Exceptional-Point-Anchored Variational Quantum Eigensolver for Non-Hermitian Many-Body Phase Diagrams: Bridging Skin-Effect Topology and Entanglement Criticality on NISQ Hardware

The paper introduces the Biorthogonal Variational Quantum Eigensolver (B-VQE), a scalable NISQ algorithm that utilizes independent variational circuits and importance sampling to efficiently simulate non-Hermitian many-body systems, accurately mapping their phase diagrams and exceptional points without costly post-selection.

The Gist

The Big Picture: A New Tool for a Weird World

Imagine you are trying to understand a complex machine, like a car engine. Usually, you assume the engine is a closed system: fuel goes in, power comes out, and nothing leaks. In standard quantum physics, this is the rule: energy is conserved, and the math is "Hermitian" (balanced and predictable).

But in the real world, systems often leak. They lose energy to the environment, or they gain energy from outside. This is called a Non-Hermitian system. Think of it like a car engine with a hole in the fuel tank or a turbocharger that sometimes adds extra fuel unpredictably. The math for these systems is messy, complex, and usually impossible to solve with standard computers because the numbers get too big too fast.

This paper introduces a new tool called B-VQE (Biorthogonal Variational Quantum Eigensolver). It is a specialized software algorithm designed to run on current, imperfect quantum computers (called NISQ devices) to solve these "leaky" or "open" quantum systems.

The Problem: The "One-Handed" Approach Doesn't Work

Standard quantum algorithms (like the original VQE) are like trying to measure a moving object with just one hand. They assume the system is balanced. If you try to use them on a "leaky" system, the math breaks down because:

1. The answers aren't just simple numbers; they can be complex (involving imaginary numbers).
2. The "input" state and the "output" state of the system are no longer the same. They drift apart.

The Solution: The "Two-Handed" Approach (B-VQE)

The authors invented a "Two-Handed" approach. Instead of one circuit trying to do everything, B-VQE uses two separate circuits working in tandem:

• The Right Hand (Right Circuit): Prepares the "forward" state of the system (what happens when you push the button).
• The Left Hand (Left Circuit): Prepares the "backward" state (what the system looks like if you rewind it).

The Analogy: Imagine trying to find the perfect balance point on a wobbly seesaw.

• A standard algorithm tries to sit on one side and guess the balance.
• B-VQE sends one person to sit on the left side and another to sit on the right side. They talk to each other, adjusting their positions until they find the exact spot where the seesaw is perfectly balanced, even if the ground beneath them is shaking.

Key Features of the New Tool

1. The "Coalescence" Detector (Finding the Tipping Point)

In these weird systems, there are special points called Exceptional Points (EPs). At an EP, two different states of the system suddenly merge into one, like two rivers flowing together to become a single, wider river. This is a critical moment where the system's behavior changes drastically.

• The Innovation: B-VQE has a built-in "EP Detector." It measures how close the "Left Hand" and "Right Hand" states are to each other. When they become identical (coalesce), the detector screams, "We found the Exceptional Point!"
• Why it matters: It allows scientists to map exactly where these dangerous, unstable tipping points are located in the system.

2. The "Importance Sampling" Trick (Avoiding the Lottery)

Usually, simulating these leaky systems on a quantum computer requires a "post-selection" step. This is like playing a lottery where you have to throw away 99% of your results because they didn't match a specific criteria. As the system gets bigger, you might have to throw away all your results, making the simulation impossible.

• The Innovation: The authors developed an "Importance Sampling" method. Instead of throwing away the "bad" results, they keep them but give them a different weight in the final calculation.
• The Analogy: Instead of only counting the lottery tickets that won the jackpot (and throwing away the rest), they count every ticket but give the jackpot winners a huge multiplier and the losers a tiny multiplier. This saves them from having to run the lottery millions of times just to get one winner. This keeps the computing cost manageable.

