---
title: 'B-VQE: Biorthogonal VQE for Non-Hermitian Systems'
url: https://www.emergentmind.com/topics/biorthogonal-variational-quantum-eigensolver-b-vqe
type: topic
---

# B-VQE: Biorthogonal VQE for Non-Hermitian Systems

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) [2606.18916]. 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 [2606.18916]. 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 [2606.18916].

## 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 [2606.18916]. 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 [2606.18916]. The detailed motivation further includes interacting regimes with NH-MBL, EP-enhanced scars, and Fermi skins, for which biorthogonal tools are required [2606.18916].

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 [2606.18916]. 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 [2606.18916].

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 [2210.04639]. 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 [2210.04639]. 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 $\langle \phi_i^L | \phi_j^R \rangle = \delta_{ij}$ [2210.04639]. 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 [2606.18916] 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
$$
\langle L_m | R_n \rangle = \delta_{mn}, \qquad \sum_n \ket{R_n}\bra{L_n} = \mathbf{1}
$$
[2606.18916]. At an EP of order two (EP2), eigenvalues and eigenvectors coalesce, and biorthogonality degenerates, $\langle L_n | R_n \rangle \to 0$ [2606.18916].

B-VQE prepares biorthogonal approximations using two independent parameterized circuits,
$$
\ket{\psi_R(\bm{\theta})} = U_R(\bm{\theta}) \ket{0}^{\otimes n}, \qquad
\ket{\psi_L(\bm{\phi})} = U_L(\bm{\phi}) \ket{0}^{\otimes n}
$$
[2606.18916]. 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 $\lambda>0$ penalizes imaginary-energy deviations [2606.18916]. In PT-unbroken phases, the penalty vanishes at convergence and the minimum of $\mathrm{Re}[E_{\mathrm{bio}}]$ yields the ground-state energy [2606.18916].

The variational architecture uses hardware-efficient layers $(R_y,R_z)$ and CNOT ladders [2606.18916]. With $p$ entangling layers on $n$ qubits per circuit, the total parameter count is $4n(p+1)$ [2606.18916]. In the reported tests, $p=3$ with $n\in\{8,10,12\}$ balances expressivity and depth [2606.18916]. Derivative-free COBYLA is reported to converge reliably, while a gradient-based variant is supported via a biorthogonal parameter-shift rule [2606.18916]. 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 [2606.18916].

The generalized left-right variational structure also matches the biorthogonal formulation in [2210.04639], which gives
$$
E(\theta_L,\theta_R)=\frac{\langle \psi_L(\theta_L)|H|\psi_R(\theta_R)\rangle}{\langle \psi_L(\theta_L)|\psi_R(\theta_R)\rangle}
$$
and corresponding left-right gradients [2210.04639]. That source emphasizes conditioning: if left/right code bases are constructed as dual frames, $\langle \phi_i^L|\phi_j^R\rangle=\delta_{ij}$, which obviates explicit $S$ inversion and improves conditioning [2210.04639]. In the non-Hermitian setting of [2606.18916], 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,
$$
H = \sum_{\alpha} c_{\alpha} P_{\alpha}, \qquad c_{\alpha} \in \mathbb{C}, \ P_{\alpha} \in \{I,X,Y,Z\}^{\otimes n},
$$
so that
$$
\langle \psi_L | H | \psi_R \rangle = \sum_{\alpha} c_{\alpha} \langle \psi_L | P_{\alpha} | \psi_R \rangle
$$
[2606.18916]. Each biorthogonal matrix element is reduced to a single-register amplitude,
$$
\langle \psi_L | P_{\alpha} | \psi_R \rangle
= \langle 0 | U_L(\bm{\phi}) \, P_{\alpha} \, U_R(\bm{\theta}) | 0 \rangle
$$
[2606.18916]. Its real and imaginary parts are measured via ancilla-based Hadamard tests or interference circuits controlling $P_{\alpha}$, avoiding any ancilla post-selection; the normalization overlap $\langle \psi_L | \psi_R \rangle$ is measured similarly [2606.18916]. Grouping commuting $P_{\alpha}$ reduces sampling overhead [2606.18916].

For gradient-based updates, the parameter-shift rule is extended to biorthogonal overlaps. For gates $R(\lambda)=e^{-i\lambda G}$ with generator $G$,
$$
\partial_{\lambda}\,\langle \psi_L(\bm{\phi}) \,||\, \psi_R(\bm{\theta}) \rangle
=
\tfrac{1}{2}\Big[ f\!\left(\lambda+\tfrac{\pi}{2}\right) - f\!\left(\lambda-\tfrac{\pi}{2}\right) \Big],
$$
where $f(\lambda)=\langle \psi_L(\bm{\phi}(\lambda)) \,||\, \psi_R(\bm{\theta}(\lambda)) \rangle$ [2606.18916]. The same rule applies to derivatives of $\langle \psi_L | P_{\alpha} | \psi_R \rangle$, enabling gradient-based optimization and NH-QGT readout with $O(|\bm{\theta}|^2)$ circuit evaluations [2606.18916].

