Quantum Information Science and Quantum Neuroscience.ppt

American Physical Society
University of New Mexico, 15 Oct 2022
Slides: http://slideshare.net/LaBlogga
Melanie Swan, PhD
Research Associate
University College London
Quantum Information
Neuroscience and Neural Signaling
“…the laws of physics present no barrier to reducing
the size of computers until bits are the size of atoms,
and quantum behavior holds dominant sway”
- Feynman, Foundations of Physics, 1985, p. 530

15 Oct 2022
Quantum Neuroscience 1
Quantum Technologies Research Program
2015 2019 2020
Blockchain Blockchain
Economics
Quantum
Computing
Quantum
Computing
for the Brain
2022
Image: Thomasian, 2021, Nat
Rev Endocrinol. 18:81-95, p. 12

15 Oct 2022
Quantum Neuroscience
Quantum Information
2
Domain Properties Top Five Properties: Quantum Matter and Quantum Computing Definition
Quantum
Matter
Symmetry Looking the same from different points of view (e.g. a face, cube, laws of physics);
symmetry breaking is phase transition
Topology Geometric structure preserved under deformation (bending, stretching, twisting, and
crumpling, but not cutting or gluing); doughnut and coffee cup both have a hole
Quantum
Computing
Superposition An unobserved particle exists in all possible states simultaneously, but once measured,
collapses to just one state (superpositioned data modeling of all possible states)
Entanglement Particles connected such that their states are related, even when separated by distance
(a “tails-up/tails-down” relationship; one particle in one state, other in the other)
Interference Waves reinforcing or canceling each other out (cohering or decohering)
Source: Swan, M., dos Santos, R.P. & Witte, F. (2022). Quantum Matter Overview. J. 5(2):232-254.
Quantum Information: the
information (physical properties)
of the state of a quantum system
Quantum Information: the
information (physical properties)
of the state of a quantum system
Nobel Prize
2022
Nobel Prize 1998
Nobel Prize 2016
2022
“groundbreaking experiments
using entangled quantum
states, where two particles
behave like a single unit even
when they are separated.
Their results have cleared the
way for new technology based
upon quantum information”
Cat

15 Oct 2022
What is Quantum?
3
QCD: Quantum Chromodynamics
Subatomic particles
Matter particles: fermions (quarks)
Force particles: bosons (gluons)
Scale Entities Physical Theory
1 1 x101 m Meter Humans Newtonian mechanics
2 1 x10-9 m Nanometer Atoms Quantum mechanics
(nanotechnology)
3 1 x10-12 m Picometer Ions, photons Optics, photonics
4 1 x10-15 m Femtometer Subatomic particles QCD/gauge theories
5 1 x10-35 m Planck scale Planck length Planck scale
Atoms Quantum objects:
atoms, ions,
photons
 “Quantum” = anything at the scale of
atomic and subatomic particles (10-9 to 10-15)
 Theme: ability to study and manipulate
physical reality at smaller scales
 Study phenomena (e.g. neurons) in the native
3D structure of physical reality

15 Oct 2022
Quantum Science Fields
4
Source: Swan, M., dos Santos, R.P. & Witte, F. (2020). Quantum Computing: Physics, Blockchains, and Deep Learning Smart
Networks. London: World Scientific.
Quantum Biology
Quantum Machine
Learning
€
$
¥
€
Quantum methods complement classical methods to study field-specific problems
Quantum
Cryptography
Quantum Space
Science Quantum Finance
Foundational
Tools
Advanced
Applications
Quantum
Chemistry

15 Oct 2022
Quantum Studies in the Academy
5
Digital
Humanities
Arts
Sciences
Quantum
Humanities
computational astronomy,
computational biology
Digital Humanities (literature & painting
analysis, computational philosophy1)
Quantum Humanities
quantum chemistry, quantum finance,
quantum biology, quantum ecology
Apply quantum methods to study field-specific problems e.g. quantum machine learning
Apply data science methods to study field-specific problems e.g. machine learning
 Data science institutes now including quantum
 What are Digital Humanities / Quantum Humanities?
1. Apply digital/quantum methods to research questions
2. Find digital/quantum examples in field subject matter
 (e.g. quantum mechanical formulations in Shakespeare)
3. Open new investigations per digital/quantum conceptualizations
Sources: Miranda, E.R. (2022). Quantum Computing in the Arts and Humanities. London: Springer. Barzen, J. & Leymann, F.
(2020). Quantum Humanities: A First Use Case for Quantum Machine Learning in Media Science. Digitale Welt. 4:102-103.
1Example of computational philosophy: investigate formal axiomatic metaphysics with an automated reasoning environment
Big Data Science
Vermeer imaging (1665-2018)
Textual
analysis

15 Oct 2022
 3d lattices: group theory not number theory (factoring)
 NIST quantum-safe cryptography (Jul 2022) (“Y2k of crypto”)
 Based on the difficulty of lattice problems (finding the shortest vector
to an arbitrary point); learning-with-errors and functions over lattices
 Quantum key distribution via quantum teleportation (Bell pair
creation, quantum networks with heralded entanglement)
 Space: hyperbolic Bloch theorem & hyperbolic space
 Flat, negative, positive curvature space
 Time: Floquet methods, discrete time crystals
 Discrete time-crystalline order (Maskara-Lukin)
 Entanglement generation in optical networks
 Standard quantum algorithms
 VQE, VAE, QAOA, RKHS, quantum amplitude
estimation; QML, QNN, Born machine, quantum walk
6
Quantum Toolkit
VQE: variational quantum eigensolver; VAE: variational autoencoder; QAOA: quantum approximate optimization algorithm;
RKHS (reproducing kernel Hilbert space); QML: quantum machine learning; QNN: quantum neural network
Sources: Swan, M., dos Santos, R.P. & Witte, F. (2022). Quantum Information Science. IEEE Internet Computing. Special Journal
Issue: Quantum and Post-Moore’s Law Computing. January/February 2022. Maskara et al. (2021). arXiv:2102.13160v1.
Hyperbolic band theory
Time-crystalline Eigenstate order

15 Oct 2022
 Quantum (neuro)biology: application of quantum methods
to investigate problems in (neuro)biology and the possible
role of quantum effects
 Brute physical processes & higher-order cognition, memory, attention
 Quantum consciousness hypothesis (microtubules)
 Research topics
 Traditional (~2010)
 Avian magneto-navigation,
photosynthesis, energy transfer
 Contemporary (Empirical vs Theoretical)
 Imaging (EEG, fMRI, etc.)
 Protein folding
 Genomics
 Collective behavior: neural signaling, swarmalator
7
Quantum Biology
Swarmalator: animal aggregations that self-coordinate in time and space
Human data: imaging (brain wave activity); Model organism data: behaving (task-driven spatiotemporal signaling data)
Source: Swan, M., dos Santos, R.P. & Witte, F. (2022). Quantum Neurobiology. Quantum Reports. 4(1):107-127.
Imaging In-cell Targeting
Connectome Parcellation

