The IQC-QuICS Math and Computer Science Seminar

Overview

This virtual seminar is jointly sponsored by Institute for Quantum Computing (University of Waterloo) and the Joint Center for Quantum Information and Computer Science (University of Maryland). We are interested in understanding the theoretical tools that underlie current results in quantum information, especially insofar as they overlap with mathematics and theoretical computer science. Talks are 50 minutes long, with additional time for Q&A and discussion.

This is a hybrid of the IQC Math and Computer Science Seminar and the QuICS Math RIT on Quantum Information.

 

QuICS Organizers: Yusuf Alnawakhtha and Carl Miller.

IQC Organizers: Daniel Grier and Hakop Pashayan.

 

Future Meetings

Date: Thursday, October 21st, 2021, 2:00-3:00pm EDT

Title: Clifford groups are not always 2-designs

Speaker: Matthew Graydon
Speaker Affiliation: University of Waterloo

Link: https://uwaterloo.zoom.us/j/93025194699?pwd=bjVJZVpIQXJkS1U2aXg2d3Q3QVhBdz09

Abstract: A group 2-design is a unitary 2-design arising via the image of a suitable compact group under a projective unitary representation in dimension d.  The Clifford group in dimension d is the quotient of the normalizer of the Weyl-Heisenberg group in dimension d, by its centre: namely U(1).  In this talk, we prove that the Clifford group is not a group 2-design when d is not prime. Our main proofs rely, primarily, on elementary representation theory, and so we review the essentials. We also discuss the general structure of group 2-designs. In particular, we show that the adjoint action induced by a group 2-design splits into exactly two irreducible components; moreover, a group is a group 2-design if and only if the norm of the character of its so-called U-Ubar representation is the square root of two. Finally, as a corollary, we see that the multipartite Clifford group (on some finite number of quantum systems) also often fails to be a group 2-design. This talk is based on joint work with Joshua Skanes-Norman and Joel J. Wallman; arXiv:2108.04200 [quant-ph].

 

Date: Thursday, November 4th, 2021, 2:00-3:00pm EDT

Title: Google's quantum experiment: a mathematical perspective

Speaker: Gail Letzter
Speaker Affiliation: National Security Agency and University of Maryland, College Park

Link: https://umd.zoom.us/j/93192622019

Abstract: In 2019, Google announced that they had achieved quantum supremacy: they performed a task on their newly constructed quantum device that could not be accomplished using classical computers in a reasonable amount of time.  In this talk, we present the mathematics and statistics involved in the set-up and analysis of the experiment, sampling from random quantum circuits.  We start with the theory of random matrices and explain how to produce a sequence of (pseudo) random unitary matrices using quantum circuits.  We then discuss how the Google team compares quantum and classical approaches using cross entropy and the Porter-Thomas distribution.  Along the way, we present other problems with potential quantum advantage and some of the latest results related to noisy near-term quantum computers. 

Date: Thursday, November 11th, 2021, 2:00-3:00pm EDT

Speaker: Adam Bene Watts
Speaker Affiliation: University of Waterloo

Past Meetings

Title: The quantum sign problem: perspectives from computational physics and quantum computer science
Speaker: Dominik Hangleiter
Speaker Affiliation: Freie Universität Berlin
Date: Tuesday, February 16th, 2021, 11:00am-12:00pm EST
Abstract: In quantum theory, whenever we make a measurement, the outcomes will be random samples, distributed according to a distribution that is determined by the Born rule. On a high level, this probability distribution arises via high-dimensional interference of paths in quantum state space. Often, this 'sign problem' is made responsible for the hardness of classical simulations on the one hand, and the power of quantum computers on the other hand. In my talk, I will provide different perspectives and results on the sign problem and ponder the question inhowfar it might serve as a delineator between quantum and classical computing. In the first part of the talk, I will motivate the emergence of the sign problem from a physics perspective, and briefly discuss how a hardness argument for sampling from the output of generic quantum computations exploits the sign problem. In the second part of the talk, I will take on a computational-physics perspective. Within the framework of Monte Carlo simulations of complex quantum systems, I will discuss the question: Can we mitigate or *ease* the sign problem computationally by finding a perhaps more suitable basis in which to describe a given system? Specifically, I will discuss various measures of the sign problem, how they are related, and how to optimize them -- practically and in principle.

