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Quantum Imaginary-Time Evolution with Polynomial Resources in Evolution Time

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Pith's one-line read A quantum algorithm prepares imaginary-time-evolved states with polynomial resource scaling in evolution time, using one ancilla qubit and an adaptive normalization factor.

desk verdict Core ITE polynomial-resource result is solid and worth refereeing; the Lindbladian extension has an unproved error-propagation step and should be treated as heuristic. read the letter →

arxiv 2507.00908 v4 pith:4QUJUY4D submitted 2025-07-01 quant-ph

Lei Zhang , Jizhe Lai , Xian Wu , Xin Wang This is my paper · ORCID
classification quant-ph MSC 81P6868Q12 PACS 03.67.Lx
keywords imaginary-timeevolutionquantumalgorithmphaseprocessingground-statepreparationLindbladiansimulationpolynomialresourcescalingsuccessprobabilityearlyfault-tolerantcomputing
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 quantum imaginary-time evolution, the standard operation that turns a state into a low-energy state by exponentiating the Hamiltonian, can be simulated on a quantum computer with resources growing polynomially in the evolution time. The core move is an adaptive normalization factor chosen slightly above the ground-state energy, which keeps the post-selection success probability near $\alpha^2\gamma^2$ instead of letting it decay as $e^{-2\tau}$. With that stabilization, the algorithm approximates the normalized evolved state to fidelity $1-O(\operatorname{poly}(\tau^{-1}))$ using $\widetilde{O}(\gamma^{-2}\tau)$ controlled queries to the evolution oracle and one ancilla qubit; if the initial overlap $\gamma$ is inverse-polynomial in system size, the whole cost is polynomial in both time and qubit number. The same primitive is then applied to ground-state preparation and energy estimation, where it shortens circuit depth by a factor $\gamma^{-1}$ at the price of more measurements, and to Lindbladian simulation, where circuit depth no longer grows with the number of dissipative terms.

What carries the argument

The object that carries the argument is an adapted target function $f_{\tau,\lambda}(x)=\alpha e^{\tau(x-\lambda)}$ on $x\in[-1,\lambda]$, extended by a smooth bounded function on $(\lambda,1]$, with $\alpha\in(e^{-1/2},1]$. A trigonometric polynomial $F$ of degree $O(\tau)$ approximates $f_{\tau,\lambda}$ with error $\epsilon=O(\operatorname{poly}(\tau^{-1}))$ via Jackson's theorem, and a quantum phase processing (QPP) circuit $V^\epsilon_f(U_H)$ implements $F(U_H)$ by interleaving controlled calls to $U_H$ and its inverse with single-qubit ancilla rotations. Post-selecting the ancilla on $|0\rangle$ applies $F(U_H)$ to the input, approximating $e^{-\tau H}$. Choosing $\lambda$ in the comfort region $[|\lambda_0|,|\lambda_0|+\tau^{-1}]$ makes the constant $C=\tau(\lambda-|\lambda_0|)$ at most $1$, so the ground-state amplitude factor $\alpha e^{-C}$ stays bounded away from zero and the success probability is stabilized at order $\alpha^2\gamma^2$.

What would settle it

Simulate the algorithm on a Hamiltonian with a known ground state, for example an antiferromagnetic Heisenberg chain, at $\tau=100$ with $\lambda$ chosen in $[|\lambda_0|,|\lambda_0|+\tau^{-1}]$, and measure the post-selection success probability and output fidelity. If the success probability falls below $\alpha^2\gamma^2-\epsilon$ or the infidelity fails to reach $O(\tau^{-1})$ with $\epsilon=O(\tau^{-2})$, Lemma 1 and hence Theorem 2 would be contradicted.

