Quantum algorithm for linear systems of equations.

@article{Harrow2009QuantumAF,
  title={Quantum algorithm for linear systems of equations.},
  author={Aram Wettroth Harrow and Avinatan Hassidim and Seth Lloyd},
  journal={Physical review letters},
  year={2009},
  volume={103 15},
  pages={
          150502
        }
}
Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b(-->), find a vector x(-->) such that Ax(-->) = b(-->). We consider the case where one does not need to know the solution x(-->) itself, but rather an approximation of the expectation value of some operator associated with x(-->), e.g., x(-->)(dagger) Mx(-->) for some matrix M. In this case, when A is sparse, N x N and has condition… Expand

Topics from this paper

Quantum algorithms for solving linear systems of equations 1
In this course we have seen a variety of quantum algorithms that are more efficient than their classical counterparts. Some of these have been solutions to toy problems, such as the Deutsch-Josza andExpand
QUBO formulations for system of linear equations
With the advent of quantum computers, many quantum computing algorithms are being developed. Solving linear systems is one of the most fundamental problems in almost all of science andExpand
Quantum Linear System Algorithm for Dense Matrices.
TLDR
A quantum algorithm is described that achieves a sparsity-independent runtime scaling of O(κ^{2}sqrt[n]polylog(n)/ε) for an n×n dimensional A with bounded spectral norm, which amounts to a polynomial improvement over known quantum linear system algorithms when applied to dense matrices. Expand
Quantum Algorithm for Solving Tri-Diagonal Linear Systems of Equations
Numerical simulations, optimisation problems, statistical analysis and computer graphics are only a few examples from the wide range of real-life applications which rely on solving large systems ofExpand
Quantum linear systems algorithms: a primer
TLDR
The Harrow-Hassidim-Lloyd quantum algorithm for sampling from the solution of a linear system provides an exponential speed-up over its classical counterpart, and a linear solver based on the quantum singular value estimation subroutine is discussed. Expand
Quantum algorithms for the polynomial eigenvalue problems
TLDR
This work attempts to solve polynomial eigenvalue problems (PEPs) in a quantum computer using the Fourier spectral method to solve ordinary differential equations (ODEs) and provides two quantum algorithms to solve PEPs by extending the quantum algorithm for GEPs. Expand
Practical Implementation of a Quantum Algorithm for the Solution of Systems of Linear Systems of Equations
TLDR
A framework for basic Quantum Arithmetic is built providing three variants of Integer Adders, two variants ofInteger Subtracters, one Integer Multiplier and one Integer Divider, and a number of improvements and extensions of algorithms presented in the literature are given, making the described algorithms function on the QX simulator and extending features. Expand
Analysis of Classical and Quantum Resources for the Quantum Linear Systems Algorithm
  • Jon Inouye
  • Computer Science
  • 2013 10th International Conference on Information Technology: New Generations
  • 2013
TLDR
The quantum algorithm by Harrow, Hassidim, and Lloyd solves a system of N linear equations and achieves exponential speedup over classical algorithms under certain conditions and is examined to examine how classical and quantum resources interact in implementation. Expand
Improving quantum linear system solvers via a gradient descent perspective
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 inExpand
Solving systems of linear algebraic equations via unitary transformations on quantum processor of IBM Quantum Experience
TLDR
It is shown that a 5-qubit superconducting quantum processor of IBM Quantum Experience is enough to solve a system of M equations for one of the variables leaving other variables unknown, provided that the matrix of a linear system satisfies certain conditions. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 54 REFERENCES
A quantum algorithm to solve nonlinear differential equations
In this paper we describe a quantum algorithm to solve sparse systems of nonlinear differential equations whose nonlinear terms are polynomials. The algorithm is nondeterministic and its expectedExpand
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
We present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse Hamiltonian H over a given time t in terms of a procedure for computing the matrix entries of H.Expand
Limit on the Speed of Quantum Computation in Determining Parity
Consider a function f which is defined on the integers from 1 to N and takes the values {minus}1 and +1 . The parity of f is the product over all x from 1 to N of f(x) . With no further informationExpand
On the Power of Quantum Computation
TLDR
This work presents a problem of distinguishing between two fairly natural classes of functions, which can provably be solved exponentially faster in the quantum model than in the classical probabilistic one, when the function is given as an oracle drawn equiprobably from the uniform distribution on either class. Expand
Topological Quantum Computation
The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones poly-Expand
Approximating fractional time quantum evolution
An algorithm is presented for approximating the arbitrary powers of a black box unitary operation, , where t is a real number and is a black box implementing an unknown unitary. The complexity ofExpand
Algorithms for quantum computation: discrete logarithms and factoring
  • P. Shor
  • Mathematics, Computer Science
  • Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
TLDR
Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored are given. Expand
The BQP-hardness of approximating the Jones Polynomial
TLDR
The universality proof of Freedman et al (2002) is extended to ks that grow polynomially with the number of strands and crossings in the link, thus extending the BQP-hardness of Jones polynomial approximations to all values to which the AJL algorithm applies, proving that for all those values, the problems are B QP-complete. Expand
Engineering functional quantum algorithms
Suppose that a quantum circuit with K elementary gates is known for a unitary matrix U, and assume that U{sup m} is a scalar matrix for some positive integer m. We show that a function of U can beExpand
Adiabatic quantum state generation and statistical zero knowledge
TLDR
The ASG approach to quantum algorithms provides intriguing links between quantum computation and many different areas: the analysis of spectral gaps and groundstates of Hamiltonians in physics, rapidly mixing Markov chains, statistical zero knowledge, and quantum random walks. Expand
...
1
2
3
4
5
...