B-VQE: Biorthogonal VQE for Non-Hermitian Systems

• B-VQE is a quantum algorithm that extends traditional VQE by independently optimizing biorthogonal left/right eigenstates for non-Hermitian many-body systems.
• It incorporates hardware-native modules such as an Exceptional-Point Detector and Non-Hermitian Quantum Geometric Tensor to identify critical phenomena and topological transitions.
• The method employs importance sampling with hardware-efficient ansätze to achieve scalable, post-selection-free simulations with energy errors below 5×10⁻³ in benchmark models.

The Biorthogonal Variational Quantum Eigensolver (B-VQE) is a quantum algorithm for simulating non-Hermitian many-body systems on noisy intermediate-scale quantum (NISQ) hardware. It is introduced as a dual-circuit variational framework that directly optimizes biorthogonal left/right eigenstates and augments VQE with two hardware-native modules: an Exceptional-Point Detector (EPD) and a Non-Hermitian Quantum Geometric Tensor (NH-QGT) (B et al., 17 Jun 2026). The method is designed for regimes in which non-Hermitian quantum matter exhibits exceptional points, parity-time symmetry breaking, and non-Hermitian skin effects, while existing quantum algorithms often rely on costly post-selection procedures and are not designed to capture biorthogonal eigenstates (B et al., 17 Jun 2026). A post-selection-free importance-sampling strategy restores polynomial overhead, and the resulting framework is presented as a scalable NISQ methodology for constructing non-Hermitian many-body phase diagrams and exploring topological and critical phenomena in open quantum systems (B et al., 17 Jun 2026).

1. Conceptual setting and motivation

B-VQE is formulated for non-Hermitian many-body systems, where $H \neq H^\dagger$ and the physically relevant eigensystem is intrinsically biorthogonal rather than orthonormal (B et al., 17 Jun 2026). The motivating phenomena include exceptional points (EPs) where eigenvalues and eigenvectors coalesce, parity-time (PT) symmetry phase transitions with real-to-complex spectra, and the non-Hermitian skin effect (NHSE) that accumulates extensive eigenstates at boundaries, breaking conventional bulk-boundary correspondence (B et al., 17 Jun 2026). The detailed motivation further includes interacting regimes with NH-MBL, EP-enhanced scars, and Fermi skins, for which biorthogonal tools are required (B et al., 17 Jun 2026).

The central algorithmic claim is that existing quantum algorithms are ill-suited to this regime. Post-selection/dilation approaches incur exponential overhead in ancillas for non-unitary dynamics and are impractical beyond small sizes; Hermitian embeddings obscure biorthogonality and EP physics; and single-circuit VQE lacks a meaningful cost when energies are complex and left/right eigenvectors differ (B et al., 17 Jun 2026). B-VQE is introduced to close this gap by preparing left and right variational states independently and by optimizing a left-right Rayleigh quotient rather than a Hermitian expectation value (B et al., 17 Jun 2026).

A related conceptual link appears in the non-orthogonal optimization literature. “Variational Quantum Non-Orthogonal Optimization” introduces a generalized variational principle in which non-orthogonal basis states produce an overlap matrix $S$, leading to a generalized Rayleigh quotient and generalized eigenvalue problem (Bermejo et al., 2022). In that setting, B-VQE is described as a generalization of VQE where two different ansätze are used—a left state and a right state—and one minimizes a left-right Rayleigh quotient (Bermejo et al., 2022). That paper also states that VQNO can be naturally reframed as a biorthogonal variational method, and that moving to biorthogonal pairs replaces the need to explicitly invert $S$ by working in a pair of dual frames where $\langle \phi_i^L | \phi_j^R \rangle = \delta_{ij}$ (Bermejo et al., 2022). This suggests a broader variational context in which B-VQE is not limited to one application class, although the non-Hermitian many-body formulation in (B et al., 17 Jun 2026) is the explicit NISQ framework developed for EPs, NHSE, topology, and entanglement criticality.

