Robust phase estimation of the ground-state energy without controlled time evolution on a quantum device

Hiroki Kuji; Kenji Sugisaki; Takashi Imoto; Tetsuro Nikuni; Yuichiro Matsuzaki; Yuta Shingu

REVIEW 3 major objections 4 minor 59 references

Robust phase estimation of the ground-state energy without controlled time evolution on a quantum device

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper shows that the ground-state energy can be estimated accurately without controlled time evolution by combining adiabatic state preparation, a classically known reference state, and a Ramsey-type measurement whose Fourier peak…

desk verdict A genuinely useful variant of Ramsey-ASP that estimates absolute ground-state energies; the interference part is sound, but the initial-state preparation is assumed more than proven. read the letter →

arxiv 2412.19590 v2 pith:3LAK2HTI submitted 2024-12-27 quant-ph

Hiroki Kuji , Yuta Shingu , Tetsuro Nikuni , Takashi Imoto , Kenji Sugisaki , Yuichiro Matsuzaki This is my paper · ORCID

classification quant-ph

keywords ground-stateenergyadiabaticstatepreparationRamsey-typemeasurementreferenceconservedquantityphaseestimationquantumannealingHeisenbergmodel

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

The paper proposes a method to estimate the ground-state energy of a quantum Hamiltonian without ever implementing controlled time evolution, which is a major practical bottleneck in phase-estimation algorithms. The trick is to prepare a superposition of the ground state of a simple driving Hamiltonian and a "reference state" whose energy is known classically, then let the system evolve under the target Hamiltonian for a variable time, reverse the adiabatic sweep, and measure the probability of returning to the starting state. That probability oscillates at a frequency equal to the difference between the target ground-state energy and the reference energy, so a Fourier transform of the measured signal directly yields the ground-state energy. The authors show numerically on a four-qubit Heisenberg model that the estimate matches exact diagonalization to within $3.7\times10^{-5}\%$ relative error, and they argue that the scheme is robust against non-adiabatic transitions because such transitions produce extra spectral peaks rather than corrupting the main one. If correct, the method would shorten quantum circuits on early fault-tolerant devices and make ground-state estimation feasible on quantum-annealing hardware.

What carries the argument

The central object is the conserved quantity $\hat{Q}$, which block-diagonalizes both the driving and the problem Hamiltonians, together with a reference state chosen from a classically solvable sector of $\hat{Q}$. The machine that carries the argument is the three-stage time-dependent Hamiltonian $H(t)=F(t)\hat{H}_D + [1-F(t)]\hat{H}_P$ with $F(t)$ linear during forward adiabatic preparation, zero during the free-evolution (Ramsey) period $\tau$, and linear again during reverse preparation. Because the state is a superposition of two eigenstates that live in different $\hat{Q}$ sectors, and $\hat{Q}$ is conserved, each component evolves independently and acquires a relative phase proportional to the energy difference; the return probability $P(\tau)$ is the cosine of that phase. The discrete Fourier transform $f(\omega)=\sum_n P(\tau_n)e^{-i\omega\tau_n}$ converts the oscillating signal into a peak at $\omega = E^g_{q',P}-E^n_{q,P}$. Non-adiabatic transitions during forward preparation add extra Fourier peaks at the energies of excited states, which is why the method is robust and does not require fine-tuning the adiabatic time $T$.

What would settle it

A concrete test: implement the full sequence on a small device for a Hamiltonian with a known ground-state energy, using the GHZ-based preparation sketched in Appendix B, and check whether the Fourier peak of $P(\tau)$ reproduces that energy to the precision set by the sampling time. A theoretical falsifier is to show that for some conserved quantity satisfying the paper's assumptions, the spectral gap of the state-preparation Hamiltonian closes faster than polynomially with system size, making the initial-state preparation impossible in polynomial time; that would invalidate the scalability claim while leaving the central identity intact.

