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Quantum Phase Transition of Non-Hermitian Systems using Variational Quantum Techniques

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2501.17003 v1 pith:BTM45K7A submitted 2025-01-28 quant-ph hep-lat

James Hancock , Matthew J. Craven , Craig McNeile , Davide Vadacchino This is my paper · ORCID
classification quant-phhep-lat
keywords non-HermitianquantummechanicsvariationaleigensolverphasetransitiontransverseIsingmodelimaginarymagneticfieldPTsymmetrysignproblemNISQalgorithms
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 6 minor

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.

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 (3)
  1. [Sec. 4, Eq. (7)] 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.
  2. [Sec. 5] 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.
  3. [Sec. 6.2] 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.
minor comments (6)
  1. [Abstract and Sec. 1] 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.
  2. [Sec. 4] 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.
  3. [Sec. 5] Typo: 'enviroment' should be 'environment'.
  4. [Sec. 6.1] The phrase 'on the hardware we used' is misleading because the results were obtained from a statevector simulation, not quantum hardware.
  5. [Fig. 2] 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.
  6. [Eq. (5)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the variational results are checked against independent exact diagonalization, and the QPT conclusions rest on exact diagonalization, not on fitted inputs or self-citations.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The non-Hermitian variational algorithm is taken from independent prior work (Ref. [25]); the cost functions C_+ and C_- (Eq. 7) are constructed from the Hamiltonian and a complex parameter E, and the paper checks the resulting eigenpairs against explicit diagonalization with the C++ Eigen library. The N=5 <M_x> curves in Fig. 1 are benchmarked against exact diagonalization, an external standard that does not depend on any parameter fitted in this paper. The central physics claims—the susceptibility peak near Gamma=1 for the real-field model and the vanishing <M_x> for the imaginary-field model—are drawn from exact diagonalization at larger system sizes (Secs. 6.1 and 6.2), not from the variational algorithm's output. There are no self-citations by the present authors, no imported uniqueness theorems, and no renamed known result. The paper explicitly acknowledges its ad hoc noise model ('does not capture the complexity of true noise') and the five-qubit resource limit, and it leaves unspecified how the 'ground state' is selected for the non-Hermitian Hamiltonian; but those are correctness and scope limitations, not circular reductions. No equation in the paper is defined in terms of the quantity it is used to predict, and no fitted parameter is relabeled as a prediction. Therefore no circular step is exhibited.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the variational optimization of E and theta, the ad hoc noise model, and a finite-size extrapolation. No new physical entities are introduced. The physics parameters (Gamma, Gamma_I) are inputs, not fitted. The main uncharged assumptions are ansatz expressibility, noise representativeness, and the extrapolation from small odd chains.

free parameters (3)
  • Complex eigenvalue search parameter E = Optimized value (not reported per data point)
    Introduced in Eq. (6); C+ is minimized over E and theta. The algorithm's output depends on this variational fit, and no uncertainty is reported.
  • Ansatz circuit parameters theta = Optimized per field value (values not reported)
    Parameters of the rotational gates and CNOTs in G(theta); the ansatz is stated to be fully or over-expressive but not specified in detail (Sec. 4).
  • Noise amplitude epsilon = 0.04
    Chosen by hand to mimic ~1000 shots; the paper states it does not capture true noise (Sec. 5).
assumptions (4)
  • domain assumption The variational ansatz G(theta) is expressive enough to represent the ground-state eigenvector, and the optimizer reaches the global minimum of C+.
    Sec. 4 asserts a fully or over-expressive ansatz is used, but no circuit depth, parameter count, or convergence guarantees are given. If the optimizer settles in a local minimum, the extracted <M_x> and chi_x are not ground-state properties.
  • ad hoc to paper The uniform noise perturbation (epsilon=0.04) on each expectation value approximates a ~1000-shot quantum measurement.
    Sec. 5 introduces this model; it is not derived from device calibration or shot-noise statistics, and the paper concedes it does not capture the complexity of true noise.
  • standard math Exact diagonalization with the C++ Eigen library provides correct reference eigenpairs.
    Used throughout Secs. 5 and 6 as the benchmark; this is a standard numerical method and is independent of the quantum algorithm.
  • domain assumption The finite-size trend of <M_x> for odd spin chains indicates the thermodynamic limit value.
    Sec. 6.2 concludes that <M_x> 'does look as though it is tending to zero' as N grows, suggesting no QPT. This is an extrapolation without a scaling analysis, and the paper itself calls for larger systems.

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Pith. "Pith review of Quantum Phase Transition of Non-Hermitian Systems using Variational Quantum Techniques." pith.science (2026). https://pith.science/paper/BTM45K7A

@misc{pith2026250117003,
  author       = {Pith},
  title        = {Pith review of: Quantum Phase Transition of Non-Hermitian Systems using Variational Quantum Techniques},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BTM45K7A}},
  note         = {Machine review of arXiv:2501.17003}
}
abstract

The motivation for studying non-Hermitian systems and the role of $\mathcal{PT}$-symmetry is discussed. We investigate the use of a quantum algorithm to find the eigenvalues and eigenvectors of non-Hermitian Hamiltonians, with applications to quantum phase transitions. We use a recently proposed variational algorithm. The systems studied are the transverse Ising model with both a purely real and a purely complex transverse field.

Figures

Figures reproduced from arXiv: 2501.17003 by the authors.

Figure 1
Figure 1. ⟨𝑀ˆ 𝑥⟩ over a change in magnetic field for the transverse Ising model and non-Hermitian transverse Ising model with 5 spins. variable uniformly distributed in the range (−𝜖, 𝜖), 𝜖 = 0.04. By running experiments on the same system using qiskit, this is comparable to using ∼ 1000 shots for five qubits but does not capture the complexity of true noise, but means we are still working in a noisy enviroment. We also used … view at source ↗
Figure 2
Figure 2. Magnetic susceptibility in the 𝜎𝑥-direction, 𝜒𝑥 (Eq. 5), for the transverse Ising model (Eq. 2), for a varying number of qubits 6.1 Results for the transverse Ising Model in real magnetic field We can use 𝜒𝑥 to get a handle on the finite volume effects by studying how the peak value moves for different system sizes. Our implementation of the quantum algorithm on the hardware we used was restricted to the maximum of … view at source ↗
Figure 3
Figure 3. Magnetic charge per spin in the 𝜎𝑥-direction, ⟨𝑀ˆ 𝑥⟩ (Eq. 4), for the non-Hermitian transverse Ising model (Eq. 3), for a varying number of qubits. 7. Conclusions We have investigated methods to study QPTs in non-Hermitian systems by using a modified algorithm for quantum computers to compute the eigenpairs proposed in Ref. [25]. We see promising indications of the ability to quantify and study the behavior of possi… view at source ↗

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