High order schemes for solving partial differential equations on a quantum computer

• URL: http://arxiv.org/abs/2412.19232v1
• Date: Thu, 26 Dec 2024 14:21:59 GMT
• Title: High order schemes for solving partial differential equations on a quantum computer
• Authors: Boris Arseniev, Dmitry Guskov, Richik Sengupta, Igor Zacharov,
• Abstract summary: We show that higher-order methods can reduce the number of qubits necessary for discretization, similar to the classical case. This result has important consequences for the practical application of quantum algorithms based on Hamiltonian evolution.
• Score: 0.0
• License:
• Abstract: We explore the utilization of higher-order discretization techniques in optimizing the gate count needed for quantum computer based solutions of partial differential equations. To accomplish this, we present an efficient approach for decomposing $d$-band diagonal matrices into Pauli strings that are grouped into mutually commuting sets. Using numerical simulations of the one-dimensional wave equation, we show that higher-order methods can reduce the number of qubits necessary for discretization, similar to the classical case, although they do not decrease the number of Trotter steps needed to preserve solution accuracy. This result has important consequences for the practical application of quantum algorithms based on Hamiltonian evolution.

Related papers

• Quantum Algorithms for Stochastic Differential Equations: A Schrödingerisation Approach [29.662683446339194]
  We propose quantum algorithms for linear differential equations. The gate complexity of our algorithms exhibits an $mathcalO(dlog(Nd))$ dependence on the dimensions. The algorithms are numerically verified for the Ornstein-Uhlenbeck processes, Brownian motions, and one-dimensional L'evy flights.
  arXiv  Detail & Related papers  (2024-12-19T14:04:11Z)
• Double-Logarithmic Depth Block-Encodings of Simple Finite Difference Method's Matrices [0.0]
  Solving differential equations is one of the most computationally expensive problems in classical computing. Despite recent progress made in the field of quantum computing and quantum algorithms, its end-to-end application towards practical realization still remains unattainable.
  arXiv  Detail & Related papers  (2024-10-07T17:44:30Z)
• 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)
• Boundary Treatment for Variational Quantum Simulations of Partial Differential Equations on Quantum Computers [1.6318838452579472]
  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.
  arXiv  Detail & Related papers  (2024-02-28T18:19:33Z)
• Hybrid quantum-classical and quantum-inspired classical algorithms for solving banded circulant linear systems [0.8192907805418583]
  We present an efficient algorithm based on convex optimization of combinations of quantum states to solve for banded circulant linear systems. By decomposing banded circulant matrices into cyclic permutations, our approach produces approximate solutions to such systems with a combination of quantum states linear to $K$. We validate our methods with classical simulations and actual IBM quantum computer implementation, showcasing their applicability for solving physical problems such as heat transfer.
  arXiv  Detail & Related papers  (2023-09-20T16:27:16Z)
• Solving Systems of Linear Equations: HHL from a Tensor Networks Perspective [39.58317527488534]
  We present an algorithm for solving systems of linear equations based on the HHL algorithm with a novel qudits methodology. We perform a quantum-inspired version on tensor networks, taking advantage of their ability to perform non-unitary operations such as projection.
  arXiv  Detail & Related papers  (2023-09-11T08:18:41Z)
• Efficient DCQO Algorithm within the Impulse Regime for Portfolio Optimization [41.94295877935867]
  We propose a faster digital quantum algorithm for portfolio optimization using the digitized-counterdiabatic quantum optimization (DCQO) paradigm. Our approach notably reduces the circuit depth requirement of the algorithm and enhances the solution accuracy, making it suitable for current quantum processors. We experimentally demonstrate the advantages of our protocol using up to 20 qubits on an IonQ trapped-ion quantum computer.
  arXiv  Detail & Related papers  (2023-08-29T17:53:08Z)
• Noisy Tensor Ring approximation for computing gradients of Variational Quantum Eigensolver for Combinatorial Optimization [33.12181620473604]
  Variational Quantum algorithms have established their potential to provide computational advantage in the realm of optimization. These algorithms suffer from classically intractable gradients limiting the scalability. This work proposes a classical gradient method which utilizes the parameter shift rule but computes the expected values from the circuits using a tensor ring approximation.
  arXiv  Detail & Related papers  (2023-07-08T03:14:28Z)
• Numerical Simulations of Noisy Quantum Circuits for Computational Chemistry [51.827942608832025]
  Near-term quantum computers can calculate the ground-state properties of small molecules. We show how the structure of the computational ansatz as well as the errors induced by device noise affect the calculation.
  arXiv  Detail & Related papers  (2021-12-31T16:33:10Z)
• 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)

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.