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

TL;DR

This paper develops transfer-matrix and quantum circuit methods to analyze semiclassical traces of the baker map, revealing nonunitarity aspects and enabling long-time calculations beyond traditional Gutzwiller summation.

Contribution

It introduces quantum circuit representations for transfer matrices of the baker map and its symmetry-reflected version, and proposes truncation schemes for long-time semiclassical trace calculations.

Findings

Quantum circuit representations exhibit typical qubit baker structure.

Nonunitarity is confined to a single qubit operator.

Truncation schemes enable long-time trace computation.

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.