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.

**Date:** Thursday, October 21^{st}, 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 4^{th}, 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 11^{th}, 2021,
2:00-3:00pm EDT

**Speaker:** Adam Bene Watts

**Speaker Affiliation:** University of Waterloo

**Title:** The quantum sign
problem: perspectives from computational physics and quantum computer science

**Speaker:** Dominik Hangleiter

**Speaker Affiliation:** Freie Universität Berlin

**Date:** Tuesday, February 16^{th}, 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 22^{nd}, 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 8^{th}, 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 26^{th}, 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 **C**^{d},
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 8^{th}, 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 15^{th}, 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 20^{th}, 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 27^{th}, 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 10^{th}, 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 9^{th},
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/

**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 14^{th},
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.

**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 7^{st},
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.