15 Oct 2022
Methods
Swarmalator: animal aggregations that self-coordinate in time and space (e.g. crickets, fish, birds)
Source: Swan, M. dos Santos, R.P., Lebedev, M.A. and Witte, F. (2022). Quantum Computing for the Brain. London: World
Scientific.
Research Topic Mathematical Physics Approaches
1 Imaging (EEG, fMRI,
MEG, etc.)
Wavefunctions: Fourier transform, Fourier slice theorem & Radon transform; QML (VQE); quantum
tomography image reconstruction (electrical and chemical (Calcium) wave forms)
2 Protein folding Lowest-energy configuration (Hamiltonian), spin glass, quantum spin liquid, Chern-Simons
Ground-state excited-state energy functions, total system energy
Qubit Hamiltonians, VQE
3 Genomics Lowest-energy knotting compaction, Chern-Simons (topological invariance)
Quantum optimization algorithms (Azure); optics; QAOA; AdS/CFT, BH, chaos, TN, MERA, RG
Quantum amplitude estimation: technique used to estimate the properties of random distributions
Collective Behavior
4 Neural Signaling Single-neuron: Hodgkin-Huxley (1963), integrate-and-fire, theta neuron
Local ensemble: FitzHugh-Nagumo, Hindmarsh-Rose, Morris-Lecor
Neural field theory: Jansen-Rit, Wilson-Cowan, Floquet, Kuramoto oscillators, Fokker-Planck equations
Neuroscience Physics: AdS/CFT, Chern-Simons, gauge theory, bifurcation & bistability
5 Swarmalator Swarmalator: phytoplankton (diffusion); krill (Brownian motion, Kuramoto oscillator); whale (clustering)
 Recurrent theme: topology (e.g. Chern-Simons)
 Solvable QFT curvature min-max = event (fold, mutation, signal)
 Quantum topological materials approach entails
 Topology: Chern-Simons, knotting, compaction
Topological data analysis: find the (n-dimensional connecting) Betti
numbers of a simplicial complex (Schmidhuber & Lloyd, 2022)

15 Oct 2022
Levels of Organization in the Brain
9
 Complex behavior spanning nine orders of
magnitude scale tiers
Level Size (decimal) Size (m) Size (m)
1 Nervous system 1 > 1 m 100
2 Subsystem 0.1 10 cm 10-1
3 Neural network 0.01 1 cm 10-2
4 Microcircuit 0.001 1 nm 10-3
5 Neuron 0.000 1 100 μm 10-4
6 Dendritic arbor 0.000 01 10 μm 10-5
7 Synapse 0.000 001 1 μm 10-6
8 Signaling pathway 0.000 000 001 1 nm 10-9
9 Ion channel 0.000 000 000 001 1 pm 10-12
Sources: Sterratt, D., Graham, B., Gillies, A., & Willshaw, D. (2011). Principles of Computational Modelling in Neuroscience.
Cambridge: Cambridge University Press. Ch. 9:226-66. Sejnowski, T.J. (2020). The unreasonable effectiveness of deep
learning in artificial intelligence. Proc Natl Acad Sci. 117(48):30033-38.
 Human brain
 86 billion neurons, 242 trillion synapses
 ~10,000 incoming signals to each neuron
 Not large numbers in the big data era

15 Oct 2022
Structure: Connectome Project Status
Fruit Fly completed in 2018
 Worm to mouse:
 10-million-fold increase in
brain volume
 Brain volume: cubic microns
(represented by 1 cm distance)
 Quantum computing technology-driven inflection point
needed (as with human genome sequencing in 2001)
 1 zettabyte storage capacity per human connectome required
vs 59 zettabytes of total data generated worldwide in 2020
Sources: Abbott, L.F., Bock, D.D., Callaway, E.M. et al. (2020). The Mind of a Mouse. Cell. 182(6):1372-76. Lichtman, J.W., Pfister,
H. & Shavit, N. (2014). The big data challenges of connectomics. Nat Neurosci. 17(11):1448-54. Reinsel, D. (2020). IDC Report:
Worldwide Global DataSphere Forecast, 2020-2024: The COVID-19 Data Bump and the Future of Data Growth (Doc US44797920).
Neurons Synapses Ratio Volume Complete
Worm 302 7,500 25 5 x 104 1992
Fly 100,000 10,000,000 100 5 x 107 2018
Mouse 71,000,000 100,000,000,000 1,408 5 x 1011 NA
Human 86,000,000,000 242,000,000,000,000 2,814 5 x 1014 NA
Connectome: map of synaptic connections
between neurons (wiring diagram), but
structure does not equal function

15 Oct 2022
Function: Motor Neuron Mapping Project Status
Multiscalar Neuroscience
11
Source: Cook, S.J. et al. (2019). Whole-animal connectomes of both Caenorhabditis elegans sexes. Nature. (571):63-89.
 C. elegans motor neuron mapping (completed 2019)
 302 neurons and 7500 synapses (25:1)
 Human: 86 bn neurons 242 tn synapses (2800:1)
 Functional map of neuronal connections

15 Oct 2022
Neural Signaling
Image Credit: Okinawa Institute of Science and Technology
NEURON: Standard computational neuroscience modeling software
Scale Number Size Size (m) NEURON Microscopy
1 Neuron 86 bn 100 μm 10-4 ODE Electron
2 Synapse 242 tn 1 μm 10-6 ODE Electron/Light field
3 Signaling pathway unknown 1 nm 10-9 PDE Light sheet
4 Ion channel unknown 1 pm 10-12 PDE Light sheet
Electrical-Chemical Signaling
Math: PDE (Partial Differential
Equation: multiple unknowns)
Electrical Signaling (Axon)
Math: ODE (Ordinary Differential
Equation: one unknown)
1. Synaptic Integration:
Aggregating thousands of
incoming spikes from
dendrites and other
neurons
2. Electrical-Chemical
Signaling:
Incorporating neuron-glia
interactions at the
molecular scale
12
Implicated in neuropathologies of Alzheimer’s, Parkinson’s, stroke, cancer
Synaptic Integration
Math: PDE (Partial Differential
Equation: multiple unknowns)

15 Oct 2022
Neural Signaling Modeling
 Example problem: integrate EEG and fMRI data
 Different time, space, and dynamics regimes
 Epileptic seizure: chaotic dynamics (straightforward)
 Resting state: instability-bifurcation dynamics (system
organizing parameter interrupted by countersignal)
 Challenging problem: collective behavior
 Neural field theories, neural gauge theories
13
Scale Models
1 Single neuron Hodgkin-Huxley, integrate-and-fire, theta neurons
2 Local ensemble FitzHugh-Nagumo, Hindmarsh-Rose, Morris-Lecor
Linear Fokker-Planck equation (FPE) (uncorrelated behavior)
Nonlinear FPE, Fractional FPE (correlated behavior)
3 Population group
(neural mass)
Neural mass models (Jansen-Rit), mean-field (Wilson-Cowan), tractography,
oscillation, network models
4 Whole brain
(neural field theories)
(neural gauge theories)
Neural field models, Kuramoto oscillators, multistability-bifurcation, directed
percolation random graph phase transition, graph-based oscillation, Floquet
theory, Hopf bifurcation, beyond-Turing instability
Sources: Breakspear (2017). Papadopoulos, L., Lynn, C.W., Battaglia, D. & Bassett, D.S. (2020). Relations between large-scale
brain connectivity and effects of regional stimulation depend on collective dynamical state. PLoS Comput Biol. 16(9). Coombes, S.
(2005). Waves, bumps, and patterns in neural field theories. Biol Cybern. 93(2):91-108.

15 Oct 2022
Neural Dynamics: Complex Statistics
14
 Collective behavior of the brain generates
unrecognized statistical distributions
 Neural ensemble: normal distribution (FPE) and
power law distribution (nonlinear FPE, fractional FPE)
 Neural mass: Wilson-Cowan, Jansen-Rit, Floquet, ODE
 Neural field theory: wavefunction, oscillation, bifurcation, PDE
FPE: Fokker-Planck equation: partial differential equation describing the time evolution of the probability density function of particle
velocity under the influence of drag forces; equivalent to the convection-diffusion equation in Brownian motion
Source: Breakspear, M. (2017). Dynamic models of large-scale brain activity. Nat Neurosci. 20:340-52.
Approach Description Statistical Distribution Neural Dynamics
1 Neural ensemble
models
Small groups of neurons,
uncorrelated states
Normal (Gaussian) Linear Fokker-Planck
equation (FPE)
2 Small groups of neurons,
correlated states
Non-Gaussian but known
(e.g. power law)
Nonlinear FPE, Fractional
FPE
3 Neural mass models Large-scale populations of
interacting neurons
Unrecognized Wilson-Cowan, Jansen-Rit,
Floquet model, Glass
networks, ODE
4 Neural field models
(whole brain)
Entire cortex as a continuous
sheet
Unrecognized Wavefunction, PDE,
Oscillation analysis

