{"id":"37f82a6a-ba1b-483b-bc90-9059fe687ba2","arxiv_id":"2501.17003","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A variational quantum algorithm for non-Hermitian Hamiltonians matches exact diagonalization on small transverse Ising chains and gives tentative evidence that the imaginary-field model has no quantum phase transition.","lead":"The authors test a variational quantum eigensolver designed for non-Hermitian Hamiltonians on small Ising spin chains, comparing its output with exact diagonalization. The quantum results agree at five spins, and the imaginary-field model shows no sign of a phase transition, though the study is preliminary.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The variational algorithm's selection of the 'ground state' among a non-Hermitian Hamiltonian's many eigenpairs is unspecified, so the N=5 agreement with exact diagonalization does not yet establish reliable QPT study.","rationale":"The reader's weakest assumption was the ad hoc uniform noise model, which is indeed a serious limitation for hardware claims. However, the paper's own Sec. 5 concedes the noise model 'does not capture the complexity of true noise,' so it is an acknowledged weakness rather than a hidden one. The more fundamental unresolved issue is the eigenstate-selection ambiguity in the variational method itself. Because C_+ is zero for every eigenpair, the variational landscape has many degenerate global minima; the optimizer must be steered to the physically relevant ground state. The paper describes the original two-stage method that targeted eigenpairs but does not explain how its modification achieves this targeting. Without that, the N=5 'good agreement' in Fig. 1 could be an artifact of accidentally landing on the matching eigenstate, and the method's utility for tracking a QPT—which requires consistent identification of the same eigenstate across a parameter sweep—is unproven. This concern is independent of machine noise and would invalidate the central claim even in a noiseless statevector simulation. I therefore recommend keeping the reader's CONDITIONAL verdict: the paper's physics conclusions are reasonable but conditional on a concrete demonstration of ground-state selection. My disagreement with the reader is partial because I identify a different, earlier weak point than the noise model, though both support a conditional acceptance.","tokens_in":6809,"tokens_out":6291,"duration_ms":62875,"concrete_test":"For the imaginary-field model (Eq. 3) at N=5 and N=7, run the variational algorithm over a range of Γ_I and record the converged E and the overlap of |φ(θ)⟩ with all exact eigenstates from explicit diagonalization. Verify that the algorithm always converges to the eigenstate with the lowest real part of the energy (or, alternatively, to the eigenstate with the largest overlap with the Γ_I→0 ground state), and that this identification is stable as Γ_I varies. If the converged state jumps between eigenstates or selects an excited state, then the method as described cannot be claimed to study the ground-state QPT.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The cost function C_+ (Eq. 7) vanishes for every right eigenpair (E, |φ⟩) of H, not just for the ground state. For a non-Hermitian Hamiltonian the eigenvalues are generally complex, so there is no natural variational ordering like the minimum energy that standard VQE exploits. The paper states (Sec. 4) that it followed a 'slightly modified version' of the two-stage optimization from Ref. [25], whose stages were designed to target a particular eigenpair, but it never specifies the selection mechanism or how the algorithm is initialized to track the physical ground state as Γ or Γ_I is varied. The reported order parameter <M_x> (Eq. 4) is defined against the 'ground state |φ0⟩', but the paper does not define which eigenstate that is for the non-Hermitian model (e.g., lowest real part, largest overlap with the Hermitian Γ→0 ground state, or something else). Without this definition and a demonstrated selection rule, the N=5 agreement with exact diagonalization could be the result of converging to a different eigenstate that happens to have a similar <M_x>, or of branch switching as parameters change. The central claim that the algorithm can 'quantify and study' quantum phase transitions requires that the same physical eigenstate be tracked consistently across the transition; the paper provides no evidence for this. This is more load-bearing than the acknowledged noise-model limitation because it affects even the noiseless, statevector-simulation results.