Quantum Information

Raussendorf introduced a powerful model of fault tolerant measurement based quantum computation, which can be understood as a layering (or “foliation”) of a multiplicity of Kitaev’s toric code.  I will discuss our generalisation of Raussendorf’s construction to an arbitrary CSS code.  We call this a Foliated Quantum Code.  Decoding this foliated construction is not necessarily straightforward, so I will discuss an example in which we foliate a family of finite-rate quantum turbo codes, and present the results of numerical simulations of the decoder performance.

If I have time, I will discuss some ongoing work (with Gavin Brennan) on relational time applied to topological quantum field theories, in particular, how anyonic systems with essentially trivial dynamics can still exhibit correlations that track “time".

In this talk I will introduce recent research into quantum clocks of finite dimension, with the focus on their accuracy, as determined by their dimension, coherence, and power consumption.

I will present arguments to bound the synchronization time of any quantum clock as a function of its dimension. In addition, quantum coherence appears to be necessary to saturate these bounds, as the synchronization time of incoherent clocks is seen to have a worse bound. In addition, I will review simple proposals for autonomous clocks built out of thermal machines, and demonstrate that the power consumption of thermal clocks determines the limit of their accuracy. Finally, I will introduce an example of a finite quantum clock that is able to control any quantum operation up to a calculable accuracy, and discuss whether it represents a best case scenario for quantum clocks.

We give a new theoretical solution to a leading-edge experimental challenge, namely to the verification of quantum computations in the regime of high computational complexity. Our results are given in the language of quantum interactive proof systems. Specifically, we show that any language in BQP has a quantum interactive proof system with a polynomial-time classical verifier (who can also prepare random single-qubit pure states), and a quantum polynomial-time prover. Here, soundness is unconditional---i.e it holds even for computationally unbounded provers. Compared to prior work achieving similar results, our technique does not require the encoding of the input or of the computation; instead, we rely on encryption of the input (together with a method to perform computations on encrypted inputs), and show that the random choice between three types of input (defining a "computational run", versus two types of "test runs") suffice. As a proof technique, we use a reduction to an entanglement-based protocol; this enables a relatively simple analysis for a situation that has previously remained ambiguous in the literature.

We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions.  A locality-preserving operation is one which maps local operators to local operators --- for example, a  constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically-local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking general properties of two-dimensional topological field theories, we find that the locality-preserving logical gates are severely limited for codes which admit non-abelian anyons; in particular, there are no locality-preserving logical gates on the torus or the sphere with M punctures if the braiding of anyons is computationally universal. Furthermore, for Ising anyons on the M-punctured sphere, locality-preserving gates must be elements of the logical Pauli group. We derive these results by relating logical gates of a topological code to automorphisms of the Verlinde algebra of the corresponding anyon model, and by requiring the logical gates to be compatible with basis changes in the logical Hilbert space arising from local F-moves and the mapping class group. 

This is joint work with Oliver Buerschaper, Robert Koenig, Fernando Pastawski and Sumit Sijher.

Quantum adiabatic optimization (QAO) slowly varies an initial Hamiltonian with an easy-to-prepare ground-state to a final Hamiltonian whose ground-state encodes the solution to some optimization problem. Currently, little is known about the performance of QAO relative to classical optimization algorithms as we still lack strong analytic tools for analyzing its performance. In this talk, I will unify the problem of bounding the runtime of one such class of Hamiltonians -- so-called stoquastic Hamiltonians -- with questions about functions on graphs, heat diffusion, and classical sub-stochastic processes. I will introduce new tools for bounding the spectral gap of stoquastic Hamiltonians and, by exploiting heat diffusion, show that one of these techniques also provides an optimal and previously unknown gap bound for particular classes of graphs. Using this intuition and combining heat diffusion with classical sub-stochastic processes, I will offer a classical adiabatic algorithm that exhibits behavior typically considered "quantum", such as tunneling.