A central practical component is the importance-sampling mitigation strategy. Let $s_j$ be an $n$-bit sample and define
$$
w_j = \frac{\left|\langle s_j|\psi_R\rangle\right|^2}{\left|\langle s_j|\psi_L\rangle\right|^2 + \varepsilon},
$$
with a small regularizer $\varepsilon$ such as $10^{-6}$ [2606.18916]. Then the IS estimator of any observable $O$ is
$$
\langle O \rangle_{\mathrm{IS}}
=
\frac{\sum_{j=1}^{M} w_j \, O(s_j)}{\sum_{j=1}^{M} w_j}
$$
[2606.18916]. This removes ancilla post-selection while retaining polynomial scaling [2606.18916]. The estimator is unbiased when left/right output distributions overlap sufficiently, which is described as an operational PT-symmetry indicator otherwise [2606.18916]. Its variance scales as
$$
\mathrm{Var}[\langle O\rangle_{\mathrm{IS}}] = O(M^{-1}\gamma_{\mathrm{NH}}^{2}),
$$
where $\gamma_{\mathrm{NH}}$ quantifies non-Hermiticity, and the $\mathcal{O}(\varepsilon)$ bias is negligible for energy-error targets of $5\times 10^{-3}$ [2606.18916].

The end-to-end workflow is stated explicitly. One initializes left/right ansatz parameters randomly; for each control parameter $\lambda$ in the scan range $\Lambda$, one iterates until convergence by preparing $U_R(\bm{\theta})$ and $U_L(\bm{\phi})$, 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 [2606.18916]. 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 [2606.18916].

## 4. Exceptional-point detection and non-Hermitian geometry

B-VQE incorporates an Exceptional-Point Detector (EPD) defined through the hardware-native coalescence metric
$$
\mathcal{C}(\lambda) = 1 - \left|\langle \psi_L(\bm{\phi}^*) | \psi_R(\bm{\theta}^*) \rangle\right|^2
$$
[2606.18916]. Here $(\bm{\theta}^*,\bm{\phi}^*)$ are converged parameters at control coupling $\lambda$ [2606.18916]. EPs are signaled when $\mathcal{C}(\lambda)\to 0$ and, in practice, the minimum of $\mathcal{C}(\lambda)$ across a parameter scan robustly locates EPs [2606.18916]. The metric is measured via simple overlap circuits, requires no post-selection, and is described as tolerant to realistic noise [2606.18916].

The second hardware-native module is the Non-Hermitian Quantum Geometric Tensor,
$$
Q^{\mu\nu}_{\mathrm{bio}}
=
\frac{\langle L | \partial_{\mu}\!\big(\mathbf{1}-|R\rangle\langle L|\big) \, | \partial_{\nu} R \rangle}{\langle L | R \rangle},
$$
where $\partial_{\mu}=\partial/\partial \lambda_{\mu}$ is a derivative with respect to a control parameter and $(1-|R\rangle\langle L|)$ projects away the ground state, maintaining gauge invariance even at EPs [2606.18916]. Its real and imaginary parts define the biorthogonal quantum metric and Berry curvature,
$$
g^{\mu\nu}_{\mathrm{bio}} = \mathrm{Re}\!\left(Q^{\mu\nu}_{\mathrm{bio}}\right), \qquad
F^{\mu\nu}_{\mathrm{bio}} = -2\,\mathrm{Im}\!\left(Q^{\mu\nu}_{\mathrm{bio}}\right)
$$
[2606.18916].

A key diagnostic distinction is the separation of state topology from band topology. The biorthogonal state Chern number is
$$
\mathcal{C}_{\mathrm{state}}
=
\frac{1}{2\pi} \int_{\mathcal{M}} dk_x\, dk_y \; F^{xy}_{\mathrm{bio}}
$$
over the occupied-state manifold $\mathcal{M}$ [2606.18916]. 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 [2606.18916]. Near EPs, the metric diverges as
$$
g^{11}_{\mathrm{bio}}(\lambda) \sim |\lambda - \lambda_{\mathrm{EP}}|^{-2},
$$
providing a quantitative EP locator complementary to $\mathcal{C}(\lambda)$ [2606.18916].

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 [2606.18916]. 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 [2606.18916].