15 Oct 2022
Biological System of the Neuron
 Neuronal Spike Integration
 Electrical
 Axonal spikes
 Dendritic NMDA spikes
 Chemical
 Dendritic sodium spikes
 Dendritic calcium spikes
15
EPSP: excitatory postsynaptic potential (contrast with IPSP: inhibitory postsynaptic potential)
Sources: Williams, S.R. & Atkinson, S.E. (2008). Dendritic Synaptic Integration in Central Neurons. Curr. Biol. 18(22). R1045-R1047.
Poirazi et al. (2022). The impact of Hodgkin–Huxley models on dendritic research. J Physiol. 0.0:1–12.
(a)
(b)
(c)
(a) Dendritic spine receives EPSP
(b) Local spiking activity along dendrite
(c) Aggregate dendritic spikes at axon
Dendritic sodium, NMDA,
calcium spikes (Poirazi)

15 Oct 2022
Glutamate (excitatory) and GABA (inhibitory)
 Post-synaptic density (PSD) proteins
16
Sources: Sheng, M. & Kim, E. (2011). The Postsynaptic Organization of Synapses. Cold Spring Harb Perspect Biol. 3(a005678):1-
20. Image: presynaptic terminal – post-synaptic density: Shine, J.M., Muller, E.J., Munn, B. et al. (2021). Computational models link
cellular mechanisms of neuromodulation to large-scale neural dynamics. Nat Neuro. 24(6):765-776.
Glutamate (Excitatory) Receptor GABA (Inhibitory) Receptor
Major proteins at Glutaminergic and GABAergic synapses

15 Oct 2022
 A physical system with a bulk volume can be described
by a boundary theory in one fewer dimensions
 A gravity theory (bulk volume) is equivalent to a gauge theory
or a quantum field theory (boundary surface)
 AdS5/CFT4 (5d bulk gravity)=(4d Yang-Mills supersymmetry QFT)
 AdS/CFT mathematics
 Metric (ds=), Operators (O=), Action (S=), Hamiltonian (H=)
AdS/CFT Correspondence (Anti-de Sitter Space/Conformal Field Theory)
17
Sources: Maldacena, J. (1998). The large N limit of superconformal field theories and supergravity. Adv Theor Math Phys.
2:231-52. Harlow, D. (2017). TASI Lectures on the Emergence of Bulk Physics in AdS/CFT. Physics at the Fundamental
Frontier. arXiv:1802.01040.
AdS/CFT Escher Circle Limits Error correction tiling
 Implications for
 Quantum gravity theories
 Black hole information paradox
 Quantum error correction
(information scrambling)
 Entropy-based short/long-range
(UV–IR) correlations (bulk structure)
 Complexity conjecture (two-sided
wormhole, thermofield double)

15 Oct 2022
 AdS/SYK (Sachdev-Yi-Kitaev) model
 Solvable model of strongly interacting fermions
 AdS/SYK: black holes and unconventional materials have
similar properties related to mass, temperature, and charge
 SYK Hamiltonian (HSYK) finds wavefunctions for 2 or 4 fermions
 Or up to 42 in a black-hole-on-a-superconducting-chip formulation
AdS/CFT Duality: Solve in either Direction
18
Sources: Sachdev, S. (2010). Strange metals and the AdS/CFT correspondence. J Stat Mech. 1011(P11022).. Pikulin, D.I. &
Franz, M. (2017). Black hole on a chip: Proposal for a physical realization of the Sachdev-Ye-Kitaev model in a solid-state
system. Physical Review X. 7(031006):1-16.
Direction Domain Known Unknown
1 Boundary-to-bulk Theoretical physics Standard quantum
field theory (boundary)
Quantum gravity (bulk)
2 Bulk-to-boundary
(AdS/SYK)
Condensed matter,
superconducting
Classical gravity (bulk) Unconventional materials
quantum field theory (boundary)
Ψ : Wavefunction
HSYK : SYK Hamiltonian
(Operator to describe evolution
and energy of system)
Bethe-Salpeter equation

15 Oct 2022
 Problem: non-local correlations driving behavior
 Each domain with own spatiotemporal and dynamics regime
 Each tier is the boundary for another bulk
AdS/Brain: Multiscalar Correspondence
19
Neuron
Network
AdS/Brain Multi-tier Holographic Correspondence
Synapse
Molecule
Tier Scale Signal AdS/Brain
1 Network 10-2 Local field potential Boundary
2 Neuron 10-4 Action potential Bulk Boundary
3 Synapse 10-6 Dendritic spike Bulk Boundary
4 Molecule 10-10 Ion docking Bulk
Scientific.

15 Oct 2022
AdS/Brain bMERA
 MERA models
 Renormalized entanglement (correlation) across system tiers
20
MERA cMERA dMERA bMERA
Continuous
spacetime MERA
Deep MERA tensor
network on NISQ devices
Multiscalar neural
field theory
Multiscalar entanglement
renormalization network
Vidal, 2007 Nozaki et al., 2012 Kim & Swingle, 2017 Swan et al., 2022
 bMERA (brainMERA)
 Renormalize system entanglement (correlation) to obtain
neural signaling action across multiple scale layers
Scientific.
MERA tensor network:
alternating layers of
isometries (triangles)
and disentanglers
(squares) (Vidal, 2007)

15 Oct 2022
 Analogy to food-web ecosystem multiscalar model
AdS/Brain: Multiscalar Correspondence
21
Neuron
Network
AdS/Brain
Synapse
Molecule
Tier Scale Neural Signaling
Event
Swarmalator
Model
Food-web
Ecosystem Event
Math Approach
1 Network 10-2 Local field potential Whale Predation Distribution
2 Neuron 10-4 Action potential Krill Swarm Lagrangian
3 Synapse 10-6 Dendritic spike Phytoplankton Availability Diffusion
4 Molecule 10-10 Ion docking Light gradient Incidence angle Advection
Scientific.
Krill
Whale
AdS/Krill
Phytoplankton
Light gradient

15 Oct 2022
Quantum Krill: 4-Tier Ecosystem Model
B. Enhanced
A. Basic
Optical analysis of light
spectrum gradient (Heggerud)
Swarmalator
hydrodynamic: O’Keeffe
(Kuramoto oscillator),
Ghosh (ring), Murphy (jet)
Lotka-Volterra predator-
prey model spiking
neuronal network
excitatory-inhibitory
model (Lagzi)
Statistical distribution (Miller)
2d Lagrangian (Hofmann)
Mathematics Diffusion (Heggerud)
Statistical analysis: 11 krill
swarm characteristics
analyzed in relation to
whale presence-absence
using Boosted regression
trees (BRTs) via a logit
(quantile function) (to
achieve local regularization
and prevent overfitting by
optimizing the number of
trees, learning rate, and
tree complexity
Quantum circuits
Random tensor network
QML: RKHS, QNN,
Quantum walk
VQE, VAE, QAOA, Quantum
amplitude estimation
Example of multiscalar system: light-phytoplankton-krill-whale
similar to neural signaling ion-synapse-neuron-network
Two species non-local
reaction-diffusion-advection
model to consider niche
differentiation via absorption
spectra separation. (rate of
change of) density of
phytoplankton species as
diffusion minus buoyancy
plus absorbed photons
minus death rate
Spatial light attenuation
through vertical water column
Ice
2d spatial Lagrangian
model based on four
random forces acting on
krill individuals:
displacement, response
to food gradients,
nearest neighbor
interaction (attraction or
repulsion), and
predation
VQE: variational quantum eigensolver; VAE: variational autoencoder; QAOA: quantum approximate optimization algorithm;
RKHS (reproducing kernel Hilbert space) (quantum kernel learning), QNN: quantum neural network
Krill swarm density (%) = forces acting on krill whale predation (death rate)
phytoplankton density –
+
light gradient +

15 Oct 2022
Conclusion
23
 Topological quantum materials: important advance
 Chern-Simons, knot theory, Wilson loops (nonlocal
observables) represented as knots (generalize to the Jones
polynomial (a knot invariant)); solve with path integrals
 Application to neural modeling
 Three-dimensional lattices; multiscalar
 Quantum circuit-ready
 Topological additions to standard quantum toolkit
 3d lattices: group theory not number theory (factoring)
(NIST), hyperbolic space, Floquet discrete time crystals,
quantum algorithms (VQE, VAE, QAOA, quantum amplitude
estimation, RKHS), QML, QNN, Born machine, quantum walk
Image: Shine, J.M., Muller, E.J., Munn, B. et al. (2021). Computational models link cellular mechanisms of neuromodulation to large-
scale neural dynamics. Nat Neuro. 24(6):765-776. Breakspear laboratory.

