Understanding how non-adiabatic terms affect quantum dynamics is fundamental to improving various protocols for quantum technologies. We present a novel approach to computing the Adiabatic Gauge Potential (AGP), which gives information on the non-adiabatic terms that arise from time dependence in the Hamiltonian. Our approach uses commutators of the Hamiltonian to build up an appropriate basis of the AGP, which can be easily truncated to give an approximate form when the exact result is intractable. We use this approach to study the AGP obtained for the transverse field Ising model on a variety of graphs, showing how the different underlying graph structures can give rise to very different scaling for the number of terms required in the AGP.
Understanding how non-adiabatic terms affect quantum dynamics is fundamental to improving various protocols for quantum technologies. We present a novel approach to computing the Adiabatic Gauge Potential (AGP), which gives information on the non-adiabatic terms that arise from time dependence in the Hamiltonian. Our approach uses commutators of the Hamiltonian to build up an appropriate basis of the AGP, which can be easily truncated to give an approximate form when the exact result is intractable. We use this approach to study the AGP obtained for the transverse field Ising model on a variety of graphs, showing how the different underlying graph structures can give rise to very different scaling for the number of terms required in the AGP.
Adiabatic gauge potential is the origin of nonadiabatic transitions. In counterdiabatic driving, which is a method of shortcuts to adiabaticity, adiabatic gauge potential can be used to realize identical dynamics to adiabatic time evolution without requiring slow change of parameters. We introduce an algebraic expression of adiabatic gauge potential. Then, we find that the explicit form of adiabatic gauge potential can be easily determined by some algebraic calculations. We demonstrate this method by using a single-spin system, a two-spin system, and the transverse Ising chain. Moreover, we derive a lower bound for fidelity to adiabatic time evolution based on the quantum speed limit. This bound enables us to know the worst case performance of approximate adiabatic gauge potential. We can also use this bound to find dominant terms in adiabatic gauge potential to suppress nonadiabatic transitions. We apply this bound to magnetization reversal of the two-spin system and to quantum annealing of the transverse Ising chain. Adiabatic gauge potential reflects structure of energy eigenstates, and thus we also discuss detection of quantum phase transitions by using adiabatic gauge potential. We find a signature of a quantum phase transition in the transverse Ising chain.
Counterdiabatic driving provides a universal route to fast quantum control, but its standard formulation presupposes a Hamiltonian generator. We develop a Hamiltonian-free theory of the adiabatic gauge potential (AGP) for parameter-dependent unitary operators Uλ. Exact and regularized versions of the AGP follow from a Hermitian, banded linear system at a cost comparable to the Lanczos construction in the Hermitian case, requiring neither diagonalization nor a matrix logarithm. We demonstrate rapid convergence of the truncated Krylov approximation for the kicked top and for an integrable brickwork circuit built from unitary Rˇ matrices, a setting in which no local effective Hamiltonian is available. Our results extend shortcuts to adiabaticity, and AGP-based diagnostics of chaos, to Floquet systems and quantum circuits.
By the adiabatic theorem, the probability of non-adiabatic transitions in a time-dependent quantum system vanishes in the adiabatic limit. The Landau-Zener (LZ) formula gives the leading functional behavior of the probability close to this limit. On the other hand, in counterdiabatic dynamics, one achieves effectively adiabatic evolution at finite driving speed by adding an extra field which suppresses non-adiabatic transitions: the adiabatic gauge potential (AGP). We investigate the mechanism by which the AGP suppresses the transition probability, changing it from the LZ formula to exactly zero. Quantitatively, we find that adding the AGP to the Hamiltonian modifies the LZ formula by a universal prefactor, independent of the adiabatic parameter, which vanishes in the counterdiabatic regime. Qualitatively, this prefactor can be understood as arising from the Aharonov-Bohm phases generated by the AGP between different paths in the complex time plane. Finally, we show that these results extend to a class of integrable time-dependent quantum Hamiltonians by proving that the AGP preserves their integrability condition.
