This is the homepage for the schedule of talks that will be held in Robert Raussendorf’s Quantum Information group.
Location: Hennings 309
Time: Tuesdays, 4 pm
Oct 19: Spectra of Graphs by Oleg
Spectral graph theory deals with problems on the intersection of graph theory and linear algebra. In this talk I will focus on what spectral graph theory can tell us about eigenvalues of matrices. Specifically, I will introduce the notions of quotient graphs, equitable partitions and the interlacing theorem. This will be based on the nice – physicist friendly – overview given in arXiv:1311.1759v4. I will finish with some of my thoughts on how to adopt these ideas to the problems of Hamiltonian spectrum in many-body systems.
Nov 2: Paper review by Robert
Review of the paper “Real-time dynamics of lattice gauge theories with a few-qubit quantum computer” by Martinez, Esteban A.; Muschik, Christine A.; Schindler, Philipp; et al.
Nov 16: A generalization of Arkhipov’s framework by Cihan
Mermin-type contextuality proofs can be generalized to magic arrangements. Arkhipov shows that an arrangement is magic if and only if the intersection graph, a certain graph associated to the arragement, is non-planar. However, there are two restrictions (1) Eigenvalues are restricted to only two values, (2) Each observable belongs to exactly two contexts. I will describe an approach that will allow us to remove these two restrictions.
Nov 23: Cavity Quantum Electrodynamics by Hirsh
Cavity Quantum Electrodynamics in the Nonperturbative Regime, PRA 97, 043820, by Bernardis, Jaako, and Rabl from TU Wien. First I’ll go over a brief derivation of perturbative cQED. Next I’ll present the paper on non-perturbative cQED and it’s nontrivial ground state. The ground state undergoes a quantum phase transition based on the strength of light-matter coupling. In these phases the dipoles spontaneously polarize and the field acquires a non-zero expectation value.
Nov 30: Efficient computation of the partial trace by Joe
I will begin my talk with a quick refresher of the partial trace, specifically with regard to the implementation used in the paper. Then the numerical computation for bipartite systems will be discussed and how it can be more efficiently calculated. This then leads to the partial trace for multi-partitions, whose computational efficiently will also be discussed.
Dec 14: Quantum computing in finance by Michael
I will be talking about an application of quantum computing in finance, namely, amplitude estimation and quantum Monte Carlo methods for options pricing. I will primarily be reviewing the paper “Quantum computational finance: Monte Carlo pricing of financial derivatives” by Patrick Rebentrost, Brajesh Gupt, and Thomas R. Bromley.
Jan 29: Topology of operator constraints by Cihan
This is a continuation of generalization of Arkhipov’s framework. We study operator relations via the topology of two dimensional cell complexes. Given the operator constraints the goal will be to construct a projective representation of the fundamental group of the cell complex.
Feb 26: A characterization of sets of n-qubit Pauli operators which are non-contextual and closed under inference by Michael
I will give a standard form for sets of n-qubit Pauli operators which are non-contextual and closed under inference and then I’ll prove that there are no sets of Pauli operators which are non-contextual, closed under inference, and not of this form.
Apr 9: Noise-robust contextuality for Kochen-Specker type scenarios: two complementary frameworks by Ravi Kunjwal
The Kochen-Specker (KS) theorem is a mathematical result that points out the inconsistency between quantum theory and any putative underlying model of it where the outcomes of a measurement are fixed prior to the act of measurement by some (possibly hidden) physical states of the system in a manner that does not depend on (operationally irrelevant) details of the measurement context, i.e., the outcome a ssignments are fixed noncontextually in the model. Thus, quantum theory admits KS-contextuality. On the other hand, within the generalized approach to contextuality proposed by Spekkens, the assumption of deterministic outcome assignments is relaxed and, instead, noncontextuality for preparation procedures is invoked to obtain constraints on the operational statistics. I will present two complementary hypergraph frameworks that accomplish the project of accommodating Kochen-Specker (KS) type scenarios within the generalized approach to contextuality due to Spekkens. One of these frameworks generalizes the graph-theoretic framework of Cabello, Severini, and Winter for statistical proofs of KS-contextuality and the other one outlines how one can obtain noise-robust noncontextuality inequalities based on logical proofs KS contextuality. Time permitting, I will speculate on possible applications of these frameworks. Based on arXiv: 1805.02083, 1709.01098, and 1708.04793, 1506.04150.