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Extended reading notes

Core claim

The paper's central assertion is Theorem 2: given a normalized Hamiltonian with negative ground-state energy, access to controlled-$U_H=e^{-iH}$ and its inverse, and an initial state with overlap $\gamma>0$ on the ground state, one can prepare the normalized imaginary-time evolved state $|\phi(\tau)\rangle$ up to fidelity $1-O(\operatorname{poly}(\tau^{-1}))$ with overall success probability $1$, using $\widetilde{O}(\gamma^{-2}\tau)$ queries to controlled-$U_H$, $O(\gamma^{-2})$ copies of the initial state, $\widetilde{O}(\tau)$ maximal query depth, and one ancilla qubit. The normalization parameter $\lambda$ is set inside $[|\lambda_0|,|\lambda_0|+\tau^{-1}]$, where $\lambda_0$ is the ground-state energy; this keeps the success-probability lower bound at $\alpha^2\gamma^2-\epsilon$ rather than $e^{-2\tau}$. When the oracle is replaced by a Trotterized Pauli decomposition and $\gamma=\Omega(\mathrm{poly}(n^{-1}))$, the cost becomes polynomial in both $n$ and $\tau$. The authors then build on this ITE primitive to give a ground-state preparation and energy-estimation algorithm with query depth reduced by a factor $\gamma^{-1}$ relative to phase-estimation-based methods, and an open-system simulation algorithm whose circuit depth is independent of the number of jump operators.

Load-bearing premise

The result collapses if the initial state's overlap $\gamma=|\langle\phi|\psi_0\rangle|$ with the ground state is exponentially small in the number of qubits, because then the guaranteed success probability scale $\alpha^2\gamma^2$ and the $O(\gamma^{-2})$ repetitions in Theorem 2 become exponentially large.

Editorial extensions

If this is right

  • Imaginary-time evolution can now be used as a rigorously analyzed subroutine in quantum algorithms, replacing heuristic Trotter or variational steps in tasks where long evolution times are needed.
  • Ground-state preparation and ground-state energy estimation can be carried out with query depth reduced by a factor $O(\gamma^{-1})$ compared with phase-estimation-based methods, at the cost of more measurement shots.
  • Lindbladian simulation in Liouville space can be implemented with circuit depth that does not grow with the number of jump operators, provided the Pauli sparsity stays fixed.
  • When the initial state has inverse-polynomial overlap with the ground state, preparing imaginary-time evolved states costs $\mathrm{poly}(n,\tau)$ gates, making long-time many-body simulation accessible in principle on early fault-tolerant hardware.

Reading between the lines

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

  • The depth-versus-measurements tradeoff suggests that on devices where depth is the scarce resource, this ITE route to ground-state problems may be preferable even when its total query count is larger; the paper notes that circuits can be run in parallel.
  • Since Theorem 2 inherits the $\widetilde{O}(\gamma^{-2}\tau)$ cost from the ground-state-energy estimation subroutine, any future improvement in spectral estimation would automatically improve ITE; the paper hints at, but does not prove, a computational equivalence between the two problems.
  • The favourable numerical success probabilities in the Lindbladian setting suggest there may exist a provable post-selection bound under a condition weaker than preserving the ground-state subspace of the dissipative Hamiltonian; finding such a condition is a concrete open problem.
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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

5 major / 5 minor

Summary. The paper introduces a QPP-based quantum algorithm for preparing normalized imaginary-time-evolved states, with an adaptive normalization parameter λ that keeps the post-selection success probability bounded away from zero over long evolution times. The central claim, Theorem 2, states that under overlap and oracle assumptions one can prepare the ITE state to fidelity 1 − O(poly(τ^{-1})) with probability 1, using Õ(γ^{-2}τ) queries and one ancilla. Theorem 3 extends this to a Trotterized Pauli-form setting. The paper then applies the ITE primitive to ground-state preparation/energy estimation (Algorithm 1, Theorem 6) and to Lindbladian simulation (Algorithm 2, Theorems 7 and 8), with numerical experiments for τ up to 50. The core adaptive-normalization idea is original and the QPP construction is appropriate, but several load-bearing proof steps in the appendices are not established as written, so the formal claims currently exceed what is proven.

