Theory of Decoherence-Free Fault-Tolerant Universal Quantum Computation

Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically relevant interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonians whose induced evolution always preserves a DFS are derived within this stabilizer formalism. Two possible collective decoherence mechanisms arising from permutation symmetries of the system-bath coupling are examined within this framework. It is shown that in both cases universal quantum computation which always preserves the DFS (*natural fault-tolerant computation*) can be performed using only two-body interactions. This is in marked contrast to standard error correcting codes, where all known constructions using one or two-body interactions must leave the codespace during the on-time of the fault-tolerant gates. A further consequence of our universality construction is that a single exchange Hamiltonian can be used to perform universal quantum computation on an encoded space whose asymptotic coding efficiency is unity. The exchange Hamiltonian, which is naturally present in many quantum systems, is thus *asymptotically universal*.Comment: 40 pages (body: 30, appendices: 3, figures: 5, references: 2). Fixed problem with non-printing figures. New references added, minor typos correcte

Approximation of L\"owdin Orthogonalization to a Spectrally Efficient Orthogonal Overlapping PPM Design for UWB Impulse Radio

In this paper we consider the design of spectrally efficient time-limited pulses for ultrawideband (UWB) systems using an overlapping pulse position modulation scheme. For this we investigate an orthogonalization method, which was developed in 1950 by Per-Olov L\"owdin. Our objective is to obtain a set of N orthogonal (L\"owdin) pulses, which remain time-limited and spectrally efficient for UWB systems, from a set of N equidistant translates of a time-limited optimal spectral designed UWB pulse. We derive an approximate L\"owdin orthogonalization (ALO) by using circulant approximations for the Gram matrix to obtain a practical filter implementation. We show that the centered ALO and L\"owdin pulses converge pointwise to the same Nyquist pulse as N tends to infinity. The set of translates of the Nyquist pulse forms an orthonormal basis or the shift-invariant space generated by the initial spectral optimal pulse. The ALO transform provides a closed-form approximation of the L\"owdin transform, which can be implemented in an analog fashion without the need of analog to digital conversions. Furthermore, we investigate the interplay between the optimization and the orthogonalization procedure by using methods from the theory of shift-invariant spaces. Finally we develop a connection between our results and wavelet and frame theory.Comment: 33 pages, 11 figures. Accepted for publication 9 Sep 201

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

Anyons in an exactly solved model and beyond

A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static $\mathbb{Z}_{2}$ gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number $\nu$. The Abelian and non-Abelian phases of the original model correspond to $\nu=0$ and $\nu=\pm 1$, respectively. The anyonic properties of excitation depend on $\nu\bmod 16$, whereas $\nu$ itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.Comment: 113 pages. LaTeX + 299 .eps files (see comments in hexagon.tex for known-good compilation environment). VERSION 3: some typos fixed, one reference adde

A magnetic model with a possible Chern-Simons phase

An elementary family of local Hamiltonians $H_{\c ,\ell}, \ell = 1,2,3, ldots$, is described for a $2-$dimensional quantum mechanical system of spin $={1/2}$ particles. On the torus, the ground state space $G_{\circ,\ell}$ is $(\log)$ extensively degenerate but should collapse under \lperturbation" to an anyonic system with a complete mathematical description: the quantum double of the $SO(3)-$Chern-Simons modular functor at $q= e^{2 \pi i/\ell +2}$ which we call $DE \ell$. The Hamiltonian $H_{\circ,\ell}$ defines a \underline{quantum} \underline{loop}\underline{gas}. We argue that for $\ell = 1$ and 2, $G_{\circ,\ell}$ is unstable and the collapse to $G_{\epsilon, \ell} \cong DE\ell$ can occur truly by perturbation. For $\ell \geq 3$, $G_{\circ,\ell}$ is stable and in this case finding $G_{\epsilon,\ell} \cong DE \ell$ must require either $\epsilon > \epsilon_\ell > 0$, help from finite system size, surface roughening (see section 3), or some other trick, hence the initial use of quotes {\l}\quad". A hypothetical phase diagram is included in the introduction.Comment: Appendix by F. Goodman and H. Wenz