## 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 $t$-$J$ model [2606.18916]. Across all models, B-VQE achieves relative energy errors below $5\times 10^{-3}$ in the abstract, and the detailed results state $\epsilon_E < 5\times 10^{-3}$ with hardware-efficient depths [2606.18916]. Exceptional points are located to within $\delta\lambda < 0.02\,t$ in noiseless simulators and remain robust under calibrated noise [2606.18916].

In the non-Hermitian Hubbard chain benchmark, four phases are resolved: ergodic, NH-MBL, PT-broken ergodic, and skin-localized [2606.18916]. The phase characterizations are given explicitly: ergodic corresponds to GUE $r\approx 0.603$ and volume-law entanglement; NH-MBL to Poisson $r\approx 0.386$ and area-law entanglement; PT-broken ergodic to a complex spectrum; and skin-localized to NHSE-dominated behavior with suppressed entanglement [2606.18916]. B-VQE energies agree with exact diagonalization within $\epsilon_E < 3\times 10^{-3}$ across sampled points, and phase boundaries are localized by EPD/NH-QGT and level-spacing/entanglement observables [2606.18916].

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 $\mathcal{C}(\lambda)$ cleanly identifies EPs [2606.18916]. Under hardware-noise simulations based on IBM Heron r2 models, the EP is located within $|\delta\gamma_{\mathrm{EP}}| < 0.08\,J$ [2606.18916].

In the two-dimensional non-Hermitian $t$-$J$ model, a Fermi skin is mapped and NH-QGT reveals state-topology with $\mathcal{C}_{\mathrm{state}}=1$ even when the band Chern number is trivial, demonstrating the state/band topology discrepancy [2606.18916]. Density accumulation at edges quantifies NHSE localization [2606.18916].

The work also reports biorthogonal entanglement entropy,
$$
S_{\mathrm{bio}} = -\sum_i \lambda_i \ln \lambda_i,
$$
with singular values $\{\lambda_i\}$ from the Schmidt decomposition weighted by the biorthogonal metric [2606.18916]. Four regimes are identified: volume law for ergodic phases, logarithmic area law for NH-MBL, EP-critical CFT-like scaling,
$$
S_{\mathrm{bio}}(\ell) =
\frac{c_{\mathrm{eff}}}{3}\,\ln\!\left[\frac{N}{\pi}\sin\!\left(\frac{\pi \ell}{N}\right)\right] + \mathrm{const},
$$
with $c_{\mathrm{eff}}\approx 0.85\pm 0.03$, and exponential suppression in PT-broken phases [2606.18916]. These scalings are stated to be consistent with non-unitary criticality at EPs [2606.18916].

## 6. NISQ resource profile, comparisons, and limitations

The reported circuit family uses hardware-efficient ansätze with depths $p\approx 3$, which achieved target accuracies at moderate gate counts [2606.18916]. In small-scale tests with $n=1$–$2$, Qiskit-reported depths are $6$–$11$ with $0$–$3$ CNOTs; for $n$ up to $12$, depths remain compatible with coherence windows [2606.18916]. For measurements, IS mitigation with $M\approx 1.6\times 10^{4}$ samples yields robust energy and EPD estimates, while NH-QGT is read out with $O(|\bm{\theta}|^2)$ evaluations using parameter-shift [2606.18916]. Under realistic Heron r2 noise, IS reduces errors by $30$–$70\%$, and B-VQE outperforms Trotterization at comparable fidelity with approximately $2$–$3\times$ shallower circuits [2606.18916]. Zero-noise extrapolation and readout mitigation can be layered atop IS, and the overall overhead scales polynomially, with empirical exponent approximately $1.55$ up to $n=16$ [2606.18916].

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 [2606.18916]. EPD and NH-QGT provide principled EP and topology detection absent in dilation approaches [2606.18916]. 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 [2606.18916].

The stated limitations follow standard variational and non-Hermitian constraints. Like all VQAs, B-VQE may encounter barren plateaus for deep circuits and large $n$; shallow hardware-efficient ansätze and local costs mitigate this up to $n\approx 16$ [2606.18916]. Warm starts and problem-inspired ansätze are identified as aids for scaling beyond $n\approx 30$ [2606.18916]. In PT-broken regimes, when left/right output distributions separate, IS variance grows and penalties must be tuned [2606.18916]. 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 [2606.18916].

A broader methodological context is provided by non-orthogonal variational optimization [2210.04639]. There, B-VQE is described as a clean Rayleigh-quotient formulation in a biorthogonal representation, improving conditioning and providing hardware-friendly cross-measurements [2210.04639]. That source also emphasizes well-posedness conditions such as requiring $D=\langle \psi_L|\psi_R\rangle>0$ and bounded away from $0$ to avoid numerical instabilities [2210.04639]. 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.

Source: https://www.emergentmind.com/topics/biorthogonal-variational-quantum-eigensolver-b-vqe