Constrained Optimization On Riemannian Manifolds
Melanie Weber (Oxford, Mathematical Institute)
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Near Optimal Sample Complexity For Matrix And Tensor Normal Models Via Geodesic Convexity
Akshay Ramachandran (University of Amsterdam)Pivot Hamiltonians: a tale of symmetry, entanglement, and quantum criticality
Nathanan Tantivasadakarn California Institute of Technology (Caltech)
Computing The Nc-Rank Via Discrete Convex Optimization On Cat(0) Spaces
Hiroshi Hirai (University of Tokyo)Norm Minimization, Invariant Theory, And Quantum Tomography
Cole Franks (MIT)Spectral analysis of product formulas for quantum simulation
Changhao Yi University of New Mexico
Regulating Loops in dS
Akhil Premkumar University of California, San Diego
Twisted bilayers: magic continuum, noncollinear magnetism and more
Zhu-Xi Luo University of California, Santa Barbara
Melonic large N limit of 5-index irreducible random tensors
Sabine Harribey Dublin Institute For Advanced Studies
Dynamics of spinning compact binaries: synergies between post-Newtonian and self-force approaches
Mohammed Khalil Perimeter Institute for Theoretical Physics
Constrained Optimization On Riemannian Manifolds
Melanie Weber (Oxford, Mathematical Institute)Many applications involve non-Euclidean data, where exploiting Riemannian geometry can deliver algorithms that are computationally superior to standard nonlinear programming approaches. This observation has resulted in an increasing interest in Riemannian methods in the optimization and machine learning community. In this talk, we consider the problem of optimizing a function on a Riemannian manifold subject to convex constraints. We introduce Riemannian Frank-Wolfe (RFW) methods, a class of projection-free algorithms for constrained geodesically convex optimization. To understand the algorithm’s efficiency, we discuss (1) its iteration complexity, (2) the complexity of computing the Riemannian gradient, and (3) the complexity of the Riemannian “linear” oracle (RLO), a crucial subroutine at the heart of the algorithm. We complement our theoretical results with an empirical comparison of RFW against state-of-the-art Riemannian optimization methods. Joint work with Suvrit Sra (MIT).Near Optimal Sample Complexity For Matrix And Tensor Normal Models Via Geodesic Convexity
Akshay Ramachandran (University of Amsterdam)The matrix normal model, the family of Gaussian matrix-variate distributions whose covariance matrix is the Kronecker product of two lower dimensional factors, is frequently used to model matrix-variate data. The tensor normal model generalizes this family to Kronecker products of three or more factors. We study the estimation of the Kronecker factors of the covariance matrix in the matrix and tensor models. We show nonasymptotic bounds for the error achieved by the maximum likelihood estimator (MLE) in several natural metrics. In contrast to existing bounds, our results do not rely on the factors being well-conditioned or sparse. For the matrix normal model, all our bounds are minimax optimal up to logarithmic factors, and for the tensor normal model our bound for the largest factor and overall covariance matrix are minimax optimal up to constant factors provided there are enough samples for any estimator to obtain constant Frobenius error. In the same regimes as our sample complexity bounds, we show that an iterative procedure to compute the MLE known as the flip-flop algorithm converges linearly with high probability. Our main tool is geodesic strong convexity in the geometry on positive-definite matrices induced by the Fisher information metric. This strong convexity is determined by the expansion of certain random quantum channels. We also provide numerical evidence that combining the flip-flop algorithm with a simple shrinkage estimator can improve performance in the undersampled regime.Pivot Hamiltonians: a tale of symmetry, entanglement, and quantum criticality
Nathanan Tantivasadakarn California Institute of Technology (Caltech)
I will introduce the notion of Pivot Hamiltonians, a special class of Hamiltonians that can be used to "generate" both entanglement and symmetry. On the entanglement side, pivot Hamiltonians can be used to generate unitary operators that prepare symmetry-protected topological (SPT) phases by "rotating" the trivial phase into the SPT phase. This process can be iterated: the SPT can itself be used as a pivot to generate more SPTs, giving a rich web of dualities. Furthermore, a full rotation can have a trivial action in the bulk, but pump lower dimensional SPTs to the boundary, allowing the practical application of scalably preparing cluster states as SPT phases for measurement-based quantum computation. On the symmetry side, pivot Hamiltonians can naturally generate U(1) symmetries at the transition between the aforementioned trivial and SPT phases. The sign-problem free nature of the construction gives a systematic approach to realize quantum critical points between SPT phases in higher dimensions that can be numerically studied. As an example, I will discuss a quantum Monte Carlo study of a 2D lattice model where we find evidence of a direct transition consistent with a deconfined quantum critical point with emergent SO(5) symmetry.