2. Biorthogonal formalism

The formal setup begins with the right and left eigenproblems

$$H \ket{R_n} = E_n \ket{R_n}, \qquad H^\dagger \ket{L_n} = E_n^* \ket{L_n},$$

together with the biorthogonality and completeness relations

$$\langle L_m | R_n \rangle = \delta_{mn}, \qquad \sum_n \ket{R_n}\bra{L_n} = \mathbf{1}$$

(B et al., 17 Jun 2026). At an EP of order two (EP2), eigenvalues and eigenvectors coalesce, and biorthogonality degenerates, $\langle L_n | R_n \rangle \to 0$ (B et al., 17 Jun 2026).

B-VQE prepares biorthogonal approximations using two independent parameterized circuits,

$$\ket{\psi_R(\bm{\theta})} = U_R(\bm{\theta}) \ket{0}^{\otimes n}, \qquad \ket{\psi_L(\bm{\phi})} = U_L(\bm{\phi}) \ket{0}^{\otimes n}$$

(B et al., 17 Jun 2026). The resulting optimization target is the biorthogonal Rayleigh quotient

$$E_{\mathrm{bio}}(\bm{\theta},\bm{\phi}) = \frac{\langle \psi_L(\bm{\phi}) | H | \psi_R(\bm{\theta}) \rangle} {\langle \psi_L(\bm{\phi}) | \psi_R(\bm{\theta}) \rangle}$$

subject to the cost

$$L(\bm{\theta},\bm{\phi}) = \mathrm{Re}\!\left[E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right] + \lambda \left[\mathrm{Im}\!\left(E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right)\right]^2,$$

where $S$0 penalizes imaginary-energy deviations (B et al., 17 Jun 2026). In PT-unbroken phases, the penalty vanishes at convergence and the minimum of $S$1 yields the ground-state energy (B et al., 17 Jun 2026).

The variational architecture uses hardware-efficient layers $S$2 and CNOT ladders (B et al., 17 Jun 2026). With $S$3 entangling layers on $S$4 qubits per circuit, the total parameter count is $S$5 (B et al., 17 Jun 2026). In the reported tests, $S$6 with $S$7 balances expressivity and depth (B et al., 17 Jun 2026). Derivative-free COBYLA is reported to converge reliably, while a gradient-based variant is supported via a biorthogonal parameter-shift rule (B et al., 17 Jun 2026). A convergence theorem and a variational lower-bound proposition are stated for PT-unbroken spectra with well-conditioned Gram matrices; near EPs, penalties and importance sampling stabilize the landscape (B et al., 17 Jun 2026).

The generalized left-right variational structure also matches the biorthogonal formulation in (Bermejo et al., 2022), which gives

$S$8

and corresponding left-right gradients (Bermejo et al., 2022). That source emphasizes conditioning: if left/right code bases are constructed as dual frames, $S$9, which obviates explicit $S$0 inversion and improves conditioning (Bermejo et al., 2022). In the non-Hermitian setting of (B et al., 17 Jun 2026), the same left-right structure is elevated from a numerical convenience to the physical representation of non-Hermitian eigenstates.

3. Measurement, optimization, and importance sampling

The Hamiltonian is decomposed into Pauli strings with complex coefficients,

$S$1

so that

$S$2

(B et al., 17 Jun 2026). Each biorthogonal matrix element is reduced to a single-register amplitude,

$S$3

(B et al., 17 Jun 2026). Its real and imaginary parts are measured via ancilla-based Hadamard tests or interference circuits controlling $S$4, avoiding any ancilla post-selection; the normalization overlap $S$5 is measured similarly (B et al., 17 Jun 2026). Grouping commuting $S$6 reduces sampling overhead (B et al., 17 Jun 2026).

For gradient-based updates, the parameter-shift rule is extended to biorthogonal overlaps. For gates $S$7 with generator $S$8,

$S$9

where $\langle \phi_i^L | \phi_j^R \rangle = \delta_{ij}$0 (B et al., 17 Jun 2026). The same rule applies to derivatives of $\langle \phi_i^L | \phi_j^R \rangle = \delta_{ij}$1, enabling gradient-based optimization and NH-QGT readout with $\langle \phi_i^L | \phi_j^R \rangle = \delta_{ij}$2 circuit evaluations (B et al., 17 Jun 2026).

