Scientific Series

Joint work with Earl Campbell (FU-Berlin) and Hussain Anwar (UCL)   Magic state distillation is a key component of some high-threshold schemes for fault-tolerant quantum computation [1], [2]. Proposed by Bravyi and Kitaev [3] (and implicitly by Knill [4]), and improved by Reichardt [4], Magic State Distillation is a method to broaden the vocabulary of a fault-tolerant computational model, from a limited set of gates (e.g. the Clifford group or a sub-group[2])  to full universality, via the preparation of mixed ancilla qubits which may be prepared without fault tolerant protection.    Magic state distillation schemes have a close relation with quantum error correcting codes, since a key step in such protocols [5] is the projection onto a code subspace. Bravyi and Kitaev proposed two protocols; one based upon the 5-qubit code, the second derived from a punctured Reed-Muller code. Reed Muller codes are a very important family of classical linear code. They gained much interest [6] in the early years of quantum error correction theory, since their properties make them well-suited to the formation of quantum codes via the CSS-construction [7]. Punctured Reed-Muller codes (loosely speaking, Reed-Muller codewords with a bit removed) in particular lead to quantum codes with an unusual property, the ability to implement non-Clifford gates transversally [8].    Most work in fault-tolerant quantum computation focuses on qubits, but fault tolerant constructions can be generalised to higher dimensions [9]  - particularly readily for prime dimensions. Recently, we presented the first magic state distillation protocols [10] for non-binary systems, providing explicit protocols for the qutrit case (complementing a recent no-go theorem demonstrating bound states for magic state distillation in higher dimensions [11]). In this talk, I will report on more recent work [12], where the properties of punctured Reed-Muller codes are employed to demonstrate Magic State distillation protocols for all prime dimensions. In my talk, I will give a technical account of this result and present numerical investigations of the performance of such a protocol in the qutrit case. Finally, I will discuss the potential for application of these results to fault-tolerant quantum computation.     This will be a technical talk, and though some concepts of linear codes and quantum codes will be briefly revised, I will assume that listeners are familiar with quantum error correction theory (e.g. the stabilizer formalism and the CSS construction) for qubits.   [1] E. Knill. Fault-tolerant postselected quantum computation: schemes, quant-ph/0402171 [2] R. Raussendorf, J. Harrington and K. Goyal, Topological fault-tolerance in cluster state quantum computation, arXiv:quant-ph/0703143v1 [3] S. Bravyi and A. Kitaev. Universal quantum computation based on a magic states distillation, quant- ph/0403025 [4] B. W. Reichardt, Improved magic states distillation for quantum universality, arXiv:quant-ph/0411036v1 [5] E.T. Campbell and D.E. Browne, On the Structure of Protocols for Magic State Distillation, arXiv:0908.0838
[6] A. Steane, Quantum Reed Muller Codes, arXiv:quant-ph/9608026 [7] Nielsen and Chuang, Quantum Information and Computation, chapter 10 [8] E. Knill, R. Laflamme, and W. Zurek, Threshold accuracy for quantum computation, quant-ph/9610011 [9] D. Gottesman, Fault-Tolerant Quantum Computation with Higher-Dimensional Systems, quant-ph/9802007 [10] H. Anwar, E.T Campbell and D.E. Browne, Qutrit Magic State Distillation, arXiv:1202.2326 [11] V. Veitch, C. Ferrie, J. Emerson, Negative Quasi-Probability Representation is a Necessary Resource for Magic State Distillation, arXiv:1201.1256v3 [12] H. Anwar, E.T Campbell and D.E. Browne, in preparation

Majorana disappeared under mysterious circumstances in 1938 and the particle that bears his name remains elusive to experiments. There is growing interests in realizing the Majorana bound state in the Laboratory because it is expected to possess unusual properties such as non-abelian statistics. I shall discuss various proposals to produce  Majorana bound states and the associated topological superconductors which support them.

It is sometimes pointed out as a curiosity that the state space of quantum theory and actual physical space seem related in a surprising way: not only is space three-dimensional and Euclidean, but so is the Bloch ball which describes quantum two-level systems. In the talk, I report on joint work with Lluis Masanes, where we show how this observation can be turned into a mathematical result: suppose that physics takes place in d spatial dimensions, and that some events happen probabilistically (dropping quantum theory and complex amplitudes altogether). Furthermore, suppose there are systems that in some sense behave as “binary units of direction information”, interacting via some continuous reversible time evolution. We prove that this uniquely determines d=3 and quantum theory, and that it allows observers to infer local spatial geometry from probability measurements.

We review the notion of a quantum state of the universe and its role in fundamental cosmology. Then we discuss recent work which points towards a profound connection, at the level of the quantum state, between (asymptotic) Euclidean AdS spaces and Lorentzian de Sitter spaces. This gives a new framework in which (a mild generalization of) AdS/CFT can be applied to inflationary cosmology.   For the specific case of the Hartle-Hawking no-boundary quantum state the ADS/ de Sitter connection yields a natural proposal for a more precise `dual' formulation of the wave function, in terms of field theories on the future de Sitter boundary that are certain relevant deformations of the CFTs that occur in AdS/CFT.

