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Quasiprobabilistic imaginary-time evolution on quantum computers

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Pith's one-line read The paper claims that Trotterized imaginary-time evolution can be simulated "in expectation" on a quantum computer by sampling from a quasiprobability decomposition of each local imaginary-time exponential, then classically re-weighting…

desk verdict Solid QPD-for-ITE idea with correct proofs, but the printed algorithms are biased because they drop the postselection indicators—fixable, but the Trotter cost claims also need tightening. read the letter →

arxiv 2505.06343 v2 pith:NRXUU3GA submitted 2025-05-09 quant-ph

Annie Ray , Esha Swaroop , Ningping Cao , Michael Vasmer , Anirban Chowdhury This is my paper · ORCID
classification quant-ph
keywords quasiprobabilitydecompositionimaginary-timeevolutionprobabilisticerrorcancellationthermalpurequantumstatemitigationTrotterizationHeisenbergmodelnear-termcomputing
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 doing imaginary-time evolution on a quantum computer without fault tolerance, by writing each local imaginary-time step $e^{-\beta H_l / r}$ as a signed, weighted sum of ordinary quantum operations and measurements, sampling those operations, and re-weighting the outcomes classically. The method targets the rescaled expectation value $\langle A \rangle_P = \operatorname{tr}[A P(\rho)] / \operatorname{tr}[P(\rho)]$ for a Trotterized evolution, requires no ancillary qubits, and can absorb a device noise model into the decomposition so that noise does not bias the estimate. The sampling cost is $O(\gamma^{2Lr}/\epsilon^2)$ circuits for a Hamiltonian with $L$ local terms and $r$ Trotter steps, where $\gamma$ is the one-norm of the quasiprobability coefficients. For thermal physics, the authors argue the method is well matched to Clifford-random thermal pure quantum states, whose expectation values converge rapidly to Gibbs-state values as the system grows, and they demonstrate this with an 8-qubit simulation of a 1D Heisenberg chain and a 2-qubit energy estimate on IBM hardware.

What carries the argument

The load-bearing object is the quasiprobability decomposition (QPD) of a $k$-local imaginary-time channel, $T_l(\rho) = e^{-\beta H_l/r} \rho e^{-\beta H_l/r} = \sum_i q_i B_i(\rho)$, with $\gamma = \sum_i |q_i|$ and $p_i = |q_i|/\gamma$, so that $T_l(\rho) = \gamma \sum_i \operatorname{sgn}(q_i)\, p_i\, B_i(\rho)$. Randomly sampling $i$ from $p_i$, applying $B_i$, and weighting the measurement by $\gamma \operatorname{sgn}(q_i)$ turns each circuit into an estimator of the numerator and denominator of the renormalized expectation value. Algorithm 2 repeats this for each of the $Lr$ local factors in a first-order Trotter decomposition, accumulating the product of weights $\gamma^R \operatorname{sgn}(q_{i_1})\cdots \operatorname{sgn}(q_{i_R})$ with $R = Lr$, so the sampling cost compounds as $\gamma^{2R}$ while the gate depth stays $O(Lr)$.

What would settle it

For a fixed $k$-local Hamiltonian, say the 2-qubit Heisenberg term, numerically compute the exact operator difference $\| e^{-\beta H} - (e^{-\beta H_1/r} e^{-\beta H_2/r})^r \|$ for increasing $r$ at fixed $\beta$, and compare the decay with the bound $O(\beta^2 L^2 e^{\beta L/r}/r)$ from Eq. (81). If the non-trace-preserving composition produces errors that do not shrink as predicted, or if the empirical deviation $|\operatorname{tr}[A T(\rho)] - \operatorname{tr}[A P(\rho)]|$ saturates or grows with $r$, then the Trotter-step choice and the total circuit count $O(\gamma^{2Lr}/\epsilon^2)$ do not follow, and the method's claimed cost would need revision.

