5), is a correct manipulation: expanding |ψ0⟩ in eigenstates and summing e^{±i(H-e_s)t} gives cos/sin filters, and the exponential suppression of off-target components follows. That part is not circular. I also find no load-bearing self-citation chain: refs. [40,41] are used as comparison baselines, not as justification for TSQES. The circularity lies in the transition from filter identity to 'direct eigenstate determination'. The algorithm's output is defined by the input e_s, and all demonstrations supply e_s from the known spectrum (and in SSH+Hubbard also the initial state from the U=0 eigenstates). Hence the numerical 'predictions' are selected by construction; they corroborate Eq. (4) but do not demonstrate that arbitrary eigenstates can be obtained without prior eigenvalue knowledge. The paper explicitly states this limitation, which is why the score is partial (6) rather than 8-10.

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

No new physical entities are introduced. The central construct is the cos/sin filter obtained from forward and backward evolution, which is mathematically elementary and present in earlier QITE and filter work; the paper adds an energy-shift parameter and a QMC estimator.

free parameters (3)
  • energy shift e_s = Per-target values: H2 -1.1 and 0; LiH -8.0 and -7.7; Kane-Mele -0.1 and 0.1; SSH from U=0 spectrum…
    Central knob selecting which eigenstate is amplified; in every application it is placed at or near the known target eigenvalue, so the output is not an independent prediction.
  • evolution time t = e.g., 1.3518 for H2; pi/5 for Kane-Mele; pi/6 for Bistritzer-MacDonald; pi/2 for LiH
    Controls filter width and convergence; chosen by hand to satisfy (E_i-e_s)t within [-pi/2, pi/2] or empirically.
  • iteration count k = 30 to 500 depending on system
    Controls how strongly the filter amplifies the target; chosen by hand until curves flatten, no adaptive stopping rule is given.
assumptions (5)
  • domain assumption The initial state has nonzero overlap with the target eigenstate (|a_s|^2 > 0).
    Invoked in Section II and in Algorithms 1 and 2; without it the filter cannot amplify the target.
  • ad hoc to paper Approximate target eigenenergy is known well enough to choose e_s.
    Stated in Section V; the method is not self-starting and the demonstrations use known spectra to set e_s.
  • ad hoc to paper All shifted energies satisfy (E_i - e_s)t in [-pi/2, pi/2], or a modified condition that preserves ordering.
    Used in Sections II and III; for a wide spectrum with e_s near the ground state, no single t may satisfy this while keeping convergence fast.
  • standard math The ratio of two Monte Carlo estimators N(O)/D is a reliable estimate of <O> under the stated sample bounds.
    QMC section and Appendix A; Hoeffding bounds assume independent samples and small variance, with no discussion of ratio estimator bias.
  • domain assumption Hamiltonian time evolution exp(-i(H-e_s)t) can be implemented coherently for the systems considered.
    Assumed throughout the LCU and QMC constructions; not demonstrated on hardware.

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Cite this review

Pith. "Pith review of A Time-Symmetric Quantum Algorithm for Direct Eigenstate Determination." pith.science (2026). https://pith.science/paper/MSROQGCI

@misc{pith2026250610283,
  author       = {Pith},
  title        = {Pith review of: A Time-Symmetric Quantum Algorithm for Direct Eigenstate Determination},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MSROQGCI}},
  note         = {Machine review of arXiv:2506.10283}
}
read the original abstract

Time symmetry in quantum mechanics, where the current quantum state is determined jointly by both the past and the future, offers a more comprehensive description of physical phenomena. This symmetry facilitates both forward and backward time evolution, providing a computational advantage over methods that rely on a fixed time direction. In this work, we present a nonvariational and \textit{time-symmetric quantum algorithm} for addressing the eigenvalue problem of the Hamiltonian, leveraging the coherence between forward and backward time evolution. Our approach enables the simultaneous determination of both the ground state and the highest excited state, as well as the direct identification of arbitrary eigenstates of the Hamiltonian. Unlike existing methods, our algorithm eliminates the need for prior computation of lower eigenstates, allowing for the direct extraction of any eigenstate and energy bandwidth while avoiding error accumulation. Its non-variational nature ensures convergence to target states without encountering the barren plateau problem. We demonstrate the feasibility of implementing the non-unitary evolution using both the linear combination of unitaries and quantum Monte Carlo methods. Our algorithm is applied to compute the energy bandwidth and spectrum of various molecular systems, as well as to identify topological states in condensed matter systems, including the Kane-Mele model and the Su-Schrieffer-Heeger model. We anticipate that this algorithm will provide an efficient solution for eigenvalue problems, particularly in distinguishing quantum phases and calculating energy bands.

