Mattie Ji: A K-theoretic Approach to Quantum Cellular Automata

A K-theoretic Approach to Quantum Cellular Automata

Algebraic K-theory, on a very high level, studies how to break and assemble objects apart linearly, which makes the field amenable to classification questions. In this work, we apply this methodology to study the classification of quantum cellular automata (QCA), which are models for the dynamics of many-particle quantum systems. Although QCAs are traditionally studied over the complex numbers, their underlying locality and tensor-product structures make sense over arbitrary commutative rings.

We first develop an algebraic theory of QCA over any commutative rings. Over a general space X and an arbitrary commutative ring R, we then use algebraic K-theory to construct a space of QCA whose deformation classes either are or refine the QCA classification groups. Furthermore, we show that the spaces of QCA on Euclidean lattices assemble to an $\Omega$-spectrum indexed by the dimension. As a corollary, we also obtain a non-connective delooping of the K-theory of Azumaya R-algebras, so the QCA classification groups may be viewed as a multiplicative version of negative K-theory. This talk is based on joint work with Bowen Yang.