Watch

Extended reading notes

Core claim

The central claim is that any Hermiticity-preserving, possibly non-trace-preserving linear map that admits a quasiprobability decomposition into implementable basis operations can be simulated in expectation on a quantum computer by sampling those operations and forming weighted measurement estimators. Applied to imaginary-time evolution, Algorithm 2 composes $r$ Trotter steps of $L$ local exponentials $e^{-\beta H_l/r}$, each decomposed with quasiprobabilities $q_i$ and normalization $\gamma = \sum_i |q_i|$, and returns an estimate of the rescaled expectation value $\langle A \rangle_P = \operatorname{tr}[A P(\rho)] / \operatorname{tr}[P(\rho)]$ using $O(\gamma^{2Lr}/\epsilon^2)$ circuits and no ancillas, with error bounded by $2\epsilon / |\operatorname{tr}[P(\rho)]|$. Because the same decomposition can fold a device noise model into the basis operations, the estimate is noise-resilient without a separate error-mitigation layer. The authors further claim this construction is naturally suited to thermal pure quantum states, where a Clifford-random state evolved under imaginary time gives Gibbs-state expectation values whose error decays rapidly with qubit number.

Load-bearing premise

The Trotter error bounds used to choose the step count $r = O(\beta^2 L^2/\epsilon)$ are standard additive bounds derived for unitary evolution, and the paper concedes these may not be well-suited to the non-trace-preserving imaginary-time operators, so the claimed step count and total cost $O(\gamma^{2Lr}/\epsilon^2)$ rest on an unverified premise.

Editorial extensions

If this is right

  • No ancilla qubits are needed: the method uses only the system qubits, avoiding the multi-qubit-controlled operations and large SWAP networks of ancilla-based Gibbs samplers.
  • Noise resilience is intrinsic: if the device noise channel is incorporated into the basis operations from calibration data, the estimate tracks the noiseless result, as the 2-qubit hardware demonstration through four Trotter steps indicates.
  • Thermal expectation values become accessible on near-term devices: Clifford-random thermal pure quantum states combined with Algorithm 2 estimate Gibbs-state expectation values for $k$-local Hamiltonians, with the error in the TPQ expectation value decaying rapidly with the number of qubits.
  • The machinery applies beyond imaginary-time evolution to any Hermiticity-preserving map that admits a QPD over implementable operations, giving a general tool for estimating rescaled expectation values of non-trace-preserving evolutions.
  • Trotter step count and sampling cost trade off against each other: $r = O(\beta^2 L^2 / \epsilon)$ controls the Trotter error while $\gamma$ depends on the local exponentials $e^{-\beta H_l/r}$, so increasing $r$ lowers each local $\gamma$ but multiplies the number of factors in the cost $O(\gamma^{2Lr}/\epsilon^2)$.

Reading between the lines

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

  • A corollary the paper does not develop: the same estimators that yield $\operatorname{tr}[A P(\rho)]$ and $\operatorname{tr}[P(\rho)]$ give direct access to the partition-function-like normalization, so the method could estimate free-energy-type quantities, not just normalized expectation values.
  • The error bound's denominator $\operatorname{tr}[P(\rho)]$ decays as $e^{-2\beta \lambda_0}$ for a Hamiltonian with smallest eigenvalue $\lambda_0$, so even when the quasiprobability cost $\gamma$ stays flat, the useful precision degrades exponentially in $\beta$ unless the initial state has good ground-state overlap; this delimits the practical temperature range.
  • The paper notes its cost analysis relies on standard additive Trotter bounds for non-trace-preserving operators; a concrete testable extension is to verify numerically for a frustrated $k$-local Hamiltonian whether the empirical deviation $|\operatorname{tr}[A T(\rho)] - \operatorname{tr}[A P(\rho)]|$ obeys the claimed $O(\beta^2 L^2 e^{\beta L/r}/r)$ scaling, since the paper only argues signal pr
  • If mid-circuit measurement with feed-forward becomes available, rejected branches of the post-selected non-trace-preserving basis operations could be terminated early, effectively reducing the per-sample cost; the paper identifies this as a hardware-level optimization but does not quantify the resulting speedup.
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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 / 5 minor