A central practical component is the importance-sampling mitigation strategy. Let $\langle \phi_i^L | \phi_j^R \rangle = \delta_{ij}$3 be an $\langle \phi_i^L | \phi_j^R \rangle = \delta_{ij}$4-bit sample and define

$\langle \phi_i^L | \phi_j^R \rangle = \delta_{ij}$5

with a small regularizer $\langle \phi_i^L | \phi_j^R \rangle = \delta_{ij}$6 such as $\langle \phi_i^L | \phi_j^R \rangle = \delta_{ij}$7 (B et al., 17 Jun 2026). Then the IS estimator of any observable $\langle \phi_i^L | \phi_j^R \rangle = \delta_{ij}$8 is

$\langle \phi_i^L | \phi_j^R \rangle = \delta_{ij}$9

(B et al., 17 Jun 2026). This removes ancilla post-selection while retaining polynomial scaling (B et al., 17 Jun 2026). The estimator is unbiased when left/right output distributions overlap sufficiently, which is described as an operational PT-symmetry indicator otherwise (B et al., 17 Jun 2026). Its variance scales as

$$H \ket{R_n} = E_n \ket{R_n}, \qquad H^\dagger \ket{L_n} = E_n^* \ket{L_n},$$0

where $$H \ket{R_n} = E_n \ket{R_n}, \qquad H^\dagger \ket{L_n} = E_n^* \ket{L_n},$$1 quantifies non-Hermiticity, and the $$H \ket{R_n} = E_n \ket{R_n}, \qquad H^\dagger \ket{L_n} = E_n^* \ket{L_n},$$2 bias is negligible for energy-error targets of $$H \ket{R_n} = E_n \ket{R_n}, \qquad H^\dagger \ket{L_n} = E_n^* \ket{L_n},$$3 (B et al., 17 Jun 2026).

The end-to-end workflow is stated explicitly. One initializes left/right ansatz parameters randomly; for each control parameter $$H \ket{R_n} = E_n \ket{R_n}, \qquad H^\dagger \ket{L_n} = E_n^* \ket{L_n},$$4 in the scan range $$H \ket{R_n} = E_n \ket{R_n}, \qquad H^\dagger \ket{L_n} = E_n^* \ket{L_n},$$5, one iterates until convergence by preparing $$H \ket{R_n} = E_n \ket{R_n}, \qquad H^\dagger \ket{L_n} = E_n^* \ket{L_n},$$6 and $$H \ket{R_n} = E_n \ket{R_n}, \qquad H^\dagger \ket{L_n} = E_n^* \ket{L_n},$$7, estimating overlaps and energy via direct interferometry or IS reweighting, evaluating the biorthogonal cost, and updating parameters by COBYLA or gradient-based steps using parameter-shift (B et al., 17 Jun 2026). After convergence, one computes the EPD coalescence metric and the NH-QGT, and then identifies EPs and classifies phases using spectral statistics, entanglement scaling, and topological indicators (B et al., 17 Jun 2026).

4. Exceptional-point detection and non-Hermitian geometry

B-VQE incorporates an Exceptional-Point Detector (EPD) defined through the hardware-native coalescence metric

$$H \ket{R_n} = E_n \ket{R_n}, \qquad H^\dagger \ket{L_n} = E_n^* \ket{L_n},$$8

(B et al., 17 Jun 2026). Here $$H \ket{R_n} = E_n \ket{R_n}, \qquad H^\dagger \ket{L_n} = E_n^* \ket{L_n},$$9 are converged parameters at control coupling $$\langle L_m | R_n \rangle = \delta_{mn}, \qquad \sum_n \ket{R_n}\bra{L_n} = \mathbf{1}$$0 (B et al., 17 Jun 2026). EPs are signaled when $$\langle L_m | R_n \rangle = \delta_{mn}, \qquad \sum_n \ket{R_n}\bra{L_n} = \mathbf{1}$$1 and, in practice, the minimum of $$\langle L_m | R_n \rangle = \delta_{mn}, \qquad \sum_n \ket{R_n}\bra{L_n} = \mathbf{1}$$2 across a parameter scan robustly locates EPs (B et al., 17 Jun 2026). The metric is measured via simple overlap circuits, requires no post-selection, and is described as tolerant to realistic noise (B et al., 17 Jun 2026).