This talk is based on arXiv:2107.04019, 2110.07599, 2110.09512Computing The Nc-Rank Via Discrete Convex Optimization On Cat(0) Spaces
Hiroshi Hirai (University of Tokyo)In this paper, we address the noncommutative rank (nc-rank) computation of a linear symbolic matrix A=A1 x1+A2 x2 +⋯+ Am xm, where each Ai is an n x n matrix over a field K, and xi (i=1,2,…,m) are noncommutative variables. For this problem, polynomial time algorithms were given by Garg, Gurvits, Oliveira, and Wigderson for K=Q, and by Ivanyos, Qiao, and Subrahmanyam for an arbitrary field K. We present a significantly different polynomial time algorithm that works on an arbitrary field K. Our algorithm is based on a combination of submodular optimization on modular lattices and convex optimization on CAT(0) spaces.Norm Minimization, Invariant Theory, And Quantum Tomography
Cole Franks (MIT)Given many identical copies of a quantum particle, what's the chance measuring their overall angular momentum yields zero? We show how to frame this problem as an optimization problem over a matrix group. We'll then discuss the relationship between this optimization problem and invariant theory, quantum tomography, and the Jacobian conjecture. This talk is based on the joint work https://arxiv.org/abs/2004.14872 with Michael Walter.Spectral analysis of product formulas for quantum simulation
Changhao Yi University of New Mexico
Trotter-Suzuki formula is a practical and efficient algorithm for Hamiltonian simulation. It has been widely used in quantum chemistry, quantum field theory and condensed matter physics. Usually, its error is quantified by the operator norm distance between the ideal evolution operator and the digital evolution operator. However, recently more and more papers discovered that, even in large Trotter step region, the quantity of interest can still be accurately simulated. These robustness phenomena imply a different approach of analyzing Trotter-Suzuki formulas. In our previous paper, by analyzing the spectral analysis of the effective Hamiltonian, we successfully established refined estimations of digital errors, and thus improved the circuit complexity of quantum phase estimation and digital adiabatic simulation.
Regulating Loops in dS
Akhil Premkumar University of California, San Diego
Perturbative QFT calculations in de Sitter are plagued by a variety of divergences. One particular kind, the secular growth terms, cause the naive perturbation expansion to break down at late times. Such contributions often arise from loop integrals, which are notoriously hard to compute in dS. We discuss an approach to evaluate such loop integrals, for a scalar field theory in a fixed de Sitter background. Our method is based on the Mellin-Barnes representation of correlation functions, which enables us to regulate divergences for scalars of any mass while preserving the symmetries of dS. The resulting expressions have a similar structure as a standard dimreg answer in flat space QFT. These features of the regulator are illustrated with two examples. Along the way, we illuminate the physical origin of these divergences and their interpretation within the framework of the dynamical renormalization group. Our calculations naturally reveal additional infrared divergences for massless scalar fields in de Sitter, that are not present in the massive case. Such loop corrections can be incorporated as systematic improvements to the Stochastic Inflation framework, allowing for a more precise description of the IR dynamics of massless fields in de Sitter.
Twisted bilayers: magic continuum, noncollinear magnetism and more
Zhu-Xi Luo University of California, Santa Barbara
Van der Waals heterostructures provide a rich venue for exotic moir\’e phenomena. In this talk, I will present a couple of unconventional examples beyond the celebrated twisted bilayer graphene. I will start by twisted bilayer of square lattice with staggered flux, which exhibits a continuum range of magic twisting angles where an exponential reduction of Dirac velocity and bandwidths occurs. Then I will discuss moir\’e magnetism arising from twisted bilayers of antiferromagnets and also ferromagnets. Despite the fact that the parent materials all exhibit collinear orderings, the bilayer system shows controllable emergent noncollinear spin textures. Time permitting, I will also discuss a theory for the potentially continuous metal-insulator transition with fractionalized electric charges in transition metal dichalcogenide moir\’e heterostructures.