I will review the construction of lattice theories which maintain one or more exact supersymmetries for non zero lattice spacing concentrating in particular on the case of N=4 super Yang-Mills. Such lattice theories may be studied using Monte Carlo techniques borrowed from lattice QCD and can be used to explore issues in holography. In three dimensions the same constructions can be used to formulate a topological theory of gravity which we argue is equivalent to Witten's Chern Simons theory.

The ground state phase of spin-1/2 J1-J2 antiferromagnetic Heisenberg model on square lattice in the maximally frustrated regime (J2 ~ 0.5J1) has been debated for decades. Here we study this model by using a recently proposed novel numerical method - the cluster update algorithm for tensor product states (TPSs). The ground state energies at finite sizes and in the thermodynamic limit (with finite size scaling) are in good agreement with the state of art exact diagonalization study, and
the energy differences between these two studies are of the order of 0.001 J1 per site. At the largest bond dimension available D (D = 9), we find a paramagnetic ground state without any valence bond solid order in the thermodynamic limit in the range of 0.5 <= J2/J1 <= 0.6, which implies the emergence of a spin-liquid phase. Furthermore, we investigate the topologically ordered nature of such a spin-liquid phase by measuring a nonzero topological entanglement entropy.

There has been some significant recent progress on the long-standing problem of identifying the conditions under which equilibrium statistical mechanics can arise from an exact quantum mechanical treatment of the dynamics. I will give an overview of this progress, describing in particular how random matrix models and the associated concentration of measure phenomena imply that equilibration is generic even for the closed system evolution of pure quantum states. I will then discuss the relevance of these models to clarifying the conditions for quantum-classical correspondence of few-body chaotic systems. In particular, I will show that the Newtonian description of the dynamics of chaotic macroscopic bodies, remarkably, does not emerge from the underlying quantum mechanical description. These results suggest, under reasonable assumptions, that pure quantum states require a statistical interpretation.

The IceCube neutrino observatory is the world's largest high-energy neutrino telescope, utilizing the deep Antarctic ice as the Cherenkov detector medium.  In December 2010 the last of the observatory's 86 strings of optical detectors was deployed, completing the approximate cubic-kilometer array.   With the addition of a low-energy extension, called DeepCore, the observatory has very high neutrino detection efficiency for energies ranging from ~10 GeV to a few EeV.  The low-energy threshold establishes the first steps towards precision neutrino measurements in the Antarctic. Discussed will be early results from this emerging particle physics program as well as initial expectations from studies of potential future detector upgrades towards creating a multi-megaton neutrino detector with O(10 MeV) energy threshold.

The curvaton scenario provides a simple explanation for the generation of the cosmological perturbations, however most works have focused on cases with rather trivial curvaton energy potentials, e.g. quadratic ones. In this talk I will present the rich phenomenology of curvatons by showing that non-quadratic curvatons exhibit new behaviors, leading to interesting signals in the resulting density perturbations. A string theory realization of the curvaton scenario will also be discussed, where D-branes located in a warped throat region of the internal space play the role of curvatons.

We propose a form of parallel computing on classical computers that is based on matrix product states. The virtual parallelization is accomplished by evolving all possible results for multiple inputs, with bits represented by matrices. The action by classical probabilistic 1-bit and deterministic 2-bit gates such as NAND are implemented in terms of matrix operations and, as opposed to quantum computing, it is possible to copy bits. We present a way to explore this method of computation to solve search problems and count the number of solutions. We argue that if the classical computational cost of testing solutions (witnesses) requires less than O(n^2) local two-bit gates acting on n bits, the search problem can be fully solved in subexponential time. Therefore, for this restricted type of search problem, the virtual parallelization scheme is faster than Grover’s quantum algorithm.

I will describe a new numerical effort to solve Einstein gravity in 5-dimensional asymptotically Anti de Sitter spacetimes (AdS). The motivation is the gauge/gravity duality of string theory, with application to scenarios that on the gravity side are described by dynamical, strong-field solutions. For example, it has been argued that certain properties of the quark-gluon plasma formed in heavy-ion collisions can be modeled by a conformal field theory, with the dual description on the gravity side provided by the collision of black holes. As a first step towards modeling such more general phenomena, we initially focus on spacetimes with SO(3) symmetry in the bulk; i.e., axisymmetric gravity, dual to states with spherical or special conformal symmetry on the boundary. For a first application we study quasi-normal ringdown of highly deformed black holes in the bulk. Even though the initial states are far from equilibrium, the boundary state is remarkably well described as a hydrodynamic flow from early times. The code is based on the generalized harmonic formulation of the field equations, and though this method has been shown to work well in many asymptotically flat scenarios, there are unique challenges that arise in obtaining regular, stable solutions in asymptotically AdS spacetimes. I will describe these challenges, and the way we have addressed them.