Commutative version of the local Hamiltonian problem and common
eigenspace problem (pp187-215)
Sergey Bravyi and Mikhail Vyalyi
doi:
https://doi.org/10.26421/QIC5.3-2
Abstracts:
We study the complexity of a problem Common
Eigenspace ---
verifying consistency of eigenvalue equations for composite quantum
systems. The input of the problem is a family of pairwise commuting
Hermitian operators H_1,\ldots,H_r on a Hilbert space (\CC^d)^{\otimes
n} and a string of real numbers \lambda=(\lambda_1,\ldots,\lambda_r).
The problem is to determine whether the common eigenspace specified by
equalities H_a|\psi\ra=\lambda_a|\psi\ra, a=1,\ldots,r has a positive
dimension. We consider two cases: (i) all operators H_a are k-local;
(ii) all operators H_a are factorized. It can be easily shown that both
problems belong to the class \QMA --- quantum analogue of \NP, and that
some \NP-complete problems can be reduced to either (i) or (ii). A
non-trivial question is whether the problems (i) or (ii) belong to \NP?
We show that the answer is positive for some special values of k and d.
Also we prove that the problem (ii) can be reduced to its special case,
such that all operators H_a are factorized projectors and all \lambda_a=0.
Key words:
Quantum complexity, quantum codes,
multipartite entanglement |