QUOXIC

This meeting took place on Thursday 25rd October at the Oxford University Computing Laboratory.

Speakers were Peter Rohde (Oxford), Andreas Doering (Imperial), Simon Perdrix (Oxford), Jamie Vicary (Imperial), and Christian Burrell (Royal Holloway).

Talk titles and abstracts appear below.

Abstracts

Scalable quantum computing requires quantum error correction protocols in order to deal with errors that inevitably arise during computations. Broadly speaking, there are two types of errors that may occur - located and unlocated error. A located error is one where we know which qubit(s) were affected by the error, while an unlocated one is one where we have no information as to its location. Quantum error correcting codes have been designed which tolerate each of these classes of error. However, little work has focussed on the problem of systems where both located and unlocated errors may occur. In this work we discuss the difference between located and unlocated errors, and derived a bound on how many of each type of error a quantum error correcting code can protect against.

I will sketch some of the recent work (with Chris Isham) on the application of topos theory to quantum physics. In particular, I will show how a certain topos is associated to each quantum system, how propositions about the quantum system and pure states are represented topos-internally, and how propositions are assigned truth-values using the internal logic of a topos. I will make some *very* preliminary remarks about how this framework might relate to the symmetric monoidal framework. Hopefully, this will lead to some discussion.

Among the models of quantum computation, the One-way Quantum Computer is one of the most promising proposals of physical realization, and opens new perspectives for parallelization by taking advantage of quantum entanglement. Since a one-way quantum computation is based on quantum measurement, which is a fundamentally nondeterministic evolution, a sufficient condition of global determinism has been introduced as the existence of a causal flow in a graph that underlies the computation. A O(n^3)-algorithm has been introduced for finding such a causal flow when the numbers of output and input vertices in the graph are equal, otherwise no polynomial time algorithm was known for deciding whether a graph has a causal flow or not. Our main contribution is to introduce a O(n^2)-algorithm for finding a causal flow, if any, whatever the numbers of input and output vertices are. This answers the open question stated by Danos and Kashefi and by de Beaudrap. Moreover, we prove that our algorithm produces an optimal flow (flow of minimal depth.)

Whereas the existence of a causal flow is a sufficient condition for determinism, it is not a necessary condition. A weaker version of the causal flow, called gflow (generalized flow) has been introduced and has been proved to be a necessary and sufficient condition for a family of deterministic computations. Moreover the depth of the quantum computation is upper bounded by the depth of the gflow. However, the existence of a polynomial time algorithm that finds a gflow has been stated as an open question. In this paper we answer this positively with a polynomial time algorithm that outputs an optimal gflow of a given graph and thus finds an optimal correction strategy to the nondeterministic evolution due to measurements.

I will describe how Fock space, along with the related machinery of creation and annihilation operators, canonical commutators and coherent states, can be formulated using the mathematics of category theory. The approach is entirely abstract: we abandon the conventional definitions rooted in the mathematics of Hilbert spaces, and instead focus on the properties of the objects we want to define. Key parts of the formalism have a very physical interpretation in terms of beam splitters and absorbers, and I will show how this arises from the underlying category theory. Finally, I will discuss the ultimate motivation: to discover new generalisations of quantum mechanics, which might be more suited for tackling the fundamental problem of quantum gravity. No knowledge of category theory required!

We investigate the propagation of information through interacting quantum spin chains with a disordered fluctuating external field. We find that it is - on average - impossible to transmit information any great distance along such a chain. This is in contrast to the ordered (noise free) case, which can be used as an effective means of transmitting quantum information.