Quantum Information

In this talk, two specific directions of research in quantum information are presented which could potentially gain from graph theory. The first is the study of quantum communication using systems of perpetually interacting qubits (or spins) as a databus. After introducing the topic through the simplest examples of linear chains of spins as transmitters of quantum information, we briefly mention existing work on quantum communication through more general graphs of spins. We then explain why the transmission of quantum information between vertices of a graph in the case of an isotropic Heisenberg interaction (between the spins placed at these vertices) depends on the Laplacian of the graph. How the quality of communication varies when starts cutting edges after starting a fully connected graph will be discussed *. Another direction is related to the entanglement naturally present in the ground state of a graph of perpetually interacting spins: various specific examples --- fully connected, star and tree will be discussed. Some easily solvable interactions obtained by putting higher spins in specific vertices of simple graphs will also be discussed. In the end we also present an example where one can get a \'graph independent\' ground state by placing qudits with exchange interactions on an arbitrary graph. (* The first part of the talk is based on ongoing work with Simone Severini, Stefano Mancini and Andrea Casaccino, while the second part is based on work with Vladimir Korepin and Christopher Hadley)

In arXiv:quant-ph/0608223, quantum network coding was proved to be no more useful than simply routing the quantum transmissions in some directed acyclic networks. This talk will connect this result, monogamity of entanglement, and graph theoretic properties of the networks involved.

It is well-known that if a graph G_1 can be obtained from another graph G_2 by removing a degree-2 vertex and combing its two neighbors, the graph state |G_1> can be obtained from |G_2> through LOCC. In this talk, I will describe how to construct a graph G\' from a given graph G so that (a) The maximum degree of G\', Delta(G\'), is no more than 3. (b) G can be obtained from G\' by a sequence of contraction operations described above. (c) The treewidth of G\', tw(G\'), is no more than tw(G)+1. (d) The construction takes exp(O(tw(G))) time. Those properties imply that a one-way quantum computation on the graph state |G> can be simulated by a randomized algorithm in time exp(O(tw(G))). Previously, it was known that such a computation can be simulated in time exp(O(tw(G) Delta(G))) [Markov and Shi, to appear in SICOMP], which is substantially more expensive than our bound when Delta(G) is large. Joint work with Igor Markov, based on arXiv:0707.3622.

It is well known that the toric code model supports abelian anyons. It can be realized on a square lattice of qubits, where the anyons are represented by the endpoints of strings of Pauli operators. We will demonstrate that the non-abelian Ising model can be realized in a similar way, where now the string operators are elements of the Clifford group. The Ising anyons are shown to be essentially superpositions of the abelian toric code ones, reproducing the required fusion, braiding and statistical properties. We propose a string framing and ancillary qubits to implement the non-trivial chirality of this model.

This talk will report recent work on two themes that relate concepts in graph theory to problems in quantum information theory. We will discuss the quantum analogue of expander graphs which prove to be of key importance when de-randomizing algorithms in classical computer science. Using powerful ideas of discrete phase space methods, efficiently implementable quantum expanders can be constructed based on an argument that barely fills three lines. We also briefly report news on novel measurement-based models of quantum computing, based on quantum systems distributed on a graph, beyond one-way computing. Work done in collaboration with D. Gross D. Gross, J. Eisert, \'Quantum Margulis expanders\', Quant. Inf. Comp. (2008), arXiv:0710.0651. D. Gross, J. Eisert, \'Quantum computational wires\', in preparation (2008). D. Gross, J. Eisert, N. Schuch, D. Perez-Garcia, \'Measurement-based quantum computation beyond the one-way model\', Phys. Rev. A 76, 052315 (2007), arXiv:0706.3401. D. Gross, J. Eisert, \'Novel schemes for measurement-based quantum computation\', Phys. Rev. Lett. 98, 220503 (2007).

I will discuss a quantum algorithm for the exact evaluation of the classical Potts partition function for a class of graphs (and hypergraphs) related to a family of classical cyclic codes. I will also present a mapping I recently constructed from quantum circuit instances to graphs and discuss some relationships to the classical Ising partition function.

