Quantum-based solution of time-dependent complex Riccati equations

• URL: http://arxiv.org/abs/2209.03504v3
• Date: Sat, 25 Mar 2023 13:57:01 GMT
• Title: Quantum-based solution of time-dependent complex Riccati equations
• Authors: D. Mart\'inez-Tibaduiza, C. Gonz\'alez-Arciniegas, C. Farina, A. Cavalcanti-Duriez and A. Z. Khoury
• Abstract summary: We show a time-dependent complex Riccati equation (TDCRE) as the solution of the time evolution operator (TEO) of quantum systems. The inherited symmetries of quantum systems can be recognized by a simple inspection of the TDCRE. As an application, but also as a consistency test, we compare our solution with the analytic one for the Bloch-Riccati equation.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Using the Wei-Norman theory we obtain a time-dependent complex Riccati equation (TDCRE) as the solution of the time evolution operator (TEO) of quantum systems described by time-dependent (TD) Hamiltonians that are linear combinations of the generators of the $\mathfrak{su}(1,1)$, $\mathfrak{su}(2)$ and $\mathfrak{so}(2,1)$ Lie algebras. Using a recently developed solution for the time evolution of these quantum systems we solve the TDCRE recursively as generalized continued fractions, which are optimal for numerical implementations, and establish the necessary and sufficient conditions for the unitarity of the TEO in the factorized representation. The inherited symmetries of quantum systems can be recognized by a simple inspection of the TDCRE, allowing effective quantum Hamiltonians to be associated with it, as we show for the Bloch-Riccati equation whose Hamiltonian corresponds to that of a generic TD system of the Lie algebra $\mathfrak{su}(2)$. As an application, but also as a consistency test, we compare our solution with the analytic one for the Bloch-Riccati equation considering the Rabi frequency driven by a complex hyperbolic secant pulse generating spin inversion, showing an excellent agreement.

Related papers

• Schrödingerization based Quantum Circuits for Maxwell's Equation with time-dependent source terms [24.890270804373824]
  This paper explicitly constructs a quantum circuit for Maxwell's equations with perfect electric conductor (PEC) boundary conditions. We show that quantum algorithms constructed using Schr"odingerisation exhibit acceleration in computational complexity compared to the classical Finite Difference Time Domain (FDTD) format.
  arXiv  Detail & Related papers  (2024-11-17T08:15:37Z)
• Calculating response functions of coupled oscillators using quantum phase estimation [40.31060267062305]
  We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer. Our proposed quantum algorithm operates in the standard $s-sparse, oracle-based query access model. We show that a simple adaptation of our algorithm solves the random glued-trees problem in time.
  arXiv  Detail & Related papers  (2024-05-14T15:28:37Z)
• Complexity and Operator Growth for Quantum Systems in Dynamic Equilibrium [1.1868310494908512]
  Krylov complexity is a measure of operator growth in quantum systems. We show that Krylov complexity can distinguish between the $mathsfPT$-symmetric and $mathsfPT$ symmetry-broken phases. Our results demonstrate the utility of Krylov complexity as a tool to probe the properties and transitions of $mathsfPT$-symmetric systems.
  arXiv  Detail & Related papers  (2023-12-25T18:58:13Z)
• Polynomial-time Solver of Tridiagonal QUBO and QUDO problems with Tensor Networks [41.94295877935867]
  We present an algorithm for solving tridiagonal Quadratic Unconstrained Binary Optimization (QUBO) problems and Quadratic Unconstrained Discrete Optimization (QUDO) problems with one-neighbor interactions. Our method is based on the simulation of a quantum state to which we will apply an imaginary time evolution and perform a series of partial traces to obtain the state of maximum amplitude.
  arXiv  Detail & Related papers  (2023-09-19T10:45:15Z)
• A hybrid quantum-classical algorithm for multichannel quantum scattering of atoms and molecules [62.997667081978825]
  We propose a hybrid quantum-classical algorithm for solving the Schr"odinger equation for atomic and molecular collisions. The algorithm is based on the $S$-matrix version of the Kohn variational principle, which computes the fundamental scattering $S$-matrix. We show how the algorithm could be scaled up to simulate collisions of large polyatomic molecules.
  arXiv  Detail & Related papers  (2023-04-12T18:10:47Z)
• Quantum Gate Generation in Two-Level Open Quantum Systems by Coherent and Incoherent Photons Found with Gradient Search [77.34726150561087]
  We consider an environment formed by incoherent photons as a resource for controlling open quantum systems via an incoherent control. We exploit a coherent control in the Hamiltonian and an incoherent control in the dissipator which induces the time-dependent decoherence rates.
  arXiv  Detail & Related papers  (2023-02-28T07:36:02Z)
• Simultaneous Stoquasticity [0.0]
  Stoquastic Hamiltonians play a role in the computational complexity of the local Hamiltonian problem. We address the question of whether two or more Hamiltonians may be made simultaneously stoquastic via a unitary transformation.
  arXiv  Detail & Related papers  (2022-02-17T19:08:30Z)
• Nonlinear Classical and Quantum Integrable Systems with PT-symmetries [0.0]
  Key feature of integrable systems is that they can be solved to obtain exact analytical solutions. We show how new models can be constructed through generalisations of some well known nonlinear partial differential equations with PT-symmetries. We develop new methods from well-known ones to obtain exact analytical soliton solutions for these new systems.
  arXiv  Detail & Related papers  (2022-01-01T01:50:53Z)
• Algebraic Compression of Quantum Circuits for Hamiltonian Evolution [52.77024349608834]
  Unitary evolution under a time dependent Hamiltonian is a key component of simulation on quantum hardware. We present an algorithm that compresses the Trotter steps into a single block of quantum gates. This results in a fixed depth time evolution for certain classes of Hamiltonians.
  arXiv  Detail & Related papers  (2021-08-06T19:38:01Z)
• Fixed Depth Hamiltonian Simulation via Cartan Decomposition [59.20417091220753]
  We present a constructive algorithm for generating quantum circuits with time-independent depth. We highlight our algorithm for special classes of models, including Anderson localization in one dimensional transverse field XY model. In addition to providing exact circuits for a broad set of spin and fermionic models, our algorithm provides broad analytic and numerical insight into optimal Hamiltonian simulations.
  arXiv  Detail & Related papers  (2021-04-01T19:06:00Z)
• Deep reinforcement learning for complex evaluation of one-loop diagrams in quantum field theory [0.0]
  We present a technique that allows for numerical analytic continuation of integrals encountered in one-loop diagrams in quantum field theory. We train a reinforcement learning agent to perform the required contour deformations. Our study shows great promise for an agent to be deployed in iterative numerical approaches used to compute non-perturbative 2-point functions.
  arXiv  Detail & Related papers  (2019-12-27T19:45:24Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.