Realizing Unitary $k$-designs with a Single Quench

Abstract

We present a single-quench protocol that generates unitary $k$-designs with minimal control. A system first evolves under a random Hamiltonian $H_1$; at a switch time $t_s \geq t_{\mathrm{Th}}$ (the Thouless time), it is quenched to an independently drawn $H_2$ from the same ensemble and then evolves under $H_2$. This single quench breaks residual spectral correlations that prevent strictly time-independent chaotic dynamics from forming higher-order designs. The resulting ensemble approaches a unitary $k$-design using only a single control operation -- far simpler than Brownian schemes with continuously randomized couplings or protocols that apply random quenches at short time intervals. Beyond offering a direct route to Haar-like randomness, the protocol yields an operational, measurement-friendly definition of $t_{\mathrm{Th}}$ and provides a quantitative diagnostic of chaoticity. It further enables symmetry-resolved and open-system extensions, circuit-level single-quench analogs, and immediate applications to randomized measurements, benchmarking, and tomography.

Figures (15)

• Figure 1: Single-quench protocol for generating unitary $k$-designs. The system evolves under $H_1$ for $0<t\leq t_s$. At the switch time $t=t_s$, a sudden quench changes the Hamiltonian to $H_2$, under which it evolves for $t_s < t \leq T$. Despite requiring only one control action, the protocol retains the design-forming power of Brownian schemes and realizes a unitary $k$-design when $t_s\geq t_{\mathrm{Th}}$(the Thouless time).
• Figure 2: Second-order frame potential $F^{(2)}$ versus total evolution time for single-quench dynamics of the complex-fermion $\mathrm{SYK}_4$ in a fixed charge sector (blue solid) and the GUE random-matrix ensemble (red dashed). Parameters: SYK—$N=8$, $q=4$, $J=1$; GUE—Hilbert-space dimension $N_{\mathrm{dim}}=70$. Curves are averages over $4000$ disorder realizations. The Haar benchmark $F^{(2)}=2$ is shown as a black dotted line. Gray dashed vertical lines mark the switch time $t_s$ in each panel: (a) $J t_s=0.2$ (before the Thouless time $t_{\mathrm{Th}}$), (b) $J t_s=2.0$ (near $t_{\mathrm{Th}}$), and (c) $J t_s=100.0$ (after $t_{\mathrm{Th}}$). At late times, the $F^{(2)}$ plateau lies above the Haar value for $t_s<t_{\mathrm{Th}}$ and saturates the Haar value for $t_s\approx t_{\mathrm{Th}}$ and $t_s>t_{\mathrm{Th}}$.
• Figure 3: Gap $\Delta_{\mathrm{FP}}(t_s)$ between the late-time plateau of the first-order frame potential $F^{(1)}$ and its Haar value for single-quench evolution of the complex SYK$_4$ model in a fixed charge sector, plotted versus the switch time $t_s$. Parameters: $N=6$, total charge $q=2$, $J=1$; averaged over $N_{\mathrm{rand}}=10^4$ disorder realizations. The Thouless time $t_{\mathrm{Th}}$ is defined as the first $t_s$ at which $\Delta_{\mathrm{FP}}(t_s)$ is consistent with zero within the statistical uncertainty (gray dashed line). The uncertainty is estimated as $\sqrt{1/[\sqrt{N_{\mathrm{rand}}}(\sqrt{N_{\mathrm{rand}}}-1)]}\simeq 0.01$.
• Figure 4: Frame potentials $F^{(k)}$ for orders $k=1,2,3,4$ versus total evolution time $Jt$ under the single-quench evolution in (a) complex SYK$_4$ model and (b) complex SYK$_2$ model. Curves for $k=1,2,3,4$ are shown in blue, orange, green, and vermilion, respectively; the Haar benchmarks $F_{\mathrm{Haar}}^{(k)}=k!$ appear as dashed lines of matching colors. The gray dotted vertical line marks the switch time $Jt_s=100$. Parameters: $N=6$ complex fermions in fixed-charge sector $q=2$, $J=1$; averages over $10^4$ disorder realizations. In (a), SYK$_4$ saturates the Haar values up to $k=4$, whereas in (b), SYK$_2$ achieves only the $k=1$ (unitary 1-design) benchmark.
• Figure 5: First-order frame potential $F^{(1)}$ versus total evolution time for single-quench dynamics of the complex-fermion SYK$_2$ model in a fixed charge sector. Parameters: $N=6$, $q=2$, $J=1$; averages over $10^4$ disorder realizations. The Haar-random benchmark $F^{(1)}=1$ is shown as a black dashed curve. The switch time $t_s$ is indicated by a gray dot vertical line in each panel: (a) $Jt_s=1.0$ (before the Thouless time $t_{\mathrm{Th}}$), (b) $Jt_s=10.0$ (near $t_{\mathrm{Th}}$), and (c) $Jt_s=100.0$ (after $t_{\mathrm{Th}}$). At late times, the plateau of $F^{(2)}$ lies above the Haar value for $t_s<t_{\mathrm{Th}}$, while it saturates the Haar value when $t_s\approx t_{\mathrm{Th}}$ or $t_s>t_{\mathrm{Th}}$.