The second hardware-native module is the Non-Hermitian Quantum Geometric Tensor,

$$\langle L_m | R_n \rangle = \delta_{mn}, \qquad \sum_n \ket{R_n}\bra{L_n} = \mathbf{1}$$3

where $$\langle L_m | R_n \rangle = \delta_{mn}, \qquad \sum_n \ket{R_n}\bra{L_n} = \mathbf{1}$$4 is a derivative with respect to a control parameter and $$\langle L_m | R_n \rangle = \delta_{mn}, \qquad \sum_n \ket{R_n}\bra{L_n} = \mathbf{1}$$5 projects away the ground state, maintaining gauge invariance even at EPs (B et al., 17 Jun 2026). Its real and imaginary parts define the biorthogonal quantum metric and Berry curvature,

$$\langle L_m | R_n \rangle = \delta_{mn}, \qquad \sum_n \ket{R_n}\bra{L_n} = \mathbf{1}$$6

(B et al., 17 Jun 2026).

A key diagnostic distinction is the separation of state topology from band topology. The biorthogonal state Chern number is

$$\langle L_m | R_n \rangle = \delta_{mn}, \qquad \sum_n \ket{R_n}\bra{L_n} = \mathbf{1}$$7

over the occupied-state manifold $$\langle L_m | R_n \rangle = \delta_{mn}, \qquad \sum_n \ket{R_n}\bra{L_n} = \mathbf{1}$$8 (B et al., 17 Jun 2026). This quantity can differ from the conventional band Chern number extracted from Bloch bands, and the distinction is explicitly stated to resolve the known state-topology/band-topology mismatch in non-Hermitian many-body systems (B et al., 17 Jun 2026). Near EPs, the metric diverges as

$$\langle L_m | R_n \rangle = \delta_{mn}, \qquad \sum_n \ket{R_n}\bra{L_n} = \mathbf{1}$$9

providing a quantitative EP locator complementary to $\langle L_n | R_n \rangle \to 0$0 (B et al., 17 Jun 2026).

The measurement protocol for NH-QGT uses the biorthogonal parameter-shift rule; each tensor element reduces to products of overlaps and their derivatives, all accessible by dual-circuit interferometry, and no density-matrix tomography is required (B et al., 17 Jun 2026). This operational role of NH-QGT is central in the article’s framing of B-VQE: it is not only an eigensolver but also a geometric and topological readout framework for non-Hermitian many-body phase diagrams (B et al., 17 Jun 2026).

5. Benchmarks, phase-diagram reconstruction, and entanglement criticality

The method is validated on three representative models: a non-Hermitian Hubbard chain, a non-Hermitian XXZ spin chain, and a two-dimensional non-Hermitian $\langle L_n | R_n \rangle \to 0$1-$\langle L_n | R_n \rangle \to 0$2 model (B et al., 17 Jun 2026). Across all models, B-VQE achieves relative energy errors below $\langle L_n | R_n \rangle \to 0$3 in the abstract, and the detailed results state $\langle L_n | R_n \rangle \to 0$4 with hardware-efficient depths (B et al., 17 Jun 2026). Exceptional points are located to within $\langle L_n | R_n \rangle \to 0$5 in noiseless simulators and remain robust under calibrated noise (B et al., 17 Jun 2026).

