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

Numerical evaluation of the fidelity error threshold for the surface code

We study how the resilience of the surface code is affected by the coupling to a non-Markovian environment at zero temperature. The qubits in the surface code experience an effective dynamics due to the coupling to the environment that induces correlations among them. The range of the effective induced qubit-qubit interaction depends on parameters related to the environment and the duration of the quantum error correction cycle. We show numerically that different interaction ranges set different intrinsic bounds on the fidelity of the code. These bounds are unrelated to the error thresholds based on stochastic error models. We introduce a definition of stabilizers based on logical operators that allows us to efficiently implement a Metropolis algorithm to determine upper bounds to the fidelity error threshold

A microscopic formulation of dynamical spin injection in ferromagnetic-nonmagnetic heterostructures

We develop a microscopic formulation of dynamical spin injection in heterostructure comprising nonmagnetic metals in contact with ferromagnets. The spin pumping current is expressed in terms of Green's function of the nonmagnetic metal attached to the ferromagnet where a precessing magnetization is induced. The formulation allows for the inclusion of spin-orbit coupling and disorder. The Green's functions involved in the expression for the current are expressed in real-space lattice coordinates and can thus be efficiently computed using recursive methods.Comment: 18 pages, 6 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

Disorder and Electronic Transport in Graphene

In this review, we provide an account of the recent progress in understanding electronic transport in disordered graphene systems. Starting from a theoretical description that emphasizes the role played by band structure properties and lattice symmetries, we describe the nature of disorder in these systems and its relation to transport properties. While the focus is primarily on theoretical and conceptual aspects, connections to experiments are also included. Issues such as short versus long-range disorder, localization (strong and weak), the carrier density dependence of the conductivity, and conductance fluctuations are considered and some open problems are pointed out.Comment: 18 pages, 5 figures, Topical Revie

The recursive Green's function method for graphene

We describe how to apply the recursive Green's function method to the computation of electronic transport properties of graphene sheets and nanoribbons in the linear response regime. This method allows for an amenable inclusion of several disorder mechanisms at the microscopic level, as well as inhomogeneous gating, finite temperature, and, to some extend, dephasing. We present algorithms for computing the conductance, density of states, and current densities for armchair and zigzag atomic edge alignments. Several numerical results are presented to illustrate the usefulness of the method.Comment: 26 pages, 15 figures; submitted to Journal of Computational Electronics (special issue on graphene

The single-electron transport in a three-ion magnetic molecule modulated by a transverse field

We study single-electron transport in a three-ion molecule with strong uniaxial anisotropy and in the presence of a transverse magnetic field. Two magnetic ions are connected to each other through a third, nonmagnetic ion. The magnetic ions are coupled to ideal metallic leads and a back gate voltage is applied to the molecule, forming a field-effect transistor. The microscopic Hamiltonian describing this system includes inter-ion hopping, on-site repulsions, and magnetic anisotropies. For a range of values of the parameters of the Hamiltonian, we obtain an energy spectrum similar to that of single-molecule magnets in the giant-spin approximation where the two states with maximum spin projection along the uniaxial anisotropy axis are well separated from other states. In addition, upon applying an external in-plane magnetic field, the energy gap between the ground and first excited states of the molecule oscillates, going to zero at certain special values of the field, in analogy to the diabolical points resulting from Berry phase interference in the giant spin model. Thus, our microscopic model provides the same phenomenological behavior expected from the giant spin model of a single-molecule magnet but with direct access to the internal structure of the molecule, thus making it more appropriate for realistic electronic transport studies. To illustrate this point, the nonlinear electronic transport in the sequential tunneling regime is evaluated for values of the field near these degeneracy points. We show that the existence of these points has a clear signature in the I-V characteristics of the molecule, most notably the modulation of excitation lines in the differential conductance.Comment: 10 pages, 13 figure

Emergent irreversibility and entanglement spectrum statistics

We study the problem of irreversibility when the dynamical evolution of a many-body system is described by a stochastic quantum circuit. Such evolution is more general than a Hamiltonian one, and since energy levels are not well defined, the well-established connection between the statistical fluctuations of the energy spectrum and irreversibility cannot be made. We show that the entanglement spectrum provides a more general connection. Irreversibility is marked by a failure of a disentangling algorithm and is preceded by the appearance of Wigner-Dyson statistical fluctuations in the entanglement spectrum. This analysis can be done at the wave-function level and offers an alternative route to study quantum chaos and quantum integrability.Comment: updated to published versio

Phonon Decoherence of a Double Quantum Dot Charge Qubit

We study decoherence of a quantum dot charge qubit due to coupling to piezoelectric acoustic phonons in the Born-Markov approximation. After including appropriate form factors, we find that phonon decoherence rates are one to two orders of magnitude weaker than was previously predicted. We calculate the dependence of the Q-factor on lattice temperature, quantum dot size, and interdot coupling. Our results suggest that mechanisms other than phonon decoherence play a more significant role in current experimental setups.Comment: RevTex, 7 pages, 5 figures. v2: appendix added, more details provided. Accepted for publication in PR

Surface code fidelity at finite temperatures

We study the dependence of the fidelity of the surface code in the presence of a single finite-temperature massless bosonic environment after a quantum error correction cycle. The three standard types of environment are considered: super-Ohmic, Ohmic, and sub-Ohmic. Our results show that, for regimes relevant to current experiments, quantum error correction works well even in the presence of environment-induced, long-range inter-qubit interactions. A threshold always exists at finite temperatures, although its temperature dependence is very sensitive to the type of environment. For the super-Ohmic case, the critical coupling constant separating high- from low-fidelity decreases with increasing temperature. For both Ohmic and super-Ohmic cases, the dependence of the critical coupling on temperature is weak. In all cases, the critical coupling is determined by microscopic parameters of the environment. For the sub-Ohmic case, it also depends strongly on the duration of the QEC cycle.Comment: 13 pages, 6 figure