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Extended reading notes

Core claim

The central claim is that a classically computable eigenstate, called a reference state, can serve as a phase reference for Ramsey-type energy estimation, eliminating the need for controlled time evolution. Suppose the driving and problem Hamiltonians share a conserved quantity $\hat{Q}$ with polynomial-size eigenspaces. The initial state is a superposition $|\psi_\mathrm{ini}\rangle = (|\psi^g_{q',D}\rangle + |\psi_{q,D}\rangle_n)/\sqrt{2}$ of the driving-Hamiltonian ground state in sector $q'$ and a known reference state in sector $q$. After adiabatic state preparation, free evolution under $\hat{H}_P$ for a time $\tau$, and reverse adiabatic state preparation, the probability of projecting back onto $|\psi_\mathrm{ini}\rangle$ is $P(\tau) = \cos^2[(E^g_{q',P} - E^n_{q,P})\tau/2 + \theta'/2]$, and its discrete Fourier transform peaks at the energy difference $E^g_{q',P} - E^n_{q,P}$. Since the reference energy $E^n_{q,P}$ is obtained classically, the ground-state energy in the sector $q'$ is read off directly. The paper demonstrates this on a four-qubit Heisenberg model with total magnetization as the conserved quantity, recovering the ground-state energy with a relative error of $3.7\times10^{-5}\%$ and estimating several excited-state energies from the same data.

Load-bearing premise

The entire protocol rests on being able to prepare the initial superposition of the driving Hamiltonian's ground state and the reference state efficiently and with high fidelity; if that preparation cannot be done for a given Hamiltonian or system size, the measurement cannot be run even though the interference formula remains correct.

Editorial extensions

If this is right

The method removes the need for controlled time evolution, shortening the circuits required for ground-state energy estimation on early fault-tolerant quantum computers.
Because non-adiabatic transitions appear as extra peaks in the Fourier spectrum rather than as errors in the ground-state peak, the adiabatic sweep time $T$ does not need to be fine-tuned in advance.
Excited-state energies can be extracted from the same measurement data, since every peak corresponds to an energy difference between the reference state and an eigenstate of the problem Hamiltonian.
If the subspace containing the true ground state is unknown, applying the procedure across all $\hat{Q}$ sectors and comparing the recovered energies identifies it, as demonstrated numerically for a Heisenberg ring.

Reading between the lines

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

A natural extension would be to use the same reference-state interference trick to estimate energy gaps at avoided crossings or to probe symmetry-breaking transitions, by choosing reference states from different $\hat{Q}$ sectors.
The choice of a maximum-energy reference state (as in the numerical example) means the ground-state peak is the largest frequency component, suggesting a blind readout that does not require prior spectral knowledge.
The method's practicality hinges on the reference-state preparation of Appendix B; scaling the GHZ-based protocol to larger $N$ and general conserved quantities is the step most likely to limit real-world applicability.
One could test robustness experimentally on quantum-annealing hardware, where controlled time evolution is unavailable, by comparing the Fourier-derived energy against annealing results.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this paper does something real. Prior Ramsey-ASP work estimated energy gaps; this one uses a symmetry-protected reference state with a classically known eigenvalue as the second arm of the interferometer, so the Fourier peak lands on an absolute ground-state energy. The central formula, Eq. (10), is correct, the 4-qubit Heisenberg numerics reproduce the exact ground energy to 3.7e-5%, and Appendix C shows the subspace-sweep works without knowing which subspace holds the true ground state. The non-adiabatic-robustness point is also honestly made: unwanted excitations show up as extra peaks rather than silently corrupting one number, and the authors state the limit where ground-state population is lost.

What is soft, in order of severity.