Title: Computability and compression of nonlocal games

Speaker: Sajjad Nezhadi
Speaker Affiliation: University of Maryland — College Park
Date: Monday, March 22nd, 2021, 10:00-11:00am EDT

Abstract:Recently, works such as the landmark MIP*=RE paper by Ji et. al. have established deep connections between computability theory and the power of nonlocal games with entangled provers. Many of these works start by establishing compression procedures for nonlocal games, which exponentially reduce the verifier's computational task during a game. These compression procedures are then used to construct reductions from uncomputable languages to nonlocal games, by a technique known as iterated compression. 

In this talk, I will introduce and contrast various versions of the compression procedure and discuss their use cases. In particular, I will demonstrate how each can be used to construct reductions from various languages in the first two levels of the arithmetical hierarchy to complexity classes defined using entangled nonlocal games. Time permitting, I will also go through a high-level overview of some ingredients involved in performing compression.

 

Title: Efficient quantum algorithm for dissipative nonlinear differential equations
Speaker: Jin-Peng Liu
Speaker Affiliation: University of Maryland — College Park
Date: Thursday, April 8th, 2021, 3:00-4:00pm EDT

Abstract: Differential equations are ubiquitous throughout mathematics, natural and social science, and engineering. There has been extensive previous work on efficient quantum algorithms for linear differential equations. However, analogous progress for nonlinear differential equations has been severely limited due to the linearity of quantum mechanics. We give the first quantum algorithm for dissipative nonlinear differential equations that is efficient provided the dissipation is sufficiently strong relative to the nonlinearity and the inhomogeneity. We also establish a lower bound showing that differential equations with sufficiently weak dissipation have worst-case complexity exponential in time, giving an almost tight classification of the quantum complexity of simulating nonlinear dynamics. Finally, we discuss potential applications of this approach to problems arising in biology as well as in fluid and plasma dynamics.
Reference: Liu, Jin-Peng, et al. "Efficient quantum algorithm for dissipative nonlinear differential equations." arXiv:2011.03185 (2020).

 

Title: Schur-Weyl duality and symmetric problems with quantum input

Speaker: Laura Mancinska

Speaker Affiliation: University of Copenhagen
Date: Monday, April 26th, 2021, 10:00-11:00am EDT

Abstract: In many natural situations where the input consists of n quantum systems, each associated with a state space Cd, we are interested in problems that are symmetric under the permutation of the n systems as well as the application of the same unitary U to all n systems. Under these circumstances, the optimal algorithm often involves a basis transformation, known as (quantum) Schur transform, which simultaneously block-diagonalizes the said actions of the permutation and the unitary groups.  I will illustrate how Schur-Weyl duality can be used to identify optimal quantum algorithm for quantum majority vote and, more generally, compute symmetric Boolean functions on quantum data.  This is based on joint work "Quantum majority and other Boolean functions with quantum inputs" with H. Buhrman, N. Linden, A. Montanaro, and M. Ozols.

 

Title: Fault-tolerant error correction using flags and error weight parities
Speaker: Theerapat Tansuwannont
Speaker Affiliation: University of Waterloo
Date: Tuesday, June 8th, 2021, 4:00-5:00pm EDT
Abstract: Fault-tolerant error correction (FTEC), a procedure which suppresses error propagation in a quantum circuit, is one of the most important components for building large-scale quantum computers. One major technique often used in recent works is the flag technique, which uses a few ancillas to detect faults that can lead to errors of high weight and is applicable to various fault-tolerant schemes. In this talk, I will further improve the flag technique by introducing the use of error weight parities in error correction. The new technique is based on the fact that for some families of codes, errors of different weights are logically equivalent if they correspond to the same syndrome and the same error weight parity, and need not be distinguished from one another. I will also give a brief summary of my works on FTEC protocols for several families of codes, including cyclic CSS codes, concatenated Steane code, and capped color codes, which requires only a few ancillas.