New and exotic phases as well as remarkable entanglement behaviors emerge in condensed matter systems (and quantum devices) living (fabricated) on graphs. To illustrate this, I will discuss the properties of Josephson junction networks fabricated on comb and star graphs and of spin models living on pertinent fiber-graphs.

Measurement-based quantum computation is unusual among quantum computational models in that it does not have an obvious classical analogue. In this talk, I shall describe some new results which shed some new light on this. In the one-way model [1], computation proceeds by adaptive single-qubit measurements on a multi-qubit entangled \'cluster state\'. The adaptive measurements require a classical computer, which processes the previous measurement outcomes to determine the correct bases for the following measurement. We shall describe a generalisation of the model where this classical \'side-computation\' plays a more central role. We shall show that this classical computer need not be classically universal, and can instead by performed by a limited power \'CNOT computer\' - a reversible classical computer whose generating gate set consists of CNOT and NOT. The CNOT computer is not universal for classical computation and is believed to be less powerful. Most notably in the context of quantum computation, it is the class of computer sufficient for the efficient simulation of Clifford group circuits [2] - a closely related result. This motivates the question of what resource states would be universal for classical computation, if the control computer is in the CNOT class. We shall answer this question with several examples. Leading from these examples, a natural question is thus, is \'classically universal measurement based computation\' possible with solely classical physics? By considering different settings, we shall answer this question both in the negative and positive, and draw some striking connections with some well-known techniques from models of generalised no-signalling theories. [1] R. Raussendorf and H.J. Briegel, Phys Rev Lett (2001) 86 5188 [2] S. Aaronson and D. Gottesman, Phys Rev A (2004) 70 052328 This is joint work with Janet Anders. We would like to acknowledge inspiring and fruitful discussions with Hans Briegel, Akimasa Miyake, Robin Blume Kohout and Debbie Leung.

Recently a simple but perhaps profound connection has been observed between the unitary solutions of the Yang-Baxter Equations (YBE) and the entangled Bell states and their higher dimensional (or more-qubit) extensions, the generalized GHZ states. We have shown that this connection can be made more explicit by exploring the relation between the solutions of the YBE and the representations of the extra-special two-groups. This relationship brings certain topological-like features to quantum information theory, and makes a connection to the well-known Jones polynomials which are topological invariants of knots and links. This emerging connection may deepen our understanding, through new representations of extra-special two-groups, of quantum error correction and topological quantum computation. This work is a collaboration with Eric Rowell, Zhenghan Wang, Molin Ge, and Yong-Zhang.

A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of N-qudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per two-qubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin\'s array and a bipartite-part and (c) a maximum independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin\'s square, and set of five mutually non-commuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) N-qubit Pauli graph is shown to be pseudo-geometric, i. e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit system is introduced; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the generalized quadrangle Q(4, 3), the dual ofW(3). Joint work with Metod Saniga. Based on arXiv:quant-ph/0701211.

We find that the overlapping of a topological quantum color code state, representing a quantum memory, with a factorized state of qubits can be written as the partition function of a 3-body classical Ising model on triangular or Union Jack lattices. This mapping allows us to test that different computational capabilities of color codes correspond to qualitatively different universality classes of their associated classical spin models. By generalizing these statistical mechanical models for arbitrary inhomogeneous and complex couplings, it is possible to study a measurement-based quantum computation with a color code state and we find that their classical simulatability remains an open problem. We complement the meaurement-based computation with the construction of a cluster state that yields the topological color code and this also gives the possibility to represent statistical models with external magnetic fields. Joint work with M.A. Martin-Delgado.

A fundamental theorem of quantum field theory states that the generating functionals of connected graphs and one-particle irreducible graphs are related by Legendre transformation. An equivalent statement is that the tree level Feynman graphs yield the solution to the classical equations of motion. Existing proofs of either fact are either lengthy or are short but less rigorous. Here we give a short transparent rigorous proof. On the practical level, our methods could help make the calculation of Feynman graphs more efficient. On the conceptual level, our methods yield a new, unifying view of the structure of perturbative quantum field theory, and they reveal the fundamental role played by the Euler characteristic of graphs. This is joint work with D.M. Jackson (UW) and A. Morales (MIT)