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A very fundamental problem in quantum statistical mechanics involves whether--and how--an isolated quantum system will thermalize at long times. In quantum systems that do thermalize, the long-time expectation value of any "reasonable" operator will match its predicted value in the canonical ensemble. The Eigenstate Thermalization Hypothesis (ETH) posits that this thermalization occurs at the level of each individual energy eigenstate; in fact, any single eigenstate in a microcanonical energy window will predict the expectation values of such operators exactly.

The momentum-space entanglement properties of several quantum spin chains are investigated. More specifically, we study the entanglement spectra, i.e. the set of eigenvalues of the reduced density matrix, between left-and right-moving particles in bosonic and fermionic formulations of quantum spin chains. We elaborate on how momentum-space entanglement spectra may support the numerical study of phase transitions and classify certain critical systems.

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The spin of a single electron confined in a quantum dot is a promising matter qubit for quantum information processing. This spin system possesses microsecond coherence time and allows picosecond timescale control using optical pulses. It is also embedded in a host semiconductor substrate that can be directly patterned to form compact integrated nanophotonic devices for photonic interfaces.

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I will discuss the quantum dissipation of a bright soliton in a quasi-one-dimensional bosonic superfluid. I will argue that due to the integrability of the original problem, usual Ohmic friction proportional to a velocity is absent. It uncovers the non-Ohmic and non-Markovian friction, which can be interpreted as the backreaction of Bogoliubov quasiparticles inelastically scattered by an accelerating soliton, which represents an analogue of the Abraham-Lorentz force known in electrodynamic.

Counterfactual communication is communication without particles in the transmission channel. It is argued that an interaction-free measurement of the presence of opaque objects can be named `counterfactual', while proposed ``counterfactual'' measurements of the absence of such objects are not counterfactual. The quantum key distribution protocols which rely only on measurements of the presence of the object are counterfactual, but quantum direct communication protocols are not.

We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method.

Ideas from topology have generated a lot of excitement in the fields of condensed matter, cold atoms, and photonics because they can give rise to intriguing phenomena such as backscattering-free edge states and fractional quantum Hall states.

For partial boolean functions, whose domain can be a subset of {0,1}^n, exponential separations are known between the number of queries a classical deterministic algorithm needs to compute a function and the number of queries a quantum algorithm needs. For a total boolean function f, whose domain is all of {0,1}^n, the situation is quite different: the quantum Q(f) and deterministic D(f) query complexities are always polynomially related, in fact D(f) = O(Q(f)^6).

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Computational complexity theory studies the classification of computational problems according to the resources required to solve them. An important problem in quantum complexity theory is the local Hamiltonian problem - given a Hamiltonian composed of local terms, determine its ground state energy up to polynomial precision.

Quantum adiabatic optimization (QAO) slowly varies an initial Hamiltonian with an easy-to-prepare ground-state to a final Hamiltonian whose ground-state encodes the solution to some optimization problem. Currently, little is known about the performance of QAO relative to classical optimization algorithms as we still lack strong analytic tools for analyzing its performance.