01855nas a2200145 4500008004100000245007300041210006900114260001500183520138800198100002201586700001901608700002001627700002501647856003701672 2023 eng d00aProjective toric designs, difference sets, and quantum state designs0 aProjective toric designs difference sets and quantum state desig c11/22/20233 a

Trigonometric cubature rules of degree t are sets of points on the torus over which sums reproduce integrals of degree t monomials over the full torus. They can be thought of as t-designs on the torus. Motivated by the projective structure of quantum mechanics, we develop the notion of t-designs on the projective torus, which, surprisingly, have a much more restricted structure than their counterparts on full tori. We provide various constructions of these projective toric designs and prove some bounds on their size and characterizations of their structure. We draw connections between projective toric designs and a diverse set of mathematical objects, including difference and Sidon sets from the field of additive combinatorics, symmetric, informationally complete positive operator valued measures (SIC-POVMs) and complete sets of mutually unbiased bases (MUBs) (which are conjectured to relate to finite projective geometry) from quantum information theory, and crystal ball sequences of certain root lattices. Using these connections, we prove bounds on the maximal size of dense Btmodm sets. We also use projective toric designs to construct families of quantum state designs. Finally, we discuss many open questions about the properties of these projective toric designs and how they relate to other questions in number theory, geometry, and quantum information.

1 aIosue, Joseph, T.1 aMooney, T., C.1 aEhrenberg, Adam1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2311.1347901752nas a2200133 4500008004100000245009000041210007000131260001400201520130500215100002201520700002001542700001901562856003701581 2023 eng d00aOn the stability of solutions to Schrödinger's equation short of the adiabatic limit0 astability of solutions to Schrödingers equation short of the adi c3/23/20233 a

We prove an adiabatic theorem that applies at timescales short of the adiabatic limit. Our proof analyzes the stability of solutions to Schrodinger's equation under perturbation. We directly characterize cross-subspace effects of perturbation, which are typically significantly less than suggested by the perturbation's operator norm. This stability has numerous consequences: we can (1) find timescales where the solution of Schrodinger's equation converges to the ground state of a block, (2) lower bound the convergence to the global ground state by demonstrating convergence to some other known quantum state, (3) guarantee faster convergence than the standard adiabatic theorem when the ground state of the perturbed Hamiltonian (H) is close to that of the unperturbed H, and (4) bound tunneling effects in terms of the global spectral gap when H is ``stoquastic'' (a Z-matrix). Our results apply to quantum annealing protocols with faster convergence than usually guaranteed by a standard adiabatic theorem. Our upper and lower bounds demonstrate that at timescales short of the adiabatic limit, subspace dynamics can dominate over global dynamics. Thus, we see that convergence to particular target states can be understood as the result of otherwise local dynamics.

1 aBringewatt, Jacob1 aJarret, Michael1 aMooney, T., C. uhttps://arxiv.org/abs/2303.1347801652nas a2200133 4500008004100000245005900041210005900100260001500159520124500174100001901419700002201438700002101460856003701481 2021 eng d00aLefschetz Thimble Quantum Monte Carlo for Spin Systems0 aLefschetz Thimble Quantum Monte Carlo for Spin Systems c10/20/20213 a

Monte Carlo simulations are often useful tools for modeling quantum systems, but in some cases they suffer from a sign problem, which manifests as an oscillating phase attached to the probabilities being sampled. This sign problem generally leads to an exponential slow down in the time taken by a Monte Carlo algorithm to reach any given level of accuracy, and it has been shown that completely solving the sign problem for an arbitrary quantum system is NP-hard. However, a variety of techniques exist for mitigating the sign problem in specific cases; in particular, the technique of deforming the Monte Carlo simulation's plane of integration onto Lefschetz thimbles (that is, complex hypersurfaces of stationary phase) has seen success for many problems of interest in the context of quantum field theories. We extend this methodology to discrete spin systems by utilizing spin coherent state path integrals to re-express the spin system's partition function in terms of continuous variables. This translation to continuous variables introduces additional challenges into the Lefschetz thimble method, which we address. We show that these techniques do indeed work to lessen the sign problem on some simple spin systems.

1 aMooney, T., C.1 aBringewatt, Jacob1 aBrady, Lucas, T. uhttps://arxiv.org/abs/2110.10699