Sydney Quantum Information Theory Workshop
(Station Q, USA)
Symmetry Enriched Topological Phases
We examine the interplay of symmetry and topological order in 2+1 dimensional topological quantum phases of matter. Analyzing the action of microscopic symmetries on topological degrees of freedom, we uncover the classification of how symmetries may fractionalize. For on-site symmetries, we can then develop a systematic theory of extrinsic symmetry defects that describes their topological properties, such as their generalized braiding exchange transformations. This yields a complete classification of the enrichment of topological phases by such symmetries.
Noise tailoring for scalable quantum computation via randomized compiling
Abstract: Quantum computers are poised to radically outperform their classical counterparts by manipulating coherent quantum systems. A realistic quantum computer will experience errors due to the environment and imperfect control. When these errors are even partially coherent, they present a major obstacle to achieving robust computation. I will describe a method for introducing independent random single-qubit gates into the logical circuit in such a way that the effective logical circuit remains unchanged. This randomization tailors the noise into stochastic Pauli errors, leading to a dramatic improvement in worst-case error rates for satisfying the fault-tolerant threshold. Moreover, this technique is provably robust to inevitable variation in the errors over the gate sets, and, perhaps most importantly, for tailored noise the worst case error rate for universal sets of gates can be directly and efficiently measured through a recent advance in randomized benchmarking protocols based on the dihedral group. Remarkably, this method enables the realization of fault-tolerant quantum computation under the error rates observed in recent experiments.
Quantum spin chains, block spin renormalization, scale invariance and Thompson’s groups F and T
Abstract: If spins in a chain are placed at dyadic rationals the group of local scale transformations is the Thompson group F, and the Thompson group T if we impose periodic boundary conditions. It seems difficult to obtain a continuum limit in this way (and we have a no-go theorem) but the renormalisation transformations and their fixed points seem to be interesting in their own right, as do the unitary representations of the Thompson groups.
The ABCs of color codes
Abstract: The color code is an example of a topological quantum code which has computationally valuable transversal (thus fault-tolerant) logical gates. Using code switching between color codes, it is possible to fault-tolerantly implement a universal gate set. Despite the importance of the color code, there are still relevant questions to be explored, such as efficient decoding.
In the talk, I will establish a connection between the color code and a well-studied model - the toric code. I will explain how one can implement a universal gate set with gauge color codes in three dimensions using techniques of code switching and gauge fixing. I will discuss decoding of the color code and a problem of finding the threshold by analyzing a phase transition in a certain statistical mechanical model.
Braid group actions on tensor spaces
Abstract: Let g be a finite dimensional complex simple Lie algebra, and let Uq = Uq(g) be the corresponding quantised enveloping algebra over the function field K := C(q) (d’aprŹs Drinfeld). For any Uq-module V, the r-string braid group Br acts on the r-fold tensor power Tr(V)=V\otimes V\otimes... \otimes V via R-matrices, and this action commutes with the Uq-action. The braid group action may (or may not) span the whole commutant of Uq on Tr(V). It always factors through a finite dimensional quotient of the group ring KBr, such as the Temperley-Lieb algebra or the BMW algebra. I shall discuss cases where these ideas together with cellularity of the quotient, may be exploited to determine the structure of Tr(V) in the case when q is specialised at a root of unity. This is joint work with Ruibin Zhang, and partly with Henning Andersen.
Abstract: Local topological quantum field theories in 2+1-dimensions associate vector spaces to 2-manifolds, and associate linear maps to cobordisms between 2-manifolds. This is the setting of the original proposal for topological quantum computation. In fact, such field theories are in one-to-one correspondence with fusion categories, which can be specified by a finite amount of linear algebraic data. We are very far from having a classification of fusion categories. I will describe the examples identified to date: all arise from finite groups, quantum groups, or 'quadratic categories', with just one known exception, coming from the extended Haagerup subfactor.
Effective conformal field theories for tensor network states
In condensed matter theory one encounters statements such as “the transverse Ising model at its critical point is an example of a 1+1d CFT with central charge c = (1/2, 1/2)”, which can be very puzzling to the newcomer (e.g., me). This is because it seems as though a discrete lattice system is somehow being equated with a continuous system. Of course, what is really meant is that the CFT is to be understood as an effective field theory (in the sense of Wilson). This is now a standard tool, and there are many methods to identify effective field theories for lattice systems. However, none of the standard approaches fit particularly well with tensor network states as they are states, and often lack a nice lagrangian description. Recently, we have worked out an approach to identify effective field theories for quantum lattice systems by identifying the observables of an effective theory as a (sequence of) special “continuous” lattice observables. In this talk I aim to explain how to apply this approach to extract some conformal data for (a sequence of) tensor network states. The construction crucially builds on an idea of Jones, who found unitary representations of a discrete analogue of the conformal group known as Thompson’s group T. Here I’ll describe how to identify the lattice observables corresponding to the (primary, secondary, …) quantum fields with fluctuation operators, which obey a lie algebra given by an Inönü-Wigner group contraction of a loop algebra giving rise, in some cases, to representations of Kac-Moody algebras. These fields induce a rather coarse topology on state space that I argue allows us to find an associated representation of the Virasoro algebra. This is done by explicitly constructing sequences of Thompson group elements converging (in the new topology) to a given element of diff(S^1) (the chiral conformal group).
