Boundary Treatment for Variational Quantum Simulations of Partial Differential Equations on Quantum Computers

• URL: http://arxiv.org/abs/2402.18619v2
• Date: Wed, 29 May 2024 16:52:30 GMT
• Title: Boundary Treatment for Variational Quantum Simulations of Partial Differential Equations on Quantum Computers
• Authors: Paul Over, Sergio Bengoechea, Thomas Rung, Francesco Clerici, Leonardo Scandurra, Eugene de Villiers, Dieter Jaksch,
• Abstract summary: The paper presents a variational quantum algorithm to solve initial-boundary value problems described by partial differential equations. The approach uses classical/quantum hardware that is well suited for quantum computers of the current noisy intermediate-scale quantum era.
• Score: 1.6318838452579472
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The paper presents a variational quantum algorithm to solve initial-boundary value problems described by second-order partial differential equations. The approach uses hybrid classical/quantum hardware that is well suited for quantum computers of the current noisy intermediate-scale quantum era. The partial differential equation is initially translated into an optimal control problem with a modular control-to-state operator (ansatz). The objective function and its derivatives required by the optimizer can efficiently be evaluated on a quantum computer by measuring an ancilla qubit, while the optimization procedure employs classical hardware. The focal aspect of the study is the treatment of boundary conditions, which is tailored to the properties of the quantum hardware using a correction technique. For this purpose, the boundary conditions and the discretized terms of the partial differential equation are decomposed into a sequence of unitary operations and subsequently compiled into quantum gates. The accuracy and gate complexity of the approach are assessed for second-order partial differential equations by classically emulating the quantum hardware. The examples include steady and unsteady diffusive transport equations for a scalar property in combination with various Dirichlet, Neumann, or Robin conditions. The results of this flexible approach display a robust behavior and a strong predictive accuracy in combination with a remarkable polylog complexity scaling in the number of qubits of the involved quantum circuits. Remaining challenges refer to adaptive ansatz strategies that speed up the optimization procedure.

Related papers

• Efficient Quantum Circuits for Non-Unitary and Unitary Diagonal Operators with Space-Time-Accuracy trade-offs [1.0749601922718608]
  Unitary and non-unitary diagonal operators are fundamental building blocks in quantum algorithms. We introduce a general approach to implement unitary and non-unitary diagonal operators with efficient-adjustable-depth quantum circuits.
  arXiv  Detail & Related papers  (2024-04-03T15:42:25Z)
• Near-Term Distributed Quantum Computation using Mean-Field Corrections and Auxiliary Qubits [77.04894470683776]
  We propose near-term distributed quantum computing that involve limited information transfer and conservative entanglement production. We build upon these concepts to produce an approximate circuit-cutting technique for the fragmented pre-training of variational quantum algorithms.
  arXiv  Detail & Related papers  (2023-09-11T18:00:00Z)
• 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)
• Efficient estimation of trainability for variational quantum circuits [43.028111013960206]
  We find an efficient method to compute the cost function and its variance for a wide class of variational quantum circuits. This method can be used to certify trainability for variational quantum circuits and explore design strategies that can overcome the barren plateau problem.
  arXiv  Detail & Related papers  (2023-02-09T14:05:18Z)
• Adiabatic quantum computing with parameterized quantum circuits [0.0]
  We propose a discrete version of adiabatic quantum computing that can be implemented in a near-term device. We compare our proposed algorithm with the Variational Quantum Eigensolver on two classical optimization problems.
  arXiv  Detail & Related papers  (2022-06-09T09:31:57Z)
• On optimization of coherent and incoherent controls for two-level quantum systems [77.34726150561087]
  This article considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schr"odinger equation with coherent control. The open system's dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation.
  arXiv  Detail & Related papers  (2022-05-05T09:08:03Z)
• Accelerated Convergence of Contracted Quantum Eigensolvers through a Quasi-Second-Order, Locally Parameterized Optimization [0.0]
  A contracted quantum eigensolver (CQE) finds a solution to the many-electron Schr"odinger equation on a quantum computer. In this work, we accelerate the convergence of the CQE and its wavefunction ansatz via tools from classical optimization theory.
  arXiv  Detail & Related papers  (2022-05-03T18:48:04Z)
• Quantum Kernel Methods for Solving Differential Equations [21.24186888129542]
  We propose several approaches for solving differential equations (DEs) with quantum kernel methods. We compose quantum models as weighted sums of kernel functions, where variables are encoded using feature maps and model derivatives are represented.
  arXiv  Detail & Related papers  (2022-03-16T18:56:35Z)
• Quantum algorithms for quantum dynamics: A performance study on the spin-boson model [68.8204255655161]
  Quantum algorithms for quantum dynamics simulations are traditionally based on implementing a Trotter-approximation of the time-evolution operator. variational quantum algorithms have become an indispensable alternative, enabling small-scale simulations on present-day hardware. We show that, despite providing a clear reduction of quantum gate cost, the variational method in its current implementation is unlikely to lead to a quantum advantage.
  arXiv  Detail & Related papers  (2021-08-09T18:00:05Z)
• Variational Quantum Optimization with Multi-Basis Encodings [62.72309460291971]
  We introduce a new variational quantum algorithm that benefits from two innovations: multi-basis graph complexity and nonlinear activation functions. Our results in increased optimization performance, two increase in effective landscapes and a reduction in measurement progress.
  arXiv  Detail & Related papers  (2021-06-24T20:16:02Z)
• Quantum Solver of Contracted Eigenvalue Equations for Scalable Molecular Simulations on Quantum Computing Devices [0.0]
  We introduce a quantum solver of contracted eigenvalue equations, the quantum analogue of classical methods for the energies. We demonstrate the algorithm though computations on both a quantum simulator and two IBM quantum processing units.
  arXiv  Detail & Related papers  (2020-04-23T18:35:26Z)

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.