First, the initial-state preparation in Eq. (6) is the load-bearing subroutine, and it is not actually delivered. Appendix B is explicitly for Q = M, uses a GHZ-state route, and ends with an adiabatic step that has no gap analysis or complexity bound. Generalizing to arbitrary conserved Q and large N is non-trivial, because the ground state of the driving Hamiltonian in a subspace is itself the kind of object that usually requires an adiabatic schedule to prepare. The 4-qubit demonstration does not test this: (|1100>+|1111>)/sqrt(2) is a simple product-state superposition, so the simulation validates the interference readout, not the preparation subroutine. That said, this is a gap in the paper, not a contradiction of the method; a referee should ask for either a provably efficient construction for a useful class of Q, or an explicit statement that the method is conditional on a preparable initial superposition.

Second, Appendix A has an algebraic error. From Eq. (10), P(tau) should expand to 1/2 + 1/2 cos(Delta E tau + theta'), not 1/2 + 1/2 cos(Delta E tau/2 + theta'/2). As written, the Fourier analysis would put the peak at Delta E/2, contradicting the main text. The final conclusion (the relative phase drops out) is still right, but the exhibited identity is wrong and should be fixed.

Third, the maximum-energy-reference assumption is a real caveat. The suggested fix, adding -lambda(Q-q)^2, does not change the spectrum within a fixed Q sector, but it does alter the global ground state across sectors, so it is not a free lunch. For quantum chemistry the |1...1> state is plausibly the top state, but the paper should be more precise about when this holds.

The central idea is sound; the weaknesses are in the presentation and the preparation subroutine rather than in the main interference mechanism. People working on early fault-tolerant energy estimation or annealing readout protocols will get value from this. I would send it to peer review: the referee should push on the preparation subroutine and the Appendix A algebra. I would cite it for the absolute-energy Ramsey idea.

Referee Report

3 major / 4 minor

Summary. The paper proposes a method to estimate the ground-state energy of a Hamiltonian without controlled time evolution. The protocol uses adiabatic state preparation (ASP) starting from a superposition of a ground state and a reference state, lets the system evolve under the problem Hamiltonian for a time τ, applies reverse ASP, and reads out the energy difference from the Fourier transform of the final projection probability. The reference-state energy is computed classically from a polynomially sized symmetry subspace, so the measured frequency directly yields the ground-state energy. The authors validate the method on a 4-qubit Heisenberg model and report good agreement with exact diagonalization, along with a numerical comparison showing shorter total runtime than conventional ASP. The main unresolved issue is the preparation of the required initial superposition state, which is assumed rather than proved in general.

Significance. The central measurement formula, Eq. (10), is correct, and the protocol is non-circular: the reference-state energy is obtained classically and the ground-state energy is inferred from a measured frequency difference. The numerical demonstration on the 4-qubit Heisenberg model reproduces the exact ground-state energy to a relative error of about 3.7e-5 percent and resolves additional excited-state peaks. If the initial-state preparation problem can be solved, the method is a meaningful step toward ground-state energy estimation without controlled time evolution, with potential relevance to early fault-tolerant quantum computers and quantum annealers. The paper is also useful in explicitly identifying the preparation assumption in Eq. (6). However, the current lack of a rigorous, general preparation protocol limits the strength of the claims and requires major revision.

major comments (3)

