QUOXIC

This meeting was held on Wednesday 16th September at Bristol Centre for Nanoscience and Quantum Information, Tyndall Avenue, Bristol.

Local information about the NSQI Centre can be found here.

The schedule and abstracts are listed below.

Schedule

• 13:00 : Sandu Popescu (Bristol)
  "TBC"
• 13:45 : Animesh Datta (Imperial)
  "Quantum effects in light-harvesting processes"
• 14:15 : Stephen Brierly (York)
  "Constructing Mutually Unbiased Bases"
• 14:45 : COFFEE BREAK
• 15:10 : Ashley Montanaro (Bristol)
  "A quantum analogue of Fourier analysis on the boolean cube"
• 15:40 : Karoline Wiesner (Bristol)
  "Quantum computational resources to generate structure"
• 16:10 : Katherine Brown (Leeds)
  "Simulating the BCS Hamiltonian on a qubus quantum computer"
• 16:40 : COFFEE BREAK
• 17:00 : Jennifer Hide (Leeds)
  "Detecting Entanglement with Jarzynski's equality"
• 17:30 : Peter Crompton (Leeds)
  "Universality of the Spin-1/2 Heisenberg Antiferromagnet on the Square Lattice"

Abstracts

We study the presence and time evolution of quantum entanglement during exciton energy transfer (EET) in a network model of the Fenna-Matthews-Olson (FMO) complex, a biological aggregate involved in the early steps of photosynthesis in sulphur bacteria. We analyze the influence of noise on entanglement generation and degradation taking account of Markovian, forms of non-Markovian as well as spatially correlated environmental noise. We also discuss the influence of different excitation injection mechanisms. We find that the time scale on which entanglement is present, as well as it magnitude, is increased in the presence of environmental correlations. In contrast with many situations in quantum information processing, where a maximal amount of entanglement tends to be beneficial, we show that near unit EET is achieved when the initial part of the evolution displays intermediate entanglement values. This optimal, albeit non-maximal, strength of quantum correlations for optimal performance is the result of the intricate interplay between coherent and noisy processes in these complex systems.

In collaboration with F. Caruso, A. W. Chin, S. F. Huelga, M. B. Plenio

Mutually unbiased (MU) bases are an important, physically motivated tool allowing the reconstruction of quantum states with optimal efficiency, however there are several unanswered questions such as the existence of complete sets in non-prime power dimensions. I will review what is known and the various attempts at attacking the problem as well as present some new results relating to low dimensions. I will show how to classify all sets of MU bases in dimensions 2 to 5 resulting in a simple proof that the known construction via finite fields is unique for d<6, and that there exists a three parameter family of triples of MU bases in dimension 4. I will then explain how these ideas can be applied to dimension 6, the first case in which the maximum number of MU bases is unknown, to add support to the conjecture that a complete set does not exist when d=6.

In recent years, Fourier analysis of functions on the boolean cube has become a powerful tool in computer science, with applications to the study of hardness of approximation, the development of efficient property testers, computational learning theory, and many other areas of research.

In this talk, I will discuss quantum (noncommutative) generalisations of some of these Fourier-analytic results, based on the use of the Pauli matrices as an analogue of the characters of the group Z_2. Many interesting classical results turn out to translate to the quantum regime with few changes to the proofs required.

A particularly useful result in the classical theory is a "hypercontractive" inequality due to Bonami, Gross and Beckner. I will give a quantum generalisation of this inequality, which is in fact a statement about the qubit depolarising channel, and mention implications for understanding the spectra of k-local operators.

This talk is based on joint work with Tobias Osborne.

One way to measure information storage in quantum systems is to ask what resources does it take to generate the system's dynamics statistically accurate. The system's dynamics is represented as a measurement sequence. Classically, this question has a well-defined answer in terms of stochastic finite-state machines. Quantum mechanically, we show how quantum finite-state machines can be used to the same end. We present a general definition of quantum finite-state machines and discuss information theoretic measures of structure present in the dynamics.

The use of quantum computers for the simulation of quantum systems was proposed by Feynman in 1982 [1] and since then has proved a popular field of research. It is hoped that using a quantum computer to simulate a quantum system will provide an exponential improvement in the resources require compared to using a classical computer [2]. It is also believed that quantum simulations will provide one of the first practical uses of a quan- tum computer since results inaccessible to a classical computer should begin to be obtained with significantly smaller systems than for other quantum algorithms.

One problem of particular interest is that of pairing Hamiltonians, specif- ically the BCS Hamiltonian. This is a relevant problem in both condensed matter and nuclear physics. While it is possible to solve certain specific cases of the Hamiltonian effciently and accurately on a classical computer [3, 4] it isn't possible to effciently solve the general class of problems. Work on simulating this problem on a quantum computer by Wu et al. [5] has shown that is should be possible to solve the general case on a quantum computer with resources which scale effiently with the size of the system.

My research focuses on simulating the BCS Hamiltonian on a qubus quantum computer. A qubus system is one in which a continuous variable subsystem is used as a method of communication between the qubits within the main system. By entangling various qubits with the bus it is possible to generate interactions between the qubits. Numerous recent proposals have suggested physical methods for creating a qubus system [6, 8, 7] and it is hoped that by allowing entanglement between distant qubits the qubus system will be more scalable than alternative systems.

My work so far has found a set of gates that can be used to simulate the BCS Hamiltonian in the general 2 qubit case and shown that it is possible to scale this up to the general n qubit case efficiently. When using the first order Trotter approximation the number of gates required for the general case scales as n^2. I have also shown that by rearranging the gates on the qubus it is possible to get a factor of 2 saving over a naïve implementation method. Both the naïve and improved method show at least a factor N improvement over previous work on simulating the BCS Hamiltonian on an NMR quantum computer [5].

References

1. R.P Feynman, Int. J of Theoret Physics 21 467-488 (1982)
2. S. Lloyd, Science 273 1073-1078 (1996)
3. R.W Richard, N. Sherman, Nuclear Physics 52 221-238 (1964)
4. J. Dukelsky, G. Sierra, Phys. Rev. Lett. 83 172-175 (1999)
5. L.A. Wu, M.S. Byrd, D.A. Lidar, Phys. Rev. Lett. 89 057904 (2002)
6. W.J.Munro et al., Phys. Rev. A 71 033819 (2005)
7. C.H Su et al., High speed quantum gates with cavity quantum electrody- namics, arXiv:0809.2133 (Sep 2008)
8. P. van Loock et al., Phys. Rev. A 78 022303 (2008)

We present a method for detecting the entanglement of a state using non-equilibrium processes. A comparison of relative entropies allows us to construct an entanglement witness. The relative entropy can further be related to the quantum Jarzynski equality, allowing non-equilibrium work to be used in entanglement detection. To exemplify our results, we consider two different spin chains.

I review recent activity on the spin-1/2 Heisenberg antiferromagnet on the square lattice, relating to developing universal entanglement entropy scaling arguments for under-doped cuprates. Unlike the quantum spin chain, which is exactly solvable, and which has a mapping onto (1+1) dimensional conformal field theories, whilst the Hopf invariant of the spin-1/2 Heisenberg antiferromagnet on the square lattice model is zero, the Hilbert space is also finite and extensive, which means that states in the system are defined by two non-orthogonal polynomials (generating anyons). This makes it far more difficult to define an algebra for the system at criticality. I give a new derivation of the Von Neumann entanglement entropy of this system, using Riemann sheets in C^2 and a polynomial ring, and present numerics.