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… 

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References

SHOWING 1-10 OF 39 REFERENCES

A quantum algorithm to solve nonlinear differential equations

A quantum algorithm to solve sparse systems of nonlinear differential equations whose nonlinear terms are polynomials and its expected resource requirements are polylogarithmic in the number of variables and exponential in the integration time.

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.

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-

On the power of quantum computation

  • Daniel R. Simon
  • Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
This work presents here a problem of distinguishing between two fairly natural classes of function, 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.

Algorithms for quantum computation: discrete logarithms and factoring

  • P. Shor
  • Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
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.

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 of

The BQP-hardness of approximating the Jones polynomial

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.

Engineering functional quantum algorithms

This work shows that a function of U can be realized on a quantum computer with at most O(mK+m{sup 2}ln m) elementary gates, and obtains efficient circuits for the fractional Fourier transform.

Space-Bounded Quantum Complexity

  • J. Watrous
  • Computer Science
    J. Comput. Syst. Sci.
  • 1999
It is shown that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in power and, furthermore, that any QTM running in space s can be simulated deterministically in NC2(2s)?DSPACE(s2)?DTIME(2O(s).

A scheme for efficient quantum computation with linear optics

It is shown that efficient quantum computation is possible using only beam splitters, phase shifters, single photon sources and photo-detectors and are robust against errors from photon loss and detector inefficiency.