Figures

Figures reproduced from arXiv: 2506.10283 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic diagram of the time-symmetric quantum eigensolver. A quantum system evolves forward in time (clockwise) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Quantum circuits for implementing time-symmetric [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Simulation of the hydrogen molecule at an inter [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Performance of our quantum algorithm in different subspaces when varying the inter-nuclear distance of a hydrogen [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Schematic diagram of the cosine function illus [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Simulation of the LiH molecule at an inter-nuclear [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Energy spectra of the LiH molecule with fixed energy [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Simulation results of the Kane-Mele model in the [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Simulation results of the eigenenergies of the SSH model with Hubbard interaction at half-filling ( [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Quantum circuit for measuring the denominator [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Quantum circuit for measuring the numerator esti [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. (a) Simulation results of the iteration processes for the Bistritizer-MacDonald model [ [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]

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Works this paper leans on

74 extracted references · 46 canonical work pages

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    The initial state is chosen as|+⟩ ⊗6, where|+⟩is an eigenvector of theσ x Pauli matrix, and the three lowest eigenenergies are calculated

    Avoiding the error-accumulation phenomenon We now apply the QESDS and FQESS algorithms [41] to a six-qubit LiH molecule at an inter-nuclear separa- tion of 1.10 Angstrom, as shown in Fig.5. The initial state is chosen as|+⟩ ⊗6, where|+⟩is an eigenvector of theσ x Pauli matrix, and the three lowest eigenenergies are calculated. We can clearly find that the...

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    In fact, the superposition of the forward and backward time evolution in the QESDS algorithm forms a cosine filtering operator, whose peak is determined by the energy-shift es

    Impact of the evolution time Furthermore, we now investigate the relationship be- tween the convergence rate and the evolution timet. In fact, the superposition of the forward and backward time evolution in the QESDS algorithm forms a cosine filtering operator, whose peak is determined by the energy-shift es. Mathematically, the absolute value of the grad...

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    We define ˆD= 1 nD nDX i=1 ˆD,(A1) ˆN= 1 nN nNX i=1 ˆN ,(A2) so it is obvious thatE ˆD=D,E ˆN=N

    Estimating the DenominatorD At first, we will discuss estimatingD. We define ˆD= 1 nD nDX i=1 ˆD,(A1) ˆN= 1 nN nNX i=1 ˆN ,(A2) so it is obvious thatE ˆD=D,E ˆN=N. For the estimation precisionϵ D of the denominatorD, usually its estimation needs to be divided into real part ℜ[·] and imaginary partℑ[·] separately, and we assume that they are selected with ...

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    = (U es f )k ′ 1−k ′ 2 , wherek ′ 1, k ′ 2 are generated separately by the binomial distributionB(k,1/2). When the initial state |ψ0⟩passes through this quantum circuit, the state will be |ΨD n ⟩= 1√ 2 (|0⟩ ⊗ |ψ0⟩+|1⟩ ⊗(Ues f )k ′ 1−k ′ 2 |ψ0⟩).(A6) Therefore, ⟨ΨD n |X|Ψ D n ⟩= Re ⟨ψ0|(U es f )k ′ 1−k ′ 2 |ψ0⟩ , ⟨ΨD n |Y|Ψ D n ⟩= Im ⟨ψ0|(U es f )k ′ 1−k ′...

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    can be obtained by calculating the expectation about (k ′ 1, k ′ 2): ˆD(k ′ 1, k ′

    work page

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    = ⟨Ψn|X|Ψ N n ⟩+i⟨Ψ n|Y|Ψ N n ⟩ ,(A8) and then D=E k′ 1,k′ 2 ˆD(k ′ 1, k ′ 2).(A9) Finally, the algorithm for estimatingDis given in Algo- rithm 1. Algorithm 1Estimating the denominatorD Require:U es f = exp (−i(H−e s)t); initial state|ψ 0⟩with nonzero overlap withj-th eigenstate of HamiltonianH: |aj|2 =|⟨E j|ψ0⟩|2 ̸= 0; the interval EL j , EU j ofE j, an...

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    6:Calculate the estimated denominator ˆDas an estimation ofD: ˆD= 1 nD PnD p=1 ˆD(k ′ 1, k ′ 2)

    according to Eq.(A8). 6:Calculate the estimated denominator ˆDas an estimation ofD: ˆD= 1 nD PnD p=1 ˆD(k ′ 1, k ′ 2)

    work page

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    Estimating the NumeratorN(O) For the estimation precisionϵ N of the numerator N(O), in a similar way as the estimation ofD, according to the Hoeffding inequality, we can obtain that Pr(|ℜ ˆN−Eℜ ˆN|⩾ϵ N )⩽2 exp − nN ϵ2 N 2∥O∥2 1 ,(A10) so the sample number of the denominator will be nN ⩾ 2·K N · ∥O∥2 1 ϵ2 N ,(A11) with a failure probability δN ⩽4 exp − KN ...

    work page

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    Note that in actual sampling, the number of samples is often much smaller than the number specified in the above formula to achieve a considerable level of accuracy

    Sampling complexity analysis Based on the analysis of the sampling complexity of the numerator and denominator, the total error estimation of the observed quantity can be performed as follows: ⟨O⟩ − ⟨ˆO⟩ = N(O) D − ˆN(O) ˆD = N(O) ˆD−D ˆN(O) D ˆD = N(O)( ˆD−D) +D ˆN(O)−N(O) D ...

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