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We study the quantum connection in positive characteristic for conical symplectic resolutions. We conjecture the equivalence of the p-curvature of such connections with (equivariant generalizations of) quantum Steenrod operations of Fukaya and Wilkins, which are endomorphisms of mod p quantum cohomology deforming the Steenrod operations. The conjecture is verified in a wide range of examples, including the Springer resolution, thereby providing a geometric interpretation of the p-curvature and a full computation of quantum Steenrod operations. The key ingredients are a new compatibility relation between the quantum Steenrod operations and the shift operators, and structural results for the mod p quantum connection recently obtained by Etingof--Varchenko.

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Zoom link: https://pitp.zoom.us/j/91010341249?pwd=QXJuMlJrWWd0dHpPdUpDUGVqVmYvZz09

Supermaps are higher-order transformations taking maps as input.  We consider quantum algorithms implementing supermaps for the input given by unknown Hamiltonian dynamics, which can be regarded as infinitely divisible unitary operations.  We first show a quantum algorithm that approximately but universally transforms black-box Hamiltonian dynamics into controlled Hamiltonian dynamics utilizing a higher-order transformation called neutralization.  Then, we present another universal algorithm that efficiently simulates linear transformations of any Hamiltonian consisting of a polynomial number of terms in system size, using only controlled-Pauli gates and time-correlated randomness.  This algorithm for implementing Hamiltonian supermaps is an instance of quantum functional programming, where the desired function is specified as a concatenation of higher-order quantum transformations. As examples, we demonstrate the simulation of negative time-evolution and time-reversal, and perform a Hamiltonian learning task.

References:

Q. Dong, S. Nakayama, A. Soeda and M. Murao, arXiv:1911.01645v3
T. Odake, Hlér Kristjánsson, A. Soeda M. Murao, arXiv:2303.09788

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Zoom Link: https://pitp.zoom.us/j/94278362588?pwd=MGszYk9uN1A3K1RTOVhYSGpkL1FQdz09

I argue that the answer is yes, by reviewing the history and current status of such a theory.  Since 1982, I have been developing a series of such theories, beginning in 1982 with an N --> infinity limit of 2+1 dimensional matrix model (the IAS model), through another N --> infinity, T --> 0 limit of a BFSS model.  During this time our work was complemented by Adler's Trace model and others.   

Beginning in 2012, Cortes and I developed a different approach to an relational quantum cosmology by adding intrinsic energy momentum to Sorkin et al's causal set models.  The addition of energy- momentum as intrinsic variables opens up a new mechanism for the emergence of space, and spacetime, plus interacting relativistic particles.  Note that the warm phase is purely algebraic, you need no prior existence of any space to get other dimensions to emerge.   In 2021 I discovered how to derive quantum non-relativistic many body theory from what has since been called the Causal Theory of Views.  Finally in papers since we report progress on the construction of special and general relativistic  Causal Theory of Views.

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Zoom link: https://pitp.zoom.us/j/95891848248?pwd=TUpWK2RWbU9GTGxZS1lMeS81QWp1dz09

The Bekenstein bound posits a maximum entropy for matter with finite energy confined to a spacetime region. It is often interpreted as a fundamental limit on the information that can be stored by physical objects. In this work, we test this interpretation by asking whether the Bekenstein bound imposes constraints on a channel's communication capacity, a context in which information can be given a mathematically rigorous and operationally meaningful definition. We first derive a bound on the accessible information and demonstrate that the Bekenstein bound constrains the decoding instead of the encoding. Then we study specifically the Unruh channel that describes a stationary Alice exciting different species of free scalar fields to send information to an accelerating Bob, who is therefore confined to a Rindler wedge and exposed to the noise of Unruh radiation. We show that the classical and quantum capacities of the Unruh channel obey the Bekenstein bound. In contrast, the entanglement-assisted capacity is as large as the input size even at arbitrarily high Unruh temperatures. This reflects that the Bekenstein bound can be violated if we do not properly constrain the decoding operation in accordance with the bound. We further find that the Unruh channel can transmit a significant number of zero-bits, which are communication resources that can be used as minimal substitutes for the classical/quantum bits needed for many primitive information processing protocols, such as dense coding and teleportation. We show that the Unruh channel has a large zero-bit capacity even at high temperatures, which underpins the capacity boost with entanglement assistance and allows Alice and Bob to perform quantum identification. Therefore, unlike classical bits and qubits, zero-bits and their associated information processing capability are not constrained by the Bekenstein bound. (This talk is based on the recent work (https://arxiv.org/abs/2309.07436) with Patrick Hayden.)

