Digitized counterdiabatic quantum critical dynamics

Abstract

We experimentally demonstrate that a digitized counterdiabatic quantum protocol reduces the number of topological defects created during a fast quench across a quantum phase transition. To show this, we perform quantum simulations of one- and two-dimensional transverse-field Ising models driven from the paramagnetic to the ferromagnetic phase. We utilize superconducting cloud-based quantum processors with up to 156 qubits. Our data reveal that the digitized counterdiabatic protocol reduces defect formation by up to 48% in the fast-quench regime -- an improvement hard to achieve through digitized quantum annealing under current noise levels. The experimental results closely match theoretical and numerical predictions at short evolution times, before deviating at longer times due to hardware noise. In one dimension, we derive an analytic solution for the defect number distribution in the fast-quench limit. For two-dimensional geometries, where analytical solutions are unknown and numerical simulations are challenging, we use advanced matrix-product-state methods. Our findings indicate a practical way to control the topological defect formation during fast quenches and highlight the utility of counterdiabatic protocols for quantum optimization and quantum simulation in material design on current quantum processors.

Figures (3)

• Figure 1: A schematic illustration of the initial and final states resulting from CD-assisted evolution and digitized annealing without CD.a, The spin system is driven across a phase transition from the paramagnetic to the ferromagnetic phase. Counterdiabatic evolution results in fewer kinks in the magnetization in the final state at time $t = T$. b, The time-dependent factors in the Hamiltonian $\mathcal{H}(\lambda) = H(\lambda) + \dot{\lambda} A_{\lambda}$. The magnitude of the coefficient $|\dot{\lambda}(t) \alpha_1(t)|$ of the CD Hamiltonian is largest at the critical point where excitations have the lowest energy cost and are most likely to occur. In one dimension, the critical point $g = J$ is crossed at $t/T = 0.5$. The scheduling function $\lambda(t) = t/T$ is chosen as linear, and the vertical lines indicate that time is discretized into steps of size $\delta t$. c, The circuit that implements the CD evolution. The colored boxes correspond to a single time step with $t_m = m \delta t$, and omitting the green boxes results in the implementation of digitized annealing. The black squares denote Hadamard gates, and $R_x(\theta)=\exp(-i\frac{\theta}{2} X)$, $R_{zz}(\theta)=\exp(-i\frac{\theta}{2} Z \otimes Z)$, $R_{yz}(\theta)=\exp(-i\frac{\theta}{2} Y \otimes Z)$, and $R_{zy}(\theta)=\exp(-i\frac{\theta}{2} Z \otimes Y)$ are single- and two-qubit gates with the Pauli matrices $X$, $Y$, and $Z$.
• Figure 2: Measured distributions of the defect density at the final evolution time $\bm{T = 0.2/J}$. We consider different geometries: a, a one-dimensional chain of length $N = 100$, b, a 2D heavy-hexagonal lattice of $156$ sites, c, a three-leg ladder of length $N_x = 15$, and d, a square lattice of size $N_x \times N_y = 6 \times 6$. In the presence of CD, the density of defects is reduced on average in all geometries. The histograms correspond to experimental data with 10000 to 20000 samples. The bin width is determined by the number of edges and thus varies. The solid lines in panel a correspond to the exact solution of the 1D transverse-field Ising model (Supplementary Note 4), while for the other geometries, the distributions cannot be computed exactly. The data without CD shows a good agreement with the exact solution. For the counterdiabatic evolution, the data is shifted to larger values due to hardware errors. The dotted lines in each panel are the normal distributions with mean $\kappa_1 = 0.5$ and variance $\kappa_2 = 0.25$ obtained in the initial state $\ket{+}^{\otimes N}$ in the infinite-size limit (Methods).
• Figure 3: Cumulants of the defect density distribution, $\bm{\kappa_1}$, $\bm{\kappa_2}$, and $\bm{\kappa_3}$, as functions of the total evolution time $\bm{T}$ with and without CD. The geometries are as in Fig. \ref{['fig:histograms']}. The markers correspond to experimental data, and the solid lines correspond to a, the exact solutions (Supplementary Note 4) or b-d, to numerical simulations with MPS (Methods). In all cases, the mean value of the density of defects $\kappa_1$ is reduced by CD. The shaded region indicates the crossover to the KZ scaling regime. a, b, The variance $\kappa_2$ of the defect density is reduced by CD while the skewness of the distribution $\kappa_3$ has a more complex behavior. c, d, The variance is slightly increased by CD, and the skewness has negative values, unlike for the 1D and heavy-hexagonal geometries. Due to increasing errors at large $T$, we only include data up to $T = 1/J$. Each experimental data point is an average of between 10000 and 20000 samples. The standard errors are smaller than the marker size.