15 Oct 2022
Risks and Limitations
24
 Quantum domain is hard to understand
 Complex, non-intuitive, alienating
 Quantum computing
 Early stage and non-starter without technical
advance in error correction (Preskill 2021)
 Substantial worldwide investment in
quantum initiatives
 Needed for next-generation quantum internet
networks, quantum cryptography
 Ability to coordinate next-tier of even larger and
more complex projects
Heidegger, The Question
Concerning Technology
+
-
Source: Preskill, J. (2021). Quantum computing 40 years later. arXiv:2106.10522.

American Physical Society
University of New Mexico, 15 Oct 2022
Slides: http://slideshare.net/LaBlogga
Melanie Swan, PhD
Research Associate
University College London
Quantum Information
Neuroscience and Neural Signaling
“…the laws of physics present no barrier to reducing
the size of computers until bits are the size of atoms,
and quantum behavior holds dominant sway”
- Feynman, Foundations of Physics, 1985, p. 530
Thank you!
Questions?

15 Oct 2022
Signal Synchrony
 Synchrony as a bulk property of the brain
 Synaptic signals arrive simultaneously but
travel different distances, so speeds must vary
 Seamless coordination of diverse signals
 Evidence: axon propagation speeds
 Electrophysiological data recorded at multiple
spatial scales
 Microscale current sources (produced by local
field potentials at membrane surfaces) modeled
in a macro-columnar structure, integrating
properties related to
 Magnitude, distribution, synchrony
26
Source: Nunez, P.L., Srinivasan, R. & Fields, R.D. (2015). EEG functional connectivity, axon delays and white matter disease. Clin
Neurophysiol. 126(1):110-20.

15 Oct 2022
Bifurcation
 Bifurcation: split; qualitative change in
system output per change in input parameter
 Fixed points, periodic orbits, chaotic attractors
 Bifurcation diagram
 Traditional model for studying neural signaling
 Cell membrane voltage changes from at-rest to
oscillatory as a result of a bifurcation
 Epilepsy, Parkinson’s disease: threshold-based
oscillatory behavior
 Ex: Hopf bifurcation: system critical point (resting-to-
firing state) at which a periodic orbit appears or
disappears due to a local change in stability
27
Source: Ermentraut, B. & Terman, D.H. (2010). Mathematical Foundations of Neuroscience. London: Springer.
www.math.pitt.edu/~bard/xpp/xpp.html
Neural bifurcation modeling
software: XPP-Aut
Traditional model of
(electrical) neuron
Bifurcation (Lagzi 2019)
Stable periodic orbit
interrupted by
negative saddle value
Toroidal limit cycle model

15 Oct 2022
Bifurcation: Human Neural States
 Critical bistability: brain’s operating at critical phase
transition between disorder and excessive order
 Bistability: two stable points within a system
 Human brain activity exhibits
 Scale-free avalanche dynamics and power-law long-range
temporal correlations (LRTCs) across the nervous system
 Bistability and LRTCs positively correlated in study data
 Resting state: moderate levels of bistability
 Epilepsy: excessive bistability
28
MEG (magnetoencephalography); SEEG (stereo-EEG)
Source: Wang, S.H., Arnulfo, G., Myrov, V. et al. (2022). Critical-like bistable dynamics in the resting-state human brain. bioRxiv
preprint doi: https://doi.org/10.1101/2022.01.09.475554. Breakspear laboratory.
MEG and SEEG cortical
parcellation data correlation with
bistability index (BiS) estimates

15 Oct 2022
Bifurcation: Lotka-Volterra Model
 Bifurcation math is advancing (neural signaling)
 Theoretical model of collective neural dynamics
 Mean-field methods
 Wilson-Cowan equations: “up-states” and “down-states”
as dynamic neuronal network states reported in striatum
 Bifurcation + Lotka-Volterra predator-prey equations
 Result: Lotka-Volterra equations provide a meaningful population-level
description of the collective behavior of spiking neuronal interaction
(via Jacobian matrix eigenvalues)
 Result: alternative low-dimensional firing rate equation for populations
of interacting spiking neurons with block-random connections
29
Source: Lagzi, F., Atay, F.A. & Rotter, S. (2019). Bifurcation analysis of the dynamics of interacting subnetworks of a spiking
network. Sci Rep. 9:11397.
dx/dt = αx – βxy
dy/dt = δxy – γxy
x: number of prey
y: number of predator
dx/dt; dy/dt: growth rate of population
α, β, δ, γ: parameters of species interaction
Lotka-Volterra predator-prey equations
Firing rate
trajectories in a 3d
state space
corresponding to
spiking network limit
cycles that bifurcate
from a starting
parameter change

15 Oct 2022
Bifurcation and Topology
 Bifurcation and topological equivalence
 Bifurcations of piecewise smooth flows
 Define topological equivalence for piecewise smooth
systems based on orbits (not segments)
 Map boundaries to boundaries based on switching
(stronger topological equivalence) and sliding (weaker
topological equivalence)
30
Source: Colombo, A., di Bernardo, M., Hogan, S.J. & Jeffrey, M.R. (2012). Bifurcations of piecewise smooth flows: Perspectives,
methodologies and open problems. Physica D.
Dynamics in a piecewise
smooth system
Local analysis of a limit
cycle at a discontinuity
Near a tangency, two regions
with different orbit topologies
Non-deterministic chaos

15 Oct 2022
Quantum Bifurcation
 Build node and network model from brain atlas data
 Canonical Wilson-Cowan neural mass models
31
Source: Coombes, S., Lai, Y.M., Sayli, M. & Thul, R. (2018). Networks of piecewise linear neural mass models. Eur. J. Appl. Math.
29(5):869–90.
Heaviside Wilson-Cowan
ring network (blue circles:
brain network eigenvalues)
Phase plane for a Wilson-Cowan node with a
Heaviside ring rate with stable periodic orbit (blue)
and unstable periodic sliding orbit (dashed magenta)
 Replace s-shaped activation with
piecewise (interval-based) function and
Heaviside (nonlinear activation) function
 Supersede usual ODEs by defining orbits
and stability as a Floquet (periodic)
network parameterized by matrix
eigenvalues and Glass networks
(biochemical periodic-aperiodic switching)
 Result: new ways to study continuous-
discontinuous, linear-nonlinear behavior
in neural spacetime networks
Key message: ability to model more sophisticated
scenarios (e.g. linear and non-linear, continuous
and discrete, smooth and non-smooth)

15 Oct 2022
Neural Operators
 Neural ODE: NN architecture whose weights are
smooth functions of continuous depth
 Input evolved to output with a trainable differential equation,
instead of mapping discrete layers (Chen 2018)
 Neural PDE: NN architecture that uses neural
operators to map between infinite-dimensional spaces
 Fourier neural operator solves all instances of
PDE family in multiple spatial discretizations
 Parameterizing the integral kernel directly in
Fourier space) (Li 2021)
 Neural RG: NN renormalization group
 Learns the exact holographic mapping between
bulk and boundary partition functions (Hu 2019)
32
Sources: Chen et al. (2018). Neural Ordinary Differential Equations. Adv Neural Info Proc Sys. Red Hook, NY: Curran Associates
Inc. Pp. 6571-83. Li et al. (2021). Fourier neural operator for parametric partial differential equations. arXiv:2010.08895v3. Hu et al.
(2019). Machine Learning Holographic Mapping by Neural Network Renormalization Group. Phys Rev Res. 2(023369).

