**Workshop dates:** June 22 -24, 2020

**Location:** Zoom

**Hosted by** the University of British Columbia, Vancouver, Canada

**ASQC4 talk videos - Click here**

*Samson Abramsky, University of Oxford:*The Logic of Contextuality*Abstract:*Kochen and Specker's seminal work on contextuality, and later work by Conway and Kochen, contain some subtle elements which have largely been overlooked in the extensive subsequent literature on the topic. We will discuss the following topics: (i) the Kochen-Specker notion of partial Boolean algebras, and how it relates to current work on contextuality such as the sheaf-theoretic and graph-theoretic approaches, (ii) the hierarchy of logical contextuality notions considered by Kochen and Specker, (iii) a general free construction extending the compatibility/commeasurability relation on a partial Boolean algebra, which has a surprisingly wide range of uses, (iv) a Logical Exclusivity Principle, and its relation to the Probabilistic Exclusivity Principle widely studied in recent work on contextuality, (v) work towards a logical characterization of the Hilbert space tensor product, (vi) Kochen-Specker paradoxes, i.e. logically contradictory statements validated by partial Boolean algebras (specifically those arising from quantum mechanics), (vii) state-dependent vs. state-independent contextuality, and ways of transforming between them.(joint work with Rui Soares Barbosa)

*Sven Bachmann (University of British Columbia):*Abelian Anyons in gapped quantum lattice systems*Abstract:*Anyons, namely quasi-particles carrying a non-trivial representation of the braid group, have long been known to be a theoretical possibility in two-dimensional quantum systems. In this talk, I will describe how Abelian anyons arise generically from microscopic quantum lattice systems under the assumptions of a spectral gap and a topologically ordered ground state space. The conserved U(1)-charge necessarily fractionalizes. In the case of the quantum Hall effect, both fractional charge and braiding statistics are intimately related to the quantization of the Hall conductance.*Ingemar Bengtsson (University of Stockholm):*DO SICS EXIST?*Abstract:*A SIC is a configuration of vectors in a finite dimensional Hilbert space. At the minimum SICs are easy to describe and perhaps useful for some engineering purposes when they exist. The question whether they do exist in all dimensions has turned out to be entwined with a problem in algebraic number theory (Hilbert's 12th). At the moment both SIC existence and Hilbert's 12th are wide open, but many things are happening. I will try to give the flavour of some of them, emphasizing number theoretical connections between Hilbert spaces of different dimensions.*Anuj Dawar (Cambridge University):*Symmetric Boolean and Arithmetic Circuit*Abstract:*In the study of classical circuit complexity, we usaually describe the complexity of Boolean functions, i.e. functions taking strings of 0s and 1s to a Boolean value. Often the input strings represent abstract combinatorial structures and this means that the functions have natural symmetries. When these symmetries are reflected in the structure of the circuits computing a function (as is the case with circuits obtained from definitions in logic) we are able to prove strong lower bounds on the size of circuits needed. In this talk, I introduce the area of study of symmetric circuits and I present a recent result on the complexity of computing the permanent with arithmetic circuits.*John DeBrota (University of Massachusetts, Boston):*Discrete Wigner Functions from Informationally Complete Quantum Measurements*Abstract:*Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, ``probability'' distributions that can go negative. Informationally complete POVMs, a subject that has been developed more recently than Wignerâ€™s time, are less familiar but provide wholly probabilistic representations of quantum theory. In this talk, we will see that the Born Rule links these two classes of structure and discuss the art of interconverting between them. In particular, we will see that discrete Wigner functions are orthogonalizations of minimal informationally complete measurements (MICs). Pushing Wigner functions to their limits, in a suitably quantified sense, reveals a new way in which the Symmetric Informationally Complete quantum measurements (SICs) are significant. Finally, we speculate that astute choices of MICs from the orthogonalization preimages of Wigner functions may in general give quantum measurements conceptually underlying the associated quasiprobability representations.(Joint work with Blake Stacey. Talk is based on forthcoming version of arXiv:1912.07554.)

*Trithep Devakul (Princeton University):*Fractalization*Abstract:*I introduce ``fractalization'', a procedure by which certain spin models may be extended to higher-dimensions while retaining many properties of the lower-dimensional model, taking as input a set of linear cellular automaton (LCA) rules. I will review the connection between LCA, fractals, and subsystem symmetries. I show how various exotic fractal models in the literature (such as fractal subsystem symmetry-protected topological phases or Yoshida's fractal spin liquids) may be thought of as the fractal extension of well-understood lower-dimensional models (the cluster state and the toric code, respectively). This is applied to the 2D Bacon-Shor subsystem code, which results in a new 3D fractal subsystem code which we conjecture to asymptotically saturate an information storage bound for locally generated subsystem codes in 3D.(These results are part of an upcoming paper with Dominic Williamson.)

*Theo Johnson-Freyd (Perimeter Institute):*SPT phases and generalized cohomology*Abstract:*A priori, classifying n-dimensional SPT phases requires understanding the homotopy type of the topological space I^n of n-dimensional invertible phases of matter. As I will explain, the spaces I^n, for different values of n, compile into a structure called an "Omega-spectrum". This provides a huge advantage: whereas topological spaces are flimsy, Omega-spectra are rigid, and algebraic topologists have developed many powerful techniques for computing with them. In particular, for each n, there is a finite set of groups G such that knowledge of the classification of n-dimensional G-SPTs for those groups determines the classification for all groups.(Based on joint work with D. Gaiotto.)

