Perimeter Institute Quantum Discussions

The study of ground spaces of local Hamiltonians is a fundamental task in condensed matter physics. In terms of computational complexity theory, a common focus in this area has been to estimate a given Hamiltonian’s ground state energy. However, from a physics perspective, it is sometimes more relevant to understand the structure of the ground space itself. In this talk, we pursue the latter direction by introducing the notion of “ground state connectivity” of local Hamiltonians.

Recently, Bravyi and Koenig have shown that there is a tradeoff between fault-tolerantly implementable logical gates and geometric locality of stabilizer codes. They consider locality-preserving operations which are implemented by a constant depth geometrically local circuit and are thus fault-tolerant by construction. In particular, they shown that, for local stabilizer codes in D spatial dimensions, locality preserving gates are restricted to a set of unitary gates known as the D-th level of the Clifford hierarchy.

A class of d-level quantum states called "magic states", whose initial purpose was to enable universal fault-tolerant computation within error-correcting codes, has a surprisingly broad range of applications. We begin by describing their structure with respect to the Clifford hierarchy, and in terms of convex geometry before proceeding to their applications. They appear to have some relevance to the search for SIC-POVMs in certain prime dimensions. A version of the CHSH non-local game, using a d-ary alphabet and Pauli measurements, has an optimal quantum strategy using magic states.

The circuit-to-Hamiltonian construction translates a dynamics (a quantum circuit and its output) into statics (the groundstate of a circuit Hamiltonian) by explicitly defining a quantum register for a clock. The standard Feynman-Kitaev construction uses one global clock for all qubits while we consider a different construction in which a clock is assigned to each point in space where a qubit of the quantum circuit resides. We show how one can apply this construction to one-dimensional quantum circuits for which the circuit Hamiltonian realizes the dynamics of a vibrating string.