15 Oct 2022
Practical Application
Brain Atlas Annotation and Deep Learning
 Machine learning smooths individual variation to
produce standard reference brain atlas
 Multiscalar neuron detection
 Deep neural network
 Whole-brain image processing
 Detect neurons labeled with genetic markers in a range
of imaging planes and modalities at cellular scale
33
Source: Iqbal, A., Khan, R. & Karayannis, T. (2019). Developing a brain atlas through deep learning. Nat. Mach. Intell. 1:277-87.

15 Oct 2022
Brain Genomics: Cortical Structure
 Genome-wide association meta-
analysis of brain fMRI (n = 51,665)
 Measurement of cortical surface area
and thickness from MRI
 Identification of genomic locations of
genetic variants that influence global
and regional cortical structure
 Implicated in cognitive function,
Parkinson’s disease, insomnia,
depression, neuroticism, and
attention deficit hyperactivity
disorder
34
fMRI: functional magnetic resonance imaging. Source: Grasby, K.L., Jahanshad, N., Painter, J.N. et al. (2020). The genetic
architecture of the human cerebral cortex. Science. 367(6484). Posthuma Laboratory.

15 Oct 2022
Alzheimer’s Disease
35
Source: Arboleda-Velasquez J.F., Lopera, F. O’Hare, M. et al. (2019). Resistance to autosomal dominant Alzheimer’s in an APOE3-
Christchurch homozygote: a case report. Nat Med. 25(11):1680-83.
 Patient case:
 Left: Subject with protective Christchurch APOE3R136S
mutation (rs121918393) A not C: heavy Aβ plaque burden
(top), but limited tau tangles (bottom), and no early onset
Alzheimer’s disease
 Right: Control case with Paisa mutation Presenilin 1
(rs63750231): low Aβ plaque burden (top), substantial tau
tangles (bottom), and early-onset Alzheimer’s
 Implication: CRISPR-based genetic cut-paste study
Plaques (top red): No
Early-onset Alzheimer’s
Tangles (bottom red):
Early-onset Alzheimer’s
 Contra indicating
plaques and
tangles

15 Oct 2022
Alzheimer’s Disease Proteome
 Cluster analysis of protein changes
 1,532 proteins changed more than 20% in Alzheimer’s disease
 Upregulation: immune response and cellular signaling pathways
 Downregulation: synaptic function pathways including long term
potentiation, glutamate signaling, and calcium signaling
36
“Omics” Field Focus Definition Completion
1 Genome Genes All genetic material of an organism Human, 2001
2 Connectome Neurons All neural connections in the brain Fruit fly, 2018
3 Synaptome Synapses All synapses in the brain and their proteins Mouse, 2020
Hotspot Clustering Analysis
Sources: Hesse et al. (2019). Comparative profiling of the synaptic proteome from Alzheimer’s disease patients with focus on the
APOE genotype. Acta Neuropath. Comm. 7(214). Minehart et al. (2021). Developmental Connectomics of Targeted Microcircuits.
Front Synaptic Neuroscience. 12(615059).

15 Oct 2022
 Alzheimer’s Disease Modeling
 Chern-Simons Neuroscience
 Quantum Chemistry
 Quantum Krill: 3-Tier Ecosystem Modeling
Appendix

15 Oct 2022
Chern-Simons theory
 Chern-Simons theory: solvable 3d
topological field theory (quantum field theory)
 Wilson loops (nonlocal observables) represented as
knots (generalize to the Jones polynomial (an
important knot invariant))
 If coarse-grained Wilson loops go to one: sufficient
condition certification of topological order (Maskara-Lukin)
 Solve with path integrals
 Provide a physical interpretation of the invariants
 Give an account of the Hilbert space structure
 Example of the AdS/CFT correspondence
 Chern-Simons theory = AdS3
 Chern-Simons theory on a 3-manifold is equivalent to
a 2D CFT on a Riemann surface (AdS3/CFT2)
38
Sources: Grabovsky, S. (2022). Chern-Simons Theory in a Knotshell. Witten, E. (1989). Quantum Field Theory and the Jones
Polynomial. Commun. Math. Phys. 121:351-399. IMAGE: van Raamsdonk, M. (2015). Gravity and Entanglement.
http://pirsa.org/15020086.

15 Oct 2022
Chern-Simons Genomics
39
QFT: quantum field theory
Source: Capozziello, S., Pincak, R., Kanjamapornkul, K. & Saridakis, E.N. (2018). The Chern-Simons current in systems of DNA-
RNA transcriptions. Annalen der Physik. 530(4): 1700271.
 Chern-Simons (topological invariance)
 Easy-to-assess min-max curvature formulation
 Interpretation: DNA mutation, folded protein
 QFT w Wilson loop observables solved as knots
 Chern-Simons current (superfluid hydrodynamics)
 Dynamical system evolution in a complex plane
 Change seen as loops in the space (Wilson loops:
measurable quantum mechanical observables)
 Sheaf cohomology (Grothendieck topology)
 Algebraic topology: geometric structure as subsets of
open sets on a space (covering sieves; sheaf of rings)
 Wide bio application: DNA, RNA, protein folding
 Cancer, drug development, disease prevention
Access non-coding regions via
retrotransposon entanglement
state of the loop space in the
Hopf fibration (3d model of 4d)

15 Oct 2022
40
 Problem: predict virus-host
genetic evolution
 Traditional method: Bayesian probability
 New method: high-d mathematical physics
 Problem-solving intuition
 DNA is chiral
 Employ right/left-handed symmetry models
 Genetic code is 4d: spatial-temporal
movement of bases
 Suggests wavefunction (particle physics)
 System evolution in a complex plane
 Tensor networks: solvable models
 Classical and quantum computing-ready
Virus docks with host cell
Wilson loop evolution of
Cd4 gene and V3 loop as
twistor exchange

15 Oct 2022
41
Transposon: sequence that can be transposed
 Virus-host genetic evolution
 Virus docks with host, inserts or
deletes code swatches into the host
genome, host responds
 Model host response
 Loops introduced to DNA environment
superspace (complex space)
 DNA code represented as fields
(energy-based state transition)
 Energy states of the coding and non-
coding fields (DNA as field)
 Focus of sheaf cohomology
(Grothendieck topology) model
Field-based viral
replication cycle (8 states)
Chern-Simons current and
Wilson loop over DNA (red dot:
knot model of DNA projection)

15 Oct 2022
42
 Result: predict time series code
evolution
 Chern-Simons current (superfluid)
transition states
 Unoriented knot, twistor states
 Knot: high-d polynomial
 Twistor: mapping of 4d space to 4d+
complex space
 Output: time series data (DNA code)
 Plotted with Chern-Simons algebra in a
complex plane
 Entanglement states modeled by Hopf
fibration (3d model of 4d) over the loop
space in the spinor field of time series DNA
Hopf fibration with S3 group
action on genetic code space

15 Oct 2022
Appendix

15 Oct 2022
Quantum Chemistry (= Molecular QM)
 Quantum Chemistry: branch of physical chemistry
applying quantum mechanics to chemical systems
 Solve classically-intractable chemistry problems
 High temperature superconductivity, solid-state/condensed matter
physics, transition metal catalysis, new compound discovery
 Biochemical reactions, molecular dynamics, protein folding
 Short-term Objectives
 Computational solutions to Schrödinger equation (approximate)
 Increase size of molecules that can be studied
Sources: Krenn et al. (2020).Self-referencing embedded strings (SELFIES): A 100% robust molecular string representation. Machine
.Learning: Sci. Tech. 1(4):045024; Kmiecik et al. (2020). Coarse-Grained Protein Models and their Applications. Chem. Rev.
116:7898−7936.