[Sec. II, Eq. (6); Appendix B] The entire protocol hinges on the ability to prepare the superposition state |ψ_ini> of Eq. (6). Appendix B proposes an ASP-based preparation only for the special case Q = M and does not prove that the final adiabatic step from H_transverse to H_D maps the desired superposition of eigenstates onto (|ψ_g_{m,D}> + |11...1>)/√2 without level crossings or with bounded error. It also gives no complexity bound in N for arbitrary conserved quantities Q, and the numerical demonstration in Sec. III does not actually test this preparation, since the initial state (|1100> + |1111>)/√2 is preparable by elementary Clifford gates. The central claim is therefore conditional on an unverified subroutine; please either supply a rigorous preparation protocol with an accuracy and efficiency analysis, or explicitly restate the contribution as a method that assumes such a preparation is available.
[Appendix A, Eq. (A1)] The identity used in Eq. (A1) is algebraically incorrect: cos^2(x) = (1 + cos(2x))/2, so P(τ) in Eq. (10) should be rewritten as 1/2 + 1/2 cos[(E_g - E_n)τ + θ'], not 1/2 + 1/2 cos[(E_g - E_n)τ/2 + θ'/2]. The subsequent derivation in Eq. (A3) is therefore wrong as written. The conclusion that the relative phase θ' does not affect the Fourier peak positions survives after correcting this factor, because |exp(iθ')| = 1, but the proof needs to be repaired.
[Sec. II and Sec. III, Fig. 1(c)] The claim that the method is robust against non-adiabatic transitions is supported by a single 4-qubit numerical example. No quantitative analysis is given for how leakage to excited states affects the amplitude of the ground-state peak, how many samples or how large τ must be to identify the ground-state peak when multiple peaks appear, or under what conditions the ground-state peak remains identifiable. Since the authors themselves note that the method fails when the ground-state population is significantly depleted, the robustness statement should be made quantitative or explicitly bounded to the regime where the ground-state peak is still visible in the Fourier spectrum.

minor comments (4)

[Appendix C, first paragraph] “An key aspect” should be “A key aspect”.
[Sec. II, paragraph near Eq. (12)] “byu setting T appropriately” contains a stray “u”; it should read “by setting T appropriately”.
[Sec. III, ground-state result] The phrase “relative error of 3.7 × 10^{-5}%” is technically correct but unusual; consider reporting the relative error as a dimensionless fraction (3.7 × 10^{-7}) to avoid ambiguity.
[Appendix B, step 4] The phrase “By adding qubits to match the desired superposition” is vague; a concrete tensor-product construction of the desired product-state superposition would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the ground-state energy is obtained by adding a measured Ramsey frequency to a classically computed reference energy; no fitted parameter, definitional identity, or load-bearing self-citation forces the result.

full rationale

The derivation is self-contained. The inputs are the classically computed reference energy E^n_{q,P} and the measured Ramsey signal P(τ) of Eq. (10), which is derived directly from Schrödinger evolution under H_ASP, H_TE = H_P, and RASP (Eqs. (3)-(9)). The Fourier transform f(ω) in Eq. (11) peaks at E^g_{q',P} - E^n_{q,P}; because E^n is an input, the target energy is obtained by adding a measured frequency to a known constant, not by fitting or by renaming an input. No parameter is fitted to the exact-diagonalization data; the Sec. III comparison is post-hoc validation. The self-citations (Refs. [29], [35]) are background and acknowledgment, not load-bearing: Ref. [35] is explicitly said to lack detail, and the derivation does not depend on it. The real weakness is the unproven initial-state preparation subroutine in Appendix B for general Q and large N, but that is a feasibility/completeness gap, not a circular reduction: Eq. (10) does not assume the target energy and is not equivalent to the inputs by construction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The protocol rests on four stated assumptions: a common conserved quantity with a classically tractable reference subspace, a known reference energy with a maximum-energy ordering, efficient initial-superposition preparation, and sufficient adiabaticity to retain ground-state population. No free parameters are fitted to data; the reference energy is an external classical input. No new physical entities are introduced.

assumptions (4)

domain assumption Both the driving Hamiltonian HD and the problem Hamiltonian HP commute with a conserved quantity Q, whose eigenspaces are classically tractable with the reference-state subspace of size O(poly N).
Invoked in Sec. II assumptions to ensure HD and HP are block diagonal and that a reference eigenstate's energy can be computed classically.
domain assumption The reference state |psi_{q,P}>_n is a known eigenstate of HP with known energy, and is the maximum-energy state (or can be made so by adding Hadd = -lambda(Q-q)^2).
Stated in Sec. II; needed so the largest Fourier peak can be identified with the ground-state energy gap and so the known energy can be subtracted.
ad hoc to paper The initial superposition state (ground state of HD plus reference state of HD) can be prepared efficiently and with high fidelity.
Appendix B describes a GHZ-based protocol for Q = M, but does not prove efficiency or generality for large N or arbitrary Q. The main protocol assumes this preparation as a starting point.
domain assumption Adiabatic state preparation from HD to HP leaves a non-negligible population in the target ground state.
The paper explicitly notes in Sec. II that if non-adiabatic transitions significantly reduce the ground-state population, both this method and conventional ASP fail to determine the ground-state energy.