3. Mapping the "Phase Diagram"

The team used this tool to map out three different complex systems:

• The Non-Hermitian Hubbard Chain: A model of electrons hopping around that can get "stuck" (localized) or flow freely.
• The XXZ Spin Chain: A model of magnetic spins that can form "scars" (special states that don't forget their past).
• The 2D t-J Model: A model showing how particles pile up at the edges of a material (the "Skin Effect").

They successfully drew maps showing where the system is stable, where it is chaotic, and where it gets "stuck" at the edges.

The Results: Does it Work?

The authors tested their tool on a simulator that mimics real, noisy quantum computers (specifically IBM's Heron processors).

• Accuracy: They found the energy levels of the systems with very high accuracy (less than 0.5% error).
• Speed: They found the "tipping points" (Exceptional Points) with a precision of about 0.02 units.
• Efficiency: Their method required much less computing power than older methods. While older methods would need exponentially more power as the system grew (like needing a supercomputer for a small problem), their method only needed a polynomial increase (like needing a slightly bigger laptop).

Summary

This paper presents a new "Two-Handed" algorithm that allows current, imperfect quantum computers to study "leaky" quantum systems. By using two circuits to track the system from both directions, and by using a clever math trick to avoid throwing away data, the authors can accurately map out where these systems become unstable and how particles behave in them. It bridges the gap between theoretical physics and what we can actually calculate on today's hardware.

Technical Summary

Technical Summary: Exceptional-Point-Anchored Variational Quantum Eigensolver for Non-Hermitian Many-Body Phase Diagrams

Problem Statement
The study of non-Hermitian quantum systems has expanded from mathematical curiosities to a frontier in physics, encompassing phenomena like exceptional points (EPs) and the non-Hermitian skin effect (NHSE). However, simulating non-Hermitian many-body systems on Noisy Intermediate-Scale Quantum (NISQ) hardware presents three fundamental challenges:

1. Complex Eigenvalues: Standard Variational Quantum Eigensolvers (VQE) rely on real-valued cost functions derived from Hermitian Hamiltonians; complex eigenvalues preclude a natural variational lower bound.
2. Biorthogonality: Non-Hermitian systems require independent preparation and measurement of left ($\langle \tilde{\psi}_L|$) and right ($|\psi_R\rangle$) eigenstates, as the standard inner product is replaced by a biorthogonal metric.
3. Resource Overhead: Simulating non-Hermitian dynamics on unitary quantum computers typically requires ancilla-based dilation or post-selection, leading to exponential overhead that renders simulations infeasible beyond $\sim 8$ qubits.

Existing proposals have addressed partial aspects (e.g., imaginary-time evolution or density-matrix dilation) but lack a unified framework capable of accessing biorthogonal ground states, detecting EPs, and mapping many-body phase diagrams simultaneously.

Methodology: The B-VQE Framework
The authors introduce the Biorthogonal Variational Quantum Eigensolver (B-VQE), a dual-circuit algorithm designed specifically for non-Hermitian Hamiltonians on NISQ hardware.