Significance. If the central Theorem 2 holds, this is a meaningful conceptual advance: it would provide the first provably polynomial-in-τ ITE algorithm, complementing the usual polynomial-in-n scaling and addressing a real gap in the literature. The adaptive normalization idea is elegant, the numerical experiments are relevant, and the comparison tables are useful. The paper also gives credit to the field by making the QPP machinery a central tool and by sharing code for the numerics. However, the significance is currently limited by the fact that the proof of Theorem 2 relies on an unproved degree-scaling step in Theorem S4 and on an asymptotic simplification in Lemma 1; and the open-system application rests on an explicitly unproved error-propagation assumption in Theorem 7. These are fixable in principle, so the work is worth serious revision rather than dismissal.

major comments (5)
  1. [Section II.D, Theorem 2 and Appendix C.1] Theorem 2 states that the ITE state is prepared 'with probability 1', but the proof invokes Theorem S5, which returns an estimate of |λ0| with failure probability e^{-τ}. If that phase-estimation subroutine fails, the chosen λ may violate λ ≥ |λ0|, and the guarantees of Lemma 1 no longer apply. No amplification to failure probability zero is described, and a finite number of repetitions cannot achieve exact probability 1. The theorem should either state success probability 1 − e^{-τ} (or 1 − δ with a stated boosting cost) or explicitly condition on the success of the QPE subroutine.
  2. [Lemma 1, Appendix C, Eq. (C.29)] The fidelity proof asserts ∥e^{-τH/2}|φ⟩∥² = O(γ² e^{-τλ0}) and ∥e^{-τH}|φ⟩∥² = O(γ² e^{-2τλ0}) directly from Assumption (iv). These bounds require that the excited-state contributions be of order γ², i.e. (1−γ²)e^{-τΔ} = O(γ²), which is not among Assumptions (i,iv,v). Without such a condition the ratio a(τ) can be larger by a factor γ^{-2}, so the claimed fidelity lower bound 1 − O(α^{-1}ε e^C) and the fidelity guarantee in Theorem 2 do not follow as stated. The lemma needs an explicit γ-dependence or an additional spectral-gap/overlap assumption.
  3. [Theorem S4, Appendix B.1] The proof claims that a trigonometric polynomial of degree L = O(τ) approximates f with error O(poly(τ^{-1})) by applying Jackson's theorem with a constant C_l that is treated as independent of τ. For g(x) = ρ(x) e^{τ(x−λ−μ)} the l-th derivative is O(τ^l), so the constant in Jackson's bound must scale as O(τ^l); then the construction L = C_l^{1/l}τ gives L = O(τ²), not O(τ), unless a new τ-independent constant is proved. Since Theorem 2's query depth and gate counts depend on the degree bound L = O(τ), the resource statement is not established. The proof should be repaired, e.g. by an analytic Chebyshev approximation argument, or the resource counts in Theorems 2 and 3 should be revised.
  4. [Theorem 7, Appendix E, Eqs. (E.18)–(E.20)] The per-step error recurrence replaces ∥V[|φ_k⟩] − V[|ζ_k⟩]∥ by ∥|φ_k⟩ − |ζ_k⟩∥, justified only by 'assuming |φ_k⟩ and |ζ_k⟩ are mainly differed by their components in ground-state subspace of H'. This is not proved. The map V includes a normalization factor (Eq. 6) and is therefore nonlinear, and the coherent rotation U_{Hc} can move population out of the ground-state subspace of H, so the effective Lipschitz constant of V is uncontrolled. Consequently the bound O(kα^{-1}ε) and the final O(t²/µN + Nε) in Theorem 7 do not follow, and Theorem 8 inherits the gap. The authors' own admission in Section III.B that no general success-probability bound is available underscores that this step is load-bearing. Please prove the simplification or explicitly restrict Theorem 7 to a setting where Hc approximately preserves the relevant subspace.
  5. [Section III.A, Assumption (x) and Theorem 6] Theorem 6 and Algorithm 1 are conditioned on Assumption (x), which states that a known B with γ²|λ0| ≥ e²B > 0 is available. The authors explicitly call this a heuristic step, and the numerical verification uses a fixed B = 1/5000 for three small Heisenberg chains; no theoretical method for obtaining B or sensitivity analysis is given. The formal complexity guarantee is therefore instance-specific and weaker than the theorem statement suggests. The assumption should appear prominently in the theorem statement, and the paper should either provide a procedure for choosing B or state the result as conditional on that promise.
minor comments (5)
  1. [Throughout] There are numerous typos and grammatical errors, e.g. 'theorectical', 'trignometric', 'postivie', 'gurantees', 'hvae', and 'sine the product'. A careful proofreading pass is needed.
  2. [Table I] The row for Theorem 3 lists expected circuit runs O(poly(n)), but the theorem statement says O(poly(n)) copies of |φ⟩; please clarify what 'circuit runs' means and whether the overall success probability is meant to be high probability rather than exactly 1.
  3. [Section III.B.1, Figure 3(c)] The caption states that 'cost' counts queries including post-selection repetitions, but the main text says 'average resource cost' without specifying the estimator. Please define the plotted cost precisely in the caption.
  4. [Appendix E, Eq. (E.35)] The notation H_≈ is used without local definition; it should be reintroduced in Appendix E or a pointer to Theorem 3's definition should be given.
  5. [Appendix C.2, Proposition S14] The step 'the operator norm of the difference of the real-time evolutions implies ∥H − H_≈∥∞ < ϵ' is stated without proof; because the logarithm is multi-valued, a short Davis–Kahan-style argument for eigenphases should be supplied.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ITE state preparation is an independent QPP/QPE construction, and the main bounds reduce to properties of exponentials rather than to the target state.

