Quantum dynamics of two $XX$ interacting PT-symmetric non-Hermitian qubits: enhancement of quantum annealing

Abstract: Quantum information platforms enable analog quantum simulations, such as quantum annealing, offering a promising route to solving complex combinatorial optimization problems. Here, we propose a quantum information architecture based on networks of interacting parity-time (PT)-symmetric non-Hermitian qubits. While the dynamics of individual PT-symmetric qubits have been experimentally demonstrated across multiple platforms including NV centers, superconducting circuits, and trapped-ion systems yet coherent dynamics in interacting systems remain largely unexplored. To address this issue we theoretically investigate stationary and time-dependent Hamiltonians relevant to quantum annealing using a minimal model of two interacting XX-coupled PT-symmetric non-Hermitian qubits. We analyze both symmetry-preserving and symmetry-broken regimes and demonstrate that adding even tiny PT-symmetric non-Hermitian terms in the qubits Hamiltonian allows to greatly enhance the probability of reaching the ground state after annealing.

• The paper demonstrates that PT-symmetric non-Hermitian terms in XX-coupled qubits can significantly enhance quantum annealing performance.
• Results show that tailored gain/loss parameters mitigate tunneling suppression and improve ground state probabilities near exceptional points.
• Numerical and analytical analyses confirm that non-Hermitian perturbations effectively lift degeneracies, optimizing both fast and slow annealing regimes.

Quantum Dynamics and Annealing Enhancement via XX-Coupled PT-Symmetric Non-Hermitian Qubits

Introduction

This paper investigates the quantum dynamics of XX-coupled PT-symmetric non-Hermitian qubits and their application to quantum adiabatic annealing (QAA). The focus is on leveraging PT-symmetric non-Hermitian terms in the qubit Hamiltonian to mitigate fundamental limitations of QAA, specifically the suppression of quantum tunneling due to vanishing energy gaps and non-unitary dissipation processes. The authors systematically explore a minimal model comprising two PT-symmetric non-Hermitian qubits, examining both stationary and time-dependent Hamiltonians relevant to analog quantum optimization.

Model and Spectral Analysis

The system considered is a chain of XX-interacting qubits subjected to staggered imaginary longitudinal fields to establish PT symmetry. The non-Hermiticity, introduced via a gain/loss parameter $y$, modifies the energy structure and quantum dynamics. For two qubits, the resulting Hamiltonian is explicitly constructed, revealing a regime-dependent spectral landscape:

• PT-Unbroken Regime: For small $y$, eigenvalues are predominantly real, with PT symmetry preserved except near exceptional points.
• PT-Broken Regime: As $y$ increases, regions with complex conjugate eigenvalues emerge, signifying PT symmetry breaking at second-order exceptional points.
• The spectral evolution is sensitive to the qubit bias parameter $\epsilon$, with qualitative changes when $\epsilon/4 > 1$, including substitution of ground-excited crossings by third-level coalescences.

This spectral structure allows for effective reduction to a PT-symmetric two-level system, most accurately near level crossings in the weak non-Hermitian limit.

Reduction to Effective Two-Level Hamiltonian

The low-energy dynamics are governed by a reduced PT-symmetric non-Hermitian $2 \times 2$ Hamiltonian, capturing the essential behavior near ground state transitions. Analytical expressions for level crossing parameters and effective Hamiltonian elements are derived, and their validity is confirmed numerically. The dynamics in the PT-unbroken regime show oscillatory population distributions between basis states, with reduced oscillation frequencies and increased amplitudes as the system approaches exceptional points. In the PT-broken regime, oscillations are suppressed, yielding exponential population convergence.

Quantum Annealing Dynamics and LZS Tunneling

The paper extends the study to adiabatic regimes, with the time-dependent parameter $s(t) = t/T$ guiding the quantum annealing process. In the Hermitian limit ($y=0$), the annealing trajectory is governed by Landau-Zener-Stückelberg (LZS) tunneling at exact energy crossings—resulting in the system ultimately occupying the excited state, even in the adiabatic limit ($T \to \infty$), a fundamental failure mode for QAA.

With nonzero $y$, the dynamics are fundamentally altered:

• LZS Tunneling Enhancement: The introduction of PT-symmetric non-Hermitian terms lifts degeneracies and enables enhanced tunneling in symmetry-broken regions, overcoming the suppression typical for Hermitian systems.
• Ground State Probability Improvement: The probability $y$0 of reaching the ground state after annealing increases significantly as $y$1 grows, reaching a maximal value of 1/2 for moderate $y$2 and annealing speeds.
• Analytical results using adapted Kayanuma-Grifoni-Hänggi formalism match numerical computations, confirming the robustness of the derived population expressions.

Distinct dynamical signatures are observed:

• Fast Annealing: Small quantum beating amplitudes post-exceptional point.
• Slow Annealing: Large amplitude quantum beats precede exceptional points, followed by equal population saturation.

Full Two-Qubit System: Numerical Demonstration

Numerical evaluation for the original four-level Hamiltonian confirms the qualitative and quantitative findings from the effective two-level model. Notably, the PT-symmetric non-Hermitian annealing protocol (with $y$3) consistently achieves higher ground state populations, substantiating the practical utility for optimization applications.

Implications and Future Directions

The study demonstrates that PT-symmetric non-Hermitian perturbations to qubit Hamiltonians provide a mechanism to circumvent the inherent limitations of conventional QAA stemming from small energy gaps or bottlenecked tunneling. Enhancement of ground state occupation in this minimal model is particularly relevant for scalability in analog quantum optimization platforms.

Theoretical implications include:

• Non-Hermitian Quantum Control: The results lay groundwork for further exploration of non-Hermitian enhancements in multi-qubit, non-integrable models.
• PT-Symmetry and Quantum Dynamics: The interplay between exceptional points, spectral topology, and dynamical transitions offers new avenues for quantum control strategies.

Practical implications are immediate for experimental platforms capable of PT-symmetric embedding, such as superconducting circuits, NV centers, and trapped ions, where gain/loss can be engineered via dilation schemes and ancilla-driven postselection.

Future research directions include:

• Generalization to larger qubit networks, addressing many-body effects.
• Clarification of optimal bias and gain/loss parameter regimes for specific optimization tasks.
• Experimental validation leveraging state tomography and control of PT-symmetry breaking, especially near exceptional points.

Conclusion

This work provides a rigorous theoretical and computational analysis of the role of PT-symmetric non-Hermitian couplings in enhancing quantum annealing. Through an explicit two-qubit model and reduction to an effective two-level system, the authors establish that non-Hermitian terms can substantially improve ground state occupation probability in QAA. Both theoretical and practical future explorations, including scaling behaviors and experimental realization, are key next steps for advancing quantum optimization leveraging non-Hermitian quantum dynamics (2605.13008).