In the non-Hermitian Hubbard chain benchmark, four phases are resolved: ergodic, NH-MBL, PT-broken ergodic, and skin-localized (B et al., 17 Jun 2026). The phase characterizations are given explicitly: ergodic corresponds to GUE $\langle L_n | R_n \rangle \to 0$6 and volume-law entanglement; NH-MBL to Poisson $\langle L_n | R_n \rangle \to 0$7 and area-law entanglement; PT-broken ergodic to a complex spectrum; and skin-localized to NHSE-dominated behavior with suppressed entanglement (B et al., 17 Jun 2026). B-VQE energies agree with exact diagonalization within $\langle L_n | R_n \rangle \to 0$8 across sampled points, and phase boundaries are localized by EPD/NH-QGT and level-spacing/entanglement observables (B et al., 17 Jun 2026).

In the non-Hermitian XXZ chain with boundary gain/loss, EP-enhanced many-body scars are observed, Loschmidt echoes exhibit strong revival amplification near EPs, and the coalescence metric $\langle L_n | R_n \rangle \to 0$9 cleanly identifies EPs (B et al., 17 Jun 2026). Under hardware-noise simulations based on IBM Heron r2 models, the EP is located within $$\ket{\psi_R(\bm{\theta})} = U_R(\bm{\theta}) \ket{0}^{\otimes n}, \qquad \ket{\psi_L(\bm{\phi})} = U_L(\bm{\phi}) \ket{0}^{\otimes n}$$0 (B et al., 17 Jun 2026).

In the two-dimensional non-Hermitian $$\ket{\psi_R(\bm{\theta})} = U_R(\bm{\theta}) \ket{0}^{\otimes n}, \qquad \ket{\psi_L(\bm{\phi})} = U_L(\bm{\phi}) \ket{0}^{\otimes n}$$1-$$\ket{\psi_R(\bm{\theta})} = U_R(\bm{\theta}) \ket{0}^{\otimes n}, \qquad \ket{\psi_L(\bm{\phi})} = U_L(\bm{\phi}) \ket{0}^{\otimes n}$$2 model, a Fermi skin is mapped and NH-QGT reveals state-topology with $$\ket{\psi_R(\bm{\theta})} = U_R(\bm{\theta}) \ket{0}^{\otimes n}, \qquad \ket{\psi_L(\bm{\phi})} = U_L(\bm{\phi}) \ket{0}^{\otimes n}$$3 even when the band Chern number is trivial, demonstrating the state/band topology discrepancy (B et al., 17 Jun 2026). Density accumulation at edges quantifies NHSE localization (B et al., 17 Jun 2026).

The work also reports biorthogonal entanglement entropy,

$$\ket{\psi_R(\bm{\theta})} = U_R(\bm{\theta}) \ket{0}^{\otimes n}, \qquad \ket{\psi_L(\bm{\phi})} = U_L(\bm{\phi}) \ket{0}^{\otimes n}$$4

with singular values $$\ket{\psi_R(\bm{\theta})} = U_R(\bm{\theta}) \ket{0}^{\otimes n}, \qquad \ket{\psi_L(\bm{\phi})} = U_L(\bm{\phi}) \ket{0}^{\otimes n}$$5 from the Schmidt decomposition weighted by the biorthogonal metric (B et al., 17 Jun 2026). Four regimes are identified: volume law for ergodic phases, logarithmic area law for NH-MBL, EP-critical CFT-like scaling,

$$\ket{\psi_R(\bm{\theta})} = U_R(\bm{\theta}) \ket{0}^{\otimes n}, \qquad \ket{\psi_L(\bm{\phi})} = U_L(\bm{\phi}) \ket{0}^{\otimes n}$$6

with $$\ket{\psi_R(\bm{\theta})} = U_R(\bm{\theta}) \ket{0}^{\otimes n}, \qquad \ket{\psi_L(\bm{\phi})} = U_L(\bm{\phi}) \ket{0}^{\otimes n}$$7, and exponential suppression in PT-broken phases (B et al., 17 Jun 2026). These scalings are stated to be consistent with non-unitary criticality at EPs (B et al., 17 Jun 2026).

