# Physics

## Secure Software Leasing Without Assumptions

Quantum cryptography is known for enabling functionalities that are unattainable using classical information alone. Recently, Secure Software Leasing (SSL) has emerged as one of these areas of interest. Given a circuit 𝐶 from a circuit class, SSL produces an encoding of 𝐶 that enables a recipient to evaluate 𝐶 and also enables the originator of the software to later verify that the software has been returned, meaning that the recipient has relinquished the possibility to further use the software. Such a functionality is unachievable using classical information alone, since it is impossible to prevent a user from keeping a copy of the software. Recent results have shown the achievability of SSL using quantum information for compute-and-compare functions (a generalization of point functions). However, these prior works all make use of setup or computational assumptions. We show that SSL is achievable for compute-and-compare circuits without any assumptions.

We proceed by studying quantum copy-protection, which is a notion related to SSL, but where the encoding procedure inherently prevents a would-be quantum software pirate from splitting a single copy of an encoding for 𝐶 into two parts each allowing a user to evaluate 𝐶. Using quantum message authentication codes, we show that point functions can be copy-protected without any assumptions against one honest and one malicious evaluator. We then show that a generic honest-malicious copy-protection scheme implies SSL. By prior work, this yields SSL for compute-and-compare functions.

This is joint work with Anne Broadbent, Stacey Jeffery, Supartha Podder, and Aarthi Sundaram.

## A hidden variable model for universal quantum computation with magic states on qubits

A central question in quantum information theory is to determine physical resources required for quantum computational speedup. In the model of quantum computation with magic states classical simulation algorithms based on quasi-probability distributions, such as discrete Wigner functions, are used to study this question. For quantum systems of odd local dimension it has been known that negativity in the Wigner function can be seen as a computational resource. The case of qubits, however, resisted a similar approach for some time since the nice properties of Wigner functions for odd dimensional systems no longer hold for qubits. In our recent work we construct a hidden variable model, which replaces the Wigner function representation, for qubit systems where any quantum state can be represented by a probability distribution over a finite state space and quantum operations correspond to Bayesian update of the probability distribution. When applied to the model of quantum computation with magic states the size of the state space only depends on the number of magic states. This is joint work with Michael Zurel and Robert Raussendorf; Phys. Rev. Lett. 125, 260404 (2020).

## Entanglement of Free Fermions on Graphs

The entanglement of free fermions on Hamming graphs will be discussed. This will be used to showcase how tools of algebraic combinatorics such as the Terwilliger algebra are well suited for this analysis. The usefulness of a Heun operator generalization will also be stressed and extensions to other association schemes will be mentioned.

## Topological superconductivity in quasicrystals

Majorana fermions -- charge-neutral spin-1/2 particles that are their own antiparticles -- have been detected in one- and two-dimensional topological superconductors. Due to the non-Abelian exchange statistics that they obey, Majorana fermions open the door to new and powerful methods of quantum information processing. Motivated by the recent experimental discovery of superconductivity in a quasicrystal, we study the possible occurrence of non-Abelian topological superconductivity (TSC) in two-dimensional quasicrystals by the same mechanism as in crystalline counterparts. We show that the TSC phase can be realised in Penrose and Ammann-Beenker quasicrystals, where the Bott index is unity. Furthermore, we confirm the existence of Majorana zero modes along the surfaces and in a vortex at the centre of the system, consistently with the bulk-boundary correspondence.

## Anomalies in (2+1)D fermionic topological phases and (3+1)D state sums for fermionic SPTs

I will describe a way to compute anomalies in general (2+1)D fermionic topological phases. First, a mathematical characterization of symmetry fractionalization for (2+1)D fermionic topological phases is presented, and then this data will be used to define a (3+1)D state sum for a topologically invariant path integral that depends on a generalized spin structure and G bundle on a 4-manifold. This path integral is a cobordism invariant and describes a (3+1)D fermion symmetry-protected topological state (SPT). The special case of time-reversal symmetry with 𝑇2=−1𝐹 gives a ℤ16 invariant of the 4D Pin+ smooth bordism group, and gives an example of a state sum that can distinguish exotic smooth structure.

**Please note, the last 3 minutes of the talk are missing from the video**

## Fractionalization and anomaly in symmetry-enriched topological phases

I will discuss recent results in the theory of symmetry-enriched topological phases, with a focus on the (2+1) case. I will review the classification of symmetry-enriched topological order and present general formula to compute relative 't Hooft anomaly for bosonic topological phases. I will also discuss partial results for fermionic topological phases and open questions.

## Classification of topological orders

Topological orders have a mathematical axiomatization in terms of their higher fusion categories of extended operators; the characterizing property of these higher fusion categories is that they are satisfy a nondegeneracy condition. After overviewing some of the higher category theory that goes into this axiomatization, I will describe what we do and don't know about the classification of topological orders in various dimensions.

## Hyperbolic band theory

The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit QED, I will present a hyperbolic generalization of Bloch theory, based on ideas from Riemann surface theory and algebraic geometry. The theory is formulated despite the non-Euclidean nature of the problem and concomitant absence of commutative translation symmetries. The general theory will be illustrated by examples of explicit computations of hyperbolic Bloch wavefunctions and bandstructures.

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## Random walks and graphs in materials, biology, and quantum information science

What does mathematics, materials science, biology, and quantum

information science have in common? It turns out, there are many

connections worth exploring. I this talk, I will focus on graphs and random

walks, starting from the classical mathematical constructs and moving on to

quantum descriptions and applications. We will see how the notions of graph

entropy and KL divergence appear in the context of characterizing

polycrystalline material microstructures and predicting their performance

under mechanical deformation, while also allowing to measure adaptation in

cancer networks and entanglement of quantum states. We will discover

unified conditions under which master equations for classical random walks

exhibit nonlocal and non-diffusive behavior and see how quantum walks allow

to realize the coveted exponential speedup in quantum Hamiltonian

simulations. Recent classical and quantum breakthroughs and open questions

will be discussed.

For other events in this series see the quanTA events website.

## The Answer to the Ultimate Question of Life, the Universe and Everything

In The Hitchhiker’s Guide to the Galaxy, by Douglas Adams, the number 42 was revealed to be the “Answer to the Ultimate Question of Life, the Universe, and Everything”. But he didn’t say what the question was! I will reveal that here. In fact it is a simple geometry question, which then turns out to be related to the mathematics underlying string theory.

#### Speaker Biography

John Baez is a leader in the area of mathematical physics at the interface between quantum field theory and category theory, and has broad interests in mathematics, and science more generally. He created one of the earliest blogs "This week's finds in Mathematical Physics" (before the term blog existed!)

Baez did his PhD at MIT, and was a Gibbs Instructor at Yale before moving to University of California, Riverside in 1988.

#### About the series

Starting in 2021, PIMS has inaugurated a high-level network-wide colloquium series. Distinguished speakers will give talks across the full PIMS network with one talk per month during the academic term. The 2021 speaker series is part of the PIMS 25th Anniversary showcase.