other seminars:

Beni Yoshida (Caltech) In this talk, I will establish the connection among classifications of gapped boundaries in topological phases of matter, bosonic symmetryprotected topological (SPT) phases and faulttolerantly implementable logical gates in quantum errorcorrecting codes. We begin by presenting constructions of gapped boundaries for the ddimensional quantum double model by using dcocycles functions (d geq 2). We point out that the system supports mdimensional excitations (m < d), which we shall call fluctuating charges, that are superpositions of pointlike electric charges characterized by mdimensional bosonic SPT wavefunctions. There exist gapped boundaries where electric charges or magnetic fluxes may not condense by themselves, but may condense only when accompanied by fluctuating charges. Magnetic fluxes and codimension2 fluctuating charges exhibit nontrivial multiexcitation braiding statistics, involving more than two excitations. The statistical angle can be computed by taking slant products of underlying cocycle functions sequentially. We find that excitations that may condense into a gapped boundary can be characterized by trivial multiexcitation braiding statistics, generalizing the notion of the Lagrangian subgroup. As an application, we construct faulttolerantly implementable logical gates for the ddimensional quantum double model by using dcocycle functions. Namely, corresponding logical gates belong to the dth level of the Clifford hierarchy, but are outside of the (d1)th level, if cocycle functions have nontrivial sequences of slant products. :: Fri 11/13, 1:30pm in 6C442 Ke Li (IBM/MIT) Suppose we are given n copies of one of the quantum states {rho_1,..., rho_r}, with an arbitrary prior distribution that is independent of n. The multiple hypothesis testing problem concerns the minimal average error probability P_e in detecting the true state. It is known that P_e~exp(En) decays exponentially to zero. However, this error exponent E is generally unknown, except for the case r=2. In this talk, I will give a solution to the problem of identifying the above error exponent, by proving Nussbaum and Szkola's conjecture that E=min_{i\neq j} C(rho_i, rho_j). The righthand side of this equality is called the multiple quantum Chernoff distance, and C(rho_i, rho_j):=max_{0 \leq s \leq 1} {log Tr rho_i^s rho_j^(1s)} has been previously identified as the optimal error exponent for testing two hypotheses, rho_i versus rho_j. The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finitedimensional, but otherwise general, quantum states. This upper bound, up to a statesdependent factor, matches the multiplestate generalization of Nussbaum and Szkola's lower bound. Specialized to the case r=2, we give an alternative proof to the achievability of the binaryhypothesis Chernoff distance, which was originally proved by Audenaert et al. :: Fri 11/20, 1:30pm in 6C442 David Gosset (Caltech) Frustrationfree local Hamiltonians are a commonly studied subclass of (general) local Hamiltonians. A frustrationfree local Hamiltonian H can be written as a sum of terms, such that any ground state of H also has minimal energy for each term. In this talk I will describe some fundamental consequences of frustrationfreeness. Firstly I will consider the scaling of the correlation length with spectral gap. Hastings proved that a ground state of a local Hamiltonian with spectral gap g has correlation length upper bounded as O(1/g). This bound cannot be improved in general. However, frustrationfree systems satisfy a stronger O(1/sqrt(g)) upper bound on correlation length. Secondly I will consider the scaling of the spectral gap with system size. For 1D translationinvariant frustrationfree systems which are gapless in the thermodynamic limit, the spectral gap of an open boundary chain of size n is upper bounded inverse quadratically with n. In contrast if we remove the frustrationfreeness restriction then the gap can scale inverse linearly with system size. Finally, I will briefly discuss an open question concerning the scaling of entanglement entropy with spectral gap in frustrationfree systems. This talk is based on a joint work with Yichen Huang (arxiv:1509.06360) and a joint work with Evgeny Mozgunov (to appear on arxiv soon). John Wright (CMU) TBA Dave Touchette (Caltech) TBA 