6. NISQ resource profile, comparisons, and limitations

The reported circuit family uses hardware-efficient ansätze with depths $$\ket{\psi_R(\bm{\theta})} = U_R(\bm{\theta}) \ket{0}^{\otimes n}, \qquad \ket{\psi_L(\bm{\phi})} = U_L(\bm{\phi}) \ket{0}^{\otimes n}$$8, which achieved target accuracies at moderate gate counts (B et al., 17 Jun 2026). In small-scale tests with $$\ket{\psi_R(\bm{\theta})} = U_R(\bm{\theta}) \ket{0}^{\otimes n}, \qquad \ket{\psi_L(\bm{\phi})} = U_L(\bm{\phi}) \ket{0}^{\otimes n}$$9–$$E_{\mathrm{bio}}(\bm{\theta},\bm{\phi}) = \frac{\langle \psi_L(\bm{\phi}) | H | \psi_R(\bm{\theta}) \rangle} {\langle \psi_L(\bm{\phi}) | \psi_R(\bm{\theta}) \rangle}$$0, Qiskit-reported depths are $$E_{\mathrm{bio}}(\bm{\theta},\bm{\phi}) = \frac{\langle \psi_L(\bm{\phi}) | H | \psi_R(\bm{\theta}) \rangle} {\langle \psi_L(\bm{\phi}) | \psi_R(\bm{\theta}) \rangle}$$1–$$E_{\mathrm{bio}}(\bm{\theta},\bm{\phi}) = \frac{\langle \psi_L(\bm{\phi}) | H | \psi_R(\bm{\theta}) \rangle} {\langle \psi_L(\bm{\phi}) | \psi_R(\bm{\theta}) \rangle}$$2 with $$E_{\mathrm{bio}}(\bm{\theta},\bm{\phi}) = \frac{\langle \psi_L(\bm{\phi}) | H | \psi_R(\bm{\theta}) \rangle} {\langle \psi_L(\bm{\phi}) | \psi_R(\bm{\theta}) \rangle}$$3–$$E_{\mathrm{bio}}(\bm{\theta},\bm{\phi}) = \frac{\langle \psi_L(\bm{\phi}) | H | \psi_R(\bm{\theta}) \rangle} {\langle \psi_L(\bm{\phi}) | \psi_R(\bm{\theta}) \rangle}$$4 CNOTs; for $$E_{\mathrm{bio}}(\bm{\theta},\bm{\phi}) = \frac{\langle \psi_L(\bm{\phi}) | H | \psi_R(\bm{\theta}) \rangle} {\langle \psi_L(\bm{\phi}) | \psi_R(\bm{\theta}) \rangle}$$5 up to $$E_{\mathrm{bio}}(\bm{\theta},\bm{\phi}) = \frac{\langle \psi_L(\bm{\phi}) | H | \psi_R(\bm{\theta}) \rangle} {\langle \psi_L(\bm{\phi}) | \psi_R(\bm{\theta}) \rangle}$$6, depths remain compatible with coherence windows (B et al., 17 Jun 2026). For measurements, IS mitigation with $$E_{\mathrm{bio}}(\bm{\theta},\bm{\phi}) = \frac{\langle \psi_L(\bm{\phi}) | H | \psi_R(\bm{\theta}) \rangle} {\langle \psi_L(\bm{\phi}) | \psi_R(\bm{\theta}) \rangle}$$7 samples yields robust energy and EPD estimates, while NH-QGT is read out with $$E_{\mathrm{bio}}(\bm{\theta},\bm{\phi}) = \frac{\langle \psi_L(\bm{\phi}) | H | \psi_R(\bm{\theta}) \rangle} {\langle \psi_L(\bm{\phi}) | \psi_R(\bm{\theta}) \rangle}$$8 evaluations using parameter-shift (B et al., 17 Jun 2026). Under realistic Heron r2 noise, IS reduces errors by $$E_{\mathrm{bio}}(\bm{\theta},\bm{\phi}) = \frac{\langle \psi_L(\bm{\phi}) | H | \psi_R(\bm{\theta}) \rangle} {\langle \psi_L(\bm{\phi}) | \psi_R(\bm{\theta}) \rangle}$$9–$$L(\bm{\theta},\bm{\phi}) = \mathrm{Re}\!\left[E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right] + \lambda \left[\mathrm{Im}\!\left(E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right)\right]^2,$$0, and B-VQE outperforms Trotterization at comparable fidelity with approximately $$L(\bm{\theta},\bm{\phi}) = \mathrm{Re}\!\left[E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right] + \lambda \left[\mathrm{Im}\!\left(E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right)\right]^2,$$1–$$L(\bm{\theta},\bm{\phi}) = \mathrm{Re}\!\left[E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right] + \lambda \left[\mathrm{Im}\!\left(E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right)\right]^2,$$2 shallower circuits (B et al., 17 Jun 2026). Zero-noise extrapolation and readout mitigation can be layered atop IS, and the overall overhead scales polynomially, with empirical exponent approximately $$L(\bm{\theta},\bm{\phi}) = \mathrm{Re}\!\left[E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right] + \lambda \left[\mathrm{Im}\!\left(E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right)\right]^2,$$3 up to $$L(\bm{\theta},\bm{\phi}) = \mathrm{Re}\!\left[E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right] + \lambda \left[\mathrm{Im}\!\left(E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right)\right]^2,$$4 (B et al., 17 Jun 2026).

