Glassy Phase of Optimal Quantum Control

We study the problem of preparing a quantum many-body system from an initial to a target state by optimizing the fidelity over the family of bang-bang protocols. We present compelling numerical evidence for a universal spin-glasslike transition controlled by the protocol time duration. The glassy critical point is marked by a proliferation of protocols with close-to-optimal fidelity and with a true optimum that appears exponentially difficult to locate. Using a machine learning (ML) inspired framework based on the manifold learning algorithm $t$-distributed stochastic neighbor embedding, we are able to visualize the geometry of the high-dimensional control landscape in an effective low-dimensional representation. Across the transition, the control landscape features an exponential number of clusters separated by extensive barriers, which bears a strong resemblance with replica symmetry breaking in spin glasses and random satisfiability problems. We further show that the quantum control landscape maps onto a disorder-free classical Ising model with frustrated nonlocal, multibody interactions. Our work highlights an intricate but unexpected connection between optimal quantum control and spin glass physics, and shows how tools from ML can be used to visualize and understand glassy optimization landscapes.

Figure 1

Bang-bang protocols $h(j\delta t)$ to control a quantum system with high fidelity (a) are equivalent to classical spin configurations $h_j$ with log fidelity playing the role of energy. (b) Using $k$-flip stochastic descent, we explore the log-fidelity landscape (c), and find a glasslike transition in the control landscape described by the effective classical model ${\mathcal{H}}_{\mathrm{eff}}$ (d).

Figure 2

Preparing states in a chain of qubits with optimal many-body fidelity $F_h(T)$ (black) features transitions from an overconstrained phase (red region) to a correlated phase (blue region) to a glasslike phase (purple region) at protocol durations $T_c^{(1)}$ and $T_c^{(2)}$. This is revealed by the nonzero fraction $f_{{\mathrm{SD}}_k}(T)$ order parameter. We used $k$-flip stochastic descent (${\mathrm{SD}}_k$) on the family of bang-bang protocols with $N_T=200$, $L=6$ and $M={10}^5$.

Figure 3

(a)–(c) $t$-SNE visualization of the control landscape above the ${\mathrm{SD}}_2$ glass critical point $T_c^{(2)}\approx 2.3$. Each data point represents a local $C_h(T)$ minimum—a bang-bang protocol embedded in a two-dimensional $t$-SNE space. Embedded protocols are colored by their fidelity in the interval $[F_{\min },F_{\max }]$ with intervals [0.919, 0.920], [0.958, 0.959], [0.992, 0.997] from (a) to (c). (a),(b) The local minima cluster are separated by extensive barriers as seen in (d),(e), the Hamming distance matrix for the local-minima protocols. ${\mathrm{dist}}_{\max }=0.5$, 0.52, 0.61 for (d),(e),(f), respectively. The protocols in the Hamming matrix are grouped by their cluster index found using density clustering (see Supplemental Material [52]). (c) At larger protocol duration ($T=3.4$) large clusters fracture in an exponential number of small clusters. The small clusters are separated by extensive barriers (f). We used ${\mathrm{SD}}_2$ with $N_T=200$, $L=6$ and sampled 5000 unique protocols.

Figure 4

Normalized density of states (DOS) of ${\mathcal{H}}_{\mathrm{eff}}$ (black line, left $y$ axis), and the distribution of the $M_h=0$ and $M_h=2$-magnetized excitations (shaded, right $y$ axis) on both sides of the glass critical point $T_c^{(2)}\approx 2.3$ for $N_T=80$, $L=6$. The position of the best obtained fidelity using ${\mathrm{SD}}_4$ is marked by the vertical dashed line.