Quantum Information

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. Often, one does not need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse and well-conditioned, with largest dimension N, the best known classical algorithms can find x and estimate x'Mx in O(N * poly(log(N))) time. In this talk I'll describe a quantum algorithm for solving linear sets of equations that runs in poly(log N) time, an exponential improvement over the best classical algorithm. This talk is based on my paper arXiv:0811.3171v2, which was written with Avinatan Hassidim and Seth Lloyd.

Entanglement is one of the most fundamental and yet most elusive properties of quantum mechanics. Not only does entanglement play a central role in quantum information science, it also provides an increasingly prominent bridging notion across different subfields of Physics --- including quantum foundations, quantum gravity, quantum statistical mechanics, and beyond. Arguably, the property of a state being entangled or not is by no means unambiguously defined. Rather, it depends strongly on how we decide to regard the whole as composed of its part or, more generally, on the restricted ways in which we are able to observe and control the system at hand. Acknowledging the implications of such an operationally constrained point of view naturally has led to a notion of 'generalized entanglement,' which is directly based on quantum observables and offers added flexibility in a variety of contexts. In this talk, I will survey some of the main accomplishments of the generalized entanglement program to date, with an eye toward recent developments and open problems.

Black holes are regions of space with gravity so strong that nothing can escape from them, not even light. This isn't science fiction - there's even a gigantic black hole at the center of our galaxy. It's hard to imagine a more effective way to irrevocably erase and destroy a computer's hard drive than to drop it into a nice big black hole. But is the information on that drive really gone forever? Paradoxically, there's a good chance that not only does the information come back, it comes back in the blink of an eye. This surprise return of the information is based on the same principles that might someday make reliable quantum computers a reality. In fact, engineers are already exploiting these principles to help distribute software and stream video over the internet. And that's where the tornadoes come in...

We study the entanglement dynamics and relaxation properties of a system of two interacting qubits in the two cases (I) two independent bosonic baths and (II) one common bath, at temperature $T$. The entanglement dynamics is studied in terms of the concurrence C(t) between the two spins and of the von Neumann entropy S(t) with respect to the bath, as a function of time. We prove that the system does thermalize. In the case (II) of a single bath, the existence of a decoherence-free (DFS) subspace makes entanglement dynamics very rich. We show that when the system is initially in a state with a component in the DFS the relaxation time is surprisingly long, showing the existence of semi-decoherence free subspaces. The equilibrium state in this case is not the Gibbs state. The entanglement dynamics for the single bath case is also studied as a function of temperature, coupling strength with the environment and strength of tunneling coupling. The case of the mixed state is finally shown and discussed.

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.

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).

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.

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.

In topological quantum computation, a quantum algorithm is performed by braiding and fusion of certain quasi-particles called anyons. Therein, the performed quantum circuit is encoded in the topology of the braid. Thus, small inaccuracies in the world-lines of the braided anyons do not adversely affect the computation. For this reason, topological quantum computation has often been regarded as error-resilient per se, with no need for quantum error-correction. However, newer work [1], [2] shows that even topological computation is plagued with (small) errors. As a consequence, it requires error-correction, too, and in the scaling limit causes a poly-logarithmic overhead similar to systems without topological error-correction. I will discuss Nussinov and Ortiz' recent result [2] that the toric code is not fault-tolerant in a Hamiltonian setting, and outline its potential implications for topological quantum computation in general. [1] Nayak, C., Simon, S. H., Stern, A. et al. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083-1159 (2008). [2] Z. Nussinov and G. Ortiz, arXiv:0709.2717 (condmat)