15 Oct 2022
Quantum Chemistry
New Materials Found for Electric Batteries
 Unsupervised machine learning
method identifies new battery
materials for electric vehicles
 Four candidates out of 300
 VAE (variational autoencoder to
compress, analyze, re-encode
high-dimensional data) used to
rank chemical combinations
 Quaternary phase fields containing two
anions (e.g. lithium solid electrolytes)
 Discovery of Li3.3SnS3.3Cl0.7
45
Source: Vasylenko, A., Gamon, J., Duff, B.D. et al. (2021). Element selection for crystalline inorganic solid discovery guided by
unsupervised machine learning of experimentally explored chemistry. Nature Communications. 12:5561.
Ranking of Synthetic Exploration
Probe structure of Li3SnS3Cl predicted
coupled anion and cation order
VAE Analyzes High-Dimensional Data

15 Oct 2022
Digital Fabrication Methods
Autonomous Robotic Nanofabrication
 Use single molecules to
produce supramolecular
structures
 Control single molecules with
the machine learning agent-
based manipulation of scanning
probe microscope actuators
 Use reinforcement learning (goal-
directed updating) to remove
molecules autonomously from the
structure with a scanning probe
microscope
46
Source: Leinen, P., Esders, M., Schutt, K.T. et al. (2020). Autonomous robotic nanofabrication with reinforcement learning. Sci. Adv.
6:eabb6987.
Subtractive manufacturing with machine
learning: molecules bind to the scanning
microscope tip; bond formation and breaking
increases or decreases the tunneling
current; new molecules are retained in the
monolayer by a network of hydrogen bonds

15 Oct 2022
Atomically-Precise Manufacturing
 Single atoms positioned to create macroscopic objects
 Applications: molecular electronics, nanomedicine, integrated
circuits, thin films, etch masks, renewable energy materials
47
STM: scanning tunneling microscope; SPM: scanning probe microscope
Source: Randall, J.N. (2021). ZyVector: STM Control System for Atomically Precise Lithography. Zyvex Labs.
https://www.zyvexlabs.com/apm/products/zyvector
Atomically-Precise Writing (Deposition) with an STM
1. Outline the structure of the design
2. Specify crystal lattice vector layout
3. Write (deposit) atoms with the STM tip
4. Finalize atomically-precise pattern
ZyVector: STM Control System
for Atomically Precise
Lithography (Zyvex Labs)

15 Oct 2022
Molecular Electronics: Quantum Circuit Design
 Molecular circuits for quantum computing, construct
 One-qubit gates using one-electron scattering in molecules
 Two-qubit controlled-phase gates using electron-electron
scattering along metallic leads
48
Source: Jensen, P.W.K., Kristensen, L.B., Lavigne, C. & Aspuru-Guzik, A. (2022). Toward Quantum Computing with Molecular
Electronics. Journal of Chemical Theory and Computation.
Electron transmission magnitude as
a function of incoming kinetic
energy for molecular hydrogen in
the 6-31G basis attached between
one input and two output leads
Electron transmission through molecular hydrogen in
STO-3G basis (the planes intersecting through the
two orbitals indicate the integration limits)

15 Oct 2022
Quantum Chemistry
System Setup: Quantum Algorithms
 Qubit Hamiltonians
 Quantum algorithm: express Hamiltonian as qubit operator
 Retain fermionic exchange symmetries
 Describe fermionic states in terms of qubit states
 Perform transformations using fermionic-to-qubit mappings (e.g.
Jordan-Wigner transformation)
 Excitation gates as Givens rotations
 Main tool: Variational Quantum Eigensolver (VQE)
 Algorithm to compute approximate system energies
 Optimize the parameters of a
quantum circuit with respect to
the expectation value of a
molecular Hamiltonian
(minimizing a cost function)
49
Source: Arrazola, J.M., Jahangiri, S., Delgado, A. et al. (2021). Differentiable quantum computational chemistry with PennyLane.
arXiv:2111.09967v2

15 Oct 2022
Quantum Chemistry
Ground and Excited-state Energies
 Ground state energy (GSE)
 Compute Hamiltonian expectation values
 Convert the system Hamiltonian to a sparse matrix
 Use the vector representation of the state to compute the expectation value
using matrix vector multiplication
 Calculate expectation values by performing single-qubit rotations
 Pauli expressions are tensor products of local qubit operators
 Manage complexity by grouping Pauli expressions into sets of mutually-
commuting operators (calculate EV from the same measurement statistics)
 Excited state energy (ESE)
 Compute excited-state energies: add penalty terms to cost function
 The lowest-energy eigenstate of the penalized system is the first
excited state of the original system
 Iterate to find k-th excited state by adjusting penalty parameters
50
Source: Arrazola, J.M., Jahangiri, S., Delgado, A. et al. (2021). Differentiable quantum computational chemistry with PennyLane.
arXiv:2111.09967v2
GSE: Minimize cost function C(θ) = 〈Ψ(θ)|H |Ψ(θ)〉
Exp Value 〈H〉 = 〈 Ψ | H | Ψ 〉
ESE: Minimize cost function C(1)(θ) = 〈Ψ(θ)|H(1) |Ψ(θ)〉
Exp Value H(1) = H + β |Ψ0 > <Ψ0|
Qubit Rotation

15 Oct 2022
Quantum Chemistry
Total System Energy
 Study total energy gradients with energy
derivatives
 Nuclear forces and geometry optimization
 Force experienced each nuclei is given by the
gradient of the total energy with respect to the
nuclear coordinates (Hellman-Feynman
theorem)
 Vibrational normal modes and frequencies in
the harmonic approximation
 Compute Hessians and vibrational modes
with expressions for higher-order energy
derivatives
51
Sources: Arrazola, J.M., Jahangiri, S., Delgado, A. et al. (2021). Differentiable quantum computational chemistry with PennyLane.
arXiv:2111.09967v2; McArdle, S., Endo, S., Aspuru-Guzik, A. et al. (2020). Quantum computational chemistry. Reviews of Modern
Physics. 92(1):015003.
Fermion to qubit mapping
for Lithium Hydride (LiH)

15 Oct 2022
Appendix

15 Oct 2022
Krill Swarm: 4-tier Food-web Ecosystem
 Largest known animal aggregation
 30,000 individuals per square meter
 Global impact
 Aggregate biomass: 500 million tons worldwide
 Food source for whales, seals, penguins, squid, fish, birds
 Distribution: dispersed patches to dense swarms (Southern Ocean)
 Remove 39 mn tons carbon from the surface ocean each year (Belcher 2020)
 Krill morphology and activity
 Zooplankton invertebrates weighing 2 grams (0.07 oz), ~5 cm long
 Eat phytoplankton (microscopic suspended plants) and under-ice algae
 Spend the day at depth, rise to ocean surface at night (traveling hundreds of meters)
 10-year lifespan if avoiding predation
 Can survive up to 200 days without food (body shrinks but not eyes)
 Reproduction: lay 10,000 eggs at a time, several times per Jan-Mar spawning season
 Eggs laid near surface, sink over a 10-day period before hatching
53
Source: BAS British Antarctic Survey: Tarling et al. (2018). Varying depth and swarm dimensions of open-ocean Antarctic krill
Euphausia superba Dana, 1850 (Euphausiacea) over diel cycles. Journal of Crustacean Biology. 38(6):716–727. Belcher-Tarling
(2020). Why krill swarms are important to the global climate. Frontiers for Young Minds. 8(518995):1–8.
Krill swarm

