An official website of the United States government
A promising avenue for the preparation of Gibbs states on a quantum computer is to simulate the physical thermalization process. The Davies generator describes the dynamics of an open quantum system that is in contact with a heat bath. Crucially, it does not require simulation of the heat bath itself, only the system we hope to thermalize. Using the state-of-the-art techniques for quantum simulation of the Lindblad equation, we devise a technique for the preparation of Gibbs states via thermalization as specified by the Davies generator.In doing so, we encounter a severe technical challenge: implementation of the Davies generator demands the ability to estimate the energy of the system unambiguously. That is, each energy of the system must be deterministically mapped to a unique estimate. Previous work showed that this is only possible if the system satisfies an unphysical 'rounding promise' assumption. We solve this problem by engineering a random ensemble of rounding promises that simultaneously solves three problems: First, each rounding promise admits preparation of a 'promised' thermal state via a Davies generator. Second, these Davies generators have a similar mixing time as the ideal Davies generator. Third, the average of these promised thermal states approximates the ideal thermal state.
more »
« less
Davies’ method for anomalous diffusions
Davies’ method of perturbed semigroups is a classical technique to obtain off-diagonal upper bounds on the heat kernel. However Davies’ method does not apply to anomalous diffusions due to the singularity of energy measures. In this note, we overcome the difficulty by modifying the Davies’ perturbation method to obtain sub-Gaussian upper bounds on the heat kernel.
more »
« less
- PAR ID:
- 10056857
- Date Published:
- Journal Name:
- Proceedings of the American Mathematical Society
- ISSN:
- 0002-9939
- Page Range / eLocation ID:
- 1
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Abstract We present residual-based a posteriori error estimates of mixed finite element methods for the three-field formulation of Biot’s consolidation model. The error estimator are upper and lower bounds of the space-time discretization error up to data oscillation. As a by-product, we also obtain a new a posteriori error estimate of mixed finite element methods for the heat equation.more » « less
-
null (Ed.)We study the sub-Laplacian of the 15-dimensional unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the octonionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of a related sub-Laplacian. As a byproduct we also obtain the spectrum of the sub-Laplacian, the small-time asymptotics of the heat kernel and explicitly compute the sub-Riemannian distance.more » « less
-
Abstract We obtain optimal estimates of the Poincaré constant of central balls on manifolds with finitely many ends. Surprisingly enough, the Poincaré constant is determined by thesecondlargest end. The proof is based on the argument by Kusuoka–Stroock where the heat kernel estimates on the central balls play an essential role. For this purpose, we extend earlier heat kernel estimates obtained by the authors to a larger class of parabolic manifolds with ends.more » « less
-
The level set estimation problem seeks to find all points in a domain where the value of an unknown function 𝑓:→ℝ exceeds a threshold 𝛼 . The estimation is based on noisy function evaluations that may be acquired at sequentially and adaptively chosen locations in . The threshold value 𝛼 can either be explicit and provided a priori, or implicit and defined relative to the optimal function value, i.e. 𝛼=(1−𝜖)𝑓(𝐱∗) for a given 𝜖>0 where 𝑓(𝐱∗) is the maximal function value and is unknown. In this work we provide a new approach to the level set estimation problem by relating it to recent adaptive experimental design methods for linear bandits in the Reproducing Kernel Hilbert Space (RKHS) setting. We assume that 𝑓 can be approximated by a function in the RKHS up to an unknown misspecification and provide novel algorithms for both the implicit and explicit cases in this setting with strong theoretical guarantees. Moreover, in the linear (kernel) setting, we show that our bounds are nearly optimal, namely, our upper bounds match existing lower bounds for threshold linear bandits. To our knowledge this work provides the first instance-dependent, non-asymptotic upper bounds on sample complexity of level-set estimation that match information theoretic lower bounds.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/proc/13324
