340,231 research outputs found
Universality of Entanglement and Quantum Computation Complexity
We study the universality of scaling of entanglement in Shor's factoring
algorithm and in adiabatic quantum algorithms across a quantum phase transition
for both the NP-complete Exact Cover problem as well as the Grover's problem.
The analytic result for Shor's algorithm shows a linear scaling of the entropy
in terms of the number of qubits, therefore difficulting the possibility of an
efficient classical simulation protocol. A similar result is obtained
numerically for the quantum adiabatic evolution Exact Cover algorithm, which
also shows universality of the quantum phase transition the system evolves
nearby. On the other hand, entanglement in Grover's adiabatic algorithm remains
a bounded quantity even at the critical point. A classification of scaling of
entanglement appears as a natural grading of the computational complexity of
simulating quantum phase transitions.Comment: 30 pages, 17 figures, accepted for publication in PR
Phase transition for cutting-plane approach to vertex-cover problem
We study the vertex-cover problem which is an NP-hard optimization problem
and a prototypical model exhibiting phase transitions on random graphs, e.g.,
Erdoes-Renyi (ER) random graphs. These phase transitions coincide with changes
of the solution space structure, e.g, for the ER ensemble at connectivity
c=e=2.7183 from replica symmetric to replica-symmetry broken. For the
vertex-cover problem, also the typical complexity of exact branch-and-bound
algorithms, which proceed by exploring the landscape of feasible
configurations, change close to this phase transition from "easy" to "hard". In
this work, we consider an algorithm which has a completely different strategy:
The problem is mapped onto a linear programming problem augmented by a
cutting-plane approach, hence the algorithm operates in a space OUTSIDE the
space of feasible configurations until the final step, where a solution is
found. Here we show that this type of algorithm also exhibits an "easy-hard"
transition around c=e, which strongly indicates that the typical hardness of a
problem is fundamental to the problem and not due to a specific representation
of the problem.Comment: 4 pages, 3 figure
Adiabatic quantum computation and quantum phase transitions
We analyze the ground state entanglement in a quantum adiabatic evolution
algorithm designed to solve the NP-complete Exact Cover problem. The entropy of
entanglement seems to obey linear and universal scaling at the point where the
mass gap becomes small, suggesting that the system passes near a quantum phase
transition. Such a large scaling of entanglement suggests that the effective
connectivity of the system diverges as the number of qubits goes to infinity
and that this algorithm cannot be efficiently simulated by classical means. On
the other hand, entanglement in Grover's algorithm is bounded by a constant.Comment: 5 pages, 4 figures, accepted for publication in PR
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