The comparison to prior non-Hermitian quantum simulation approaches is explicit. Versus post-selection VQE or Trotterization, B-VQE avoids exponential ancilla overhead, directly targets biorthogonal eigenpairs, and measures complex expectations without post-selection (B et al., 17 Jun 2026). EPD and NH-QGT provide principled EP and topology detection absent in dilation approaches (B et al., 17 Jun 2026). Versus Hermitian special cases or PT-symmetric algorithms, B-VQE handles generic non-Hermitian many-body systems, including PT-broken phases and NHSE, and separates state topology from band topology via NH-QGT (B et al., 17 Jun 2026).

The stated limitations follow standard variational and non-Hermitian constraints. Like all VQAs, B-VQE may encounter barren plateaus for deep circuits and large $$L(\bm{\theta},\bm{\phi}) = \mathrm{Re}\!\left[E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right] + \lambda \left[\mathrm{Im}\!\left(E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right)\right]^2,$$5; shallow hardware-efficient ansätze and local costs mitigate this up to $$L(\bm{\theta},\bm{\phi}) = \mathrm{Re}\!\left[E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right] + \lambda \left[\mathrm{Im}\!\left(E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right)\right]^2,$$6 (B et al., 17 Jun 2026). Warm starts and problem-inspired ansätze are identified as aids for scaling beyond $$L(\bm{\theta},\bm{\phi}) = \mathrm{Re}\!\left[E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right] + \lambda \left[\mathrm{Im}\!\left(E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right)\right]^2,$$7 (B et al., 17 Jun 2026). In PT-broken regimes, when left/right output distributions separate, IS variance grows and penalties must be tuned (B et al., 17 Jun 2026). Time-dependent or non-Markovian open systems, Floquet non-Hermitian phases, multi-parameter phase diagrams, and improved measurement or ansatz designs such as symmetry-preserving layers and commuting-group measurement are listed as open directions (B et al., 17 Jun 2026).

A broader methodological context is provided by non-orthogonal variational optimization (Bermejo et al., 2022). There, B-VQE is described as a clean Rayleigh-quotient formulation in a biorthogonal representation, improving conditioning and providing hardware-friendly cross-measurements (Bermejo et al., 2022). That source also emphasizes well-posedness conditions such as requiring $$L(\bm{\theta},\bm{\phi}) = \mathrm{Re}\!\left[E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right] + \lambda \left[\mathrm{Im}\!\left(E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right)\right]^2,$$8 and bounded away from $$L(\bm{\theta},\bm{\phi}) = \mathrm{Re}\!\left[E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right] + \lambda \left[\mathrm{Im}\!\left(E_{\mathrm{bio}}(\bm{\theta},\bm{\phi})\right)\right]^2,$$9 to avoid numerical instabilities (Bermejo et al., 2022). A plausible implication is that this conditioning perspective complements the physical motivation in non-Hermitian many-body simulation: in B-VQE, the left-right structure is simultaneously a representation of non-Hermitian spectral theory and a strategy for stabilizing variational optimization.