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript investigates the use of the variational quantum algorithm of Xie, Xue, and Zhang (Ref. [25]) for computing eigenpairs of non-Hermitian Hamiltonians, with the goal of studying quantum phase transitions. Two one-dimensional spin models are considered: the transverse Ising model with a real transverse field (Eq. 2) and a non-Hermitian variant with an imaginary transverse field (Eq. 3). The authors use a statevector simulator of a five-qubit circuit with a uniform noise perturbation on expectation values, compare the resulting <M_x> values with exact diagonalization, and then use exact diagonalization for larger system sizes to examine finite-size behavior. They report agreement for N=5 and use the susceptibility peak to identify the real-field transition near Gamma=1, while for the imaginary-field model the order parameter appears to vanish with increasing odd system size, suggesting no quantum phase transition.","tokens_in":7095,"tokens_out":4891,"duration_ms":40288,"significance":"If the central feasibility claim were fully established, this would be a valuable step toward using quantum computers for non-Hermitian systems and sign-problem-affected lattice field theories. The paper's strengths include the direct comparison against exact diagonalization as an external benchmark, the use of the established algorithm from Ref. [25], and a conservative interpretation of the imaginary-field results. The physics conclusion for the real-field Ising model is consistent with known results. However, the lack of a defined eigenstate-selection rule and the ad hoc noise model leave the main claim only partially supported.","major_comments":[{"comment":"The cost function C+(theta,E) vanishes for every right eigenpair (E,|phi>) of the non-Hermitian Hamiltonian, not just for the ground state. Since non-Hermitian eigenvalues are generally complex, there is no natural variational ordering analogous to the minimum energy in Hermitian VQE. The manuscript states in Sec. 4 that a 'slightly modified version' of the two-stage optimization from Ref. [25] is used, but it never specifies how the algorithm selects a particular eigenpair or how it tracks the physical ground state as Gamma or Gamma_I is varied. The order parameter <M_x> in Eq. (4) is defined with respect to 'the ground state |phi0>', but for the non-Hermitian model it is not defined which eigenstate this is (e.g., lowest real part, largest overlap with the Hermitian Gamma->0 ground state, or some other rule). Without this definition and a demonstrated selection rule, the N=5 agreement with exact diagonalization could reflect convergence to different eigenpairs at different parameter values, and the central claim that the algorithm can 'quantify and study' quantum phase transitions is not yet supported. Please specify the selection mechanism and provide evidence that a single physical state is tracked across the transition.","section":"Sec. 4, Eq. (7)"},{"comment":"The noisy-quantum-simulation results rest on an ad hoc noise model: each expectation value is perturbed by a uniform random variable in (-0.04,0.04), claimed to be comparable to about 1000 shots. This model does not reproduce the binomial statistics of actual measurement shots, does not include gate errors or decoherence, and the manuscript itself concedes that it 'does not capture the complexity of true noise.' Moreover, no error bars are shown on the noisy data points in Fig. 1, so the reader cannot judge whether the observed agreement with exact diagonalization is statistically meaningful. As a result, the statement in Sec. 7 that the method works 'in a noisy environment' is not substantiated. Please replace this with shot-based sampling or at least show statistical uncertainties, and temper the noisy-environment claim accordingly.","section":"Sec. 5"},{"comment":"The conclusion that the imaginary-field Ising model has no quantum phase transition is based on the observation that <M_x> for odd numbers of spins appears to tend to zero as N increases (Fig. 3). This is a qualitative reading of the plot; there is no quantitative finite-size scaling analysis, no extrapolation to the thermodynamic limit, and no discussion of the even-odd discrepancy beyond citing boundary effects. Since a negative claim (absence of a transition) requires more care than a positive one, the conclusion 'there is no quantum phase transition' is premature. Please add a quantitative scaling analysis or explicitly reframe the conclusion as a provisional statement.","section":"Sec. 6.2"}],"minor_comments":[{"comment":"The abstract says PT-symmetry is discussed, but the two models studied (Eqs. 2-3) do not respect PT-symmetry; consider making the connection between the introductory discussion and the actual models more explicit.","section":"Abstract and Sec. 1"},{"comment":"The 'slightly modified version' of the optimization strategy is not described in enough detail for reproduction. Please give the number of stages, the parameter update rules, and the convergence criteria used.","section":"Sec. 4"},{"comment":"Typo: 'enviroment' should be 'environment'.","section":"Sec. 5"},{"comment":"The phrase 'on the hardware we used' is misleading because the results were obtained from a statevector simulation, not