Quantum critical dynamics and emergent universality in decoherent digital quantum processors

TL;DR

The work investigates how decoherence, modeled as continuous quantum nondemolition measurements of the instantaneous Hamiltonian, reshapes the quantum Kibble-Zurek scaling during quenches through a quantum critical point in the one-dimensional transverse-field Ising model. It combines analytical, numerical, and experimental (IBM quantum hardware) approaches to reveal that universal dynamical scaling can persist under decoherence, but with exponents that differ from ideal QKZ predictions and from simplified noise models, indicating a distinct noise-influenced universality class. The study uses a density-operator Bloch-ball framework to compute equal-time correlators and spin correlations, and employs data-collapse analyses to extract scaling exponents, observing crossovers from noiseless to strongly decoherent regimes. The findings suggest universal dynamical scaling as a high-level descriptor of quantum hardware, offering a complementary perspective to gate-level metrics and motivating future two-dimensional explorations and hardware-based universality classifications.

Abstract

Understanding how noise influences nonequilibrium quantum critical dynamics is essential for both fundamental physics and the development of practical quantum technologies. While the quantum Kibble-Zurek (QKZ) mechanism predicts universal scaling during quenches across a critical point, real quantum systems exhibit complex decoherence that can substantially modify these behaviors, ranging from altering critical scaling to completely suppressing it. By considering a specific case of nondemolishing noise, we first show how decoherence can reshape universal scaling and verify these theoretical predictions using numerical simulations of spin chains across a wide range of noise strengths. Then, we study linear quenches in the transverse-field Ising model on IBM superconducting processors where the noise model is unknown. Using large system sizes of 80-120 qubits, we measure equal-time connected correlations, defect densities, and excess energies across various quench times. Surprisingly, unlike earlier observations where noise-induced defect production masked universal behavior at long times, we observe clear scaling relations, pointing towards persistent universal structure shaped by decoherence. The extracted scaling exponents differ from both ideal QKZ predictions and analytic results for simplified noise models, suggesting the emergence of a distinct noise-influenced universality regime. Our results, therefore, point toward the possibility of using universal dynamical scaling as a high-level descriptor of quantum hardware, complementary to conventional gate-level performance metrics.

Figures (8)

