**Workshop dates:** May 22 - 26, 2017

**Location:** University of British Columbia, Vancouver, Canada

*Juan Bermejo-Vega, Freie Universitaet Berlin, Berlin, Germany:*Architectures for quantum simulation showing quantum supremacy*Abstract:*One of the main aims in the field of quantum simulation is to achieve what is called "quantum supremacy", referring to the experimental realization of a quantum device that computationally outperforms classical computers. In this work, we show that one can devise versatile and feasible schemes of two-dimensional dynamical quantum simulators showing such a quantum supremacy, building on intermediate problems involving IQP circuits. In each of the schemes, an initial product state is prepared, potentially involving an element of randomness as in disordered models, followed by a short time evolution under a translationally invariant Hamiltonian with nearest-neighbor interactions and a mere sampling measurement in a fixed basis. The final state preparation in each scheme is fully efficiently certifiable. We discuss experimental necessities and possible physical architectures, inspired by platforms of cold atoms in optical lattices and a number of others, as well as specific assumptions that enter the complexity-theoretic arguments. This work shows that benchmark settings exhibiting a quantum advantage may require little control in contrast to universal quantum computing.Based on arXiv:1703.00466

*Lorenzo Catani, University College London, London, UK:*Spekkens' toy model as a unifying framework for state-injection schemes with contextuality as resource*Abstract:*The idea of this project is to show that Spekkens' toy model - a non-contextual phase-space inspired hidden variable model with a restriction on what an observer can know about the reality - can be used to unify frameworks of state-injection schemes, in all dimensions, where contextuality is an injected resource to reach universal quantum computation. I will apply this idea, formalised in the notion of "Spekkens' circuits", to the popular examples of Howard et al. regarding qudits (odd prime dimensions) and Delfosse et al. regarding rebits. At the end I will also compare our approach to the recent one by Bermejo-Vega et al. . Our framework, in addition to this application in quantum computation, would also answer the open question about which are the maximal sub theories of Spekkens' model that are consistent (operationally equivalent) with QM.*Nicolas Delfosse, UC Riverside + Caltech, CA, USA:*Optimal decoding of surface codes for qubit loss, and a little bit more*Abstract:*Surface codes are among the best candidates to ensure the fault-tolerance of a quantum computer. In order to avoid the accumulation of errors during a computation, it is crucial to have at our disposal a fast decoding algorithm to quickly identify and correct errors as soon as they occur. I will describe a decoding algorithm for dealing with qubit loss that is optimal both in terms of performance and speed. Then, I will talk about realistic applications of this decoding strategy.Based on joint work with Gilles Zemor, https://arxiv.org/abs/1703.01517 .