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Zoom link: https://pitp.zoom.us/j/98778081764?pwd=WktjNU84R3NWRXNyVmt1eDVMK2JnUT09

Abstract TBA

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Zoom link: https://pitp.zoom.us/j/97133792880?pwd=d1pxMThTd1R3YjZUVENiOWRLOUZpZz09

I will introduce a new picture of quantum dynamics that might be thought of as "gauging" Schrodinger's picture that results in many "local" Hilbert spaces [1]. Truncating the dimensions of the local Hilbert spaces in this new picture yields an exciting new kind of tensor network whose computational cost does not increase with increasing spatial dimension (for fixed bond dimension) [2]. More detail: Although quantum dynamics are local for local Hamiltonians, the locality is not explicit in the Schrodinger picture since the wavefunction amplitudes do not obey a local equation of motion. In the first part of this talk, I will introduce a new picture of quantum dynamics—the gauge picture—which is similar to Schrodinger's picture, but with the feature that spatial locality is explicit in the equations of motion. In a sense, the gauge picture might be thought of as the result of "gauging" the global unitary symmetry of quantum dynamics into a local symmetry[1]. In the second part of the talk, I discuss a new kind of tensor network ansatz that is inspired from the gauge picture. In the gauge picture, different regions of space are associated with different Hilbert spaces, which are related by gauge connections. By relaxing the unitary constraint on the gauge connections, we can truncate the Hilbert space dimensions associated with different regions to obtain an approximate description of quantum dynamics. This truncated gauge picture, which we dub "quantum gauge network", is intriguingly similar to a classical lattice gauge theory coupled to a Higgs field (which are "local" wavefunctions in the gauge picture), but with non-unitary connections. In one spatial dimension, a quantum gauge network can be easily mapped to a matrix product density operator, and a matrix product state can be mapped to a quantum gauge network. Unlike tensor networks such as PEPS, quantum gauge networks boast the advantage that for fixed bond dimension, the computational cost does not increase with the number of spatial dimensions! Encoding fermionic wavefunctions is also remarkably straightforward. We provide a simple algorithm for approximately simulating quantum dynamics of bosonic or fermionic Hamiltonians in any spatial dimension. We compare the new quantum dynamics algorithm to exact methods for fermion systems in up to three spatial dimensions [2]. [1] The Gauge Picture of Quantum Dynamics. arXiv:2210.09314 [2] Quantum Gauge Networks: A New Kind of Tensor Network. arXiv:2210.12151

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Zoom link: https://pitp.zoom.us/j/94596192271?pwd=MytzNUx4ZEZEemkvcEEzbllWM1J6QT09

The onset of the formation of structure in the early universe was marked by the monolithic collapse of smooth peaks in the initial density field. This process creates prompt rho ~ r^-1.5 density cusps of dark matter, which persist largely unaltered through the subsequent growth of dark matter halos around them. Consequently, in the standard collisionless dark matter paradigm, these prompt cusps are expected to be enormously abundant, and one resides at the center of every halo and subhalo. Prompt cusps present new opportunities to test the nature of dark matter. In annihilating dark matter models, the abundance of these features and the high density inside them greatly influence the intensity and morphology of the annihilation signal. For example, if the Galactic Center gamma-ray excess is due to annihilating dark matter, then a matching signal from unresolved prompt cusps should be detectable elsewhere. Moreover, the properties of prompt cusps are closely linked to details of the primordial density field. In warm dark matter models, prompt cusps are expected to be large enough to influence stellar motions within galaxies at detectable levels.

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Zoom link: https://pitp.zoom.us/j/98307421845?pwd=V3BqZmtyQ09XcjBwNEltTzFPTHJPUT09