A Decision-Making Framework for Simulating Open Quantum Systems on Quantum Computers via Hermitianization

Abstract

The advancement of quantum hardware is fundamentally limited by the challenge of decoherence—the unavoidable interaction of qubits with their environment. This "open system" dynamic is described by non-Hermitian mathematics, which is incompatible with the unitary nature of quantum computation. To build more robust quantum technologies, from better error-correcting codes to more sensitive quantum sensors, we must first be able to accurately and efficiently simulate these open system effects.

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GitHub Repo : https://github.com/Apoth3osis-ai/e-dqas_qiskit_open_systems

Research Gate: https://www.researchgate.net/publication/392787695_A_Decision-Making_Framework_for_Simulating_Open_Quantum_Systems_on_Quantum_Computers_via_Hermitianization

This paper introduces a comprehensive software framework developed by Apoth3osis that serves as a decision-making toolkit for simulating non-Hermitian systems on quantum computers. Our approach is centered on Hermitianization, a class of techniques that formally map a non-Hermitian problem onto a larger, effective Hamiltonian that is fully Hermitian and thus executable on quantum hardware. We present a rigorous, side-by-side implementation and comparison of multiple leading Hermitianization methods, including those based on Schur decomposition, warped phase transforms, and isometric embeddings.

Our framework is validated on canonical models of open systems, such as the Damped Quantum Oscillator and the Hatano-Nelson model. The results provide a clear analysis of the trade-offs between different techniques in terms of simulation accuracy, resource requirements, and numerical stability. Crucially, we demonstrate the platform's ability to model exotic non-Hermitian phenomena like the skin effect and exceptional points—features essential for designing next-generation quantum sensors. By establishing a clear methodology for choosing the optimal simulation strategy for a given problem, this work provides the necessary foundation for engineering robust, noise-aware quantum applications and hardware.

1 Introduction: The Open System Imperative

In the current NISQ era, every quantum processor is an open quantum system. The performance of any quantum algorithm is ultimately dictated by the complex, non-unitary interactions between the core computational system (the qubits) and its external environment. These interactions, which lead to energy dissipation and dephasing, are the root cause of decoherence and computational error. To build fault-tolerant quantum computers and high-precision quantum sensors, we must move beyond treating this environment as an adversary to be suppressed and begin modeling it as a fundamental component of the system itself.

The dynamics of such open systems are mathematically described by non-Hermitian Hamiltonians. However, quantum computers, by their very nature, can only execute unitary operations, which correspond to evolution under Hermitian Hamiltonians. This creates a fundamental mismatch between the physics we need to simulate and the tools we have to simulate it.

To bridge this gap, Apoth3osis has developed a comprehensive software framework to systematically implement and evaluate Hermitianization techniques. These methods provide a mathematically rigorous pathway to map a non-Hermitian system in a smaller Hilbert space to an equivalent, larger Hermitian system. By simulating this expanded system on a quantum computer, we can effectively reproduce the non-unitary dynamics of the original open system.

This paper presents our framework not as a single, monolithic algorithm, but as a decision-making toolkit. We have implemented and benchmarked several distinct Hermitianization methods to create a quantitative guide for selecting the optimal simulation strategy based on the specific goals of the task—be it high-fidelity modeling of decoherence for error correction, or the exploration of exotic non-Hermitian physics for quantum sensing.

2 Methodology: A Toolkit for Non-Hermitian Simulation

Our platform is designed to provide a complete workflow, from defining a non-Hermitian problem to analyzing the results of its quantum simulation.

2.1. The Principle of Hermitianization

The core concept is to embed a non-Hermitian operator H, which acts on an N-dimensional Hilbert space, into a larger Hermitian operator Heff that acts on a 2N-dimensional (or larger) space. A common construction, known as a dilation, takes the form:

Heff A, B, B†, C

where A, B, and C are matrices constructed from the Hermitian and anti-Hermitian parts of the original H. While the evolution of the full system under Heff is unitary, the dynamics of the original N-dimensional subspace, when projected out, are non-unitary and faithfully reproduce the desired open system behavior.

2.2. A Comparative Analysis of Techniques

Our framework implements several distinct methods to construct Heff, each with unique trade-offs:

Schur Decomposition: A numerically stable method based on decomposing H into Q T Q†, where Q is unitary and T is upper-triangular. This method demonstrated the highest accuracy in preserving the system's energy spectrum in our tests.

Warped Phase Transform: A sophisticated technique that is particularly insightful for certain classes of problems but can introduce minor numerical instabilities. Our implementation includes an enhanced version that enforces exact Hermiticity post-transformation.

Isometric Embedding: A direct approach that embeds H and its conjugate transpose H† into the off-diagonal blocks of H_eff. This method is often the fastest computationally but may be less accurate for systems with strong non-Hermitian effects.

PT-Symmetric Methods: A specialized approach for systems that respect Parity-Time (PT) symmetry, a property that can lead to purely real energy spectra even in non-Hermitian systems.

