Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
Quantum computation and quantum information
Quantum computation and quantum information
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Succinct quantum proofs for properties of finite groups
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Classical and Quantum Computation
Classical and Quantum Computation
The Complexity of the Local Hamiltonian Problem
SIAM Journal on Computing
3-local Hamitonian is QMA-complete
Quantum Information & Computation
Adptive quantum computation, constant depth quantum circuits and arthur-merlin games
Quantum Information & Computation
Consistency of local density matrices is QMA-Complete
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Complexity of commuting Hamiltonians on a square lattice of qubits
Quantum Information & Computation
A note about a partial no-go theorem for quantum PCP
Quantum Information & Computation
Commuting quantum circuits: efficient classical simulations versus hardness results
Quantum Information & Computation
Product-state approximations to quantum ground states
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Guest column: the quantum PCP conjecture
ACM SIGACT News
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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 H1,..., Hr on a Hilbert space (Cd)⊗n and a string of real numbers λ =(λ1,...,λr). The problem is to determine whether the common eigenspace specified by equalities Ha|Ø〉 = λa|Ø〉, a = 1, ..., r has a positive dimension. We consider two cases: (i) all operators Ha are k-local; (ii) all operators Ha 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 Ha are factorized projectors and all λa = 0.