Quantum algorithm for linear differential equations with exponentially improved dependence on precision
TL;DR: This work presents a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients that produces a quantum state that is proportional to the solution at a desired final time using a Taylor series.
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Abstract: We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability.
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Citations
Efficient quantum algorithm for dissipative nonlinear differential equations.
Jin-Peng Liu,Herman Oie Kolden,Herman Oie Kolden,Hari Krovi,Nuno Loureiro,Konstantina Trivisa,Andrew M. Childs +6 more
TL;DR: A lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations is provided, showing that the problem is intractable for $R \ge \sqrt{2}$.
217
Quantum gradient descent for linear systems and least squares
TL;DR: In this article, a quantum method for performing gradient descent when the gradient is an affine function is proposed, which can be used for solving positive semidefinite linear systems and for stochastic gradient descent for the weighted least-squares problem with reduced quantum memory requirements.
Solving nonlinear differential equations with differentiable quantum circuits
TL;DR: A hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions is described, and how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space is shown.
Quantum algorithm for simulating the wave equation
TL;DR: In this paper, a quantum algorithm for simulating the wave equation under Dirichlet and Neumann boundary conditions is presented, which uses Hamiltonian simulation and quantum linear system algorithms as subroutines.
127
Time-dependent Hamiltonian simulation with $L^1$-norm scaling
Dominic W. Berry,Andrew M. Childs,Yuan Su,Xin Wang,Xin Wang,Nathan Wiebe,Nathan Wiebe,Nathan Wiebe +7 more
- 20 Apr 2020
TL;DR: Two new techniques are introduced: a classical sampler of time-dependent Hamiltonians and a rescaling principle for the Schrodinger equation that is nearly optimal with respect to all parameters of interest, whereas the sampling-based approach is easier to realize for near-term simulation.
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Universal Quantum Simulators
TL;DR: Feynman's 1982 conjecture, that quantum computers can be programmed to simulate any local quantum system, is shown to be correct.
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TL;DR: This work exhibits a quantum algorithm for estimating x(-->)(dagger) Mx(-->) whose runtime is a polynomial of log(N) and kappa, and proves that any classical algorithm for this problem generically requires exponentially more time than this quantum algorithm.
Quantum Amplitude Amplification and Estimation
TL;DR: In this article, the amplitude amplification algorithm was proposed to find a good solution after an expected number of applications of the algorithm and its inverse which is proportional to a factor proportional to 1/a.
Optimal Hamiltonian Simulation by Quantum Signal Processing.
Guang Hao Low,Isaac L. Chuang +1 more
TL;DR: It is argued that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation.
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Hamiltonian Simulation by Qubitization
Guang Hao Low,Isaac L. Chuang +1 more
- 12 Jul 2019
TL;DR: The Hamiltonian is presented, where the Hamiltonian of a unit is the cause of error and the time-evolution operator is approximate.
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