By benchmarking these methods side-by-side, our platform allows a user to select the optimal tool for their specific problem, balancing the need for accuracy, speed, and numerical robustness.

2.3. From Operator to Circuit

Once the effective Hermitian Hamiltonian Heff is constructed, its time-evolution operator U(t) exp(-i Heff t) must be compiled into a quantum circuit. Our framework uses first-order Suzuki-Trotter decomposition to approximate this evolution, breaking the Hamiltonian down into a sum of simple Pauli terms and translating each term into a sequence of native quantum gates. This process is designed to be NISQ-friendly, with analysis tools to monitor and control circuit depth and resource requirements.

3 Experimental Validation and Benchmarking

To validate our toolkit, we applied it to two canonical non-Hermitian systems: the Damped Quantum Oscillator and the 1D Hatano-Nelson model. We compared the results of the quantum simulations against exact classical solutions.

3.1. Key Findings

The results of our comprehensive analysis provided a clear performance profile for each technique:

Spectral Accuracy: The Schur Decomposition method proved most effective at accurately reproducing the complex energy spectrum of the original non-Hermitian Hamiltonian.

Simulation Fidelity: All methods demonstrated a comparable ability to simulate the system's state evolution over time, achieving moderate fidelity against the exact solution. The remaining fidelity gap is primarily attributable to Trotter error, which can be systematically reduced by increasing the number of Trotter steps at the cost of deeper circuits.

Resource Efficiency: The Isometric Embedding method was consistently the fastest in terms of classical preprocessing time, while all methods produced quantum circuits of similar depth and complexity for the systems under study.

3.2. A Decision-Making Toolkit in Action

These results allow us to frame our framework as a strategic toolkit. For a user who needs to:

Precisely model the energy levels of a noisy system, the Schur Decomposition method is the recommended choice.

Rapidly simulate system dynamics where exact energy levels are less critical, the faster Isometric Embedding method is preferable.

Investigate PT-symmetric phenomena, the specialized PT-Symmetric Hermitianizer provides the correct physical constraints.

This ability to match the simulation strategy to the problem's requirements is a core feature of the Apoth3osis platform.

4 Applications in Quantum Technology Design

The ability to accurately simulate open quantum systems is not an academic exercise; it is a critical enabling capability for engineering the next generation of quantum technologies.

Quantum Sensor Design: Many proposed high-precision quantum sensors operate near "exceptional points" (EPs)—a feature unique to non-Hermitian systems where energy levels coalesce, leading to an enhanced response to external perturbations. Our framework provides the necessary tools to simulate, analyze, and design devices that leverage EPs, potentially leading to sensors with sensitivities far beyond what is achievable with standard Hermitian systems.

Hardware-Specific Error Correction: All quantum hardware is subject to decoherence. By using our framework to create a high-fidelity simulated "digital twin" of a specific quantum device—one that captures its unique, non-Hermitian noise profile—we can design and test bespoke, hardware-specific error mitigation and correction protocols. This moves beyond generic error correction to create solutions tailored to the reality of a particular machine.

Engineered Dissipation: In some algorithms, dissipation is not a bug but a feature. It can be used to rapidly initialize qubits or to steer a system towards a desired steady state. Our platform provides the means to simulate and control these dissipative processes, opening the door to new classes of quantum algorithms.

5 Projected Future Work

This work provides the foundation for a suite of powerful simulation and design tools. Our immediate roadmap includes:

Hardware Integration and Validation: The most critical next step is to partner with a hardware provider to apply this toolkit to a real quantum device. By modeling the device's specific noise channels and comparing our simulations to experimental results, we can create a high-fidelity digital twin and demonstrate hardware-specific error mitigation.

Automated Method Selection: We plan to develop a machine learning layer on top of the toolkit that, given a new non-Hermitian problem, automatically recommends the optimal Hermitianization technique and simulation parameters.

Scalability to Larger Systems: We will integrate advanced techniques, such as tensor network methods, to extend our simulation capabilities to larger and more complex multi-qubit open systems.

6 Implications and Conclusion

The narrative of near-term quantum computing is inextricably linked to the challenge of decoherence. The Apoth3osis E-DQAS framework represents a significant step towards mastering this challenge. By providing a rigorous, verifiable, and flexible toolkit for simulating open quantum systems, we empower researchers and engineers to move from fighting noise to understanding, predicting, and ultimately leveraging it.

The implications are twofold. For the scientific community, this provides a powerful new tool for investigating the rich and often counter-intuitive physics of non-Hermitian systems. For the quantum industry, it provides a direct path to designing more robust and powerful technologies.

Apoth3osis has developed a production-ready engine for modeling the precise dynamics of real-world quantum devices. We are now seeking a strategic hardware partner to apply this platform to their system. A collaboration would enable us to co-design customized, noise-resilient algorithms and error-correction schemes that are uniquely tailored to your device's architecture. This partnership would not only validate the performance of your hardware at a new level of precision but also accelerate the development of practical, high-value quantum applications on your platform.