Topological Phase Transitions in a Constrained Two-Qubit Quantum Control Landscape

TL;DR

This work reveals a new class of control landscape phase transitions (CLPTs) in a constrained two-qubit state-preparation problem, where the global topology of the optimal level set changes discontinuously as protocol duration $T$ crosses key thresholds. By sampling optimal protocols with Langevin-Monte Carlo and analyzing pairwise protocol distances, the authors identify events where the number of connected components $b_0(T)$ jumps, including mergers of components at $T_{t-}$ and the birth of a new component from a local trap at $T_{t+}$, all occurring beyond the quantum speed limit $T_{QSL}$. They demonstrate the symmetry-independence of these topological transitions and propose experimentally accessible signatures via single-qubit reduced density matrices and Bloch-sphere trajectories. The results provide a global, topology-based lens on quantum-control landscapes, enabling detection and characterization of CLPTs and offering guidance for robust protocol design in multi-qubit systems.

Abstract

In optimal quantum control, control landscape phase transitions (CLPTs) indicate sharp changes occurring in the set of optimal protocols, as a physical model parameter is varied. Here, we demonstrate the existence of a new class of CLPTs, associated with changes in the topological properties of the optimal level set in a two-qubit state-preparation problem. In particular, the distance distribution of control protocols sampled through stochastic homotopic dynamics reveals discontinuous changes in the number of connected components in the optimal level set, as a function of the protocol duration. We demonstrate how topological CLPTs can be detected in modern-day experiments.

Figures (11)

• Figure 1: Topological transitions in the two-qubit quantum control landscape in Eq. \ref{['eq:H']}. a) Piecewise-constant protocols $s(t)$ control a system $H_0$ by modulating the external drive $H_1$ for a total duration $T$ (inset panel (b)). b) Illustration of the optimization landscape $I_T[s]$ for a fixed protocol duration $T$. Depending on $T$, the landscape possesses a different number of basins of attraction; in this schematic, two basins are associated with the connected components $C_+,C_-$ of the infinite-dimensional optimal level set and one with the local trap $s_3$. c) Sketch of the optimal level set $\{\mathrm{arg\,min}_sI_T[s]\}$ as a function of $T$. Topological transitions at $T_\text{sb}$, $T_\text{t+}$, and $T_\text{t--}$ mark sudden change in the number of connected components $b_0(T)$. d) Topological CLPTs correspond to jump-discontinuities of a topological invariant known as the $0^\text{th}$ Betti number $b_0(T)$ as a function of $T$; the $T_\text{QSL}$ transition does not affect $b_0(T)$.
• Figure 2: Analysis of the connected components of the optimal level set $\{\mathrm{arg\,min}_sI_T[s]\}$, before and after the quantum speed limit $T_\text{QSL}$ (black dashed line). a) Density distribution $P(d_\text{avg})$ of the average distance $d_\text{avg}$ in Eq. \ref{['eq:d_a']}, for different protocol durations $T$. Changes in the number of peaks in $P(d_\text{avg})$ mark topological transitions (red dashed lines). b) Graph summarizing structure of optimal level set for $T_\text{t+}{\le}T{\le}T_\text{t--}$. Connected components $C_+,C_-,C_3$ are represented by nodes while edges represent the relative distances $d(C_i,C_j),i{=}+,-,3$. A peak in the distance distribution $P(d_\text{avg})$ in (a) corresponds to at least one edge in the graph. c) The order parameter $\min_n \abs{m(T)[s_n]}$ in Eq. \ref{['eq:m']} identifies with high precision the location $T_\text{t--}$, associated with the merging $C_+ C_-{\to}C_\pm$. LMC parameters: $L{=}64,\beta{=}10^6,\sigma{=}10^{-2}$.
• Figure 3: Topological transitions are experimentally accessible in modern-day experiments. The figure shows Bloch sphere trajectories in the reduced density matrix obtained from protocols sampled by the Langevin-Monte Carlo (LMC) algorithm. The green (red) arrow indicates the initial (target) state. Solid lines represent averages over three LMC runs, each sampling one of the three connected components $C_+,C_-,C_3$ of the optimal level set (colored in blue, orange and purple, respectively); low-opacity dotted trajectories represent single protocols and serve to visualize protocol fluctuations. a)$T{=}2.6{>}T_\text{sb}$: blue and orange trajectories correspond to the isolated optimal protocols $s_+,s_-$; the purple curve $s_3$ is a local trap. b)$T{=}3.0{>}T_\text{QSL}$: optimal level set consists of two connected components $C_+,C_-$, each containing infinitely many optimal protocols (cf. blue and orange clouds surrounding the solid lines). c)$T{=}3.4{>}T_\text{t+}$: the orange and blue clouds are still well-separated; the purple protocol, now also optimal, gives rise to the third connected component $C_3$. d)$T{=}3.6{>}T_\text{t--}$: orange and blue trajectories overlap and the corresponding clouds are indistinguishable: the previously distinct connected components $C_+,C_-$ have merged and formed $C_\pm$. Insets: the purple trajectories reach the largest entanglement entropy as the associated norm $\abs{n(t)}$ drops to ${\approx}0.5$ during the evolution.
• Figure 4: A local trap $s_3$ appears in the landscape for $T{\in}[2.6,3.2]$. a) Infidelity associated with the locally stable protocols $s_+,s_-,s_3$ found via LMC (in blue, orange and purple, respectively) shows that $s_3$ is a local trap. b,c) Visualization of protocols $s_+,s_-,s_3$ at $T{=}2.80$. Contrary to $s_+$ and $s_-$, $s_3$ does not break the control problem symmetry $s(t){\leftrightarrow} {-}s(T{-}t)$.
• Figure 5: $\textbf{a)}$ Maximum, minimum and mean infidelity obtained at the end of the optimization performed by the Langevin-Monte Carlo optimization with simulated annealing, for $T\in[2.0,4.0]$. $\textbf{b)}$ Difference between the maximum and minimum infidelity obtained. For this analysis, we consider a set of $2^{10}$ independent LMC runs at each $T$. We see that the local trap $s_3$ impacts the optimization performance around the quantum speed limit, $T_\text{QSL}\approx3.0$.