full rationale

I walked the claimed derivation chain. The target state is defined in Eq. (1) as |phi(tau)> = e^{-tau H}|phi>/||e^{-tau H}|phi>||. The algorithm implements, via QPP, a polynomial F approximating f_{tau,lambda}(x) = alpha e^{tau(x-lambda)} on the spectrum of H (Eqs. (3)-(4), Theorem S4), and post-selects the ancilla. Lemma S6 and Lemma 1 then bound the success probability and fidelity by explicitly comparing F with f on the eigenvalue interval; the bounds depend on the overlap gamma, the approximation error epsilon, and the chosen lambda. None of these quantities is fitted to the target state: lambda is obtained from an independent ground-state-energy estimation subroutine (Theorem S5, from Ref. [18] and standard QPE variants), not from the ITE output itself. Theorem 2 merely combines that independent lambda estimate with the QPP circuit, and the resource count is the sum of the QPE and ITE parts. The self-citation of Ref. [18] for QPP/QPE is central, but that cited result is parameter-free, does not assume ITE-state preparation, and is also paralleled by the alternative QSVT framework cited as Ref. [17], so it does not raise the circularity score. The main equations reduce to known properties of exponentials and Fourier/Jackson approximation, not to the desired answer. I also flag two non-circular rigor gaps, per the review rule. First, the proof of Theorem 7 in Appendix E (around Eqs. (E.18)-(E.23)) asserts: 'by assuming |phi_k> and |zeta_k> are mainly differed by their components in ground-state subspace of H, one can simplify the second term as ||V[|phi_k>] - V[|zeta_k>]|| approx |||phi_k> - |zeta_k>||'. This is an unproved error-propagation simplification for the Lindbladian claims, and the paper itself later states that no general success-probability lower bound is obtained for Algorithm 2. That is a correctness risk for the open-system results, not a definitional circularity. Second, in the proof of Theorem 2, the paper says 'with precision tau^{-1/2}' and then 'By adding this value by tau^{-1/2}, we obtain an estimation lambda in [|lambda0|, |lambda0|+tau^{-1}]'; the arithmetic does not obviously place lambda in that interval, but this is a technical slip, not an equivalence between input and output. The central imaginary-time-evolution claim is self-contained and does not reduce to its inputs by construction.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The main new ingredient is not an entity but a choice of target function and a search procedure. The cost estimates depend on spectral assumptions that are standard for ground-state problems; Assumption (x) for Algorithm 1 is explicitly heuristic. No new particles, mediators, forces, or conserved quantities are introduced.

