Transfer Matrices and Circuit Representation for the Semiclassical Traces of the Baker Map

Abstract

Because of a formal equivalence with the partition function of an Ising chain, the semiclassical traces of the quantum baker map can be calculated using the transfer-matrix method. We analyze the transfer matrices associated with the baker map and the symmetry-reflected baker map (the latter happens to be unitary but the former is not). In both cases simple quantum-circuit representations are obtained, which exhibit the typical structure of qubit quantum bakers. In the case of the baker map it is shown that nonunitarity is restricted to a one-qubit operator (close to a Hadamard gate for some parameter values). In a suitable continuum limit we recover the already known infinite-dimensional transfer-operator. We devise truncation schemes allowing the calculation of long-time traces in regimes where the direct summation of Gutzwiller's formula is impossible. Some aspects of the long-time divergence of the semiclassical traces are also discussed.