*Iman Marvian, MIT, USA:*Symmetry-Protected Topological Entanglement*Akimasa Miyake, University of New Mexico, Albuquerque, USA:*Measurement-based quantum computation and genuine 2D symmetry-protected topological orders*Abstract:*After a brief introduction to symmetry-protected topological orders, I discuss novel algebraic structures of measurement-based quantum computation when the resource multipartite entanglement is genunine (or stronger form of) 2D symmetry-protected topologically ordered states. This talk is based on recent works (arXiv:1508.02695, arXiv:1612.08135, arXiv:1703.11002) in collaboration with Jacob Miller.*Hendrik Poulsen Nautrup, University of Innsbruck, Austria:*Fault-tolerant Interface between quantum memories and processors*Abstract:*Topological error correction codes are promising candidates to protect quantum computations from the deteriorating effects of noise. While some codes provide high noise thresholds suitable for robust quantum memories, others allow straightforward gate implementation needed for data processing. To exploit the particular advantages of different topological codes for fault-tolerant quantum computation, it is necessary to be able to switch between them. In my talk I present a practical solution, subsystem lattice surgery, which requires only two-body nearest neighbor interactions in a fixed layout in addition to the indispensable error correction. This method can be used for the fault-tolerant transfer of quantum information between arbitrary topological subsystem codes in two dimensions. As an example which is of practical interest, I consider a simple interface, a quantum bus, between noise resilient surface code memories and flexible color code processors.*Cihan Okay, University of Western Ontario, London, Canada:*Topological methods and contextuality*Abstract:*I will talk about some applications of topological methods in quantum computation. More specifically the topological part will involve group cohomology and various constructions suitable for studying contextuality in quantum mechanics. The aim of the talk will be towards finding possible applications of structural results for such constructions which arise in topology. For a recent work along these lines see arXiv:1701.01888.*Robert Raussendorf, University of British Columbia, Vancouver, Canada**David Stephen, University of British Columbia, Vancouver, Canada**Emily Tyhurst, University of British Columbia, Vancouver, Canada:*Separating state-independent and state-dependent contextuality in the context of Mermin's square*Abstract:*Connections between negativity in quasiprobability distributions and quantum contextuality as a resource for computation are well-established in local Hilbert space dimension greater than two. However, for qubits the separation between state-independent and state-dependant contextuality complicates matters. In this talk I will speak about the canonical example of Mermin's square, and a simple contextual hidden variable model that allows for a classical simulation of the system. The simulation method deliberately separates costs due to state-independent contextuality, provided by two different hidden variable models; and the costs due to state-dependent contextuality, provided by a quasiprobability distribution over the hidden variable models. The quasiprobability distribution is in part due to work by Howard and Campbell in arXiv:1609.07488 [quant-ph]*Mark van Raamsdonk, University of British Columbia, Vancouver, Canada:*Locally Maximally Entangled States of Multipart Quantum Systems*Abstract:*For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement, i.e. the reduced density matrix for each elementary subsystem is a multiple of the identity matrix. In this talk, we first show that information about the representations of arbitrary finite and compact groups can be used to construct a special class of ``stabilizer'' LME states. We then review how the space of LME states up to local unitary transformations has two very natural geometrical descriptions, one as a symplectic manifold (i.e. with the structure of a phase space in Hamiltonian classical mechanics) and one as a complex manifold. The equivalence of these descriptions shows that the space of LME states up to local unitary transformations is actually equivalent to the space of all states with ``generic'' entanglement up to SLOCC equivalence. Using this geometrical viewpoint, we are able to provide necessary and sufficient conditions on the subsystem dimensions (d_1, d_2, ... , d_n) for the existence of LME states and compute the dimension of the space of such states when they exist.*Dongsheng Wang, University of British Columbia, Vancouver, Canada:*Topological qubits from valence bond solids*Tzu-Chieh Wei, University of Stony Brook, Stony Brook, USA:*Symmetry-protected topologically ordered states for universal quantum computation*Abstract:*Measurement-based quantum computation (MBQC) is a model for quantum information processing utilizing local measurement on suitably entangled states for the implementation of quantum gates. The cluster state on the 2D square lattice was first discovered to enable universal quantum computation. However, complete characterization for universal resource states is still missing. Recent development in condensed matter physics on symmetry-protected topological (SPT) order has provided an intriguing link and new perspective for the quest of novel resource states. The 2D AKLT states are special points in the so-called valence-bond solid phase, which is regarded as a SPT phase, but it requires translational invariance to be imposed, in addition to the internal spin rotation symmetry. Here I will show that certain types of fixed-point wave functions in generic nontrivial 2D SPT phases (without requiring translational invariance) are indeed universal for MBQC. (These fixed-point results can be generalized to higher dimensions.) Moreover, I will discuss whether we can extend the universality beyond the fixed points by examples. What would be a potential breakthrough is to establish an entire SPT phase supporting universal MBQC, but this is still an open question.*Kohei Kishida, University of Oxford, Oxford, UK*

*Abstract:* I propose an order parameter for the Symmetry-Protected
Topological (SPT) phases which are protected by Abelian on-site symmetries. This
order parameter, called the "SPT-entanglement", is defined as the entanglement
between A and B, two distant regions of the system, given that the total charge
(associated with the symmetry) in a third region C is measured and known, where C
is a connected region surrounded by A, B and the boundaries of the system. In the
case of 1-dimensional systems I prove that in the limit where A and B are large and
far from each other compared to the correlation length, the SPT-entanglement remains
constant throughout a SPT phase, and furthermore, it is zero for the trivial phase
while it is nonzero for all the non-trivial phases. Moreover, I show that the
SPT-entanglement is invariant under the low-depth quantum circuits which respect
the symmetry, and hence it remains constant throughout a SPT phase in the higher
dimensions as well. Also, I show that there is an intriguing connection between
SPT-entanglement and the Fourier transform of the string order parameters, which
are the traditional tool for detecting SPT phases. This leads to a new algorithm
for extracting the relevant information about the SPT phase of the system from the
string order parameters. Finally, I discuss implications of these results in the
context of measurement-based quantum computation.

Juan Bermejo-Vega (FU Berlin, Germany),

Robert Raussendorf (UBC),

Tzu-Chieh Wei (Stony Brook, USA)