211 research outputs found

### Exact results on the quench dynamics of the entanglement entropy in the toric code

We study quantum quenches in the two-dimensional Kitaev toric code model and
compute exactly the time-dependent entanglement entropy of the non-equilibrium
wave-function evolving from a paramagnetic initial state with the toric code
Hamiltonian. We find that the area law survives at all times. Adding disorder
to the toric code couplings makes the entanglement entropy per unit boundary
length saturate to disorder-independent values at long times and in the
thermodynamic limit. There are order-one corrections to the area law from the
corners in the subsystem boundary but the topological entropy remains zero at
all times. We argue that breaking the integrability with a small magnetic field
could change the area law to a volume scaling as expected of thermalized states
but is not sufficient for forming topological entanglement due to the presence
of an excess energy and a finite density of defects.Comment: 14 pages, 7 figures, published versio

### Optimal control for unitary preparation of many-body states: application to Luttinger liquids

Many-body ground states can be prepared via unitary evolution in cold atomic
systems. Given the initial state and a fixed time for the evolution, how close
can we get to a desired ground state if we can tune the Hamiltonian in time?
Here we study this optimal control problem focusing on Luttinger liquids with
tunable interactions. We show that the optimal protocol can be obtained by
simulated annealing. We find that the optimal interaction strength of the
Luttinger liquid can have a nonmonotonic time dependence. Moreover, the system
exhibits a marked transition when the ratio $\tau/L$ of the preparation time to
the system size exceeds a critical value. In this regime, the optimal protocols
can prepare the states with almost perfect accuracy. The optimal protocols are
robust against dynamical noise.Comment: 4 pages, 4 figures, extended results on robustness, to appear in PR

### Renyi entropies as a measure of the complexity of counting problems

Counting problems such as determining how many bit strings satisfy a given
Boolean logic formula are notoriously hard. In many cases, even getting an
approximate count is difficult. Here we propose that entanglement, a common
concept in quantum information theory, may serve as a telltale of the
difficulty of counting exactly or approximately. We quantify entanglement by
using Renyi entropies S(q), which we define by bipartitioning the logic
variables of a generic satisfiability problem. We conjecture that
S(q\rightarrow 0) provides information about the difficulty of counting
solutions exactly, while S(q>0) indicates the possibility of doing an efficient
approximate counting. We test this conjecture by employing a matrix computing
scheme to numerically solve #2SAT problems for a large number of uniformly
distributed instances. We find that all Renyi entropies scale linearly with the
number of variables in the case of the #2SAT problem; this is consistent with
the fact that neither exact nor approximate efficient algorithms are known for
this problem. However, for the negated (disjunctive) form of the problem,
S(q\rightarrow 0) scales linearly while S(q>0) tends to zero when the number of
variables is large. These results are consistent with the existence of fully
polynomial-time randomized approximate algorithms for counting solutions of
disjunctive normal forms and suggests that efficient algorithms for the
conjunctive normal form may not exist.Comment: 13 pages, 4 figure

### Virtual Parallel Computing and a Search Algorithm using Matrix Product States

We propose a form of parallel computing on classical computers that is based
on matrix product states. The virtual parallelization is accomplished by
representing bits with matrices and by evolving these matrices from an initial
product state that encodes multiple inputs. Matrix evolution follows from the
sequential application of gates, as in a logical circuit. The action by
classical probabilistic one-bit and deterministic two-bit gates such as NAND
are implemented in terms of matrix operations and, as opposed to quantum
computing, it is possible to copy bits. We present a way to explore this method
of computation to solve search problems and count the number of solutions. We
argue that if the classical computational cost of testing solutions (witnesses)
requires less than $O(n^2)$ local two-bit gates acting on $n$ bits, the search
problem can be fully solved in subexponential time. Therefore, for this
restricted type of search problem, the virtual parallelization scheme is faster
than Grover's quantum algorithmComment: 4 pages, 1 figure (published version

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