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Quantum algorithm
for matrix functions by Cauchy's integral formula
(pp14-36)
Souichi Takahira, Asuka Ohashi, Tomohiro Sogabe, and Tsuyoshi
S. Usuda
doi:
https://doi.org/10.26421/QIC20.1-2-2
Abstracts:
For matrix A,
vector b and
function f,
the computation of vector f(A)b arises
in many scientific computing applications. We consider the problem of
obtaining quantum state |f> corresponding
to vector f(A)b.
There is a quantum algorithm to compute state |f> using
eigenvalue estimation that uses phase estimation and Hamiltonian
simulation e^{\im
A t}.
However, the algorithm based on eigenvalue estimation needs \poly(1/\epsilon) runtime,
where \epsilon is
the desired accuracy of the output state. Moreover, if matrix A is
not Hermitian, \e^{\im
A t} is
not unitary and we cannot run eigenvalue estimation. In this paper, we
propose a quantum algorithm that uses Cauchy's integral formula and the
trapezoidal rule as an approach that avoids eigenvalue estimation. We
show that the runtime of the algorithm is \poly(\log(1/\epsilon)) and
the algorithm outputs state |f> even
if A is
not Hermitian.
key words:
Quantum algorithm,
Matrix functions,
HHL algorithm, Cauchy's integral formula |
