Quenching, Fast and Slow: Breaking Kibble-Zurek Universal Scaling by Jumping along Geodesics

TL;DR

The paper tackles the adiabatic quantum computing limitation imposed by the energy-gap constraint, where traditional runtimes scale as $T \propto 1/(\Delta E)^2$. It introduces a geodesic-based strategy, including a discrete geo-jump protocol using $\pi$ pulses, to fast-forward dynamics without adding extra Hamiltonians. Analytically and numerically, the authors demonstrate a rate-independent defect plateau (DRIP) in the 1d XY model and related Ising-like systems, with the leading DRIP defect density given by $n_{\rm defect}^{\rm geo.jump} \simeq \frac{\pi^4}{32}\,\gamma^2\,(h_f-h_i)^2$. They also derive a generalized adiabatic condition and connect geodesic control to the Fubini–Study metric and quantum speed limits, outlining potential experimental implementations and limitations. The findings reveal a new universality class of nonequilibrium dynamics beyond Kibble–Zurek and anti-KZ scenarios, with implications for fast quantum control and robust state preparation under minimal excitation.

Abstract

A major drawback of adiabatic quantum computing (AQC) is fulfilling the energy gap constraint, which requires the total evolution time to scale inversely with the square of the minimum energy gap. Failure to satisfy this condition violates the adiabatic approximation, potentially undermining computational accuracy. Recently, several approaches have been proposed to circumvent this constraint. One promising approach is to use the family of adiabatic shortcut procedures to fast-forward AQC. One caveat, however, is that it requires an additional Hamiltonian that is very challenging to implement experimentally. Here, we investigate an alternate pathway that avoids any extra Hamiltonian in the evolution to fast-forward the adiabatic dynamics by traversing geodesics of a quantum system. We find that jumping along geodesics offers a striking mechanism to highly suppress the density of excitations in many-body systems. Particularly, for the spin-$1/2$ XY model, we analytically prove and numerically demonstrate a rate-independent defect plateau, which contrasts with well-established results for the Kibble-Zurek and anti-Kibble-Zurek mechanisms.

Figures (9)

• Figure 1: (a) An artistic impression of a quantum geodesic trajectory. The dashed arrows indicate the shortest paths, geodesics, in a complex projective space endowed with a Hermitian form. When working in a parametrization, here, $x$'s are associated with a quantum system's external control parameters, and $s$ is associated with the evolution time $t$. $\Gamma$'s are the Christoffel symbols that are functions of Fubini-Study metrics: $g_{\alpha \beta}$. A train of pulses indicates the proposed means to traverse the geodesic to minimize quantum excitations in a relatively short evolution time without needing to fulfill the standard adiabatic theorem. Two quantum systems we studied in this paper. (b) A Stern-Gerlach type experiment where one would be able to observe the Landau-Zener system by varying the external magnetic field. (c) XY quantum spin model with $N$ quantum spins with periodic boundary conditions and external field $h_z$.
• Figure 2: Unitary evolution under the LZ Hamiltonian, Eq.\ref{['eq:LZ_Ham']}. The initial state is the ground state of $H_{LZ}(x_i)$ and the final target state is the ground state of $H_{LZ}(x_f)$, using three different dynamical strategies: lin, geo and geo-jump, accordingly. Fidelity means the overlap squared between the target final state and the instantaneous state evolved under the respective strategies. (a-c) corresponds to three different total evolution times from large to small ones. (d) shows the instantaneous energy gap between the ground and first excited states for the lin strategy ($\Delta E_l$) and the geo strategy ($\Delta E_g$), for the specific total evolution time seen in (b). The explicit dependence on $\Delta E$ is clearly seen in all the figures (a-c). We note that $\Delta E_l$ in (a) is bigger than $\Delta E_l$ in (b) and (c). Thus, we see a small bump in the lin fidelity in (a) around $T\approx 11$ while such signal is missing in both (b-c). The geo strategy in (a-b) works well due to the presence of a much larger energy gap $\Delta E_g$ as compared to $\Delta E_l$. However, as we reduce the total evolution time in (c), the geo strategy does not give rise to near unit fidelity value. Geo-jump strategy works across three different cases presented here. As the green lines suggest, it is discretely sampling the continuous fidelity curve generated by the geo strategy. The locations of the fidelity jump correspond to where $\pi$ pulse is being applied.
• Figure 3: Clockwise, from top to bottom: fidelity plots vs time, real and imaginary components of the dynamical phase difference vs time, error (Eq.\ref{['eq:new_condition']}) vs time and instantaneous energy gap of the LZ system vs time, are presented here. Refer to the main text for the detailed descriptions. Here, $\ket{\psi(t_f)}$ refers to the ground state of the final Hamiltonian. $\ket{\psi(t)}$ refers to the instantaneous time evolved states and $\ket{E_1 (t)}$ corresponds to the instantaneous excited state, obtained by direct diagonalization of the Hamiltonian at that time instance.
• Figure 4: Density of excitations/defects $n_{ex}$ generated along $N=250$ spins $1d$$XY$ model is plotted against quench rate $\nu$ in log-log plot. The canonical $XY$ model is very rich in physics and there are many interesting critical points based on the choice of system parameters. (a) corresponds to the anisotropy line with $|h|< |J|$ with varying $\gamma(t)$ from $\gamma_i = -1$ and $\gamma_f =1$. (b) is the gapless line with $h/J =1$ with varying $\gamma(t)$ from $\gamma_i = -1$ and $\gamma_f =1$. (c) is the Ising line with fixed $\gamma =1$ and varying $h(t)$ from $h_i = 10$ to $h_f =0.$ In all three cases, we recover the critical exponents in linear ramps. From the figures, it is apparent that density of excitations does not depend on the quench rate $\nu$ in all cases.
• Figure 5: $p_k$ vs $k$ plots. (a) Linear strategy (b) Geo strategy (c) Geo-jump strategy for the XY model with $\gamma_i =-1, \gamma_f=1, h=0.5$ for three different total evolution time or quench rate ($v=1/T$). Here, $T=[0.5, 1, 5]$ stands for the blue, red, and green lines. In our numerics, we use a square-pulse width equal to the time vector spacing $\delta t$, namely, $\Delta t = dt=0.001\,({\rm a.u.})$. The latter assures the pulses behave like single-sample kicks.

Theorems & Definitions (1)

• Proposition 1