Scientific Series

When an interacting quantum many-body system is cooled down to its ground state, there can be discrete "topological invariants" that characterize the properties of such ground states. This leads to the concept of "topological phases of matter" distinguished by these topological invariants. Experimental manifestations of these topological phases of matter include the integer and fractional quantum Hall effect, as well as topological insulators.

In this talk, after a general overview of topological phases of matter, I will explain how to define topological invariants that are specific to the ground states of regular crystals, i.e. systems that are periodic in space. I will discuss the physical manifestations of the resulting "crystalline topological phases", including implications for the properties of crystalline defects such as dislocations and disclinations. Then, I will explain how these ideas can be generalized to quasicrystals, which are a different class of materials that have long-range spatial order without exact periodicity. These ideas ultimately lead to a general classification principle for crystalline and quasicrystalline topological phases of matter.

Classical shadow tomography provides an efficient method for predicting functions of an unknown quantum state from a few measurements of the state. It relies on a unitary channel that efficiently scrambles the quantum information of the state to the measurement basis. However, it is quite challenging to realize deep unitary circuits on near-term quantum devices, and an unbiased reconstruction map is non-trivial to find for arbitrary random unitary ensembles. In this talk, I will discuss our recent progress on combining classical shadow tomography with quantum chaotic dynamics. Particularly, I will introduce two new families of shadow tomography schemes: 1) Hamiltonian-driven shadow tomography and 2) Classical shadow tomography with locally scrambled quantum dynamics. In both works, I’ll derive the unbiased reconstruction map, and analyze the sample complexity. In the Hamiltonian-driven scheme, I will illustrate how to use proper time windows to achieve a more efficient tomography. In the second work, I will demonstrate advantages of shadow tomography in the shallow circuit region. Then I’ll conclude by discussing approximate shadow tomography with local Hamiltonian dynamics, and demonstrate that a single quench-disordered quantum spin chain can be used for approximate shadow tomography.

References:
[1] Hong-Ye Hu, Yi-Zhuang You. “Hamiltonian-Driven Shadow Tomography of Quantum States”. arXiv:2102.10132 (2021)
[2] Hong-Ye Hu, Soonwon Choi, Yi-Zhuang You. “Classical Shadow Tomography with Locally Scrambled Quantum Dynamics”. arXiv: 2107.04817 (2021)

Zoom Link: https://pitp.zoom.us/j/99011187936?pwd=OVU3VkpyZ21YcXRCOW5DOHlnSWlVQT09

I will first give a brief overview of my research in the field of out-of-equilibrium quantum many-
body physics, ranging from the theory of many-body localization, to the recent application of Tensor

Processing Units for accelerating simulations of quantum dynamics. I’ll then focus on (1) the

experimental observation and theoretical explanation of subdiffusive dynamics in a “tilted” Fermi-
Hubbard system [PRX 10, 011042 (2020)], and (2) a “freezing” phase transition between weak and

strong ergodicity breaking in systems with particles that are immobile by themselves, but undergo
coordinated pair hopping [PRB 101, 214205 (2020)]. These topics contain the common thread of
either an emergent or microscopic conservation of the dipole moment (center of mass of the particle
distribution), and I will provide simple pictures for how this leads to the subdiffusion and ergodicity
breaking.

Paleo-Detectors are natural minerals which record damage tracks from nuclear recoils over geological timescales. Minerals commonly found on Earth are as old as a billion years, and modern microscopy techniques may allow to reconstruct damage tracks with nanometer scale spatial resolution. Thus, paleo-detectors would constitute a technique to achieve keV recoil energy threshold with exposures comparable to a kiloton-scale conventional "real-time" detector. In this talk, I will discuss the potential of paleo-detectors for the direct detection of dark matter as well as for detecting low-energy neutrinos as are e.g. emitted by core collapse supernovae or our Sun. Furthermore, the age of the minerals provides the ability to look back across gigayear-timescales, giving paleo detectors the unique ability to probe changes in the cosmic ray rate or the galactic supernova rate over such timescales as well as dark matter substructure Earth might have encountered during its past few trips around our Galaxy.

