Abstract
Whether one is interested in quantum state preparation or in the design of efficient heat engines, adiabatic (reversible) transformations play a pivotal role in minimizing computational complexity and energy losses. Understanding the structure of these transformations and identifying the systems for which such transformations can be performed efficiently and quickly is therefore of primary importance. In this paper we focus on finding optimal paths in the space of couplings controlling the system's Hamiltonian. More specifically, starting from a local Hamiltonian we analyze directions in the space of couplings along which adiabatic transformations can be accurately generated by local operators, which are both realizable in experiments and easy to simulate numerically. We consider a non-integrable 1D Ising model parametrized by two independent couplings, corresponding to longitudinal and transverse magnetic fields. We find regions in the space of couplings characterized by a very strong anisotropy of the variational adiabatic gauge potential (AGP), generating the adiabatic transformations, which allows us to define optimal adiabatic paths. We find that these paths generally terminate at singular points characterized by extensive degeneracies in the energy spectrum, splitting the parameter space into adiabatically disconnected regions. The anisotropy follows from singularities in the AGP, and we identify special robust weakly-thermalizing and non-absorbing many-body "dark" states which are annihilated by the singular part of the AGP and show that their existence extends deep into the ergodic regime.
Highlights
With the rapid progress of quantum technologies, the design of efficient protocols to control and numerical methods to describe quantum systems quickly moved to the forefront of current research
VI and Appendix G, where we discuss the perturbative expansion of the adiabatic gauge potential (AGP) for small values of g, we show that new singularities emerge in correspondence with the degenerate points along the line g = 0 when increasing the support of the variational adiabatic gauge potential (VAGP) ansatz
In this final section we present a derivation of the divergences appearing in the VAGP by developing a perturbative expansion of the exact AGP in small g near g = 0, i.e., near the classical Ising limit, using the integral representation of the AGP given by Eq (15)
Summary
With the rapid progress of quantum technologies, the design of efficient protocols to control and numerical methods to describe quantum systems quickly moved to the forefront of current research. Our results can have a broad range of applications in various problems, beyond finding optimal paths for annealing or state preparation They can be used to find efficient local conservation laws and corresponding “most-integrable” directions, to find the nearest integrable (simple) points that are locally connected to a Hamiltonian of interest, to define most efficient ways of obtaining effective low-energy theories starting from a noninteracting model, and so on. We demonstrate that both for conventional adiabatic driving and for the approximate CD protocols state preparation along the optimal paths shows a much better performance than along the orthogonal directions.
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