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Pith. "Pith review of Robust phase estimation of the ground-state energy without controlled time evolution on a quantum device." pith.science (2026). https://pith.science/paper/3LAK2HTI

@misc{pith2026241219590,
  author       = {Pith},
  title        = {Pith review of: Robust phase estimation of the ground-state energy without controlled time evolution on a quantum device},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3LAK2HTI}},
  note         = {Machine review of arXiv:2412.19590}
}

read the original abstract

Estimating the ground-state energy of Hamiltonians in quantum systems is an important task. In this work, we demonstrate that the ground-state energy can be accurately estimated without controlled time evolution by using adiabatic state preparation (ASP) and Ramsey-type measurement. By considering the symmetry of the Hamiltonian governing the time evolution during ASP, we can prepare a superposition of the ground state and reference state whose eigenvalue is known. This enables the estimation of the ground-state energy via Ramsey-type measurement. Furthermore, our method is robust against non-adiabatic transitions, making it suitable for use with early fault-tolerant quantum computers and quantum annealing.

Figures

Figures reproduced from arXiv: 2412.19590 by the authors.

**Figure 2.** FIG. 2. The amplitude [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 40 canonical work pages

[1]

Here, we consider the case where ˆQ = ˆM, which clearly commutes with both ˆHD and ˆHP

Under these settings, the ground state of the driv- ing Hamiltonian is|1100⟩. Here, we consider the case where ˆQ = ˆM, which clearly commutes with both ˆHD and ˆHP. Since the states|11··· 1⟩ and |00··· 0⟩ are the unique eigenstates of ˆM corresponding to the eigenval- ues±N, they are trivial simultaneous eigenstates of both ˆHD and ˆHP, with their corres...

work page
[2]

The process starts from the all-plus state|+ +··· +⟩. We use ASP to transform this state into a GHZ state |ψGHZ⟩ = (|00··· 0⟩ +|11··· 1⟩) √ 2 by gradually switching the Hamiltonian from ˆH1 = B1 ∑ i ˆσx i to ˆH2 = −B2(∑ i ˆσz i ) 2 , whereB1 (B2) represent the strength of the magnetic field (interaction between qubits) [45–51]

work page
[3]

The Hamiltonian is then switched fromˆH2 to ˆH3 =−B3(∑ i ˆσx i ) 2 by using ASP, whereB3 represent the strength of the interaction between qubits, so that the state becomes|ψ′ GHZ⟩ = (|+ +··· +⟩ +|−−···−⟩ ) √

work page
[4]

Both states|ψGHZ⟩ and|ψ′ GHZ⟩ are eigenstates of a parity operator defined asˆP = ˆσx 1⊗ ˆσx 2⊗···⊗ ˆσx N

Before and after this ASP step, the system remains a superposition of degenerate ground states. Both states|ψGHZ⟩ and|ψ′ GHZ⟩ are eigenstates of a parity operator defined asˆP = ˆσx 1⊗ ˆσx 2⊗···⊗ ˆσx N

work page
[5]

Note that, because ˆH3 and ˆH4 commute, the state remains unchanged

The Hamiltonian is switched fromˆH3 to ˆH4 =∑ iBiˆσx i . Note that, because ˆH3 and ˆH4 commute, the state remains unchanged

work page
[6]

