Linear response theory and Green's functions provide a universal framework for understanding how macroscopic and strongly correlated systems respond to weak external perturbations. While the theoretical foundation for non-Hermitian linear response theory has been recently established to describe open quantum systems, generalizing these predictions onto practical quantum computers remains a formidable algorithmic challenge due to the non-unitary nature of the dynamics. In this work, we present a systematic algorithmic mapping that transforms the non-unitary multi-time correlation functions into a unitary form viable for quantum hardware. By mapping the vectorization of the Lindblad master equation into a unitary Schrödinger-like equation using the continuous-variable Schrödingerization technique, we show that generalized non-Hermitian Green's functions can be systematically extracted. This approach bridges the gap between the established physical theory of non-Hermitian linear response and quantum simulation, achieving optimal state preparation cost.
Linear response theory and Green's functions provide a universal framework for understanding how macroscopic and strongly correlated systems respond to weak external perturbations. While the theoretical foundation for non-Hermitian linear response theory has been recently established to describe open quantum systems, generalizing these predictions onto practical quantum computers remains a formidable algorithmic challenge due to the non-unitary nature of the dynamics. In this work, we present a systematic algorithmic mapping that transforms the non-unitary multi-time correlation functions into a unitary form viable for quantum hardware. By mapping the vectorization of the Lindblad master equation into a unitary Schrödinger-like equation using the continuous-variable Schrödingerization technique, we show that generalized non-Hermitian Green's functions can be systematically extracted. This approach bridges the gap between the established physical theory of non-Hermitian linear response and quantum simulation, achieving optimal state preparation cost.
The rapid advancement of quantum computing has created new opportunities for solving computationally intensive problems across diverse fields such as optimization, physical system simulation, and machine learning. However, practical implementation on Noisy Intermediate-Scale Quantum (NISQ) devices remains challenging due to hardware limitations including noise, decoherence, and restricted qubit connectivity. A fundamental constraint further complicating this effort is that all quantum operations must be unitary, while many real-world problems are naturally expressed in non-unitary forms. This discrepancy necessitates novel techniques for transforming non-unitary mathematical structures into unitary or Hamiltonian representations compatible with quantum hardware.This thesis proposes a unified framework for converting structured, sparse, or otherwise non-unitary classical matrices into unitary forms that can be executed efficiently on NISQ devices. The methodology introduces general transformation strategies such as linear combinations of unitaries, structured embeddings, and resource-aware circuit construction, enabling the deployment of quantum algorithms without incurring prohibitive overhead. Each technique is developed with an emphasis on preserving problem structure, minimizing circuit depth, and ensuring compatibility with IBM’s NISQ hardware. The proposed frameworks are validated through theoretical analysis and quantum simulations, with performance benchmarks based on gate count, circuit fidelity, convergence behavior, and scalability. This work demonstrates how non-unitary formulations arising in various computational contexts can be systematically adapted for quantum execution, advancing the practical utility of NISQ-era devices.
Physical Unclonable Functions (PUFs) provide hardware-level security by exploiting intrinsic randomness to produce device-unique responses. However, machine learning and side-channel attacks increasingly undermine their classical assumptions, calling for new approaches to ensure unforgeability. Quantum mechanics naturally supports this goal through intrinsic randomness and the no-cloning theorem, motivating the study of Quantum Physical Unclonable Functions (QPUFs). Yet, existing QPUF models often assume ideal unitary dynamics, neglecting non-unitary effects such as decoherence and dissipation that arise in real quantum devices. This work introduces a new class of non-unitary QPUFs that leverage open quantum system dynamics as a foundation for security. Three architectures are proposed: the Dissipative QPUF (D-QPUF), which uses amplitude damping as an entropy source; the Measurement-Feedback QPUF (MF-QPUF), which employs mid-circuit measurements and conditional unitaries; and the Lindbladian QPUF (L-QPUF), which models Markovian noise via the Lindblad master equation and Trotter-Suzuki decomposition. Simulation results show that these non-unitary designs achieve strong uniqueness, uniformity, and unforgeability, with controllable reliability trade-offs from stochastic noise. The L-QPUF, in particular, exhibits exponential modeling resistance under limited challenge-response access. By reframing environmental noise as a constructive resource, this work establishes a framework for noise-aware quantum hardware authentication and highlights non-unitary evolution as a viable foundation for post-quantum security.