Title: Fermion Sampling: a robust quantum computational advantage scheme using fermionic linear optics and magic input states
Speaker: Michał Oszmaniec
Speaker Affiliation: Center for Theoretical Physics, Polish Academy of Sciences
Date: Tuesday, June 15th, 2021, 10:00am-11:00am EDT
Abstract: Fermionic Linear Optics (FLO) is a restricted model of quantum computation which in its original form is known to be efficiently classically simulable. We show that, when initialized with suitable input states, FLO circuits can be used to demonstrate quantum computational advantage with strong hardness guarantees. Based on this, we propose a quantum advantage scheme which is a fermionic analogue of Boson Sampling: Fermion Sampling with magic input states.
We consider in parallel two classes of circuits: particle-number conserving (passive) FLO and active FLO that preserves only fermionic parity and is closely related to Matchgate circuits introduced by Valiant. Mathematically, these classes of circuits can be understood as fermionic representations of the Lie groups U(d) and SO(2d). This observation allows us to prove our main technical results. We first show anticoncentration for probabilities in random FLO circuits of both kind. Moreover, we prove robust average-case hardness of computation of probabilities. To achieve this, we adapt the worst-to-average-case reduction based on Cayley transform, introduced recently by Movassagh, to representations of low-dimensional Lie groups. Taken together, these findings provide hardness guarantees comparable to the paradigm of Random Circuit Sampling.
Importantly, our scheme has also a potential for experimental realization. Both passive and active FLO circuits are relevant for quantum chemistry and many-body physics and have been already implemented in proof-of-principle experiments with superconducting qubit architectures. Preparation of the desired quantum input states can be obtained by a simple quantum circuit acting independently on disjoint blocks of four qubits and using 3 entangling gates per block. We also argue that due to the structured nature of FLO circuits, they can be efficiently certified.
Reference: Oszmaniec, Michał, et al. "Fermion Sampling: a robust quantum computational advantage scheme using fermionic linear optics and magic input states." arXiv preprint arXiv:2012.15825 (2020).

Title: Quantum coding with low-depth random circuits
Speaker: Michael Gullans
Speaker Affiliation: University of Maryland — College Park
Date: Tuesday, July 20th, 2021, 4:00pm-5:00pm EDT
Abstract: We study quantum error correcting codes generated by local random circuits and consider the circuit depth required to achieve high-performance against local error models. Notably, we find that random circuits in D spatial dimensions generate high-performing codes at depth at most O(log N) independent of D. Our approach to quantum code design is rooted in arguments from statistical physics and establishes several deep connections between random quantum coding and critical phenomena in phase transitions. In addition, we introduce a method of targeted measurements to achieve high-performance coding at sub-logarithmic depth above one dimension. These latter results provide interesting connections to the topic of measurement-induced entanglement phase transitions.
Reference: Gullans, Michael J., et al. "Quantum coding with low-depth random circuits." arXiv preprint arXiv:2010.09775 (2020).

Title: Lower Bounds on Stabilizer Rank
Speaker: Dr. Ben Lee Volk
Speaker Affiliation: The University of Texas at Austin
Date: Tuesday, July 27th, 2021, 4:00pm-5:00pm EDT
Abstract: The stabilizer rank of a quantum state ψ is the minimal integer r such that ψ can be written as a linear combination of r stabilizer states. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n-th tensor power of single-qubit magic states. In this talk we'll present a recent improved lower bound of Ω(n) on the stabilizer rank of such states, and an Ω(sqrt{n}/log n) lower bound on the rank of any state which approximates them to a high enough accuracy. Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of the boolean cube, along with some tools from computational complexity theory.
Reference: Peleg, Shir, Amir Shpilka, and Ben Lee Volk. "Lower Bounds on Stabilizer Rank." arXiv preprint arXiv:2106.03214 (2021).

