Quantum Information

Topological order is a new kind of collective order which appears in two-dimensional quantum systems such as the fractional quantum Hall effect and brings about rather unusual particles: unlike bosons or fermions these anyons obey exotic statistics and can be exploited to perform quantum computation. Topological order also implies that quantum states at low energies exhibit a very subtle, yet intricate inner structure. Remarkably, both phenomena can be studied in relatively simple spin systems (like Kitaev's quantum double models and the ubiquitous toric code) which in fact capture the essential properties of entire topological phases of matter in many important cases. What is the relationship between these topological phases? Reviewing recent work I will explain how they arrange themselves in a landscape of dualities and hierarchies. In particular, I will focus on two aspects: first, a duality between electric and magnetic quasiparticles in generalized quantum double models and second, a hierarchy construction of quantum states which are related by the condensation of topological charges.

The counter-intuitive phenomena in quantum mechanics are often based on the counter-factual (or virtual) processes. The famous example is the Hardy paradox, which has been recently solved in two independent experiments. Also, the delayed choice experiment and one of quantum descriptions of the closed time like curves can be also examples of the counter-intuitive phenomena. The counter-factual processes can be characterized by the weak value initiated by Yakir Aharonov and his colleagues. In this talk, I will introduce the weak value from the probability theory and the connection to the counter-factual processes in these examples.

In this talk I present recent work on combining game theory, statistics, and control theory. This combination provides new techniques for predicting / controlling any system comprising humans, human groups (e.g., firms, tribes), and / or adaptive automated systems (e.g., reinforcement learning robots). As illustrations, I will focus on three projects: 1) Suppressing flutter in an airplane wing by controlling a set of autonomous micro-flaps at its trailing edge. 2) First raising taxes in a human economy, and then lowering them back to the starting values, to steer the economy to a Pareto superior equilibrium. 3) Designing a dynamic auction of arrival slots at a weather-impacted airport, to maximize social welfare.

In this talk (based on arXiv:1001.0354) we give a quantum statistical interpretation for the Kauffmann bracket polynomial state sum <K> for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to all values of the polynomial variable that lie on the unit circle in the complex plane. Letting C(K) denote the Hilbert space for this model, there is a natural unitary transformation U from C(K) to itself such that <K> = <F|U|F> where |F> is a sum over basis states for C(K). The quantum algorithm arises directly from this formula via the Hadamard Test. We then show that the framework for our quantum model for the bracket polynomial is a natural setting for Khovanov homology. The Hilbert space C(K) of our model has basis in one-to-one correspondence with the enhanced states of the bracket state summmation and is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers. We show that for the Khovanov boundary operator d defined on C(K) we have the relationship dU + Ud = 0. Consequently, the unitary operator U acts on the Khovanov homology, and we therefore obtain a direct relationship between Khovanov homology and this quantum algorithm for the Jones polynomial. The formula for the Jones polynomial as a graded Euler characteristic is now expressed in terms of the eigenvalues of U and the Euler characteristics of the eigenspaces of U in the homology. The quantum algorithm given here is inefficient, and so it remains an open problem to determine better quantum algorithms that involve both the Jones polynomial and the Khovanov homology.

Multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols. However obtaining a generic, structural understanding of entanglement in N-qubit systems is still largely an open problem. Here we show that multipartite quantum entanglement admits a compositional structure. The two SLOCC-classes of genuinely entangled 3-qubit states, the GHZ-class and the W-class, exactly correspond with the two kinds of commutative Frobenius algebras on C^2, namely `special' ones and `anti-special' ones. Within the graphical language of symmetric monoidal categories, the distinction between `special' and `anti-special' is purely topological, in terms of `connected' vs.~`disconnected'. These GHZ and W Frobenius algebras form the primitives of a graphical calculus which is expressive enough to generate and reason about representatives of arbitrary N-qubit states. This calculus induces a generalised graph state paradigm for measurement-based quantum computing, and refines the graphical calculus of complementary observables due to Duncan and one of the authors [ICALP'08], which has already shown itself to have many applications and admit automation. References: Bob Coecke and Aleks Kissinger, http://arxiv.org/abs/1002.2540

