Quantum Simulation of Two-Level $PT$-Symmetric Systems Using Hermitian Hamiltonians

Technical Summary: Quantum Simulation of Two-Level PT-Symmetric Systems Using Hermitian Hamiltonians

Problem Statement
Simulating open quantum systems (OQSs) on unitary-gate-based quantum computers presents a fundamental challenge because OQSs often exhibit non-unitary time propagators, which cannot be directly mapped to the unitary framework of current quantum hardware. Parity-Time (PT)-symmetric systems are a specific class of OQSs characterized by non-Hermitian Hamiltonians that, in their "unbroken" phase, possess real energy spectra. While these systems offer unique properties for quantum sensing and entanglement generation—such as faster evolution times between states compared to Hermitian systems—their non-unitary dynamics complicate direct simulation on quantum processors. Existing methods for open quantum simulation, including imaginary time evolution and block encodings, require efficient approaches specifically tailored to PT-symmetric dynamics to leverage their unique advantages.

Methodology
The authors propose a method to simulate PT-symmetric dynamics by leveraging similarity transformations to map non-Hermitian PT-symmetric Hamiltonians onto equivalent Hermitian Hamiltonians. This mapping is valid in the unbroken PT-symmetric regime where the Hamiltonian is pseudo-Hermitian. The core of the approach involves a similarity transformation matrix, $\tau = \sqrt{\eta}$, which connects the non-Hermitian PT-symmetric Hamiltonian ($\hat{H}_{PT}$) to a Hermitian equivalent ($\hat{h}$). The time evolution is then expressed as:
$$e^{-i\hat{H}_{PT}t} = \tau^{-1} e^{-i\hat{h}t} \tau$$
This formulation allows the simulation to be decomposed into three steps: initialization via $\tau$, unitary evolution via $e^{-i\hat{h}t}$, and a final transformation via $\tau^{-1}$.

The paper introduces two distinct algorithms to implement this framework:

1. Hybrid Classical-Quantum Algorithm: This approach performs the initialization and unitary time evolution on a single qubit. The final step, applying $\tau^{-1}$, is handled via classical post-processing. This requires full quantum state tomography to reconstruct the density matrix before the classical multiplication by $\tau^{-1}$ is performed.
2. Dilation Algorithm: To avoid the scalability limitations of full tomography, this method implements $\tau^{-1}$ as a dilated unitary operator using an ancillary qubit. By utilizing the spectral decomposition of the Hermitian matrix $\tau$, the non-unitary operator is embedded into a larger unitary space. The evolution is conditioned on the ancilla qubit being measured in the $|0\rangle$ state. This approach eliminates the need for full density matrix tomography.

To extend these methods to larger systems, the authors apply first-order perturbation theory to simulate two weakly interacting PT-symmetric systems. They construct a new similarity transformation based on the perturbed eigenstates of the coupled system, allowing the dynamics to be mapped onto a Hermitian evolution suitable for the hybrid algorithm.

Key Results
The algorithms were validated through simulations and experiments on the IBM-Sherbrooke quantum processing unit (QPU) and noisy simulators:

• Single Qubit Dynamics: Both algorithms successfully reproduced the ground- and excited-state population dynamics of a single PT-symmetric system near an exceptional point. The experimental data (circles) and noise-model simulations (dashed lines) showed strong agreement with exact classical solutions (solid lines).
• Two-Qubit Entanglement: Using the hybrid algorithm and perturbation theory, the authors simulated two weakly coupled PT-symmetric qubits. They successfully demonstrated the generation of entanglement, quantified by concurrence, which reached a maximum when the populations of the basis states became approximately equal.
• Error Analysis: The study analyzed the fidelity of both algorithms under varying measurement errors.
  • The hybrid algorithm was found to be highly sensitive to measurement errors, particularly in the fast evolution regime where errors are amplified by the classical post-processing step. Fidelity dropped significantly (to ~20%) at high error rates.
  • The dilation algorithm, while suffering from increased circuit volume and two-qubit gate errors, demonstrated greater robustness against measurement errors, maintaining higher fidelity (~70%) at high error rates compared to the hybrid approach. However, it remains sensitive to coherence limitations inherent in two-qubit gates.

Significance and Claims
The paper claims to offer a scalable approach for simulating PT-symmetric systems and pseudo-Hermitian operators on unitary gate-based quantum computers. By constructing equivalent Hermitian Hamiltonians, the authors demonstrate that non-unitary PT-symmetric dynamics can be simulated using real-time evolution rather than individual non-unitary gates.

The work highlights that:

• PT-symmetric systems can be simulated on current NISQ (Noisy Intermediate-Scale Quantum) devices using either hybrid or fully quantum (dilation) approaches.
• The methods can be extended to weakly coupled multi-qubit systems via perturbation theory, enabling the study of entanglement generation in non-Hermitian systems.
• The unique error characteristics of PT-symmetric simulations—specifically the amplification of errors in fast evolution regimes—mirror the unique physical behavior of these systems near exceptional points, suggesting potential utility for quantum sensing applications.

The authors conclude that this work opens an avenue for further investigation of non-Hermitian Hamiltonians on quantum computers, providing a foundation for future work aimed at enhancing algorithmic efficiency and exploring practical applications of these non-Hermitian systems.