While non-Hermitian Hamiltonians enable exotic dynamical phenomena, implementing their nonunitary time evolution on near-term quantum devices remains challenging. We propose a hardware-friendly protocol that simulates non-Hermitian dynamics without ancillas, controlled time evolution, or continuous monitoring. The protocol combines a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) evolution via classical Gaussian white-noise averaging with stochastic quantum error mitigation (QEM) to cancel the jump contribution at the level of expectation values. The mitigation layer uses only single-qubit operations. We validate the method through numerical simulations of an asymmetric-hopping model and an open {\it XXZ} spin chain with non-Hermitian boundary fields. Our work provides a programmable and ancilla-free framework for investigating exotic dynamics beyond the class of completely positive and trace-preserving dynamics using QEM.
Linear response theory is the theoretical foundation of modern experiments. In particular, it plays a vital role in measuring quantum matters. Its main idea is to take the external probe signal of the physical system as a perturbation and use the correlation function in the unperturbed equilibrium state to depict the response to the observable in system. In recent half century, the linear response theory for the closed quantum system has achieved great success in experiments on quantum matters. In recent years, with the tremendous progress of the precise manipulation of the light-matter interaction, the ultracold atom experiments can precisely control dissipative quantum many-body systems. With the discovery of many novel phenomena, dissipative quantum systems and non-Hermitian physics have attracted extensive attention in theory and experiment. We developed a linear response theory, named non-Hermitian linear response theory, to deal with open quantum systems. This theory takes the non-Hermitian term and quantum noise, which are induced by dissipation, as an external perturbative input, to detect the properties of the quantum system, and relates the time evolution of the observable with the correlation function in the unperturbed state of the system. The non-Hermitian linear response theory provides a new method for distinguishing the exotic quantum phase from the normal phase. The theoretical predictions are highly consistent with the recent experimental results of cold atom systems. This paper will review the non-Hermitian linear response theory and discuss its applications in quantum many-body and time-reversal symmetric quantum systems.
Linear response functions are a cornerstone concept in physics as they enable efficient estimation of many dynamical properties. In addition to predicting dynamics of observables under perturbations without resimulating the system, these response functions lead to electric conductivity, magnetic susceptibility, dielectric constants, etc. Estimating two-time correlation functions is a key ingredient of measuring linear response functions. However, for open quantum systems, simulating the reduced density operator with a quantum master equation only yields \emph{one-point} observables and is insufficient for this task. In this paper, we develop a memoryless, system-only formulation of two-point correlations for open quantum systems that extends the standard quantum regression theorem (QRT) beyond the Markov limit. We further incorporate the spectral property of the bath and express the time propagators in the response function as the memoryless generators in Lindblad-type forms. The resulting expressions recast the total response function into evolutions generated by time-dependent Hamiltonian and Lindblad primitives together with the more challenging propagation of commutators and anti-commutators. In addition to the derivation of the new QRT, we present quantum algorithms for these primitives and obtain an estimator for two-time correlations whose cost scales poly-logarithmically in the system dimension and 1/ε1.25 in the target accuracy ε. The framework removes the separability (Born-Markov) assumption and offers a pathway to efficient computation of nonequilibrium properties from open quantum systems.
Despite rapid advances in quantum hardware, noise remains a central obstacle to deploying quantum algorithms on near-term devices. In particular, random coherent errors that accumulate during circuit execution constitute a dominant and fundamentally challenging noise source. We introduce a noise-resilient framework for Quantum Signal Processing (QSP) that mitigates such coherent errors without increasing circuit depth or ancillary qubit requirements. Our approach uses ensembles of noisy QSP circuits combined with measurement-level averaging to suppress random phase errors in Z rotations. Building on this framework, we develop robust QSP algorithms for implementing polynomial functions of Hermitian matrices and for estimating observables, with applications to Hamiltonian simulation, quantum linear systems, and ground-state preparation. We analyze the trade-off between approximation error and hardware noise, which is essential for practical implementation under the stringent depth and coherence constraints of current quantum hardware. Our results establish a practical pathway for integrating error mitigation seamlessly into algorithmic design, advancing the development of robust quantum computing, and enabling the discovery of scientific applications with near- and mid-term quantum devices.