73,743 research outputs found
Quantum annealing for systems of polynomial equations
Numerous scientific and engineering applications require numerically solving
systems of equations. Classically solving a general set of polynomial equations
requires iterative solvers, while linear equations may be solved either by
direct matrix inversion or iteratively with judicious preconditioning. However,
the convergence of iterative algorithms is highly variable and depends, in
part, on the condition number. We present a direct method for solving general
systems of polynomial equations based on quantum annealing, and we validate
this method using a system of second-order polynomial equations solved on a
commercially available quantum annealer. We then demonstrate applications for
linear regression, and discuss in more detail the scaling behavior for general
systems of linear equations with respect to problem size, condition number, and
search precision. Finally, we define an iterative annealing process and
demonstrate its efficacy in solving a linear system to a tolerance of
.Comment: 11 pages, 4 figures. Added example for a system of quadratic
equations. Supporting code is available at
https://github.com/cchang5/quantum_poly_solver . This is a post-peer-review,
pre-copyedit version of an article published in Scientific Reports. The final
authenticated version is available online at:
https://www.nature.com/articles/s41598-019-46729-
Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations
We construct quantum algorithms to compute the solution and/or physical
observables of nonlinear ordinary differential equations (ODEs) and nonlinear
Hamilton-Jacobi equations (HJE) via linear representations or exact mappings
between nonlinear ODEs/HJE and linear partial differential equations (the
Liouville equation and the Koopman-von Neumann equation). The connection
between the linear representations and the original nonlinear system is
established through the Dirac delta function or the level set mechanism. We
compare the quantum linear systems algorithms based methods and the quantum
simulation methods arising from different numerical approximations, including
the finite difference discretisations and the Fourier spectral discretisations
for the two different linear representations, with the result showing that the
quantum simulation methods usually give the best performance in time
complexity. We also propose the Schr\"odinger framework to solve the Liouville
equation for the HJE, since it can be recast as the semiclassical limit of the
Wigner transform of the Schr\"odinger equation. Comparsion between the
Schr\"odinger and the Liouville framework will also be made.Comment: quantum algorithms,linear representations,noninea
Variational quantum simulation of general processes
Variational quantum algorithms have been proposed to solve static and dynamic
problems of closed many-body quantum systems. Here we investigate variational
quantum simulation of three general types of tasks---generalised time evolution
with a non-Hermitian Hamiltonian, linear algebra problems, and open quantum
system dynamics. The algorithm for generalised time evolution provides a
unified framework for variational quantum simulation. In particular, we show
its application in solving linear systems of equations and matrix-vector
multiplications by converting these algebraic problems into generalised time
evolution. Meanwhile, assuming a tensor product structure of the matrices, we
also propose another variational approach for these two tasks by combining
variational real and imaginary time evolution. Finally, we introduce
variational quantum simulation for open system dynamics. We variationally
implement the stochastic Schr\"odinger equation, which consists of dissipative
evolution and stochastic jump processes. We numerically test the algorithm with
a six-qubit 2D transverse field Ising model under dissipation.Comment: 18 page
Hybrid algorithms to solve linear systems of equations with limited qubit resources
The solution of linear systems of equations is a very frequent operation and
thus important in many fields. The complexity using classical methods increases
linearly with the size of equations. The HHL algorithm proposed by Harrow et
al. achieves exponential acceleration compared with the best classical
algorithm. However, it has a relatively high demand for qubit resources and the
solution is in a normalized form. Assuming that the
eigenvalues of the coefficient matrix of the linear systems of equations can be
represented perfectly by finite binary number strings, three hybrid iterative
phase estimation algorithms (HIPEA) are designed based on the iterative phase
estimation algorithm in this paper. The complexity is transferred to the
measurement operation in an iterative way, and thus the demand of qubit
resources is reduced in our hybrid algorithms. Moreover, the solution is stored
in a classical register instead of a quantum register, so the exact
unnormalized solution can be obtained. The required qubit resources in the
three HIPEA algorithms are different. HIPEA-1 only needs one single ancillary
qubit. The number of ancillary qubits in HIPEA-2 is equal to the number of
nondegenerate eigenvalues of the coefficient matrix of linear systems of
equations. HIPEA-3 is designed with a flexible number of ancillary qubits. The
HIPEA algorithms proposed in this paper broadens the application range of
quantum computation in solving linear systems of equations by avoiding the
problem that quantum programs may not be used to solve linear systems of
equations due to the lack of qubit resources.Comment: 22 pages, 6 figures, 6 tables, 48 equation
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