• Dual-Circuit Ansatz: Unlike standard VQE, B-VQE employs two independent parameterized unitaries, $U_R(\theta)$ and $U_L(\phi)$, to approximate the right eigenstate $|\psi_R^\theta\rangle$ and the left eigenstate $|\tilde{\psi}_L^\phi\rangle$, respectively.
• Biorthogonal Cost Function: The algorithm optimizes a cost function $L_{B-VQE}$ defined as:
  $$L_{B-VQE}(\theta, \phi) = \text{Re}[E_{bio}(\theta, \phi)] + \lambda [\text{Im}(E_{bio}(\theta, \phi))]^2$$
  where $E_{bio}$ is the biorthogonal Rayleigh quotient. The second term penalizes imaginary energy deviations, driving the system toward real eigenvalues in the unbroken Parity-Time (PT) symmetry phase.
• Exceptional-Point Detector (EPD): An operational module identifies EPs by measuring the coalescence metric $C_\lambda = 1 - |\langle \tilde{\psi}_L^\phi | \psi_R^\theta \rangle|^2$. At an EP, the left and right eigenstates coalesce ($C_\lambda \to 0$), providing a hardware-native signature.
• Non-Hermitian Quantum Geometric Tensor (NH-QGT): The framework extends the quantum geometric tensor to the biorthogonal setting. This allows for the separate characterization of state topology (via the biorthogonal Berry curvature) and band topology, resolving discrepancies specific to non-Hermitian systems.
• Importance-Sampling (IS) Mitigation: To overcome the exponential overhead of post-selection, the authors propose a classical reweighting scheme. By sampling bitstrings from the quantum circuits and applying classical weights based on the ratio of right and left amplitudes, the method recovers polynomial overhead without requiring ancilla post-selection.

Key Results
The framework was validated on three paradigmatic models using Qiskit-Aer simulations calibrated to IBM Heron r2 hardware noise models:

1. Non-Hermitian Hubbard Chain: B-VQE successfully mapped a phase diagram containing four distinct phases: ergodic, non-Hermitian many-body localized (NH-MBL), PT-broken ergodic, and skin-localized. The algorithm achieved relative energy errors $\epsilon_E < 5 \times 10^{-3}$ and correctly identified the MBL transition via level-spacing ratios and entanglement entropy scaling.
2. Non-Hermitian XXZ Spin Chain: The method detected exceptional points with an accuracy of $\delta\lambda < 0.02t$. It also characterized "exceptional-point-enhanced quantum many-body scars," observing that scar coherence times are exponentially prolonged near EPs. The biorthogonal entanglement entropy revealed a conformal field theory (CFT) scaling with an effective central charge $c_{eff} \approx 0.85$ at the EP critical point.
3. 2D Non-Hermitian t-J Model: The algorithm resolved a many-body Fermi skin, where states accumulate at the boundary. It successfully measured the topological winding number, demonstrating a sharp transition from trivial ($W=0$) to topological ($W=1$) phases coinciding with the onset of macroscopic skin accumulation.

Performance and Scalability

• Accuracy: Across all models, B-VQE maintained energy errors below $5 \times 10^{-3}$ on noise-free simulators and remained robust under calibrated hardware noise (e.g., IBM Kingston, Fez, Marrakesh).
• EP Detection: The EPD module located exceptional points within $\delta\lambda < 0.02t$ on noise-free simulators and $\delta\lambda < 0.08t$ under realistic noise conditions.
• Resource Efficiency: Compared to Trotterization, B-VQE required approximately 2x shallower circuits to achieve equivalent state fidelity. The IS mitigation scheme reduced the scaling of overhead from exponential (standard post-selection) to polynomial ($O(n^{1.55})$), enabling simulations up to $n=16$ qubits with success probabilities $>0.4$.

Significance and Claims
The paper positions B-VQE as a scalable, complete NISQ methodology for non-Hermitian many-body physics. Its primary contributions are:

1. Algorithmic Gap Resolution: It provides the first variational framework that simultaneously accesses biorthogonal ground states, systematically detects exceptional points, and maps many-body phase diagrams.
2. Topological Resolution: By utilizing the NH-QGT, it resolves the state-topology/band-topology discrepancy inherent in non-Hermitian systems, a feature inaccessible to standard Hermitian VQE or band-structure-only approaches.
3. Practical Feasibility: The importance-sampling mitigation scheme eliminates the exponential post-selection barrier, making the simulation of interacting non-Hermitian systems feasible on current and near-future hardware.

The authors conclude that B-VQE establishes a pathway for hardware-native exploration of EP-enhanced quantum sensing, skin-effect-based transport, and non-unitary quantum criticality, effectively bridging the gap between theoretical non-Hermitian many-body physics and experimental quantum simulation.