• Corpus ID: 237452490

Improving quantum linear system solvers via a gradient descent perspective

  title={Improving quantum linear system solvers via a gradient descent perspective},
  author={Sander Gribling and Iordanis Kerenidis and D'aniel Szil'agyi},
Solving systems of linear equations is one of the most important primitives in quantum computing that has the potential to provide a practical quantum advantage in many different areas, including in optimization, simulation, and machine learning. In this work, we revisit quantum linear system solvers from the perspective of convex optimization, and in particular gradient descent-type algorithms. This leads to a considerable constant-factor improvement in the runtime (or, conversely, a several… 

Figures and Tables from this paper

A Survey of Quantum Computing for Finance

A comprehensive summary of the state of the art of quantum computing for financial applications, with particular emphasis on stochastic modeling, optimization, and machine learning, describing how these solutions, adapted to work on a quantum computer, can potentially help to solve financial problems more efficiently and accurately.

Fourier-based quantum signal processing

An algorithm for Hermitian-operator function design from an oracle given by the unitary evolution with respect to that operator at a fixed time is presented, which implements a Fourier approximation of the target function based on the iteration of a basic sequence of single-qubit gates, for which it is proved the expressibility.



Quantum algorithms for Second-Order Cone Programming and Support Vector Machines

Experimental evidence is presented that in this case the quantum algorithm exhibits a polynomial speedup over the best classical algorithms for solving general SOCPs that run in time O(nω+0.5) (here, ω is the matrix multiplication exponent, with a value of roughly 2.37 in theory, and up to 3 in practice).

Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions

A quantum primitive called fast inversion is introduced, which can be used as a preconditioner for solving quantum linear systems, and two efficient approaches for computing matrix functions, based on the contour integral formulation and the inverse transform respectively are introduced.

Quantum algorithm for linear systems of equations.

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 singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

A new “Quantum singular value transformation” algorithm is developed that can directly harness the advantages of exponential dimensionality by applying polynomial transformations to the singular values of a block of a unitary operator.

Efficient phase-factor evaluation in quantum signal processing

This work presents an optimization based method that can accurately compute the phase factors using standard double precision arithmetic operations and demonstrates the performance of this approach with applications to Hamiltonian simulation, eigenvalue filtering, and the quantum linear system problems.

Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision

The algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation, and allows the quantum phase estimation algorithm, whose dependence on $\epsilon$ is prohibitive, to be bypassed.

Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters

An algorithm for sparse Hamiltonian simulation whose complexity is optimal (up to log factors) as a function of all parameters of interest is presented, and a new lower bound is proved showing that no algorithm can have sub linear dependence on tau.

The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation

The framework of block-encoding, introduced by Low and Chuang, is applied to the study of quantum machine learning algorithms and general results are derived that are applicable to a variety of input models, including sparse matrix oracles and matrices stored in a data structure.

Quantum lower bounds by polynomials

This work examines the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model and gives asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings.

Variable time amplitude amplification and quantum algorithms for linear algebra problems

This paper generalizes quantum amplitude amplification to the case when parts of the algorithm that is being amplified stop at different times, and applies it to give two new quantum algorithms for linear algebra problems.