**Coogee16**

**Sydney Quantum Information Theory Workshop**

**Parsa**** Bonderson**

**(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.

**Joseph
Emerson**

**(Waterloo,
Canada)**

*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.

**Vaughan
Jones **

**(Vanderbilt, USA)**

*Quantum
spin chains, block spin renormalization, scale invariance and Thompsons 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.

**Aleksander**** Kubica **

**(Caltech,
USA)**

*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.

Based on http://arxiv.org/abs/1410.0069, http://arxiv.org/abs/1503.02065 and
recent work with F. Brandao, K. Svore
and N. Delfosse.

**Gus
Lehrer**

**(Sydney,
Australia)**

*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) (daprs
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.

**Scott
Morrison **

**(ANU,
Australia)**

*Fusion
Categories*

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.

**Tobias
Osborne**

**(Hannover,
Germany)**

*Effective
conformal field theories for tensor network states*

Abstract:

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
Thompsons group T. Here Ill 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 Inn-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).

Further
reading:

https://github.com/tobiasosborne/What-is-a-quantum-field-state-

https://github.com/tobiasosborne/Continuous-Limits-of-Quantum-Lattice-Systems

http://arxiv.org/abs/1412.7740

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)

**David
Poulin **

**(Sherbrooke, Canada)**

*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.

**Terry
Rudolph**

**(Imperial
College, UK)**

**Norbert
Schuch **

**(Munich,
Germany)**

*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.

**Frank
Verstraete **

**(Vienna,
Austria)**

*Quantum
Tensor Network Intersections between topology, quantum many-body physics and
quantum information*

**Guifre**** Vidal **

**(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.

**Beni**** Yoshida **

**(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 (d2). 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.