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.

Semiclassical approach to fidelity amplitude

“…As the dimension of U i is increased, the largest eigenvalue (in modulus) tends to the unit circle. In the limit of infinite dimensional matrices, all the spectrum is contained in the unit disc [43,47]. We have not attempted an analytical study of the spectral properties of random matrices with the baker structure.…”

“…Secondly, even for completely random perturbations, if M is finite dimensional, then the leading eigenvalue will be in general outside the unit circle. It is true that as the dimension N of M is increased, the leading eigenvalue tends to the unit circle, but it does so very slowly, possibly like N −1/3 [43]. So, in practice, some finite deviations from the Lyapunov decay rate should not be ruled out.…”

“…As the dimension of U i is increased, the largest eigenvalue (in modulus) tends to the unit circle. In the limit of infinite dimensional matrices, all the spectrum is contained in the unit disk [43,47].…”

“…The transfer matrix method is a standard tool in statistical mechanics for the calculation of the partition function of Ising lattices [37][38][39]. Due to the analogy between Ising partition functions (for imaginary temperature) and semiclassical periodic orbit sums, the method has also found applications in the quantum chaos domain [40][41][42][43].…”