quantum hardware.","section":"Sec. 6.1"},{"comment":"The caption says quantum-simulator results are also reported for five spins, but the figure does not distinguish them from the exact-diagonalization points; add distinct markers and a legend.","section":"Fig. 2"},{"comment":"The notation 'approx' for the susceptibility is vague; it should be stated that this is a zero-temperature fluctuation measure rather than a standard thermodynamic response function.","section":"Eq. (5)"}],"recommendation":"major_revision","confidential_remarks":"This is a proceedings-style contribution for LATTICE2024, so the technical depth is appropriately brief. The most serious issue is the unspecified eigenstate selection in the variational algorithm, which is not merely a presentation matter: it affects the interpretation of every numerical result in Figs. 1-3. The authors should be encouraged to address it and to add error estimates to the noisy simulation points. I see no signs of misconduct or missing references; the paper is honest about its limitations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a small, honest proceedings paper. The new bit is applying the Xie-Xue-Zhang non-Hermitian VQE to the imaginary-transverse-field Ising model, plus a tentative observation that <M_x> tends to zero in the thermodynamic limit, suggesting no QPT. The N=5 statevector results agree with exact diagonalization for both models, which is a valid feasibility check. The susceptibility peak for the real-field model near Gamma=1 is consistent with known physics. Credit where due: the authors do not oversell, and they flag their own limitations about the noise model and the need for larger systems.\n\nThe soft spots are real but not disqualifying. The stress-test concern is the more load-bearing one: C_+ vanishes for every eigenpair, and the paper never specifies how the optimization selects the 'ground state' among the complex spectrum, nor how it tracks that state as Gamma or Gamma_I varies. Saying they use a 'slightly modified version' of Ref [25] is not enough, because the selection mechanism is precisely what makes the order parameter physically meaningful. Without a defined rule, the N=5 agreement could in principle come from branch switching to a different eigenstate. However, the smooth curves and agreement with exact diagonalization across two models suggest they are tracking something consistent; I would not call the paper wrong, just underspecified on this point.\n\nThe noise model is admittedly crude (uniform perturbation of expectation values, not shot statistics), and no error bars are shown on the noisy points. That is a weakness, but it does not undercut the noiseless statevector result, which is the actual demonstration. Also, the physics conclusion for the thermodynamic limit comes from exact diagonalization, not from the quantum algorithm, so the paper is really two things: a proof-of-principle for the algorithm on five spins, and a separate classical finite-size study. Both are fine, but the connection between them is loose.\n\nWho is this for? Anyone working on non-Hermitian quantum simulation or lattice field theory with sign problems. It is a useful sanity check that an existing algorithm transfers to a new model, and the tentative no-QPT observation is worth having in the literature. It deserves a serious referee: yes, send it out. A referee should ask the authors to define the eigenstate selection rule and show that the same state is tracked across the phase diagram. The paper is not a breakthrough, but it is a legitimate, readable contribution that does what it sets out to do.","headline":"A modest but honest proceedings paper: known algorithm applied to a new non-Hermitian Ising model, with a real but fixable gap in specifying which eigenstate the variational search targets.","tokens_in":7637,"tokens_out":1484,"would_cite":false,"duration_ms":15703,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The non-Hermitian variational quantum eigensolver, applied to the transverse Ising model with real and imaginary transverse fields, reproduces exact-diagonalization magnetization on five qubits and gives a finite-size susceptibility peak…","keywords":["non-Hermitian quantum mechanics","variational quantum eigensolver","quantum phase transition","transverse Ising model","imaginary magnetic field","PT symmetry","sign problem","NISQ algorithms"],"falsifier":"Run the same non-Hermitian VQE on actual five-qubit hardware (or a shot-level simulator) for the real-field Ising chain near $\\Gamma = 1$ and compare the resulting $\\langle M_x \\rangle$ and $\\chi_x$ with the paper's exact-diagonalization curves; agreement outside the $\\pm 0.04$ uniform band would show the noise model is not representative. Alternatively, compute the shot-noise variance of $\\langle M_x \\rangle$ under the cost function; if it is inconsistent with a uniform bound, the noiseless-plus-uniform model is