39 research outputs found
The quantum version of the shifted power method and its application in quadratic binary optimization
In this paper, we present a direct quantum adaptation of the classical
shifted power method. The method is very similar to the iterative phase
estimation algorithm; however, it does not require any initial estimate of an
eigenvector and as in the classical case its convergence and the required
number of iterations are directly related to the eigengap. If the amount of the
gap is in the order of , then the algorithm can converge to the
dominant eigenvalue in time. The method can be potentially used
for solving any eigenvalue related problem and finding minimum/maximum of a
quantum state in lieu of Grover's search algorithm. In addition, if the
solution space of an optimization problem with parameters is encoded as the
eigenspace of an dimensional unitary operator in time and
the eigengap is not too small, then the solution for such a problem can be
found in . As an example, using the quantum gates, we show how to
generate the solution space of the quadratic unconstrained binary optimization
as the eigenvectors of a diagonal unitary matrix and find the solution for the
problem
Improvements in Quantum SDP-Solving with Applications
Following the first paper on quantum algorithms for SDP-solving by Brandão and Svore [Brandão and Svore, 2017] in 2016, rapid developments have been made on quantum optimization algorithms. In this paper we improve and generalize all prior quantum algorithms for SDP-solving and give a simpler and unified framework. We take a new perspective on quantum SDP-solvers and introduce several new techniques. One of these is the quantum operator input model, which generalizes the different input models used in previous work, and essentially any other reasonable input model. This new model assumes that the input matrices are embedded in a block of a unitary operator. In this model we give a O~((sqrt{m}+sqrt{n}gamma)alpha gamma^4) algorithm, where n is the size of the matrices, m is the number of constraints, gamma is the reciprocal of the scale-invariant relative precision parameter, and alpha is a normalization factor of the input matrices. In particular for the standard sparse-matrix access, the above result gives a quantum algorithm where alpha=s. We also improve on recent results of Brandão et al. [Fernando G. S. L. Brandão et al., 2018], who consider the special case w
Variational Quantum Singular Value Decomposition
Singular value decomposition is central to many problems in engineering and
scientific fields. Several quantum algorithms have been proposed to determine
the singular values and their associated singular vectors of a given matrix.
Although these algorithms are promising, the required quantum subroutines and
resources are too costly on near-term quantum devices. In this work, we propose
a variational quantum algorithm for singular value decomposition (VQSVD). By
exploiting the variational principles for singular values and the Ky Fan
Theorem, we design a novel loss function such that two quantum neural networks
(or parameterized quantum circuits) could be trained to learn the singular
vectors and output the corresponding singular values. Furthermore, we conduct
numerical simulations of VQSVD for random matrices as well as its applications
in image compression of handwritten digits. Finally, we discuss the
applications of our algorithm in recommendation systems and polar
decomposition. Our work explores new avenues for quantum information processing
beyond the conventional protocols that only works for Hermitian data, and
reveals the capability of matrix decomposition on near-term quantum devices.Comment: 23 pages, v3 accepted by Quantu
Fast quantum subroutines for the simplex method
We propose quantum subroutines for the simplex method that avoid classical
computation of the basis inverse. For an constraint matrix with at
most nonzero elements per column, at most nonzero elements per column
or row of the basis, basis condition number , and optimality tolerance
, we show that pricing can be performed in
time, where the
notation hides polylogarithmic factors. If the ratio is
larger than a certain threshold, the running time of the quantum subroutine can
be reduced to . The steepest edge pivoting rule also admits a quantum
implementation, increasing the running time by a factor .
Classically, pricing requires
time in the worst case using the fastest known algorithm for sparse matrix
multiplication, and with steepest
edge. Furthermore, we show that the ratio test can be performed in
time, where
determine a feasibility tolerance; classically, this requires time in
the worst case. For well-conditioned sparse problems the quantum subroutines
scale better in and , and may therefore have a worst-case asymptotic
advantage. An important feature of our paper is that this asymptotic speedup
does not depend on the data being available in some "quantum form": the input
of our quantum subroutines is the natural classical description of the problem,
and the output is the index of the variables that should leave or enter the
basis.Comment: Added discussion on condition number and infeasibilitie
A Quantum Interior Point Method for LPs and SDPs
We present a quantum interior point method with worst case running time
for
SDPs and for LPs, where the output of our algorithm is a pair of matrices
that are -optimal -approximate SDP solutions. The factor
is at most for SDPs and for LP's, and is
an upper bound on the condition number of the intermediate solution matrices.
For the case where the intermediate matrices for the interior point method are
well conditioned, our method provides a polynomial speedup over the best known
classical SDP solvers and interior point based LP solvers, which have a worst
case running time of and respectively. Our results
build upon recently developed techniques for quantum linear algebra and pave
the way for the development of quantum algorithms for a variety of applications
in optimization and machine learning.Comment: 32 page