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Quantum algorithm for non-homogeneous linear partial differential equations
Quantum algorithm for non-homogeneous linear partial differential equations
- Publication Year :
- 2018
- Publisher :
- arXiv, 2018.
-
Abstract
- We describe a quantum algorithm for preparing states that encode solutions of non-homogeneous linear partial differential equations. The algorithm is a continuous-variable version of matrix inversion: it efficiently inverts differential operators that are polynomials in the variables and their partial derivatives. The output is a quantum state whose wavefunction is proportional to a specific solution of the non-homogeneous differential equation, which can be measured to reveal features of the solution. The algorithm consists of three stages: preparing fixed resource states in ancillary systems, performing Hamiltonian simulation, and measuring the ancilla systems. The algorithm can be carried out using standard methods for gate decompositions, but we improve this in two ways. First, we show that for a wide class of differential operators, it is possible to derive exact decompositions for the gates employed in Hamiltonian simulation. This avoids the need for costly commutator approximations, reducing gate counts by orders of magnitude. Additionally, we employ methods from machine learning to find explicit circuits that prepare the required resource states. We conclude by studying an example application of the algorithm: solving Poisson's equation in electrostatics.<br />Comment: 9 pages, 6 figures
- Subjects :
- Physics
Quantum Physics
Differential equation
FOS: Physical sciences
Poisson distribution
Differential operator
01 natural sciences
010305 fluids & plasmas
symbols.namesake
Quantum state
0103 physical sciences
symbols
Partial derivative
Applied mathematics
Quantum algorithm
010306 general physics
Wave function
Hamiltonian (quantum mechanics)
Quantum Physics (quant-ph)
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....8eb90c4ea33ca91644402de9536ca64c
- Full Text :
- https://doi.org/10.48550/arxiv.1809.02622