falsified.","tokens_in":6587,"feed_emoji":"⚛️","tokens_out":7933,"duration_ms":70450,"temperature":0.7,"pith_summary":"This paper tests whether a recently proposed variational quantum algorithm for non-Hermitian Hamiltonians can locate quantum phase transitions. Using five qubits, it computes the ground-state transverse magnetization of the transverse Ising chain for both a real and an imaginary transverse field and compares the results with exact diagonalization. The real-field runs reproduce the magnetization curve and show a susceptibility peak at $\\Gamma = 1$ that matches the known transition. The imaginary-field runs also match exact diagonalization, but the magnetization tends to zero as system size grows, suggesting no ground-state quantum phase transition in that model. The paper frames the work as a first step toward using quantum computers on sign-problem systems.","feed_headline":"Five-qubit runs match exact Ising magnetization in a non-Hermitian VQE","feed_subtitle":"Five-spin runs match exact diagonalization; the susceptibility peak sits at Gamma = 1, and the imaginary-field model shows none.","key_machinery":"The load-bearing object is the operator pair $M_\\pm(E; H) = (H^\\dagger - E^*)(H - E)$ and $(H - E)(H^\\dagger - E^*)$, which are Hermitian and positive semi-definite even when $H$ is not. Their expectation values $C_\\pm$ vanish exactly when the parametrized state $|\\phi(\\theta)\\rangle$ and the complex shift $E$ form a right- or left-hand eigenpair, respectively. Minimizing $C_+$ with a fully expressive circuit built from single-qubit rotations and CNOTs turns the non-Hermitian eigenproblem into a VQE-style optimization, with a modified multi-stage gradient strategy adapted from Ref. [25]. The paper also uses the decomposition of any matrix into Pauli strings to feed $H$ and $H^\\dagger$ into the cost function, and uses the transverse magnetization $\\langle M_x \\rangle$ and the zero-temperature susceptibility $\\chi_x \\approx \\langle M_x^2 \\rangle - \\langle M_x \\rangle^2$ as the order parameters for phase-transition searches.","core_discovery":"On its own terms, the paper's central claim is that the non-Hermitian variational quantum eigensolver of Ref. [25] can serve as a practical probe of quantum phase transitions in non-Hermitian spin systems. The authors construct cost functions $C_\\pm$ from the positive semi-definite operators $M_\\pm = (H^\\dagger - E^*)(H - E)$ and $(H - E)(H^\\dagger - E^*)$, minimize $C_+$ with an over-expressive ansatz, and add a uniform noise of $\\pm 0.04$ to every expectation value to mimic a roughly 1000-shot measurement. With five spins, the resulting $\\langle M_x \\rangle$ values agree with explicit diagonalization for both the real-field Ising chain and the non-Hermitian chain with an imaginary transverse field. The susceptibility $\\chi_x$ computed from exact diagonalization develops a peak that moves with system size and is consistent with the known transition at $\\Gamma = 1$, while for the imaginary-field model the size trend of $\\langle M_x \\rangle$ points toward zero in the thermodynamic limit, suggesting no ground-state quantum phase transition. The paper explicitly leaves the imaginary-field conclusion tentative, noting that larger systems are needed.","pith_inferences":["This suggests a sharper test: rerun the same five-qubit pipeline with actual shot-noise sampling instead of uniform perturbations; if the cost-function noise is not approximately uniform, the reported agreement will not transfer to hardware.","The absence of a transition is established only for the $\\langle M_x \\rangle$ order parameter; a transition might still appear in the complex spectrum itself, such as a gap closing or exceptional-point structure, which the paper does not scan.","The same $M_+$ construction applies to $\\mathcal{PT}$-symmetric chains with staggered complex fields, so the method could be extended to the class of models studied in earlier numerical diagonalization work.","A definitive thermodynamic-limit answer for the imaginary-field model would come from pushing exact diagonalization or the statevector simulator to roughly $N = 20$ and extrapolating the odd-spin $\\langle M_x \\rangle$ curve; the paper's current five-spin quantum data alone cannot separate a true zero from a slow decay."],"forward_implications":["If the five-qubit agreement carries over to larger circuits, the non-Hermitian VQE gives a concrete quantum route to the ground-state properties of sign-problem Hamiltonians, not just the two Ising variants tested here.","The susceptibility peak in the real-field model reproduces the known $\\Gamma = 1$ transition from finite-size data, so $\\chi_x$ can be used to extrapolate transition locations in larger non-Hermitian systems.","The even-odd spin alternation in $\\langle M_x \\rangle$ for the imaginary-field model is a finite-size artifact that appears to vanish in the thermodynamic limit, supporting a