Zoom Link: https://pitp.zoom.us/j/99322296758?pwd=WUNGcE1JS3FpZ1VxbklsSCtYTEJVdz09
Melonic large N limit of 5-index irreducible random tensors
Sabine Harribey Dublin Institute For Advanced Studies
The main feature of tensor models is their melonic large N limit, leading to applications ranging from random geometry and quantum gravity to many-body quantum mechanics and conformal field theories. However, this melonic limit is lacking for tensor models with ordinary representations of O(N) or Sp(N). We demonstrate that random tensors with sextic interaction transforming under rank-5 irreducible representations of O(N) have a melonic large N limit. This extends the recent proof obtained for rank-3 models with quartic interaction. After giving an introduction to random tensors, I will present the main ideas of our proof relying on recursive bounds derived from a detailed combinatorial analysis of the Feynman graphs.
Zoom Link: https://pitp.zoom.us/j/94691275506?pwd=RGFaN0NZR0FScFdOTXFzeFVXaXUvUT09
Dynamics of spinning compact binaries: synergies between post-Newtonian and self-force approaches
Mohammed Khalil Perimeter Institute for Theoretical Physics
Accurate waveform models are crucial for gravitational-wave (GW) data analysis, and since numerical-relativity waveforms are computationally expensive, it is important to improve the analytical approximations for the binary dynamics. The post-Newtonian (PN) approximation is most suited for describing the inspiral of comparable-mass binaries, which are the main sources for ground-based GW detectors. In this talk, I discuss a method for deriving PN results valid for arbitrary mass ratios from first-order self-force results, by exploiting the simple mass dependence of the scattering angle in the post-Minkowskian expansion. I present results for the spin-orbit dynamics up to the fourth-subleading PN order (5.5PN) and the spin-spin dynamics up to the third-subleading PN order (5PN). I also discuss implications for the first law of binary mechanics.
Zoom Link: https://pitp.zoom.us/j/92861625861?pwd=cHpXUlM1d01pc09mNGhhQVZxRHBiQT09
Complex critical points and curved geometries in four-dimensional Lorentzian spinfoam quantum gravity
This talk focuses on the semiclassical behavior of the spinfoam quantum gravity in 4 dimensions. There has been long-standing confusion, known as the flatness problem, about whether the curved geometry exists in the semiclassical regime of the spinfoam amplitude. The confusion is resolved by the present work. By numerical computations, we explicitly find curved Regge geometries from the large-j Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam amplitudes on triangulations. These curved geometries are with small deficit angles and relate to the complex critical points of the amplitude. The dominant contribution from the curved geometry to the spinfoam amplitude is proportional to e^{i I}, where I is the Regge action of the geometry plus corrections of higher order in curvature. As a result, the spinfoam amplitude reduces to an integral over Regge geometries weighted by e^{i I} in the semiclassical regime. As a byproduct, our result also provides a mechanism to relax the cosine problem in the spinfoam model. Our results provide important evidence supporting the semiclassical consistency of the spinfoam quantum gravity.
Zoom Link: https://pitp.zoom.us/j/93699343757?pwd=RnpFeitaTU5qTktzNjlXQW45K1gvQT09
Life is What?
Sara Walker Arizona State University
Currently, no general theory exists that explains what life is. While many definitions for life do exist, these are primarily descriptive, not predictive, and they have so far proved insufficient to explain the origins of life, or to provide rigorous constraints on what properties we might expect all examples of life to share (e.g., in our search for life in alien environments). In this talk I discuss new approaches to understanding what universal principles might explain the nature of life and elucidate the mechanisms of its origins, focusing on recent work in our group elucidating regularities and law-like behavior of biochemical networks on Earth from the scale of individual organisms to the planetary scale.
Zoom Link: https://pitp.zoom.us/j/91944267625?pwd=QzBmTzRKK0k3YXhXWnQ3WjNBSDR2UT09