Shortcuts to adiabaticity (STA) reproduce adiabatic dynamics in finite time, but their counterdiabatic implementation relies on the adiabatic gauge potential (AGP), which is difficult to compute and implement in many-body systems and whose extension to open and non-Hermitian settings has remained largely model-specific. Here, we develop a general, diagonalization-free framework for engineering STA in non-Hermitian systems by representing the AGP in Krylov space. Starting from an integral representation of the counterdiabatic control, we recast the AGP as a nested-commutator series with controlled locality and generate the associated Krylov basis using the bi-Lanczos and Arnoldi algorithms. This reduces the exact or truncated AGP to a sparse tridiagonal or upper-Hessenberg matrix equation that generalizes the Hermitian construction. We demonstrate the method on a decaying two-level atom, where it recovers the exact drive and signals the exceptional point; on the interacting Hatano-Nelson model, where truncated controls rapidly suppress nonadiabatic excitations; and on a PT-symmetric Heisenberg chain, whose AGP norm detects the PT-symmetry-breaking transition. Throughout, the expansion converges with only a small fraction of the full Krylov space, offering a practical route to fast, accurate control of many-body non-Hermitian systems.
The Adiabatic Gauge Potential (AGP) is the generator of adiabatic deformations between quantum eigenstates. There are many ways to construct the AGP operator and evaluate the AGP norm. Recently, it was proposed that a Gram-Schmidt-type algorithm can be used to explicitly evaluate the expression of the AGP. We employ a version of this approach by using the Lanczos algorithm to evaluate the AGP operator in terms of Krylov vectors and the AGP norm in terms of the Lanczos coefficients. It has the advantage of minimizing redundancies in evaluating the nested commutators in the analytic expression for the AGP operator. The algorithm is used to explicitly construct the AGP operator for some simple systems. We derive an integral transform relation between the AGP norm and the autocorrelation function of the deformation operator. We present a modification of the Variational approach to derive the regulated AGP norm. Using this, we approximate the AGP to varying degrees of success. Finally, we compare and contrast the quantum-chaos-probing capacities of the AGP and K-complexity in view of the Operator Growth Hypothesis.
In the last few decades, the study of many-body quantum systems far from equilibrium has risen to prominence, with exciting developments on both experimental and theoretical physics fronts. In this dissertation, we will focus particularly on the adiabatic gauge potential (AGP), which is the generator of adiabatic deformations between quantum eigenstates and also related to "fidelity susceptibility", as our lens into the general phenomenon. In the first two projects, the AGP is studied in the context of counter-diabatic driving protocols which present a way of generating adiabatic dynamics at an arbitrary pace. This is quite useful as adiabatic evolution, which is a common strategy for manipulating quantum states, is inherently a slow process and is, therefore, susceptible to noise and decoherence from the environment. However, obtaining and implementing the AGP in many-body systems is a formidable task, requiring knowledge of the spectral properties of the instantaneous Hamiltonians and control of highly nonlocal multibody interactions. We show how an approximate gauge potential can be systematically built up as a series of nested commutators, remaining well-defined in the thermodynamic limit. Furthermore, the resulting counter-diabatic driving protocols can be realized up to arbitrary order without leaving the available control space using tools from periodically-driven (Floquet) systems. In the first project, this driving protocol was successfully implemented on the electronic spin of a nitrogen vacancy in diamond as a proof of concept and in the second project, it was extended to many-body systems, where it was shown the resulting Floquet protocols significantly suppress dissipation and provide a drastic increase in fidelity. In the third project, the AGP is studied in the context of quantum chaos wherein it is found to be an extremely sensitive probe. We are able to detect transitions from non-ergodic to ergodic behavior at perturbation strengths orders of magnitude smaller than those required for standard measures. Using this alternative probe in two generic classes of spin chains, we show that the chaotic threshold decreases exponentially with system size and that one can immediately detect integrability-breaking (chaotic) perturbations by analyzing infinitesimal perturbations even at the integrable point. In some cases, small integrability-breaking is shown to lead to anomalously slow relaxation of the system, exponentially long in system size. This work paves the way for further studies in various areas such as quantum computation, quantum state preparation and quantum chaos.