zero-order-parameter ground state.","Adapting the optimization strategy (adding a parameter stage, using a stochastic optimizer) preserves accuracy under the paper's noise model, which is a direct requirement for any NISQ implementation."],"supporting_citations":[{"why":"Supplies the non-Hermitian variational algorithm, the $M_\\pm$ cost functions, and the optimization strategy that the paper adapts.","marker":"[25]"},{"why":"Provides the standard VQE framework that the non-Hermitian algorithm extends and that motivates the variational approach.","marker":"[24]"},{"why":"Defines quantum phase transitions and gives the known infinite-volume transition at $\\Gamma = 1$ used as the benchmark.","marker":"[22]"},{"why":"Provides a prior numerical-diagonalization study of a non-Hermitian transverse-field Ising model, giving context for this system class.","marker":"[23]"},{"why":"Establishes the sign problem that motivates using quantum computers for non-Hermitian systems.","marker":"[18]"},{"why":"Supplies the decomposition of a general matrix into Pauli strings, which lets the algorithm represent $H$ and $H^\\dagger$ on qubits.","marker":"[21]"},{"why":"Provides the classical exact-diagonalization comparison that the quantum algorithm's results are checked against.","marker":"[31]"}],"fun_headline_variants":["Non-Hermitian VQE matches exact Ising magnetization at 5 qubits","Imaginary-field Ising shows no transition in VQE runs","Non-Hermitian VQE reproduces Ising criticality at Gamma=1","Five-spin VQE probes non-Hermitian Ising transition","Variational quantum eigensolver tracks non-Hermitian phase change"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that adding a uniform random error in $(-0.04, 0.04)$ to every expectation value adequately represents the noise of a real roughly-1000-shot quantum measurement, even though the paper concedes this does not capture the complexity of true noise.","fun_headline_variants_meta":{"raw":{"variants":["Non-Hermitian VQE matches exact Ising magnetization at 5 qubits","Imaginary-field Ising shows no transition in VQE runs","Non-Hermitian VQE reproduces Ising criticality at Gamma=1","Five-spin VQE probes non-Hermitian Ising transition","Variational quantum eigensolver tracks non-Hermitian phase change"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00043,"raw_usage":{"total_tokens":2154,"prompt_tokens":861,"completion_tokens":1293,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":477,"completion_tokens_details":{"reasoning_tokens":1196}},"tokens_in":477,"tokens_out":1293,"duration_ms":10398,"temperature":1.0,"reasoning_tokens":1196,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T05:14:25.214685+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same non-Hermitian VQE on actual five-qubit hardware (or a shot-level simulator) for the real-field Ising chain near $\\Gamma = 1$ and compare the resulting $\\langle M_x \\rangle$ and $\\chi_x$ with the paper's exact-diagonalization curves; agreement outside the $\\pm 0.04$ uniform band would show the noise model is not representative. Alternatively, compute the shot-noise variance of $\\langle M_x \\rangle$ under the cost function; if it is inconsistent with a uniform bound, the noiseless-plus-uniform model is falsified.","supporting_citations":[{"cited_title":"Variational quantum algorithms for scanning the complex spectrum of non-hermitian systems,","cited_arxiv_id":null,"evidence_quote":"Supplies the non-Hermitian variational algorithm, the $M_\\pm$ cost functions, and the optimization strategy that the paper adapts."},{"cited_title":"TheVariationalQuantumEigensolver: Areviewofmethodsandbestpractices,","cited_arxiv_id":null,"evidence_quote":"Provides the standard VQE framework that the non-Hermitian algorithm extends and that motivates the variational approach."},{"cited_title":"Sachdev, Quantum Phase Transitions , 2nd ed","cited_arxiv_id":null,"evidence_quote":"Defines quantum phase transitions and gives the known infinite-volume transition at $\\Gamma = 1$ used as the benchmark."},{"cited_title":"Quantum phase transitions in non-hermitian PT-symmetric transverse-field ising spin chains,","cited_arxiv_id":null,"evidence_quote":"Provides a prior numerical-diagonalization study of a non-Hermitian transverse-field Ising model, giving context for this system class."},{"cited_title":"Computational complexity and fundamental limitations to fermionic quantum monte carlo simulations,","cited_arxiv_id":null,"evidence_quote":"Establishes the sign problem that motivates using quantum computers for non-Hermitian systems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the decomposition of a general matrix into Pauli strings, which lets the algorithm represent $H$ and $H^\\dagger$ on qubits."},{"cited_title":"Guennebaud, B","cited_arxiv_id":null,"evidence_quote":"Provides the classical exact-diagonalization comparison that the quantum algorithm's results are checked against."}],"review_version":1}