• Figure 1: Digital quantum simulations of the quantum Kibble-Zurek mechanism. (a) Schematic of a continuous linear quench of the one-dimensional transverse-field Ising model (TFIM). The Hamiltonian is ramped according to $J(t) = ( 1 + t / \tau_Q )$ and $h(t) = ( 1 - t/\tau_Q )$, with $\tau_Q$ denoting the characteristic quench time. The evolution starts from the paramagnetic phase at $t = -\tau_Q$, crosses the quantum critical point at $t=0$, and reaches the Ising Hamiltonian at $t=\tau_Q$. (b) Near the critical point, the control parameter becomes vanishingly small, $|\epsilon(t)| \propto |t / \tau_Q |$ as $t \to 0$ (dashed red line), and the correlation time, $\xi_t \propto |\epsilon|^{-z \nu}$, diverges (solid blue line). The Kibble-Zurek freeze-out regime associated with the quench ramp is indicated by the light-blue shaded region. In this regime, the correlation length, $\xi \propto |\epsilon|^{-\nu}$, grows rapidly, but as a result of the critical slowing down finite-sized domains become effectively frozen and the system loses its ability to suppress defect formation. (c) An example qubit layout on the ibm_fez Heron processor for a chain with 120 sites. (d) Trotterization of the quench protocol for the TFIM. The unitary time-evolution operator is implemented using a first-order Suzuki-Trotter decomposition with discrete time step $dt$. The on-site transverse-field term maps to a layer of single-qubit $R_X$ rotation gates, while the nearest-neighbor Ising interaction is realized using a standard construction consisting of an $R_Z$ rotation gate sandwiched between two CNOT gates. To reduce qubit idling and enable parallel execution, the interaction gates are applied in sequential odd and even sublayers.
• Figure 2: Local magnetization and connected correlations during a quench of the transverse-field Ising model. Experimental measurements and theoretical simulations are shown for a quench from the paramagnetic ground state into the ferromagnetic phase, implemented using $N=20$ qubits and 16 Trotter steps of duration $dt=0.25$ ($-2 \le t \le 2$). (a) Measurements along the $x$ axis yield the local magnetization $\langle \sigma_i^x \rangle_t$ at each discrete time step (blue squares). The experimental data closely follow the theoretical evolution of a closed system undergoing the Trotterized quench (purple circles), with residual deviations attributable to hardware noise. (b) Measurements along the $z$ axis yield the equal-time connected spin--spin correlation function $|C(t,x)|$. Experimental results are shown in the right panel and zero-noise theoretical simulations in the left panel. The dashed white lines indicate the light-cone trajectories for the spread of correlations, computed in the thermodynamic limit for the continuous quench protocol shown in Fig. \ref{['fig:Experimental_setup']}(a).
• Figure 3: Scaling behavior at the quantum critical point under quantum nondemolition noise. Numerical simulations of $N=512$ spins subject to the continuous quench protocol to the QCP, shown in \ref{['fig:Experimental_setup']}(a), for various quench times and QND noise strength values. Rows (a)-(c) show results after solving \ref{['eq:Master_eq']} for quenches performed with $\lambda=0,1,100$ respectively. The correlation function, \ref{['eq:site_averaged_connected_2pt_fn']}, is depicted in column 1 (values below $5\times 10^{-4}$, as well as the $x=0$ point, are masked throughout). In column 2 we show RMSE results for different candidate scaling exponents using the search protocol described in the main text. The scaling exponents predicted by the standard (noiseless) QKZ mechanism are marked with white stars and those found by the search algorithm under decoherence strength $\lambda$ are marked with red diamonds. Using the exponent pair with the lowest RMSE, in column 3 we present the rescaled correlation function data, $C(0,x) \tau_Q^{b_\mathrm{fit}}$ vs $x / \tau_Q^{a_\mathrm{fit}}$. In the two extreme limits of zero- and large-QND noise strength, rows (a) and (c) respectively, the exponent pair $(a,b)$ that corresponds to the lowest RMSE converge (within the grid precision of $0.025$) to the theoretical predictions. In row (a), we obtain $(a_\mathrm{fit},b_\mathrm{fit})=(0.5, 0.125)$ in agreement with the QKZ hypothesis and in row (c) we get $(a_\mathrm{fit},b_\mathrm{fit})=(0.325, 0.075)$ which are close to the scaling exponents $(a_\mathrm{QND},b_\mathrm{QND})=(0.333\dots, 0.083\dots)$ extracted from \ref{['eq:QND_freezeout']}. In both cases, there is a noticeable data collapse in the rescaled correlation functions. (b) At intermediate QND noise strengths, where scaling relations are not predicted to hold, the fitted exponents do not yield a clear data collapse.
• Figure 4: Scaling behavior at the quantum critical point in digital quantum computers. (a) Numerical simulations of $N=120$ spins undergoing a Trotterized quench to the QCP for various quench times ($\tau_Q = d t \cdot \mathrm{steps}$ with $d t = 0.2$). (b) and (c) Digital quantum simulations of the Trotterized quenches in (a) performed on ibm_fez using 120 and 100 qubits respectively. Similar to \ref{['fig:QND_QKZ']}, column 1 shows the correlation function data as function of distance, column 2 the search for scaling exponents (on a grid with spacing $0.025$), and column 3 the rescaled correlation function data using the best-fit exponent pair. Because of statistical shot noise, correlation function data below $10^{-3}$ is omitted, as is the $x=0$ point where QKZ scaling does not apply. (a) In the theory simulation, Trotterization and finite-size effects lead to small deviations between the fitted scaling exponents $(a_\mathrm{fit},b_\mathrm{fit})=(0.45, 0.15)$ (red diamond) from QKZ theory $(a_\mathrm{QKZ},b_\mathrm{QKZ})=(0.5, 0.125)$ (white star). (b)-(c) Using IBM Quantum hardware, we observe significantly stronger deviations from QKZ scaling predictions due to noise. Rather than correlations increasing with longer quench times, the accumulation of gate errors causes noise to dominate, leading to a suppression of correlations. As expected, the extracted best-fit scaling exponents differ markedly from the QKZ hypothesis yielding $(0.025,\,0.475)$ for $N=120$ and $(0.025,\,0.325)$ for $N=100$. However, despite the deviation, the fitted curves still exhibit a data collapse, consistent with the emergence of universal scaling behavior shaped by quantum decoherence.
• Figure 5: Magnetization, defect density, and excess energy following a full quench of the transverse-field Ising model. Digital quantum simulations were performed using $N=80, 100,$ and $120$ qubits on ibm_fez. The system is measured at the end of the quench, $t=\tau_Q$, while varying the total number of Trotter steps between 8 and 32 at fixed step size $dt=0.25$. (a,b) Local magnetization profiles measured along the $x$ and $z$ axes, respectively, for different quench times in the $N=100$-qubit chain. The experimental data reveal deviations from translational invariance induced by hardware noise, as well as spatial structure in the longitudinal magnetization consistent with the formation of finite-sized domains. (c,d) Defect density $n_{\mathrm{def}}$ and excess energy $\varepsilon_{\mathrm{exc}}$ as functions of the quench time $\tau_Q$ for different system sizes (blue upward-pointing triangles). Power-law fits yield $n_{\mathrm{def}}(\tau_Q)\propto \tau_Q^{-\beta}$ with $\beta \approx - 0.3$ and $\varepsilon_{\mathrm{exc}}(\tau_Q)\propto \tau_Q^{-\gamma}$ with $\gamma \approx - 0.6$. These results are contrasted with theoretical simulations of the same Trotterized quench in a noiseless system (purple downward-pointing triangles), for which the defect density closely follows $n_{\mathrm{def}}(\tau_Q)\propto \tau_Q^{-1/2}$, in agreement with the exact result for a linear quench in the thermodynamic limit.