Title: Linear growth of quantum circuit complexity
Speaker: Jonas Haferkamp
Speaker Affiliation: Freie Universität Berlin
Date: Tuesday, August 10th, 2021, 10:00am-11:00am EDT
Abstract: Quantifying quantum states’ complexity is a key problem in various subfields of science, from quantumcomputing to black-hole physics. We prove a prominent conjecture by Brown and Susskind about how randomquantum circuits’ complexity increases. Consider constructing a unitary from Haar-random two-qubit quantumgates. Implementing the unitary exactly requires a circuit of some minimal number of gates - the unitary’sexact circuit complexity. We prove that this complexity grows linearly in the number of random gates, withunit probability, until saturating after exponentially many random gates. Our proof is surprisingly short, giventhe established difficulty of lower-bounding the exact circuit complexity. Our strategy combines differentialtopology and elementary algebraic geometry with an inductive construction of Clifford circuits.
Link: https://umd.zoom.us/j/91765577450?pwd=T1h1OXBmSVRvVVAreGZsK3pHRHpqUT09
Reference: Haferkamp, Jonas, et al. "Linear growth of quantum circuit complexity." arXiv preprint arXiv:2106.05305 (2021).

Date: Thursday, September 9th, 2:00-3:00pm EDT

Title: Trapdoor claw-free functions in quantum cryptography

Speaker: Carl Miller

Speaker Affiliation: University of Maryland

Link:  https://umd.zoom.us/j/91375701575

Abstract: Trapdoor claw-free functions (TCFs) are central to a recent wave of groundbreaking work in quantum cryptography that was originated by U. Mahadev and other authors.  TCFs enable protocols for cryptography that involve quantum computers and classical communication.  In this expository talk I will present the definition of a TCF and its variants, and I will discuss quantum applications, including the recent paper "Quantum Encryption with Certified Deletion, Revisited: Public Key, Attribute-Based, and Classical Communication" by T. Hiroka et al. (arXiv:2105.05393).

 

Date: Tuesday, September 14th, 2021, 4:00-5:00pm EDT

Title: How to perform the coherent measurement of a curved phase space

Speaker: Dr. Christopher Sahadev Jackson

Speaker Affiliation: Sandia National Laboratories

Link:  https://uwaterloo.zoom.us/j/91715806521?pwd=QjBuWTBDODZOVitFVkV5WXU2dHhlUT09

Abstract: In quantum optics, the Hilbert space of a mode of light corresponds to functions on a plane called the phase space (so called because it reminded Boltzmann of oscillators in 2-d real space.)  This correspondence offers three important features:  it can autonomously handle quantum theoretical calculations, it allows for the infinite-dimensional Hilbert space to be easily visualized, and it is intimately related to a basic experimental measurement (the so-called heterodyne detection).  Continuous phase space correspondences exist naturally for many types of Hilbert space besides this particular infinite-dimensional one.  Specifically, the two-sphere is a natural phase space for quantum spin systems.  Although well studied on the theoretical and visualization fronts, the corresponding measurement (theoretically referred to as the spin-coherent-state positive-operator-valued measure or SCS POVM) has yet to find a natural way to be experimentally performed.  In this talk, I will review the history of phase space, it’s connection to representation theory, quantization, coherent states, and continuous measurement.  Finally, I will explain how the SCS POVM can be simply performed, independent of the quantization.  Such a demonstration is a fundamental contribution to the theory of continuous quantum measurement which revives several differential-geometric ideas from the classical and modern theory of complex semisimple Lie groups.

 

Date: Thursday, October 7st, 2021, 10:00-11:00am EDT

Title: Bounding quantum capacities via partial orders and complementarity

Speaker: Christoph Hirche
Speaker Affiliation: Technische Universität München and National University of Singapore

Link:  https://umd.zoom.us/j/91230236736?pwd=Uk5zalFSTnZUQnhZQkFaN0ZnL05XUT09

Abstract: Calculating quantities such as the quantum or private capacity of a quantum channel is a fundamental, but unfortunately a generally very hard, problem. A well known class of channels for which the task simplifies is that of degradable channels, and it was later shown that the same also holds for a potentially bigger class of channels, the so called less noisy channels. Based on the former, the concept of approximately degradable channels was introduced to find bounds on capacities for general channels. We discuss how the idea can be transferred to other partial orders, such as less noisy and more capable channels, to find potentially better capacity bounds. Unfortunately these are not necessarily easy to compute, but we show how they can be used to find operationally meaningful bounds on capacities that are based on the complement of the quantum channel and might give a deeper understanding of phenomena such as superadditivity. Finally, we discuss how the framework can be transferred to quantum states to bound the one-way distillable entanglement and secret key of a bipartite state.