*Shane Mansfield (Sorbonne, Paris):*Quantifying Quantum Advantage with Sequential Contextuality*Abstract:*I will introduce a notion of contextuality for sequences of transformations. Like other forms of contextuality, it can be characterised as the absence of a hidden variable model that preserves the operational (in this case sequential) structure. I will discuss two kinds of informational task, both of which involve memory-restrictions. The first kind relates to the computation of nonlinear functions, while the second has to do with information retrieval tasks including random access codes. In both cases I will demonstrate the use of sequential contextuality as a quantifier of quantum advantage.*Cihan Okay (University of British Columbia):*Stable homotopy and quantum contextuality*Abstract:*Linear constraint systems (LCS) provide instances of Kochen-Specker type contextuality proofs generalizing the well-known example of Peres-Mermin square. A LCS is specified by a linear system of equations. Solutions in the unitary group are called quantum solutions. In the contextual case no solution exists in the group of scalar matrices. Homotopy theory has proved to be useful in detecting contextual LCS and extending earlier results such as Arkhipov's graph theoretic characterization of contextuality. In the present work we extend the homotopical approach to classify quantum solutions in terms of homotopy classes of maps. For this we introduce a topological version of quantum solutions which uses classifying spaces tailored for contextuality. These classifying spaces can be ``stabilized'' in a way similar to the stabilization of vector bundles to obtain topological K-theory. This brings in a stable notion of contextuality detected by a generalized cohomology theory that is a commutative variant of topological K-theory. This procedure is in close analogy with the classification of symmetry-protected topological phases via generalized cohomology theories. We apply our machinery to prove various results about LCS.(This talk is based on arXiv:2006.07542)

*William Slofstra (IQC Waterloo):*Operator solutions for linear systems mod p*Abstract:*Any linear system over a finite field can be turned into a contextuality scenario (and also into a nonlocal game). The resulting scenario has a local hidden variables model if and only if the system has a classical solution, and a state-independent quantum model if and only if the system has an operator solution. The Mermin-Peres magic square is the prototypical example of a binary linear system with an operator solution, but no classical solution. In this case, the quantum model is a tensor product of Pauli matrices. We now know many other examples of binary linear systems with an operator solution, but no classical solution. However, examples over finite fields of order p > 2 have been harder to find. Recently, Qassim and Wallman have shown that when p > 2, a linear system cannot be solved with a tensor product of Pauli matrices unless it has a classical solution.In this talk, I'll show how to construct linear systems with an operator solution, but no classical solution, for p > 2, using small cancellation conditions. Unfortunately, while the resulting systems can be solved with infinite-dimensional operators, we do not yet know whether they can be solved with finite-dimensional operators.

Contains joint work with Luming Zhang, and with Connor Paddock, Vincent Russo, and Turner Silverthorne.

*David Stephen (Max Planck Institute for Quantum Optics, Garching/Munich):*Subsystem symmetry-enriched topological order*Abstract:*In this talk I will introduce a new type of topological order called subsystem symmetry-enriched topological (SSET) order. This type of order lives in three dimensions, and is defined by the fractionalization of planar subsystem symmetries on loop-like topological excitations. To help understand how this model is constructed, I will first review a physically intuitive procedure that can be used to create symmetry enriched topological order in 2D, in which a 2D topological order is decorated with 1D symmetry-protected topological (SPT) orders. Then, I will extend this construction to 3D topological order decorated by 2D subsystem SPT order to derive a simple model of SSET order. Finally, I will discuss some of the properties of SSET order, including extensive degeneracy of loop-like excitations, an increased value of the topological entanglement entropy, and the effect of gauging the subsystem symmetries. This model, and those proximate to it via gauging, combine features of subsystem symmetries (relevant for universal measurement-based quantum computation), and of topological order (relevant to the robust storage of quantum information). Thus, it seems hopeful that these models may find use for the purposes of robust quantum computation.(This talk is based on arXiv:2004.04181.)

*Nathanan Tantivasadakarn (Harvard University):*Jordan-Wigner Dualities for Translation-Invariant Hamiltonians in Any Dimension: Emergent Fermions in Fracton Topological Order*Abstract:*I will present a framework for an exact bosonization, which locally maps a translation-invariant model of spinless fermions to a gauge theory of Pauli spins. The duality exists for a fermion system protected by subsystem or fractal fermion parity symmetries, allowing the dual spin model to exhibit fracton topological order. Using this, I will demonstrate how to use known translation-invariant CSS codes to construct new ``twisted'' stabilizer codes where the emergent excitations are transmuted into fermions instead of bosons, and how to bosonize Majorana codes into new Pauli codes distinct from their doubled CSS codes. If time permits, I will give an example of a bosonized code that is also dual to a cluster state protected by fractal symmetries in 3D.(This talk is based on arXiv:2002.11345)

*Dominic Williamson (Stanford University):*Topological measurement-based quantum computation with the 3-fermion model*Abstract:*I will describe a scheme for topological measurement-based quantum computation using the 3-fermion Walker-Wang model. Since the 3-fermion theory does not admit a gapped boundary to vacuum, we utilize an encoding of qubits into pointlike symmetry defects, or twists. The Clifford gates can be implemented via braiding these twist defects. I will go on to explain how we introduce the symmetry defects into the 3-fermon Walker-Wang model and how they are related to the model's 2-group symmetry.*Michael Zurel (University of British Columbia):*A hidden variable model for universal quantum computation with magic states on qubits*Abstract:*I will present a hidden variable model for quantum computation with magic states on qubits. With this model, every quantum computation can be described by Bayesian update of a probability distribution on a finite state space. Negativity in a quasiprobability function is not required, neither in the states nor in the operations.(joint work with Cihan Okay and Robert Raussendorf, talk based on arXiv:2004.01992)

Cihan Okay (UBC Vancouver),

Robert Raussendorf (UBC Vancouver)