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This Semester


:: Fri 11/6, 1:30pm in 6C-442

Beni Yoshida (Caltech)
Gapped boundaries, group cohomology and fault-tolerant logical gates

In this talk, I will establish the connection among classifications of gapped boundaries in topological phases of matter, bosonic symmetry-protected topological (SPT) phases and fault-tolerantly implementable logical gates in quantum error-correcting codes. We begin by presenting constructions of gapped boundaries for the d-dimensional quantum double model by using d-cocycles functions (d geq 2). We point out that the system supports m-dimensional excitations (m < d), which we shall call fluctuating charges, that are superpositions of point-like electric charges characterized by m-dimensional 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 codimension-2 fluctuating charges exhibit non-trivial multi-excitation 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 multi-excitation braiding statistics, generalizing the notion of the Lagrangian subgroup. As an application, we construct fault-tolerantly implementable logical gates for the d-dimensional quantum double model by using d-cocycle functions. Namely, corresponding logical gates belong to the dth level of the Clifford hierarchy, but are outside of the (d-1)th level, if cocycle functions have non-trivial sequences of slant products.

:: Fri 11/13, 1:30pm in 6C-442

Discriminating quantum states: the multiple Chernoff distance

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 right-hand 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^(1-s)} 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 finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and Szkola's lower bound. Specialized to the case r=2, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al.

:: Fri 11/20, 1:30pm in 6C-442

David Gosset (Caltech)
Some consequences of frustration-freeness

Frustration-free local Hamiltonians are a commonly studied subclass of (general) local Hamiltonians. A frustration-free 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 frustration-freeness. 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, frustration-free 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 translation-invariant frustration-free 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 frustration-freeness 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 frustration-free 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).

:: Fri 12/4, 1:30 in 6C-442

John Wright (CMU)
Random words, longest increasing subsequences, and quantum PCA


:: Fri 12/11, 1:30pm in 2-105

Dave Touchette (Caltech)



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