Exponential Quantum Speed-ups are Generic

A central problem in quantum computation is to understand which quantum circuits are useful for exponential speed-ups over classical computation. We address this question in the setting of query complexity and show that for almost any sufficiently long quantum circuit one can construct a black-box problem which is solved by the circuit with a constant number of quantum queries, but which requires exponentially many classical queries, even if the classical machine has the ability to postselect. We prove the result in two steps. In the first, we show that almost any element of an approximate unitary 3-design is useful to solve a certain black-box problem efficiently. The problem is based on a recent oracle construction of Aaronson and gives an exponential separation between quantum and classical bounded-error with postselection query complexities. In the second step, which may be of independent interest, we prove that linear-sized random quantum circuits give an approximate unitary 3-design. The key ingredient in the proof is a technique from quantum many-body theory to lower bound the spectral gap of local quantum Hamiltonians.Comment: 24 pages. v2 minor correction

Sharing or gambling? On risk attitudes in social contexts

This paper investigates experimentally whether risk attitudes are stable across social contexts. In particular, it focuses on situations where some resource (for instance, a position, decision power, a bonus) has to be allocated between two parties: the decision maker can either opt for sharing the resource or for using a random device that allocates the entire prize to one of the two parties. By varying the relative situation of the decision maker with respect to the other party, we show that risk attitude is strongly affected by social contexts: participants in the experiment seem to be relatively risk seeking when they possess a relatively weaker position than the other party and risk averse when the opposite is true. Our main average results seem to be driven by the behavior of around a quarter of subjects whose choices appear to be fully determined by social comparisons. Various interpretations of the behavior are provided linking our results to preferences under risk with a social reference point and on status-seeking preferences

Delocalization of weakly interacting bosons in a 1D quasiperiodic potential

We consider weakly interacting bosons in a 1D quasiperiodic potential (Aubry-Azbel-Harper model) in the regime where all single-particle states are localized. We show that the interparticle interaction may lead to the many-body delocalization and we obtain the finite-temperature phase diagram. Counterintuitively, in a wide range of parameters the delocalization requires stronger cou- pling as the temperature increases. This means that the system of bosons can undergo a transition from a fluid to insulator (glass) state under heating

Efficient Quantum Pseudorandomness

Randomness is both a useful way to model natural systems and a useful tool for engineered systems, e.g. in computation, communication and control. Fully random transformations require exponential time for either classical or quantum systems, but in many case pseudorandom operations can emulate certain properties of truly random ones. Indeed in the classical realm there is by now a well-developed theory of such pseudorandom operations. However the construction of such objects turns out to be much harder in the quantum case. Here we show that random quantum circuits are a powerful source of quantum pseudorandomness. This gives the for the first time a polynomialtime construction of quantum unitary designs, which can replace fully random operations in most applications, and shows that generic quantum dynamics cannot be distinguished from truly random processes. We discuss applications of our result to quantum information science, cryptography and to understanding self-equilibration of closed quantum dynamics.Comment: 6 pages, 1 figure. Short version of http://arxiv.org/abs/1208.069

(Quantumness in the context of) Resource Theories

We review the basic idea behind resource theories, where we quantify quantum resources by specifying a restricted class of operations. This divides the state space into various sets, including states which are free (because they can be created under the class of operations), and those which are a resource (because they cannot be). One can quantify the worth of the resource by the relative entropy distance to the set of free states, and under certain conditions, this is a unique measure which quantifies the rate of state to state transitions. The framework includes entanglement, asymmetry and purity theory. It also includes thermodynamics, which is a hybrid resource theory combining purity theory and asymmetry. Another hybrid resource theory which merges purity theory and entanglement can be used to study quantumness of correlations and discord, and we present quantumness in this more general framework of resource theories.Comment: review articl

Finite-Temperature Fluid-Insulator Transition of Strongly Interacting 1D Disordered Bosons

We consider the many-body localization-delocalization transition for strongly interacting one- dimensional disordered bosons and construct the full picture of finite temperature behavior of this system. This picture shows two insulator-fluid transitions at any finite temperature when varying the interaction strength. At weak interactions an increase in the interaction strength leads to insulator->fluid transition, and for large interactions one has a reentrance to the insulator regime