15 Oct 2022
Quantum Krill Ecosystem Model
B. Enhanced
A. Basic
Optical analysis of light
spectrum gradient (Heggerud)
Swarmalator
hydrodynamic: O’Keeffe
(Kuramoto oscillator),
Ghosh (ring), Murphy (jet)
Lotka-Volterra predator-
prey model spiking
neuronal network
excitatory-inhibitory
model (Lagzi)
Statistical distribution (Miller)
2d Lagrangian (Hofmann)
Mathematics Diffusion (Heggerud)
Statistical analysis: 11 krill
swarm characteristics
analyzed in relation to
whale presence-absence
using Boosted regression
trees (BRTs) via a logit
(quantile function) (to
achieve local regularization
and prevent overfitting by
optimizing the number of
trees, learning rate, and
tree complexity
Quantum circuits
Random tensor network
QML: RKHS, QNN,
Quantum walk
VQE, VAE, QAOA, Quantum
amplitude estimation
Example of multiscalar system: phytoplankton-
krill-whale parallel to synapse-neuron-network
Two species non-local
reaction-diffusion-advection
model to consider niche
differentiation via absorption
spectra separation. (rate of
change of) density of
phytoplankton species as
diffusion minus buoyancy
plus absorbed photons
minus death rate
Spatial light attenuation
through vertical water column
Ice
2d spatial Lagrangian
model based on four
random forces acting on
krill individuals:
displacement, response
to food gradients,
nearest neighbor
interaction (attraction or
repulsion), and
predation
VQE: variational quantum eigensolver; VAE: variational autoencoder ; QAOA i: quantum approximate optimization algorithm; QAOA
ii: quantum alternating operator ansatz (guess); RKHS (reproducing kernel Hilbert space) (quantum kernel learning), QNN: quantum
neural network
Krill swarm density (%) = forces acting on krill whale predation (death rate)
phytoplankton density –
+

15 Oct 2022
Ice
Phytoplankton
Whales
Krill swarm
Krill distribution
Whale distribution
Phytoplankton distribution
Multiscalar System: 4-tier Food-web Ecosystem
Southern Ocean: Phytoplankton – Krill Swarm – Whale
Primary factors: light, nutrients
Secondary factors: temperature
Primary factors: daylight (solar elevation,
radiation), proximity to Antarctic continental slope
Secondary factors: current velocities & gradients
Primary factors: foraging availability,
distance to neighbors
Secondary factors: predation, light,
physiological stimuli, reproduction
HSO = f (P1, K1, W1,
s,
)
∂s
∂P1
∂s
∂K1
∂s
∂W1
, ,
f (P, K, W, s) + g (P, K, W, s) + h (P, K, W, s) = i (P, K, W, s)
∂s
∂W
∂s
∂K
∂s
∂P
Mathematical Model by Ecosystem Tier
 Phytoplankton: Reaction-diffusion-advection per light
spectrum differentiation, coupled plankton-oxygen dynamics,
fluid dynamics and Brownian motion (Heggerud, 2021)
 Krill swarm: Lagrangian (Brownian motion, spatial distribution)
(Hofmann, 2004); hydrodynamic signal per drafting within
front neighbor propulsion jet (Murphy, 2019); Kuramoto
oscillator for time and space synchrony (O’Keeffe, 2022)
 Krill-whale relation: hotspot clustering, statistical field theory
(Miller, 2019)
Light Spectrum Differentiation

15 Oct 2022
Phytoplankton: Diffusion (Heggerud)
56
∂tu1 = D1∂xu1 – α1∂xu1 + [g1 (γ1 (x,t)) – d1(x)]u1
∂tu2 = D2∂xu2 – α2∂xu2 + [g2 (γ2 (x,t)) – d2(x)]u2
D1, D2 > 0 Turbulence diffusion coefficients
Sinking/buoyancy coefficients (constants)
α1, α2 ϵ ℝ
γ1 (x,t) Number of absorbed photons
Death rate of the species at depth x and maximum L
d1(x) ϵ C [0,L]
γ1 (x,t) = a1(λ) k1(λ) I(λ, x)dλ
ʃ
u1 (x, t)
x Vertical depth in the water column
Density of phytoplankton species1,2 (depth x, time t)
(rate of change of) Density of Phytoplankton species =
Diffusion – Buoyancy + (Absorbed Photons – Death Rate)
D1u1 = D2u2
Source: Heggerud, C.M., Lam, K.-Y. & Wang, H. (2021). Niche differentiation in the light spectrum promotes coexistence of
phytoplankton species: a spatial modelling approach. arXiv:2109.02634v1.
Absorption spectra
k1(λ)
Action spectrum (proportion of absorbed photons used for photosynthesis)
a1(λ)
I(λ, x)dλ Incident light spectrum (wavelength intensity) of sunlight entering water column (Lambert-Beer’s Law)
Growth rate of species as a function of absorbed photons
g1 (γ1 (x,t))
Ice
2
2
No outcompeting species in the basic model
Enhanced model: attenuation of light through
the vertical water column, spatially explicit
diffusivity of phytoplankton and potential for
system buoyancy regulation (advection)

15 Oct 2022
Krill: 2d Lagrangian (Forces) (Hofmann)
57
*Enhanced model: additional variable (equation not included)
Source: Hofmann, E.E., Haskell, A.G.E., Klinck, J.M. & Lascara, C.M. (2004). Lagrangian modelling studies of Antarctic krill
(Euphausia superba) swarm formation. ICES Journal of Marine Science. 61:617e631.
____
β
D
dXi =
dt
X, Y Two horizontal spatial dimensions
dYi
dt
=
____ Krill swarm formation factors:
D: Random displacement
F: Response to food gradients
N: Nearest neighbor interaction
attraction-repulsion
P: Predation
Vf (food,t) Foraging speed
Direction coefficient
local
P = P0(1-e-γρ )
ρ swarm density*
ρlocal < ρtarget
ρlocal < ρrepulsive
ρtarget < ρlocal < ρrepulsive
Diffusion motion
F Foraging motion
N Neighbor-induced motion
α Foraging angle
mFA Minimum turning angle
λFR Increased turning due to food
γ Predation rate constant
P Predation rate
λ Random turning modifier*
Lagrangian model to simulate Antarctic krill swarm formation
κ Neighbor response coefficient
ζ
δ Turning potential*
Sensing distance*
Turning threshold*
Ψ

15 Oct 2022
Order, Disorder, Chaos
 Order (arrangement), disorder (confusion), chaos
(self-organization: confusion gives way to order)
 Flocking: 3D orientation vis-à-vis 5-10 neighbors
 Swarmalators: self-synchronization in time and space
 Krill self-position in propulsion jet of nearest front neighbor (draft) as
a hydrodynamic communication channel that structures the school
(via metachronal stimulation of individual krill pleopods (~fins))
58
Source: Murphy et al. (2019). The Three-Dimensional Spatial Structure of Antarctic Krill Schools in the Laboratory. Scientific
Reports. 9(381):1-12.
Krill swarm: 30,000 individuals per square meter
Flocking: 3D orientation vis-a-vis 5-10 nearest neighbors
Black holes, quasi-
particles, quantum
spin liquids,
schooling, flocking,
swarming
Hydrodynamic jet orientation
vis-à-vis nearest neighbor

15 Oct 2022
 Quantum (Neuro)Biology
 Topological Quantum Materials
 Short-range protected materials (local symmetry protection)
 Topological insulators, superconductors, semimetals
 Long-range entangled materials
 Fractional quantum Hall states, quantum spin liquid, entropy
 Neural Signaling
 Neural signaling modeling
 Bifurcation, Lotka-Volterra, Ads/Brain, Chern-Simons
Agenda

15 Oct 2022
Topological Quantum Materials
 2016 Nobel Prize
 Theoretical discoveries of topological phase transitions and
topological phases of matter (materials, condensed matter physics)
 Topological quantum materials: novel matter phases at zero-temperature
described by topology (global invariants) vs symmetry breaking (local)
60
Source: Schine, N.A., Chalupnik, M., Can, T. & Gromov, A. (2019). Measuring Electromagnetic and Gravitational Responses of
Photonic Landau Levels. Nature. 565(7738).
Semimetals and
quantum spin liquids
 Symmetry (classical materials)
 Looks the same from
different points of view
 Phase transition due to
local symmetry breaking
 Topology (quantum materials)
 Global properties preserved
despite local deformation
 Bending, stretching, twisting
 Phase transition not
described by Landau
symmetry
breaking