Summary. The paper proposes a quasiprobabilistic imaginary-time evolution algorithm for quantum computers. The central idea is to decompose a Trotterized imaginary-time evolution into a probabilistic linear combination of native operations, including non-trace-preserving ones, and to estimate the rescaled expectation value tr[AT(ρ)]/tr[T(ρ)] by separately estimating numerator and denominator with weighted samples. The authors claim an O(γ^{2Lr}/ε^2) circuit complexity, no ancilla qubits, and natural noise resilience when device noise is included in the decomposition. They illustrate the method with classical simulations of thermal pure quantum state preparation for 1D Heisenberg Hamiltonians on up to 8 qubits and with a 2-qubit IBM quantum computer demonstration. The proof appendix gives the intended unbiased estimators.

Significance. If the central claim holds, the work is a useful near-term tool: it extends quasiprobability decomposition techniques from error mitigation to algorithmic simulation of non-unitary evolution, avoids ancilla qubits, and connects the construction to thermal pure quantum states. The paper's strengths are its explicit local QPD formulation, the analytic estimator proofs in Appendix D, the concrete sampling-cost analysis, and the fact that no parameter is fitted to target expectation values in the demonstrations. However, the printed algorithms do not match the proved estimators, and the Trotter-error analysis rests on bounds that the authors themselves flag as potentially invalid for non-unitary evolution. These issues are central to the paper's correctness claims and must be resolved before the results can be relied upon.

major comments (4)
  1. [Algorithm 2, line 13 and Algorithm 1, line 12] The printed estimators omit the postselection indicators. In Algorithm 2, M_j is set to γ^R sgn(q_{j1})...sgn(q_{jR}) A_j, while the proof in Appendix D defines F = γ^R sgn(q_{i1}...q_{iR}) L_{iR}...L_{i1} Y. Because the A measurement is executed even when an intermediate non-TP outcome is rejected, the empirical average of M_j includes contributions from the rejected branches, so E[M_j] is not tr[AE(ρ)]. The same omission appears in Algorithm 1, where M_j should be γ sgn(q_i) I_j A_j. This is a direct correctness failure of the central algorithm as printed, independent of the Trotter-error gaps, and it affects the hardware demonstration and the TPQ simulations if they were run according to the pseudocode. The fix is local: multiply by the indicators (or postselect before measuring A and set A_j=0 on rejection), and Appendix D shows that the intended estimator is then unbiased.
  2. [Algorithm 2, line 11] The instruction 'Measure A' appears inside the k-loop, so A is measured after each Trotter layer and the variable A_j is overwritten; only the last measurement is retained. The estimator in Lemma 2 is defined for an observable measured after all R applications of T. Unless A commutes with every intermediate map, this procedure yields a different expectation value. The measurement of A should be moved to after the loop and A_j should be defined from that single measurement.
  3. [Appendix E, Eqs. (80)-(86); Section 3.2, Eqs. (32)-(34)] The Trotter-error claim is made by applying Theorem 6 of Ref. [65], whose hypotheses concern unitary evolution generated by Hermitian terms, to the non-unitary, non-trace-preserving maps e^{-βH_l/r}. Section 3.2 concedes that these bounds 'may not be well-suited for non-trace-preserving imaginary-time evolution.' The choice r = O(β^2L^2/ε) and the total circuit count O(γ^{2Lr}/ε^2) rest on this unverified premise. The authors should either prove a non-unitary version of the Trotter bound or benchmark the operator-norm error numerically for small systems; without this, the stated complexity of the algorithm is not established.
  4. [Section 3, first paragraph; Appendix C, Proposition 1] The statement 'We may assume without loss of generality that H≥0' requires knowing the ground-state energy λ0 in order to shift H by αI. Appendix C acknowledges that determining α is challenging in general and that the method's useful behavior is tied to frustration-free Hamiltonians. Because the signal strength tr[P(ρ)] and hence the denominator of the ratio estimator depend on the shift, this should be presented in the main text as an algorithmic condition or a limitation, not as a free assumption.
minor comments (5)
  1. [Figure 1 caption] There is a typo: 'thermal expecation value' should be 'thermal expectation value'.
  2. [Section 3.1 and Figure 5 caption] The text says the noiseless simulation implements Algorithm 1, while the Figure 5 caption says the simulation implements Algorithm 2; please make the references consistent.
  3. [Section 3.2, Eq. (34)] The expression O(γ^{2Lr}/ε^2) assumes that all local decomposition costs γ_l are equal; the paper should state this assumption explicitly and give the general form O(∏_l γ_l^{2r}/ε^2).
  4. [References] Reference [58] is a duplicate of reference [25]; consolidate them.
  5. [Algorithm 1, comment on lines 7-12] For clarity, the pseudocode should specify that A is measured only if I_j=1, or that A_j is discarded when I_j=0; otherwise the reader might reasonably follow the printed instruction and measure A on every circuit.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the QPD derivation is self-contained, with separate correctness caveats outside this pass.

