Depth analysis of variational quantum algorithms for heat equation

• URL: http://arxiv.org/abs/2212.12375v2
• Date: Fri, 5 May 2023 09:08:42 GMT
• Title: Depth analysis of variational quantum algorithms for heat equation
• Authors: N. M. Guseynov, A. A. Zhukov, W. V. Pogosov, A.V. Lebedev
• Abstract summary: We consider three approaches to solve the heat equation on a quantum computer. An exponential number of Pauli products in the Hamiltonian decomposition does not allow for the quantum speed up to be achieved. The ansatz tree approach exploits an explicit form of the matrix what makes it possible to achieve an advantage over classical algorithms.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Variational quantum algorithms are a promising tool for solving partial differential equations. The standard approach for its numerical solution are finite difference schemes, which can be reduced to the linear algebra problem. We consider three approaches to solve the heat equation on a quantum computer. Using the direct variational method we minimize the expectation value of a Hamiltonian with its ground state being the solution of the problem under study. Typically, an exponential number of Pauli products in the Hamiltonian decomposition does not allow for the quantum speed up to be achieved. The Hadamard test based approach solves this problem, however, the performed simulations do not evidently prove that the ansatz circuit has a polynomial depth with respect to the number of qubits. The ansatz tree approach exploits an explicit form of the matrix what makes it possible to achieve an advantage over classical algorithms. In our numerical simulations with up to $n=11$ qubits, this method reveals the exponential speed up.

Related papers

• Quantum Algorithm For Solving Nonlinear Algebraic Equations [0.0]
  We build upon the classical Newton method and some recent works on quantum algorithm plus block encoding from quantum singular value transformation. A detailed analysis are then carried out to reveal that our method polylogarithmic time in relative to the number of variables. In particular, we show that our method can be modified with little effort to deal with various types, thus implying the generality of our approach.
  arXiv  Detail & Related papers  (2024-04-04T21:20:56Z)
• Hamiltonian simulation for time-evolving partial differential equation by scalable quantum circuits [1.7453899104963828]
  Hamiltonian simulation is a potential and promising approach to achieve this purpose. This paper presents a method that enables us to explicitly implement the quantum circuit for Hamiltonian simulation. We show that the space and time complexity of the constructed circuit is exponentially smaller than that of all classical algorithms.
  arXiv  Detail & Related papers  (2024-02-28T15:17:41Z)
• 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 Algorithm For Estimating Eigenvalue [0.0]
  We provide a quantum algorithm for estimating the largest eigenvalue in magnitude of a given Hermitian matrix. Our quantum procedure can also yield exponential speedup compared to classical algorithms that solve the same problem.
  arXiv  Detail & Related papers  (2022-11-11T13:02:07Z)
• Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints [62.72309460291971]
  We introduce a variational quantum algorithm for Goemans-Williamson algorithm that uses only $n+1$ qubits. Efficient optimization is achieved by encoding the objective matrix as a properly parameterized unitary conditioned on an auxilary qubit. We demonstrate the effectiveness of our protocol by devising an efficient quantum implementation of the Goemans-Williamson algorithm for various NP-hard problems.
  arXiv  Detail & Related papers  (2022-06-30T03:15:23Z)
• Near-term quantum algorithm for computing molecular and materials properties based on recursive variational series methods [44.99833362998488]
  We propose a quantum algorithm to estimate the properties of molecules using near-term quantum devices. We test our method by computing the one-particle Green's function in the energy domain and the autocorrelation function in the time domain.
  arXiv  Detail & Related papers  (2022-06-20T16:33:23Z)
• Alternatives to a nonhomogeneous partial differential equation quantum algorithm [52.77024349608834]
  We propose a quantum algorithm for solving nonhomogeneous linear partial differential equations of the form $Apsi(textbfr)=f(textbfr)$. These achievements enable easier experimental implementation of the quantum algorithm based on nowadays technology.
  arXiv  Detail & Related papers  (2022-05-11T14:29:39Z)
• Adiabatic Quantum Graph Matching with Permutation Matrix Constraints [75.88678895180189]
  Matching problems on 3D shapes and images are frequently formulated as quadratic assignment problems (QAPs) with permutation matrix constraints, which are NP-hard. We propose several reformulations of QAPs as unconstrained problems suitable for efficient execution on quantum hardware. The proposed algorithm has the potential to scale to higher dimensions on future quantum computing architectures.
  arXiv  Detail & Related papers  (2021-07-08T17:59:55Z)
• Variational quantum algorithm based on the minimum potential energy for solving the Poisson equation [7.620967781722716]
  We present a variational quantum algorithm for solving the Poisson equation. The proposed method defines the total potential energy of the Poisson equation as a Hamiltonian. Because the number of terms is independent of the size of the problem, this method requires relatively few quantum measurements.
  arXiv  Detail & Related papers  (2021-06-17T09:01:53Z)
• Synthesis of Quantum Circuits with an Island Genetic Algorithm [44.99833362998488]
  Given a unitary matrix that performs certain operation, obtaining the equivalent quantum circuit is a non-trivial task. Three problems are explored: the coin for the quantum walker, the Toffoli gate and the Fredkin gate. The algorithm proposed proved to be efficient in decomposition of quantum circuits, and as a generic approach, it is limited only by the available computational power.
  arXiv  Detail & Related papers  (2021-06-06T13:15:25Z)
• Combinatorial optimization through variational quantum power method [0.0]
  We present a variational quantum circuit method for the power iteration. It can be used to find the eigenpairs of unitary matrices and so their associated Hamiltonians. The circuit can be simulated on the near term quantum computers with ease.
  arXiv  Detail & Related papers  (2020-07-02T10:34:16Z)

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.