Apr 16: On Construction of Irreducible Representations of Von Neumann Matrix Algebras or How to Simultaneously Block-Diagonalize a Bunch of Matrices by Oleg
It is not possible to simultaneously diagonalize non commuting matrices. Nevertheless, it is possible to maximally block diagonalize any set of matrices using the structure of Von Neumann matrix algebras (VNMA). VNMAs (a.k.a -algebras) are becoming a common tool in physical literature with application ranging from quantum codes to the study of dynamics, decoherence and super selection rules. In this talk I will introduce the notion of VNMA and the motivation behind it, and give an overview of an algorithmic construction of irreducible representations of VNMA from generators. At the very least, we will learn how to block diagonalize a bunch of non commuting matrices.
Sept 13 @11am: Generative training of quantum Boltzmann machines with hidden units by Nathan Wiebe
We provide the first method for fully quantum generative training of quantum Boltzmann machines with both visible and hidden units while using quantum relative entropy as an objective. This is significant because prior methods were not able to do so due to mathematical challenges posed by the gradient evaluation. We present two novel methods for solving this problem. The first proposal addresses it, for a class of restricted quantum Boltzmann machines with mutually commuting Hamiltonians on the hidden units, by using a variational upper bound on the quantum relative entropy. The second one uses high-order divided difference methods and linear-combinations of unitaries to approximate the exact gradient of the relative entropy for a generic quantum Boltzmann machine. Both methods are efficient under the assumption that Gibbs state preparation is efficient and that the Hamiltonian are given by a sparse row-computable matrix.
Sept 17 @5pm: Problems in topology of contextuality by Cihan
I will present a topological approach to studying contextuality following this paper. In this work first homotopy group of a space plays a crucial role in determining whether a given arrangement of observables is contextual. I will sketch this approach and end by listing some open problems.
Sept 23: No group meeting
Oct 1: Finding Quantum Wire in Hypergraph States by Amelia
We consider hypergraph states supported on sets $\Omega$ of qubits. For such a state $|H\rangle$, we divide $\Omega$ into three disjoint subsets, $\Omega=I\cup M\cup O$. $I$ is an ”input region”, $M$ a ”transmimssion region” and $O$ an ”output region”. We set out to find hypergraph states $|H\rangle$ on $\Omega$ where there exists Bell-type entanglement between regions $I$ and $O$ after the qubits in M have been measured in the local $X$-basis. This talk goes through the procedure of preparing the hypergraph state, performing the measurement, extracting qubits from the state and determining entanglement and suitability for quantum wire. We examine some of these results with the current stabilizer argument and find some of the state’s post-measurement stabilizers.
Oct 8: Symmetry reduction for phase space simulation methods (Part 1) by Michael
Phase space methods are a promising approach for classically simulating quantum computation but in even dimensional Hilbert spaces there is an obstacle to practically using these methods. In even dimensional Hilbert spaces, the Wigner function is not unique. The best Wigner function to use for simulation is the solution to a convex optimization problem and the size of this convex optimization problem scales exponentially with the number of qubits. Here I will present some techniques for using simultaneous symmetries of states and the phase space to reduce this convex optimizaton problem to one which scales polynomially with the number of qubits. The main references I will use are: arXiv:1905.05374 and Quantum 3, 132 (2019).
Oct 15: Symmetry reduction for phase space simulation methods (Part 2) by Michael
continuation…
Nov 5: Towards 1D MBQC with Fermionic Systems by Paul
The search for a physically realizable cluster state is ongoing and is motivated by the potential of quantum computers to significantly outperform their classical counterparts for certain important classes of problems, such as prime factoring and quantum many-body simulations. In this report I analyze a system of fermions to determine whether or not it can be used as a resource for MBQC. Fermions are a class of particles characterized by the Pauli exclusion principle, which causes systems of fermions to be naturally entangled and suggests that they might be a good candidate for a physically realizable cluster state. However, some significant obstacles to creating a fermionic cluster state remain unresolved. For instance, a cluster state must have entanglement only between neighboring particles, but for free fermions the entanglement extends across the entire system, since in principle any two fermions could swap positions regardless of the distance between them. In 2012 Feder proposed confining non-interacting fermions in a double-well potential lattice and showed that its ground state is equivalent to a 1D cluster state. My research extends that work by considering the effect of adding interactions between fermions as well as an extra potential on one of the lattice sites. The goal is to determine whether or not these new features make it possible to implement fermionic MBQC. Specifically, is it possible to teleport a universal set of qubit rotations using measurements of individual lattice sites?