free parameters (3)
  • lambda (normalization factor) = estimated via QPE, constrained to [|lambda_0|, |lambda_0|+tau^{-1}]
    The target function f_{tau,lambda} depends on lambda; the success probability and fidelity bounds rely on lambda being near the ground-state energy. The paper obtains lambda from a phase-estimation subroutine, so it is data-dependent but not hand-fitted.
  • alpha = 0.85 in numerics; alpha=e^{-tau mu} in [e^{-1},1]
    A free scale in the target function. The lower bound alpha>e^{-1/2} is needed for Proposition S25; numerical experiments set alpha=0.85. It affects constants, not asymptotic scaling.
  • B (Algorithm 1) = 1/5000 in numerical experiments
    Assumption (x) requires a known lower bound B with gamma^2 |lambda_0| >= e^2 B. The energy-estimation theorem and measurement count 8L Lambda^2 B^{-2} tau^3 depend on B. The paper calls the choice a heuristic guess, and it is not derived from the problem.
assumptions (8)
  • domain assumption Normalized Hamiltonian with eigenvalues in [-1,1] and lambda_0 < 0 (Assumption i)
    WLOG by rescaling and shifting; stated in Section II.A.
  • domain assumption Oracle access to controlled-U_H and its inverse (Assumption ii), or Pauli decomposition of H (Assumption iii)
    Both access models are stated in Section II.A and used in Theorem 2 and Theorem 3 respectively.
  • domain assumption Nonzero initial overlap gamma > 0, and for polynomial-in-n scaling gamma = Omega(poly(n^{-1})) (Assumptions v and vii)
    Standard in ground-state algorithms; the success probability lower bound is proportional to gamma^2 in Lemma 1.
  • domain assumption Non-degeneracy and distinguishable gap: Delta > 0 and Delta = Omega(tau^{-1} log poly(tau)) (Assumptions viii and ix)
    Needed for the Trotterized version of the algorithm and for ground-state convergence in Lemma 5 and Theorem 3.
  • ad hoc to paper Ad hoc Assumption (x): known B with gamma^2 |lambda_0| >= e^2 B
    Heuristic; Theorem 6 and Algorithm 1 depend on it, and numerical experiments simply guess B=1/5000.
  • standard math QPP existence theorem from Ref [18] and Jackson's theorem for smooth functions (Theorems S1, S2)
    External mathematical results used to realize polynomial transformations of U_H with O(tau) queries and super-polynomial approximation accuracy.
  • standard math Davis-Kahan theorem and first-order Trotter error bound (Theorems S12, S13)
    Used in the Trotterized resource analysis of Theorem 3 to transfer ground-state closeness from U_H to H.
  • ad hoc to paper Unproved simplification in Theorem 7 that errors are dominated by ground-state-subspace components of H during Lindbladian propagation
    Stated in Appendix E without proof; this is a gap in the open-system analysis.

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Cite this review

Pith. "Pith review of Quantum Imaginary-Time Evolution with Polynomial Resources in Evolution Time." pith.science (2026). https://pith.science/paper/4QUJUY4D

@misc{pith2026250700908,
  author       = {Pith},
  title        = {Pith review of: Quantum Imaginary-Time Evolution with Polynomial Resources in Evolution Time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4QUJUY4D}},
  note         = {Machine review of arXiv:2507.00908}
}
read the original abstract

Imaginary-time evolution is fundamental for analyzing quantum many-body systems, with applications spanning quantum chemistry, condensed matter physics, and quantum field theory, yet classical simulation requires exponentially growing resources in both system size and evolution time. While quantum approaches reduce the system-size scaling, existing methods rely on heuristic techniques with measurement precision or success probability that deteriorates as evolution time increases. We present a quantum algorithm that prepares normalized imaginary-time evolved states using an adaptive normalization factor to maintain a stable success probability over long imaginary-time intervals. Our algorithm approximates the target state with error polynomially small in the inverse imaginary time using a polynomial number of elementary quantum gates and a single ancilla qubit, with success probability close to one. When the initial state has reasonable overlap with the ground state, this algorithm also achieves polynomial query complexity in the system size. To our knowledge, this is the first quantum algorithm for imaginary-time evolution with provably polynomial resource scaling in evolution time. Numerical experiments validate our theoretical analysis for evolution time up to 50, demonstrating the algorithm's effectiveness for long-time evolution. Building on this technique, we further develop imaginary-time-evolution-based algorithms for ground-state-related problems and for simulating open quantum systems. These algorithms can reduce circuit depth in certain regimes compared with existing methods, at the expense of higher total query complexity, advancing the practical feasibility of quantum simulation on early fault-tolerant devices.