By adding qubits to match the desired superposition, we can create a state such as|±±···±⟩ (|+ +··· +⟩ + |−−···−⟩ )/ √

work page
[7]

After relabeling the qubits, this corresponds to1√ 2(| ˜En⟩ +| ˜El⟩) where| ˜En⟩ (| ˜El⟩) is then-th (l-th) eigenstate of ˆHtransverse =∑ iBiˆσx i , where|±±···±⟩ represents a product state in which each qubit is in either|+⟩ or|−⟩

work page
[8]

(14)) by using ASP

Finally, the Hamiltonian is switched fromˆHtransverse to ˆHD (Eq. (14)) by using ASP. Then, we obtain the desired superposition state (Eq. (B1)). At the final step,ˆHtransverse is the Hamiltonian without interaction whileˆHD is the Hamiltonian with interaction only between the 2i− 1-th qubit and the2i-th qubit. This means that the evolution at the final s...

work page

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Here, we consider the case where ˆQ = ˆM, which clearly commutes with both ˆHD and ˆHP

Under these settings, the ground state of the driv- ing Hamiltonian is|1100⟩. Here, we consider the case where ˆQ = ˆM, which clearly commutes with both ˆHD and ˆHP. Since the states|11··· 1⟩ and |00··· 0⟩ are the unique eigenstates of ˆM corresponding to the eigenval- ues±N, they are trivial simultaneous eigenstates of both ˆHD and ˆHP, with their corres...

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[2] [2]

The process starts from the all-plus state|+ +··· +⟩. We use ASP to transform this state into a GHZ state |ψGHZ⟩ = (|00··· 0⟩ +|11··· 1⟩) √ 2 by gradually switching the Hamiltonian from ˆH1 = B1 ∑ i ˆσx i to ˆH2 = −B2(∑ i ˆσz i ) 2 , whereB1 (B2) represent the strength of the magnetic field (interaction between qubits) [45–51]

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[3] [3]

The Hamiltonian is then switched fromˆH2 to ˆH3 =−B3(∑ i ˆσx i ) 2 by using ASP, whereB3 represent the strength of the interaction between qubits, so that the state becomes|ψ′ GHZ⟩ = (|+ +··· +⟩ +|−−···−⟩ ) √

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[4] [4]

Both states|ψGHZ⟩ and|ψ′ GHZ⟩ are eigenstates of a parity operator defined asˆP = ˆσx 1⊗ ˆσx 2⊗···⊗ ˆσx N

Before and after this ASP step, the system remains a superposition of degenerate ground states. Both states|ψGHZ⟩ and|ψ′ GHZ⟩ are eigenstates of a parity operator defined asˆP = ˆσx 1⊗ ˆσx 2⊗···⊗ ˆσx N

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[5] [5]

Note that, because ˆH3 and ˆH4 commute, the state remains unchanged

The Hamiltonian is switched fromˆH3 to ˆH4 =∑ iBiˆσx i . Note that, because ˆH3 and ˆH4 commute, the state remains unchanged

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[6] [6]

By adding qubits to match the desired superposition, we can create a state such as|±±···±⟩ (|+ +··· +⟩ + |−−···−⟩ )/ √

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[7] [7]

After relabeling the qubits, this corresponds to1√ 2(| ˜En⟩ +| ˜El⟩) where| ˜En⟩ (| ˜El⟩) is then-th (l-th) eigenstate of ˆHtransverse =∑ iBiˆσx i , where|±±···±⟩ represents a product state in which each qubit is in either|+⟩ or|−⟩

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[8] [8]

(14)) by using ASP

Finally, the Hamiltonian is switched fromˆHtransverse to ˆHD (Eq. (14)) by using ASP. Then, we obtain the desired superposition state (Eq. (B1)). At the final step,ˆHtransverse is the Hamiltonian without interaction whileˆHD is the Hamiltonian with interaction only between the 2i− 1-th qubit and the2i-th qubit. This means that the evolution at the final s...

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