15 Oct 2022
61
 Short-range protected materials
 Local topology/symmetry protection, entanglement not relevant
 Topological insulators
 Superconductors
 Semimetals
 Entanglement as central non-local ordering parameter to
describe interactions and correlations, symmetry not relevant
 Fractional quantum Hall states
 Quantum spin liquids
 Entanglement entropy
Quantum Matter: novel matter
phases at zero-temperature

15 Oct 2022
 Bulk wavefunctions lead to surface states
 Energy band theory (allowed energy tiers)
 Classification: insulator, semiconductor, semimetal, metal
 Floquet engineering: band reshaping
 Materials with a conducting surface and an insulating interior
 Symmetry-protected surface states (time-reversal, particle-hole, chiral symmetry)
 Superconductors
 Materials that conduct electricity without resistance (heat)
 Semimetals
 Materials with tunable surface states due to overlapping region
between conduction and valence bands
 Adjust material thickness, defects (doping), state degeneracy
Short-range Protected Materials

15 Oct 2022
Semimetals
 Dirac semimetals
 Arc-like surface states: complex bulk-
boundary relation between surface
Fermi arcs and bulk Dirac points
 Require time-reversal and crystal
lattice (rotation or reflection) symmetry
 Weyl semimetals
 Arc-like surface states: 1d Fermi arcs
from bulk 3d Weyl points
 Require translation symmetry
 Nodal-line semimetals
 Energy band-touching manifolds at 1d
nodal lines or rings in the bulk
63
Dirac/Weyl and Nodal-line Semimetals

15 Oct 2022
64
 Short-range protected materials
 Local topology/symmetry protection, entanglement not relevant
 Superconductors
 Semimetals
 Entanglement as central non-local ordering parameter to
describe interactions and correlations, symmetry not relevant
 Fractional quantum Hall states
 Quantum spin liquids
 Entanglement entropy
Quantum Matter: novel matter
phases at zero-temperature

15 Oct 2022
Fractional Quantum Hall Effect
 1998 Nobel prize:
 The discovery of a new form of
quantum fluid with fractionally charged excitations
 Hall effect: voltage difference by applying a magnetic
field to a current of electrons in a thin conducting strip
 The Lorentz force perpendicular to the current causes a
build-up of charge on the edge of the strip that induces a
voltage across the width of the strip (Edwin Hall 1879)
 Fractional quantum Hall effect: quantized
plateaus at fractional values of charge
 Gives rise to quasiparticles (collective states)
in which electrons bind magnetic flux lines to
make new quasiparticles that have a fractional
charge and obey anyonic statistics
65
Source: Liu, C.-X., Zhang, S.-C. & Qi, X.-L. (2015). The quantum anomalous Hall effect: Theory and experiment. Ann. Rev. Cond.
Matt. Phys. 7:301–321.
Fractional Quantum Hall Effect (FQHE)

15 Oct 2022
Quantum Spin Liquid and Magnetics
66
Spin `“glass” by analogy to window glass with irregular atomic bond structure vs uniform crystal lattice bonds
Source: Carleo, G. & Troyer, M. (2017). Solving the Quantum Many-Body Problem with Artificial Neural Networks. Science.
355(6325):602-26.
Classical Quantum
 Ferromagnet
 Magnetic spins aligned in the
same direction (ordered)
 Antiferromagnet
 Magnetic spins aligned in
opposite directions (ordered)
 Spin glass
 Magnetic spins not aligned in
a regular pattern (disordered)
 Quantum spin glass
 Spin glass phase transition at
zero-temperature per quantum
(not thermal) fluctuations
 Ising model
 Statistical mechanical model
of phase transition studying
ferromagnetism as lattice-
based spins
 Ising (basic)
 Heisenberg (extensive)
Spin Glass ->
Quantum Spin Liquid

15 Oct 2022
Quantum Spin Liquid (QSL)
 Quantum spin liquid: novel phase in
condensed matter physics in which strong
quantum fluctuations prevent long-range
magnetic order from being established
 Electron spins do not form an ordered pattern but
remain liquid-like even at absolute zero temperature
 Exotic properties
 Long-range entanglement
 Fractional (anyon) excitations
 Topological order
 Use: quantum communication and computation
 Matter phase of quantum spins interacting in magnetic
materials, manipulable quasi-disordered ground state
67
Quantum spin liquid:
electron spins (blue
arrows) show no long-
range ordering even at
low temperatures
Source: Wen, J., Yu, S.-L., Li, S. et al. (2019). Experimental identification of quantum spin liquids. npj Quantum Materials.
4:12.

15 Oct 2022
Quantum Spin Liquids
Natural and Synthesized
 Herbertsmithite
 Mineral with quantum spin liquid
magnetic properties
 Magnetic particles with constantly
fluctuating scattered orientations
on a regular kagome (triangle-
hexagon) lattice
68
 Made in the Lab
 Kitaev honeycomb (2015)
 Superconducting circuit (2021)
 Optical atom array (2021)
 2021 result
 Engineering the topological
order known as the toric code
 Archetypical 2d lattice model
that exhibits the exotic
properties of topologically
ordered states
 Proposed for quantum error
correction
Sources: Satzinger et al. (2021). Realizing topologically ordered states on a quantum processor. Science. 374(6572):1237–1241.
Semeghini et al. (2021). Probing topological spin liquids on a programmable quantum simulator. Science. 374(6572):1242–1247.
ZnCu3(OH)6Cl2
Zinc, Copper, Oxygen, Hydrogen, Chlorine
Chile, Arizona, Iran, Greece

15 Oct 2022
Topological Entanglement Entropy
 Topology-based measure of entanglement entropy
 Problem: the phases on each side of a quantum critical point
have different topological order
 Need a non-local parameter to distinguish phases
 Use long-range entanglement entropy
 Computation methods
69
Entanglement: Quantum property
of correlated physical attributes
among particles (position,
momentum, spin, polarization)
Entropy: number of possible
system microarrangements
Entanglement Entropy: measure of
quantum correlations in a many-
body system
Source: Kitaev, A. & Preskill, J. (2006). Topological entanglement entropy. Phys Rev Lett. 96(110404).
1. Take the logarithm of the total quantum
dimension of the quasiparticle
excitations of the many-body state
2. Compare the von Neumann entropy
between a spatial block and the rest of
the system
 Using tripartite information (information
measure of two time, one space dimensions)

15 Oct 2022
 Quantum (Neuro)Biology
 Topological Quantum Materials
 Short-range protected materials (local symmetry protection)
 Topological insulators, superconductors, semimetals
 Fractional quantum Hall states, quantum spin liquid, entropy
 Neural Signaling
 Neural signaling modeling
 Bifurcation, Lotka-Volterra, Ads/Brain, Chern-Simons
Agenda

15 Oct 2022
 Multiscalar renormalization scheme (tensor networks)
 Flow from boundary surface (UV) to bulk (IR) and back up to
boundary to discover hidden correlations in both
AdS/MERA
71
MERA: Multiscale Entanglement Renormalization Ansatz (guess)
Source: Vidal, G. (2007). Entanglement renormalization. Phys Rev Lett. 99(220405).
Boundary
Bulk
Boundary
Vidal, 2007
Swingle, 2012
McMahon, 2020
Vidal, 2007
Renormalization: physical system viewed at different scales
Tensor network: mathematical tool for the efficient representation
of quantum states (high-dimensional data in the form of tensors);
tensor networks factor a high-order tensor (a tensor with a large
number of indices) into a set of low-order tensors whose indices
can be summed (contracted) in the form of a network

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