The original motivation to build a quantum computer came from Feynman who envisaged a machine capable of simulating generic quantum mechanical systems, a task that is intractable for classical computers. Such a machine would have tremendous applications in all physical sciences, including condensed matter physics, chemistry, and high energy physics. Part of Feynman's challenge was met by Lloyd who showed how to approximately decompose the time-evolution operator of interacting quantum particles into a short sequence of elementary gates, suitable for operation on a quantum computer. However, this left open the more fundamental problem of how to simulate the equilibrium and static properties of quantum systems. This requires the preparation of ground and Gibb's states on a quantum computer. For classical systems, this problem is solved by the ubiquitous Metropolis algorithm, a method that basically acquired a monopoly for the simulation of interacting particles. In this talk, I will demonstrate that the corresponding quantum problem can be solved by a quantum Metropolis algorithm. This validates the quantum computer as a universal simulator, and proves that the so-called sign problem occurring in quantum Monte Carlo methods can be resolved with a quantum computer.

In quantum information, entanglement has often been viewed as a resource. But in this talk, I will look at (pure bipartite) entanglement through the lens of superselection rules. The idea is that it requires quantum communication not only to create entanglement, but also to destroy it in a way that doesn't leak information to the environment. As a result, when communication is scarce, superpositions of different numbers of EPR pairs can be difficult to obtain. This constraint is not a strict superselection rule, but rather an approximate version that gives rigorous bounds on achievable fidelities. After describing the general phenomenon, I will show how it relates to communication complexity, information theory and fairy tales about princesses. This talk is based on 0803.3066, 0909.1557, 0912.5537 and other unpublished work.

I'll discuss some work-in-progress about the computational complexity of simulating the extremely "simple" quantum systems that arise in linear optics experiments. I'll show that *either* one can describe an experiment, vastly easier than building a universal quantum computer, that would test whether Nature is performing nontrivial quantum computation, or else one can give surprising new algorithms for approximating the permanent. Audience feedback as to which of these possibilities is the right one is sought. Joint work with Alex Arkhipov.

Alongside the effort underway to build quantum computers, it is important to better understand which classes of problems they will find easy and which others even they will find intractable. Inspired by the success of the statistical study of classical constraint optimization problems, we study random ensembles of the QMA$_1$-complete quantum satisfiability (QSAT) problem introduced by Bravyi. QSAT appropriately generalizes the NP-complete classical satisfiability (SAT) problem. We show that, as the density of clauses/projectors is varied, the ensembles exhibit quantum phase transitions between phases that are product satisfiable, entangled satisfiable and unsatisfiable. Remarkably, almost all instances of QSAT for a fixed interaction graph exhibit the same dimension of the satisfying manifold. This establishes the generic QSAT decision problem as equivalent to a purely graph theoretic property and that the hardest typical instances are likely to be localized in a bounded range of clause density. Based on papers: C.R. Laumann, R. Moessner, A. Scardicchio, and S.L. Sondhi. Phase transitions and random quantum satisfiability. QIC 10 (1/2), (2009). arXiv:0903.1904 C.R. Laumann, A.M. Lauchli, R. Moessner, A. Scardicchio, and S.L. Sondhi. On product, generic and random generic quantum satisfiability. arXiv:0910.2058

If a pure quantum state is drawn at random, this state will almost surely be almost maximally entangled. This is a well-known example for the "concentration of measure" phenomenon, which has proved to be tremendously helpful in recent years in quantum information theory. It was also used as a new method to justify some foundational aspects of statistical mechanics. In this talk, I discuss recent work with David Gross and Jens Eisert on concentration in the set of pure quantum states with fixed mean energy: We show typicality in this manifold of quantum states, and give a method to evaluate expectation values explicitly. This involves some interesting mathematics beyond Levy's Lemma, and suggests potential applications such as finding stronger counterexamples to the additivity conjecture.

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.

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)