Quantum fluctuations in inflation provide the seeds for the large scale distribution of matter today. According to the standard paradigm, these fluctuations induce density perturbations that are adiabatic and Gaussian distributed. In this limit, all the information is contained within the two-point correlation function, or equivalently, the power spectrum. Today, the distribution of matter is far from Gaussian, with structures forming across a vast range of scales. Despite this, almost all analyses of observational data are performed using two-point functions. This begs the question: what information lies in higher-point statistics? 

In this seminar, I will present a pedagogical overview of the non-Gaussian correlation functions, and demonstrate how they can be used both to sharpen constraints on known physical parameters, and to provide stringent tests of new physics occurring in the early Universe. One of the major barriers to constraining cosmology from the higher-point functions is computational: measuring the statistics with conventional techniques is infeasible for current and future datasets. I will discuss new methods capable of reducing the computational cost by orders of magnitude, and show how this facilitates a number of exciting new tests of the cosmological model.

Quantum computers are expected to dramatically outperform classical computers for certain computational problems. While there has been extensive previous work for linear dynamics and discrete models, for more complex realistic problems arising in physical and social science, engineering, and medicine, the capability of quantum computing is far from well understood. One fundamental challenge is the substantial difference between the linear dynamics of a system of qubits and real-world systems with continuum, stochastic, and nonlinear behaviors. Utilizing advanced linear algebra techniques and nonlinear analysis, I attempt to build a bridge between classical and quantum mechanics, understand and optimize the power of quantum computation, and discover new quantum speedups over classical algorithms with provable guarantees. In this talk, I would like to cover quantum algorithms for scientific computational problems, including topics such as linear, nonlinear, and stochastic differential equations, with applications in areas such as quantum dynamics, biology and epidemiology, fluid dynamics, and finance.

Reference:
Quantum spectral methods for differential equations, Communications in Mathematical Physics 375, 1427-1457 (2020), https://arxiv.org/abs/1901.00961
High-precision quantum algorithms for partial differential equations, Quantum 5, 574 (2021), https://arxiv.org/abs/2002.07868
Efficient quantum algorithm for dissipative nonlinear differential equations, Proceedings of the National Academy of Sciences 118, 35 (2021), https://arxiv.org/abs/2011.03185
Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance, Quantum 5, 481 (2021), https://arxiv.org/abs/2012.06283

In thermal equilibrium, 1d bosonic systems (e.g. spin- or qubit- chains) cannot support intrinsically topological phases without symmetry protection. For example, the edge states of the Haldane spin chain are fragile to magnetic fields, in contrast to the absolutely stable Majorana edge states of a topological superconducting wire of fermionic electrons. This fragility is a serious drawback to harnessing topological edge states as protected quantum memories in existing AMO and qubit platforms for quantum simulation and information processing. In this talk, I will present evidence for a non-equilibrium topological phase of quasiperiodically-driven trapped ion chains, that exhibits topological edge states that are protected purely by emergent dynamical symmetries that cannot be broken by microscopic perturbations. This represents both the first experimental realization of a non-equilibrium quantum phase, and the first example of a 1d bosonic topological phase that does not rely on symmetry-protection.

With the rapid development of programmable quantum simulators, the quantum states can be controlled with unprecedented precision. Thus, it opens a new opportunity to explore the strongly correlated phase of matter with new quantum technology platforms. In quantum simulators, one can engineer interactions between the microscopic degree of freedom and create exotic phases of matter that presumably are beyond the reach of natural materials. Moreover, quantum states can be directly measured instead of probing physical properties indirectly via optical and electrical responses of material as done in traditional condensed matter. Therefore, it is pressing to develop new approaches to efficiently prepare and characterize desired quantum states in the novel quantum technology platforms.