Figures

Figures reproduced from arXiv: 2507.00908 by the authors.

Figure 1
Figure 1. The performance of the ITE circuit V ϵ fτ,λ (UH) with different choices of λ (horizontal axis) and τ = 20. The blue line shows the state infidelity between the ITE state |ϕ(τ )⟩ and the output state |ϕe(τ )⟩. The orange line shows the success probability of obtaining the output state. The vertical axis is scaled by a logarithm of 10 for better visibility. |ϕ(τ )⟩ is large. This infidelity behavior is expected, as th… view at source ↗
Figure 2
Figure 2. Experiment results for applying our ITE-based algorithms to ground-state problems for antiferromagnetic Heisenberg (AFM) chains. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Numerical performance of Algorithm 2 for Lindbladian simulation. (a,b) Infidelity and resource cost as a function of evolution time t = 1, . . . , 20 with N = t 3 steps. Panel (a) shows results for four 4-qubit TFIM instances with jump operators from [75–78]. Panel (b) shows an antiferromagnetic Heisenberg chain in Equation (12) with five random sets of jump operators. In each panel, the blue curve (log scale) is th… view at source ↗

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

  1. Ground state preparation in $(2+1)$-dimensional pure $\mathbb{Z}_2$ lattice gauge theory via deterministic quantum imaginary time evolution

    hep-lat 2026-04 unverdicted novelty 6.0 of 10

    Deterministic QITE with a Gauss-law-reduced Pauli pool reproduces DMRG ground-state energies of (2+1)-D pure Z2 lattice gauge theory to within 0.1% for ladders of up to 32 qubits and coupling λ ∈ [0.5, 5].

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    Similar to Ref

    Quantum phase estimation Given an eigenstate |ψ⟩ of a unitary U and its evolution operator U, the problem of quantum phase estimation is to estimate the corresponding eigenvalue x such that U |ψ⟩ = eix|ψ⟩. Similar to Ref. [19, 34], QPP can simulate the STEP function f (x − a) ...

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    Proof of Lemma 1, Theorem 2 and Corollary 4 Lemma 1 Let C ≥ τ (λ − |λ0|) ≥ 0. Under Assumptions (i,iv,v), the output state | eϕ(τ )⟩ from the ITE circuit V ϵ fτ,λ (UH ) is obtained with success probability lower bounded by α2γ2e−2C − ϵ. Moreover, the state fidelity between the...

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    Resource analysis for Trotter case In this section, we analyze the resource complexity when UH is now realized by its Trotter decomposition. It is hard to implement U (t) directly, so generally Hamiltonians of interest will be written as the sum of L Pauli matrices: U (t) = ex...

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    Performance guarantee of sampling measurements Lemma S17 Let eω(λ) be the expectation value of the quantum state V ϵ fτ,λ (UH )(|0⟩ ⊗ |ϕ⟩) with respect to bH = |0⟩ ⟨0| ⊗H, eω(λ) = (⟨0| ⊗ ⟨ϕ|)V ϵ fτ,λ (UH )† · bH · V ϵ fτ,λ (UH )(|0⟩ ⊗ |ϕ⟩). (D.1) Then under Assumption (i), the...

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    Location of the starting point The overall idea is to use binary search to locate the region where|ω(λ)| > B, and then use ternary search combined with the Algorithm 1 to determine braking. Algorithm 6: Binary Search Input : τ, |ϕ⟩, Has defined in Section II, lower bound B in ...

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    Resource cost of Algorithm 4 Proposition S22 Let δ ≥ 0, k ≥ 0 and λ ≥ −λ0. Under Assumption (v,viii), when λ − δ ≥ −λ0, bω(λ − δ) = e2τ δbω(λ) ; (D.16) when −λ0 > λ− δ, bω(λ − δ) = e2τ δ− R(λ; δ) bω(λ) , (D.17) where the remain term R(λ; δ) is given as R(λ; δ) = X j:−λj >λ−δ |...

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