Quantum Physics

In topological quantum computation the geometric details of a particle trajectory become irrelevant; only the topology matters. This is one reason for the inherent fault tolerance of topological quantum computation. I will speak about a model in which this idea is taken one step further. Even the topology is irrelevant. The computation is determined solely by the permutation of the particles. Unlike topological quantum computation, which requires anyons confined to two dimensions, permutational quantum computations can in principle be performed by permuting a set of ordinary spin-1/2 particles with definite total angular momentum in three dimensions. The resulting model of computation appears to be intermediate in power between classical computation (P) and standard quantum computation (BQP). The model may be equivalently defined in terms of spin networks, which are an important concept in loop quantum gravity.

Quantum mechanics is a non-classical probability theory, but hardly the most general one imaginable: any compact convex set can serve as the state space for an abstract probabilistic model (classical models corresponding to simplices). From this altitude, one sees that many phenomena commonly regarded as ``characteristically quantum' are in fact generically ``non-classical'. In this talk, I'll show that almost any non-classical probabilistic theory shares with quantum mechanics a notion of entanglement and, with this, a version of the so-called measurement problem. I'll then discuss what's required for an abstract probabilistic theory to admit a somewhat simplified version of Everett's response to this problem -- an exercise that turns out to be instructive in several ways.

Suppose we are given two probability distributions on some N-element set. How many samples do we need to test whether the two distributions are close or far from each other in the L_1 norm? This problem known as Statistical Difference has been extensively studied during the last years in the field of property testing. I will describe quantum algorithms for Statistical Difference problem that provide a polynomial speed up in terms of the query complexity compared to the known classical lower bounds. Specifically, I will assume that each distribution can be generated by querying an oracle function on a random uniformly distributed input string. It will be shown that testing whether distributions are orthogonal requires approximately N^{1/2} queries classically and approximately N^{1/3} queries quantumly. Testing whether distributions are close requires approximately N^{2/3} queries classically and O(N^{1/2}) queries quantumly. This is a joint work with Aram Harrow (University of Bristol) and Avinatan Hassidim (The Hebrew University).

Our universe is unquestionably quantum in nature. What does this mean? Why does it matter? What’s in it for me? Join us for a fun and fascinating session on “what you need to know about the quantum.” Find out why we can’t live without it. Discover what’s so unbelievably quirky about it. And learn how it empowers amazing technologies, from present day (e.g. every electronic device on the planet) to future possibilities including quantum computing and global quantum communication. The half hour multimedia presentation will be followed by a half hour question and answer session with a leading scientist in the field. The future is quantum. Find out why.

A standard canonical quantization of general relativity yields a time-independent Schroedinger equation whose solutions are static wavefunctions on configuration space. Naively this is in contradiction with the real world where things do change. Broadly speaking, the problem how to reconcile a theory which contains no concept of time with a changing world is called 'the problem of time'. In this seminar we shall study this problem using a reformulation of Newtonian mechanics due to Jacobi (Jacobi's timeless mechanics) which allows one to study the problem of time without all the technical difficulties present in quantized general relativity. We show explicitly that Jacobi's timeless mechanics is a straightforward counterexample to the claim that all first class constraints generate gauge transformations, i.e. physically indistinguishable states. The implications of this is unclear. By making use of deBroglie-Bohm trajectories we derive a necessary and sufficient condition for a time-dependent Schroedinger equation to emerge for subsystems. The importance of 'strong' entanglement between subsystem and environment for the emergence of time is stressed.

Recent results have shown that quantum computers can approximate the value of a tensor network efficiently. These results have initiated a search for tensor networks which contract to computationally interesting quantities. Topological Lattice Field Theories (TLFTs) are one source of such networks; when defined appropriately, networks arising from TLFTs contract to give topological invariants. In this elementary talk, we will define and classify TLFTs which lead to invariants of surfaces, and sketch out the corresponding quantum algorithm. Our exposition will be targetted at a general mathematically-inclined audience; no previous knowledge of field theories is required.

Condensed matter systems at ultra low-temperature that show purely quantum phenomena – even on big scales (superfluidity, superconductivity, Bose-Einstein condensates); measurements in the quantum world; probing the foundations of quantum theory.

We introduce a two-body quantum Hamiltonian model of spin-1/2 on a 2D spatial lattice with exact topological degeneracy in all coupling regimes. There exists a gapped phase in which the low-energy sector reproduces an effective color code model. High energy excitations fall into three families of anyonic fermions that turn out to be strongly interacting. The model exhibits a Z_2xZ_2 gauge group symmetry and string-net integrals of motion, which are related to the existence of topological charges that are invisible to moving high-energy fermions.

The essential ingredients of a quantum theory are usually a Hilbert space of states and an algebra of operators encoding observables. The mathematical operations available with these structures translate fairly well into physical operations (preparation, measurement etc.) in a non-relativistic world. This correspondence weakens in quantum field theory, where the direct operational meaning of the observable algebra structure (encoded usually through commutators) is lost. The situation becomes even worse when we want to give a more dynamical role to spacetime as for example in attempts to formulate a quantum theory of gravity. I argue that a revision of the structures that we think of as fundamental in a quantum theory is in order. I go on to outline a proposal in this direction, based on the so called 'general boundary formulation', emphasizing the operational meaning of the ingredients. If time permits I will also comment on the relation to the framework of algebraic quantum field theory.

Both classical probability theory and quantum theory lend themselves to a Bayesian interpretation where probabilities represent degrees of belief, and where the various rules for combining and updating probabilities are but algorithms for plausible reasoning in the face of uncertainty. I elucidate the differences and commonalities of these two theories, and argue that they are in fact the only two algorithms to satisfy certain basic consistency requirements. In order to arrive at this result I develop an over-arching framework for plausible reasoning that incorporates both classical probability and quantum theory as special cases.