arXiv:2605.02293

Description and error analysis of quantum alghorithms in the projection evolution model -- the Deutsch algorithm case

Krzysztof Lider, Marek Góźdź

quant-phSubmitted May 4, 2026• Announced May 5, 2026• 0 citations

Abstract

This work demonstrates that the Deutsch algorithm can be effectively modelled using a two-level harmonic oscillator within the second quantization formalism. By adopting this framework, evolution operators are derived. We present a projection evolution model that accurately characterizes the physical state transformation within quantum gates. This approach provides a systematic method for finding evolution operators, enabling the complete description and prediction of state evolution - including projection errors - in quantum algorithms.

Related papers

Excited states by quantum Monte Carlo methods: Imaginary time evolution with projection operators

D. Blume, Marius Lewerenz, Parhat Niyaz +1 more

We present a Monte Carlo algorithm suitable for the calculation of excited state energies of multidimensional quantum systems. Energies are extracted from a maximum entropy analysis of the imaginary time evolution of a state prepared by application of a projection operator on an initial wave function. The imaginary time evolution is computed with a pure diffusion Monte Carlo algorithm. The method is demonstrated on a harmonic oscillator and several Morse oscillator test problems.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsMar 1, 199764 citations

Elementary derivation of the quantum propagator for the harmonic oscillator

Jiushu Shao

Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.

American Journal of PhysicsSep 21, 20169 citations

Spectral analysis of three-state quantum walks with general coin matrices

Chusei Kiumi, Jirô Akahori, Takuya Watanabe +1 more

Mathematical analysis of the spectral properties of the time-evolution operator in quantum walks is essential for understanding key dynamical behaviors such as localization and long-term evolution. The inhomogeneous three-state case, in particular, poses substantial analytical challenges due to its higher internal degrees of freedom and the absence of translational invariance. We develop a general framework for the spectral analysis of three-state quantum walks on the one-dimensional lattice with arbitrary time-evolution operators. Our approach is based on a transfer-matrix formulation that reduces the infinite-dimensional eigenvalue problem to a tractable system of two-dimensional recursions, enabling exact characterization of eigenstates. This framework applies broadly to space-inhomogeneous models, including those with finite defects and two-phase structures. We rigorously derive necessary and sufficient conditions for the existence of point spectrum, along with a complete description of the corresponding eigenvalues and eigenstates, which are known to underlie quantum localization phenomena. Furthermore, we give a complete spectral decomposition—discrete spectrum, flat-band eigenvalues (of infinite multiplicity), and absolutely continuous spectrum—with explicit characterization of each component. Using this method, we perform exact numerical analyses of the Fourier walk with spatial inhomogeneity, revealing the emergence of localization despite its delocalized nature in the homogeneous case. Our results provide mathematical tools and physical insights into the structure of quantum walks, offering a systematic path for identifying and characterizing localized quantum states in complex quantum systems.

arXiv:2311.06468Physical Review ResearchApr 2, 20260 citations

Projection evolution of quantum states

ANDRZEJ GÓŹDŹ, Marek Góźdź, Aleksandra Pȩdrak

We discuss the problem of time in quantum mechanics. In the traditional formulation time enters the model as a parameter, not an observable, which follows from the famous Pauli theorem. It is now known, that Pauli's assumptions were too strong and that by removing some of them time can be represented as a quantum observable. In this case, instead of the unitary time evolution, other operators which map the space of initial states into the space of final states at each step of the evolution can be used. This allows to treat time as a quantum observable in a consistent way. We present the projection evolution model and show how the traditional Schroedinger evolution can be obtained from it. We propose the form of the time operator which satisfies the energy-time uncertainty relation. As an example, we discuss the temporal version of the double-slit experiment.

Oct 24, 20191 citation

Non-Markovian noise sources for quantum error mitigation

Doyeol Ahn, Byeongyong Park

Reducing the impact of errors and decoherence in near-term quantum computers, such as noisy intermediate-scale quantum (NISQ) devices, is critical for their practical implementation. These factors significantly limit the applicability of quantum algorithms, necessitating a comprehensive understanding of their physical origins to establish effective error mitigation strategies. In this study, we present a non-Markovian model of quantum state evolution and a quantum error mitigation cost function tailored for NISQ devices interacting with an environment represented by a set of simple harmonic oscillators as a noise source. Employing the projection operator formalism and both advanced and retarded propagators in time, we derive the reduced-density operator for the output quantum states in a time-convolutionless form by solving the quantum Liouville equation. We examine the output quantum state fluctuations for both identity and controlled-NOT (CNOT) gate operations in two-qubit operations using a range of input states. Subsequently, these results are compared with experimental data from ion-trap and superconducting quantum computing systems to estimate the crucial parameters of the cost functions for quantum error mitigation. Our findings reveal that the cost function for quantum error mitigation increases as the coupling strength between the quantum system and its environment intensifies. This study underscores the significance of non-Markovian models in understanding quantum state evolution and highlights the practical implications of the quantum error mitigation cost function when assessing experimental results from NISQ devices.

arXiv:2302.05053Feb 10, 20230 citations

Quantum State Evolution Models: A Comprehensive Approach

Jincheng Zhang

Quantum state evolution is a fundamental aspect of quantum mechanics, particularly concerning the dynamics of quantum systems over time. This paper explores the development of a novel model for quantum state evolution, integrating principles of quantum field theory and advanced computational methods. The core claim is to construct a model capable of accurately simulating the non-linear time evolution of quantum states, offering predictive capabilities for complex environments. We introduce a "Quantum Field Theory Extension" concept, linking quantum states to the non-linear dynamic changes inherent in field theories, and establish a tailored algorithm for precise prediction. This work represents a significant advancement in quantum simulation, aiming to address limitations of existing models by incorporating the core dynamics of quantum evolution.

Sep 7, 20260 citations