full rationale

The paper's central derivation is not circular. The target quantity is tr[AT(ρ)]/tr[T(ρ)] for a map T, and the algorithm is built directly from a stated quasiprobability decomposition T = Σ_i q_i B_i (Eq. 1). The estimators in Appendix D are proved unbiased by linearity of expectation: E[F] = tr[AT(ρ)] and E[G] = tr[T(ρ)] (Eqs. 54, 57, 73, 74). No parameter is fitted to the expectation values being reported: the QPD coefficients are obtained by solving a linear system or linear program for the known map T (Section 2.4), γ is computed from those coefficients, and the TPQ checks in Fig. 1 compare against independently computed ideal-ITE values. The author-overlap citations ([14], [28]) are background results on Gibbs sampling and thermal-state complexity; they are not load-bearing for Algorithm 2 or the error bounds. The paper itself flags the main genuine gap: standard additive Trotter error bounds "may not be well-suited for non-trace-preserving imaginary-time evolution" (Section 3.2, after Eq. 36), and it defers a detailed complexity analysis to future work. That is an acknowledged accuracy/cost limitation, not a circular reduction of outputs to inputs. Separately, there is a discrepancy between the indicator-carrying estimator F in Appendix D (Eq. 71) and Algorithm 2 as printed, whose line 13 sets M_j = γ^R sgn(q_{j1})...sgn(q_{jR}) A_j without multiplying by the postselection indicators I_jk, and whose line 11 measures A inside the k-loop. That is an algorithmic correctness issue, not a case where the claimed prediction is equivalent to its inputs by construction. Accordingly, the appropriate circularity finding is no significant circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central algorithm rests on standard QPD/PEC machinery (axioms 1-2), plus three application-specific assumptions: accurate and stable noise characterization, validity of Trotter bounds for non-TP maps, and the ability to shift H to known PSD form. The only hand-set numeric input is the identity shift for the Hamiltonian. No new physical entities are introduced.

