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Many recent advances may be viewed as controlling the expansion of a graph, a group, or a manifold, by local quantities which depend only on small pieces of it. For instance: Eldan’s stochastic localization process is a way to control the Poincare constant...
In quantum state learning, one is given samples from a quantum state, and the goal is to output an estimate which is close to the quantum state. Quantum state learning is a fundamental problem in both theory and practice, and the last 10 years have seen substantial progress in the design of sample-optimal algorithms for this task.
In this talk, I will give a survey of recent results in this area, with a focus on a recent set of unbiased estimators for quantum state learning which I have developed with my collaborators. These unbiased estimators have allowed us to give a unified and conceptually simpler framework for achieving optimal sample complexities for a number of important quantum state learning applications.
Based on joint work with Angelos Pelecanos, Thilo Scharnhorst, Jack Spilecki, Ewin Tang, and Mark Zhandry.
Isomorphism of tensors is a basic notion in algebraic complexity. Recently, Grochow and Qiao uncovered connections between tensor isomorphism and isomorphism problems for groups, algebras, and polynomials, leading to the formulation of the Tensor Isomorphism complexity class. By varying the groups, actions, and underlying fields or rings, tensor isomorphism problems give rise to many connections and exhibit a range of complexity behaviours. For example:
1. Over finite fields, tensor isomorphism, like Graph Isomorphism, lies in NP ∩ coAM. Complexity reductions around tensor isomorphism play a key role in recent progress on Group Isomorphism (Ivanyos–Mendoza–Qiao–Sun–Zhang, FOCS 2024).
2. Over C, tensor isomorphism is in AM (Koiran, J. Complex.). Tensor isomorphism under unitary and orthogonal group actions naturally appears as an orbit-closure intersection problem.
3. Over Z, tensor isomorphism is decidable (Grunewald–Segal, Ann. Math., 1980). The 2×2×2 case over Z is central to Bhargava’s formulation of higher Gauss composition laws (Ann. Math., 2004), which in turn leads to a quantum polynomial-time algorithm via Hallgren’s algorithm for principal ideal testing (J. ACM, 2007).
Based on joint works with Josh Grochow, Kate Stange, Xiaorui Sun.
The purpose of this event is for participants to work together intensively in a group on a specific problem for one day. (1) We will begin with short 10min talks in the morning by volunteers each explaining a specific open problem to work on. The problem...
In this talk I will discuss covering numbers of real algebraic varieties and applications to computational mathematics. Specifically, we control the number of ell_2 balls of radius epsilon needed to cover a real variety, image of a polynomial map, and semialgebraic set in Euclidean space, in terms of the degrees of the relevant polynomials and number of variables. The bound improves upon the best known general bound, and its proof is much more straightforward. The result is applicable to computational problems in linear and multilinear algebra, and I will discuss consequences for CP tensor decomposition as well as randomized (polynomial) optimization.
Beyond linear spaces and Riemannian manifolds, geodesic metric spaces of nonpositive curvature, also known as Hadamard spaces when complete, provide a suitable framework for developing convex optimization. Familiar examples include Hilbert spaces and complete simply connected Riemannian manifolds of nonpositive sectional curvature, but also other spaces that are not manifolds such as CAT(0) cubical complexes. In the particular latter setting, by decomposing into Euclidean cubes, geodesically convex optimization problems can be addressed by appropriately applying some standard Euclidean convex optimization techniques. In general Hadamard spaces, geodesically convex optimization methods based on proximal steps have been developed in recent years, but they are rarely implementable. While Riemannian subgradient algorithms make use of tangent space constructions and local linearization, understanding subgradients and the algorithms relying on them is less straightforward in Hadamard spaces. By using horoballs and Busemann functions, we explore subgradient-style methods for objectives satisfying convexity conditions that are distinct from geodesic convexity. The algorithms are framed within the underlying space and the derived complexity bounds match those typically obtained in Euclidean space.
Spectral gaps play a fundamental role in many areas of mathematics, computer science, and physics. In quantum mechanics, the spectral gap of Schrödinger operators has a long history of study due to its physical relevance, while in quantum computing spectral gaps are an important proxy for efficiency, such as in the quantum adiabatic algorithm. Motivated by convex optimization, we study Schrödinger operators associated with self-concordant barriers over convex domains and prove non-asymptotic lower bounds on the spectral gap for this class of operators. Significantly, we find that the spectral gap does not display any condition-number dependence when the usual Laplacian is replaced by the Laplace--Beltrami operator, which uses second-order information of the barrier and hence can take the curvature of the barrier into account. As an algorithmic application, we construct a novel quantum interior point method that applies to arbitrary self-concordant barriers and shows no condition-number dependence. To achieve this, we combine techniques from semiclassical analysis, convex optimization, and quantum annealing.