This work is currently “not without mathematical rigour”, and does rely on some physical arguments to carry it through (which I hope one day to improve).
Algebraic structure of quantum fluctuations, B. Momont , A. Verbeure, V. A. Zagrebnov, Journal of Statistical Physics November 1997, Volume 89, Issue 3, pp 633-653
Lie algebra of anomalously scaled fluctuations, M. Broidioi, B. Momont and A. Verbeure, J. Math. Phys. 36, 6746 (1995)
Detecting topological order in the Heisenberg picture
Abstract: We introduce a numerical method for identifying topological order in two-dimensional models based on one-dimensional bulk operators. The idea is to identify approximate symmetries supported on thin strips through the bulk that behave as string operators associated to an anyon model. We can express these ribbon operators in matrix product form and define a cost function that allows us to efficiently optimize over this ansatz class. We test this method on spin models with abelian topological order by finding ribbon operators for Z_d quantum double models with local fields and Ising-like terms. In addition, we identify ribbons in the abelian phase of Kitaev's honeycomb model. We further identify the topologically encoded qubit in the quantum compass model, and show that despite this qubit, the model does not support topological order.
(Imperial College, UK)
Topological phase transitions in tensor networks: A holographic perspective
Abstract: We investigate topological phases and phase transitions in the framework of tensor network models. We discuss the role of symmetries in this description, and show how it allows to relate topological phases and transitions between them to symmetry broken and symmetry protected phases exhibited by the transfer operator of the system, i.e., at the boundary. This is accomplished by translating the string-like topological excitations in the 2D bulk to string order parameters characterizing the different phases under symmetry at the boundary. We show that by taking into account the constraints arising from complete positivity of the transfer operator, which restricts the possible phases at the boundary, this yields a complete characterization of all possible ways in which topological phase transitions can occur through condensation and confinement of anyons.
Quantum Tensor Network Intersections between topology, quantum many-body physics and quantum information
(Perimeter Institute, Canada)
Tensor network renormalization: scale invariance on the lattice
I will discuss how to define scale transformations on 1d quantum (or 2d classical) systems on the lattice. At a quantum (or thermal) critical point, one expects scale invariance in the continuum. On the lattice, however, scale invariance is explicitly broken by the lattice spacing. Nevertheless, we will see that one can still define global scale transformations on the lattice, under which the critical 1d ground state (2d partition function) is explicitly invariant and, more generally, an RG flow is produced with the correct structure of fixed points. Moreover, one can similarly define local scale transformations on the lattice, and observe an emergent local scale invariance/covariance. On the practical side, such transformations yield accurate numerical estimates of the universal data characterizing the phase transition --- namely the central charge, scaling dimensions, conformal spins, and operator product expansion coefficients of the underlying conformal field theory.
Based on work with Glen Evenbly et al, arXiv:1412.0732(prl), arXiv:1502.05385(prl), arXiv:1510.00689, and arXiv:1510.07637.
(Perimeter Institute, Canada)
Transversal logical gate, group cohomology and gapped boundary
Abstract: Finding/classifying transversal logical gates in quantum error-correcting codes is a long-standing problem which is at the heart of fault-tolerant quantum computing. In this talk, I will establish the connection among transversal logical gates, symmetry-protected topological (SPT) phases and gapped boundaries in the context of topological quantum codes.
We begin by presenting constructions of gapped boundaries for the d-dimensional quantum double model by using d-cocycles functions (d≥2). We point out that the system supports m-dimensional excitations (m<d), which we shall call fluctuating charges, that are superpositions of point-like electric charges characterized by m-dimensional bosonic SPT wavefunctions. There exist gapped boundaries where electric charges or magnetic fluxes may not condense by themselves, but may condense only when accompanied by fluctuating charges. Magnetic fluxes and codimension-2 fluctuating charges exhibit non-trivial multi-excitation braiding statistics, involving more than two excitations. The statistical angle can be computed by taking slant products of underlying cocycle functions sequentially. We find that excitations that may condense into a gapped boundary can be characterized by trivial multi-excitation braiding statistics, generalizing the notion of the Lagrangian subgroup. As an application, we construct fault-tolerantly implementable logical gates for the d-dimensional quantum double model by using d-cocycle functions. Namely, corresponding logical gates belong to the dth level of the Clifford hierarchy, but are outside of the (d_1)th level, if cocycle functions have non-trivial sequences of slant products.