Nov 19: Quantum to classical transition and its relation to fundamental scales by Oleg
Canonical commutation relations of position and momentum is one of the defining properties of quantum mechanics. As a result, the non-vanishing Planck constant sets the scale at which quantum effects such as measurement disturbance and uncertainty relations emerge. In this talk I will demonstrate that on a lattice, the scale at which quantum to classical transition occur is given by the square root of the size of the lattice. By taking the continuum limit we will recover the quantum-classical transition scale set by the Planck constant and see how it relates to the fundamental scales of minimal and maximal lengths.
Jan 28: Quantum Time Travel using Closed Timelike Curves (CTCs) by Arnab
In this talk, we will discuss two nonequivalent quantum models of time travel using CTCs. We will also look into some computational and cryptographic schemes that use CTCs as resources to perform tasks that are impossible to do in a world without CTCs. We would conclude the talk by having a closer look at causality and consistency constraints in such schemes.
Feb 25: Overview of variational quantum optimization algorithms by Oleg
I will introduce the basic principles behind VQE and QAOA algorithms and discuss the challenges and applications.
Mar 10: Renormalizing Entanglement in Tensor Networks by Paul
I will review the multi-scale entanglement renormalization ansatz (MERA), a type of tensor network which is designed to preserve local operators and entanglement. Ref: https://arxiv.org/pdf/quant-ph/0610099.pdf
Mar 24: Stable homotopy theory and quantum contextuality by Cihan
I will describe an approach to study quantum solutions to linear constraint systems which uses methods from stable homotopy theory.
Apr 14: Paper review by Robert
Review of the paper.
May 5: Our new paper, and Robert Spekkens ideas about contextuality by Michael
The talk will be in two parts. In the first part I will review our new paper (arXiv:2004.01992). I’ll briefly summarize the main results and discuss their implications. In the second part I will review Robert Spekkens ideas about contextuality and I will relate them to the hidden variable model defined in the new paper. The main reference for the second part of the talk is the paper “Contextuality for preparations, transformations, and unsharp measurements” by R.W. Spekkens (Physical Review A 71, 052108 (2005)).
May 19: Geometry of quantum probabilities by Cihan
I will try to give more intuition on the structure of the generalized phase space (arXiv:2004.01992) using different coordinates. There arises questions related to the interaction between probabilities describing measurement statistics and the geometry of the partially ordered set of projectors.
June 2: Quantum Thermodynamics: A primer for information theorists by Arnab
In this group presentation, I will be reviewing the emerging field of quantum thermodynamics. The talk is aimed as an introduction to the field dealing with the interplay between thermodynamics and information theory. The presentation is divided into two parts: First, I will talk about demons, engines and the thermodynamic cost of (quantum) computation. In the latter part of the talk, I will use a resource theory framework to show the inadequacy of Von-neumann entropy in characterising extractable work.
July 28: Review of Stabilizer extent is not multiplicative by Paul
The Gottesman-Knill theorem states that a Clifford circuit acting on stabilizer states can be simulated efficiently on a classical computer. Recently, this result has been generalized to cover inputs that are close to a coherent superposition of logarithmically many stabilizer states. The runtime of the classical simulation is governed by the stabilizer extent, which roughly measures how many stabilizer states are needed to approximate the state. An important open problem is to decide whether the extent is multiplicative under tensor products. An affirmative answer would yield an efficient algorithm for computing the extent of product inputs, while a negative result implies the existence of more efficient classical algorithms for simulating large-scale quantum circuits. Here, we answer this question in the negative. Our result follows from very general properties of the set of stabilizer states, such as having a size that scales sub-exponentially in the dimension, and can thus be readily adapted to similar constructions for other resource theories.