In this talk, I will introduce our recent works on the characterization of the topological invariants from a ground state wave function of the topological order phase and the implementation in noisy intermediate quantum devices. First, using topological field theory and tensor network simulations, we demonstrate how to extract the many-body Chern number (MBCN) given a bulk of a fractional quantum Hall wave function [1]. We then propose an ancilla-free experimental scheme for measuring the MBCN without requiring any knowledge of the Hamiltonian. Specifically, we use the statistical correlations of randomized measurements to infer the MBCN of a wave function [2]. Finally, I will present an unbiased numerical optimization scheme to systematically find the Wilson loop operators given a ground state wave function of a gapped, translationally invariant Hamiltonian on a disk. We then show how these Wilson loop operators can be cut and glued through further optimization to give operators that can create, move, and annihilate anyon excitations. We then use these operators to determine the braiding statistics and topological twists of the anyons, yielding a way to fully characterize topological order from the bulk of a ground state wave function [3]. 

[1] H. Dehghani, Z.P. Cian, M. Hafezi, and M. Barkeshl, Phys. Rev. B 103, 075102
[2] Z.P. Cian, H. Dehghani, A. Elben, B. Vermersch, G. Zhu, M. Barkeshli, P. Zoller, and M. Hafezi, Phys. Rev. Lett. 126, 050501
[3] Z.P. Cian, M. Hafezi, and M. Barkeshl, Manuscript in preparation.

Entropic Dynamics (ED) is a framework in which Quantum Mechanics is derived as an application of entropic methods of inference. In ED the dynamics of the probability distribution is driven by entropy subject to constraints that are codified into a quantity later identified as the phase of the wave function. The challenge is to specify how those constraints are themselves updated.

The important ingredients are two: the cotangent bundle associated to the probability simplex inherits (1) a natural symplectic structure from ED, and (2) a natural metric structure from information geometry.

The requirement that the dynamics preserves both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow) leads to a Hamiltonian dynamics of probabilities in which the linearity of the Schrödinger equation, the emergence of a complex structure, Hilbert spaces, and the Born rule, are derived rather than postulated.

Self-gravitating quantum matter may exist in a wide range of cosmological and astrophysical settings: from the very early universe through to present-day boson stars. Such quantum matter arises in UltraLight Dark Matter (ULDM): an exciting axion-like particle candidate which keeps the successes of CDM on large scales but alleviates tensions on small scales. This small scale behavior is due to characteristic cores in ULDM called solitons, which also correspond to the ground state of the self-gravitating quantum system governing ULDM. We calculate the full spectrum of eigenstates and decompose simulations of ULDM into these states, allowing us to precisely track the evolution of the tell-tale soliton cores and the surrounding halo “skirt”. Using this formalism, we investigate formation of halos through binary soliton collisions and the dependence of the final halo product on initial parameters. We further link characteristic ULDM halo behavior—such as the soliton “breathing mode” and random walk of the center of mass—to the presence of certain modes. Finally, we comment on the relationship between eigenenergies and oscillatory timescales present in the system, as well as future directions for understanding ULDM through the language of its eigenstates.

 I will report on my progress, joint with David Reutter, to construct and analyze the algebraic closure of nVec – in other words, the universal n-category of framed nD TQFTs. The invertibles are Pontryagin dual to the stable homotopy groups of spheres. The Galois group is almost, but not quite, the stable PL group. An invertible TQFT can be condensed from the vacuum if and only if it trivializes on (possibly-exceptional) spheres.
 

For quantum gravity states associated to open spin network graphs, we study how the boundary (the set of open edges, which carries spin degrees of freedom) is affected by the bulk, specifically by its combinatorial structure and by the quantum correlations among the intertwiners. In particular, we determine under which conditions certain classes of quantum gravity states map bulk degrees of freedom into boundary ones isometrically (which is a necessary condition for holography). We then look at the entanglement entropy of the boundary and recover, for slightly entangled intertwiners, the Ryu-Takayanagi formula with corrections induced by the entanglement entropy of the bulk state. We also show that the presence of a region with highly entangled intertwiners deforms the minimal-area surface, which is prevented from entering that region when the entanglement entropy of the latter exceeds a certain bound, a mechanism which thus leads to the rise of a black hole-like region in the bulk.

Zoom Link: https://pitp.zoom.us/j/96356007543?pwd=U2VrRlhyOThMODdMYllDMnB6VjlZQT09