free parameters (1)
  • Hamiltonian identity shift alpha = alpha=1 for the 2-qubit Heisenberg demo; otherwise requires lambda_0
    Added to e^{-beta H} to make H positive semidefinite and control the sampling cost; Appendix C shows the optimal choice depends on the unknown ground-state energy.
assumptions (6)
  • standard math Hermiticity-preserving linear maps admit a quasiprobability decomposition over a complete basis of CP operations, with coefficients obtainable by linear programming.
    Used in Section 2.4 and Algorithms 1-2; relies on Refs [39,59,62,85].
  • domain assumption Non-trace-preserving basis elements are physically implemented by a POVM followed by postselection, with success probability tr[B_i(rho)].
    Section 2.1, Eq 8 and Algorithm 1; the whole estimator depends on the ability to project and postselect.
  • domain assumption Device noise can be modeled by a single gate-independent channel N obtained from calibration data, and this channel is stable during the experiment.
    Section 3.1, hardware demo; the authors attribute the third Trotter step outlier to noise drift, showing the assumption can fail.
  • domain assumption First-order Trotter error bounds from Childs et al. apply to non-unitary imaginary-time exponentials.
    Appendix E, Eqs 80-86; flagged in Section 3.2 as possibly ill-suited for non-trace-preserving ITE.
  • ad hoc to paper The Hamiltonian can be shifted by a multiple of identity so that H>=0 with known ground-state energy lambda_0=0.
    Section 3 states H>=0 without loss of generality, but Appendix C shows the optimal shift requires knowledge of lambda_0 and cannot be found locally for frustrated Hamiltonians.
  • standard math Random Clifford unitaries form a 3-design sufficient for thermal pure quantum states.
    Section 3.2.1, Eq 39 and Refs [72,74]; this underpins the TPQ application.

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Cite this review

Pith. "Pith review of Quasiprobabilistic imaginary-time evolution on quantum computers." pith.science (2026). https://pith.science/paper/NRXUU3GA

@misc{pith2026250506343,
  author       = {Pith},
  title        = {Pith review of: Quasiprobabilistic imaginary-time evolution on quantum computers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NRXUU3GA}},
  note         = {Machine review of arXiv:2505.06343}
}
read the original abstract

Imaginary-time evolution plays an important role in algorithms for computing ground-state and thermal equilibrium properties of quantum systems, but can be challenging to simulate on classical computers. Many quantum algorithms for imaginary-time evolution have resource requirements that are prohibitive for current quantum devices and face performance issues due to noise. Here, we propose a new algorithm for computing imaginary-time evolved expectation values on quantum computers, inspired by probabilistic error cancellation, an error-mitigation technique. Our algorithm works by decomposing a Trotterization of imaginary-time evolution into a probabilistic linear combination of operations, each of which is then implemented on a quantum computer. The measurement data is then classically post-processed to obtain the expectation value of the imaginary-time evolved state. Our algorithm requires no ancillary qubits and can be made noise-resilient without additional error mitigation. It is well-suited for estimating thermal expectation values by making use of the notion of a thermal pure quantum state. We demonstrate our algorithm by performing numerical simulations of thermal pure quantum state preparation for the 1D Heisenberg Hamiltonian on 8 qubits, and by using an IBM quantum computer to estimate the energy of the same Hamiltonian on 2 qubits. We observe promising results compared to the exact values, illustrating the potential of our algorithm for probing the physics of quantum many-body systems on current hardware.

Figures

Figures reproduced from arXiv: 2505.06343 by the authors.

Figure 1
Figure 1. Thermal expectation value estimation using ITE. For each data point we average the error in the thermal [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. A schematic of a linear map T implemented by sampling from its quasiprobability distribution (see Eq. (4), Algorithm 1). Given a quasiprobability decomposition of the map T , basis circuits are randomly sampled according to the distribution |qi|/γ, and their measurement outcomes are weighted by the factors sgn(qi)γ. In particular, we consider the task of estimating the expectation value of some observable A (having … view at source ↗
Figure 3
Figure 3. Example of an operation T ′ = T1T2T3 that we implement. Here T1, T2, T3, are 2-qubit operations applied on nearest-neighbour qubits at different locations. In the analysis of Lemma 2, we assume for simplicity that T1 = T2 = T3 = T . We now explain how to implement a sequence of linear maps T1, T2, . . . , TR where each map Ti acts non-trivially on a constant k number of qubits (see [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Scaling of the QPD cost γ with inverse temperature β for the decomposition of e −βH using different basis sets, where H is the 2-qubit Heisenberg Hamiltonian H = −XX − Y Y − ZZ + II. The Takagi basis (orange) outperforms the EBL basis (blue), illustrating the utility o…
Figure 5
Figure 5. Figure 5: Energy estimation using ITE. We estimate the energy [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

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