Toward a Theory of Phase Transitions in Quantum Control Landscapes

Abstract

Control landscape phase transitions (CLPTs) occur as abrupt changes in the cost function landscape upon varying a control parameter, and can be revealed by non-analytic points in statistical order parameters. A prime example are quantum speed limits (QSL) which mark the onset of controllability as the protocol duration is increased. Here we lay the foundations of an analytical theory for CLPTs by developing Dyson, Magnus, and cumulant expansions for the cost function that capture the behavior of CLPTs with a controlled precision. Using linear and quadratic stability analysis, we reveal that CLPTs can be associated with different types of instabilities of the optimal protocol. This allows us to explicitly relate CLPTs to critical structural rearrangements in the extrema of the control landscape: utilizing path integral methods from statistical field theory, we trace back the critical scaling of the order parameter at the QSL to the topological and geometric properties of the set of optimal protocols, such as the number of connected components and its dimensionality. We verify our predictions by introducing a numerical sampling algorithm designed to explore this optimal set via a homotopic stochastic update rule. We apply this new toolbox explicitly to analyze CLPTs in the single- and two-qubit control problems whose landscapes are analytically tractable, and compare the landscapes for bang-bang and continuous protocols. Our work provides the first steps towards a systematic theory of CLPTs and paves the way for utilizing statistical field theory techniques for generic complex control landscapes.

Figures (26)

• Figure 1: In quantum control problems, there are two types of physical degrees of freedom: the controlled system (e.g., qubits) and the control variables defining the control protocol (e.g., modulation of electric or magnetic fields, or electromagnetic radiation, external forces, etc.). The figure of merit that measures the quality of control protocols defines the control landscape. Control phase transitions refer to critical changes in the structure of the control landscape, as a model parameter (e.g., the protocol duration $T$) is varied. Note that the control landscape can present a many-body problem in the control variables, even when the controlled system consists of as few as a single degree of freedom, see Sec. \ref{['sec:model']}.
• Figure 2: Control Landscape Phase Transitions (CLPTs) in the single- (a) and two-qubit (b) control problems. The curves shown are obtained from Stochastic Descent (SD) algorithm in the exact infidelity landscape, searching in the space of bang-bang protocols. The minimum infidelity $\min_s I(T)[s]$ and the order parameter $q_\text{BB}(T)$ (cf. Eq. \ref{['eq:q_BB']}) are shown as a function of the protocol duration $T$ for a fixed number of bang-bang steps. Non-analytic points in the order parameter $q_\text{BB}(T)$ mark control landscape phase transitions (dashed vertical lines). Similar transitions appear in both control problems: at $T{=}T_c$, and at $T{=}T_\text{QSL}$ (where the infidelity vanishes). The two-qubit problem possesses an additional symmetry-breaking transition at $T{=}T_\text{sb}$ related to the separation of the set of optimal protocols into two separated subsets which break the $s(t) {\leftrightarrow} {-}s(T{-}t)$ symmetry of the control problem Bukov18_Broken.
• Figure 3: Comparison of the order parameter $q_\text{BB}(T)$ (cf. Eq. \ref{['eq:q_BB']}) obtained from Stochastic Descent (SD) in the landscape defined by the infidelity expansions introduced in Sec. \ref{['sec:expansions']}, in the single- and two-qubit control problem. Control protocols are here restricted to the bang-bang class. The black curve refers to the exact infidelity landscape in Eq. \ref{['eq:infid-def']}. The letters D, M, and C in the legend stand for Dyson, Magnus, and Cumulant expansion while the adjacent integer specifies the order of truncation. Single-qubit: The $T{=}T_c{\simeq}0.98$ transition is approximately captured by all three methods. The second transition at $T{=}T_\text{QSL}{\simeq}2.51$ is only visible in the Magnus and Cumulants results. Two-qubit: The $T{=}T_c{\simeq}0.56$ and $T{=}T_\text{sb}{\simeq}1.57$ transitions are approximately captured by all three methods. The non-analytic point at $T{=}T_\text{QSL}{\simeq}2.95$ is less sharp than the corresponding point in the single-qubit problem and only the Magnus expansion truncated at third order is able to reproduce the qualitative behavior of the exact curve.
• Figure 4: Two different one-dimensional functions (blue and red curves) illustrating the stability conditions for optimal points in Eq. \ref{['eq:lin_stab_cond']}. The minimum of the red curve at $x{=}+0.5$ (red dot), lying inside the allowed domain $[-1,1]$, satisfies $f'(x){=}0,f"(x){>}0$. The minimum of the blue curve at $x{=}{-}1$ (blue dot), lying at the edge of the allowed domain $[-1,1]$, satisfies $f'(x){>}0$.
• Figure 5: Stability analysis in the single-qubit control problem. The plots show the linear ($b_t$, orange curve) and second-order terms ($J_{t_1t_2}$, colormap insets) of the infidelity Taylor expansion centered around optimal piecewise continuous protocols ($s_t$, blue curve) found by the adiabatic tracing method (cf. Sec. \ref{['sec:stability']}), for different value of the duration $T$ of the quantum evolution (CLPTs in $T_\text{c}{\simeq}0.98,\,T_\text{QSL}{\simeq}2.51$). Linear term:$b_t$. For $T{\in}[0,T_c]$ the optimal protocol is $s_0$ in Eq. \ref{['eq:stab_s0']}. (a) At $T{=}0.25$, $s_0$ is anti-aligned with $b_t$ and is linearly stable. (b) For $T{=}T_\text{c}$, $s_0$ becomes linearly unstable around $t{=}T/2$ as shown by the horizontal inflection of $b_t$ around $t{=}T/2$. (c) For $T{\in}[T_c,T_\text{QSL}]$ the optimal protocol is $s_{\Delta_0(T)}$ in Eq. \ref{['eq:stab_s_Delta']}. At $T{=}2.0$, $s_{\Delta}$ is stable provided $\Delta$ is chosen as Eq. \ref{['eq:stab_s_Delta']} (cf. Fig. \ref{['fig:stab_1q_Delta']}a). (d) At $T{=}T_\text{QSL}$, the first-order term is exactly zero and it flips sign for $T{=}T_\text{QSL}^+$: $s_{\Delta_0(T)}$ becomes linearly unstable. Quadratic term:$J_{t_1t_2}$. In the language of statistical mechanics, the second-order term represents the effective two-body interactions of the field $s_t$. In this case, interactions are antiferromagnetic (AFM) and long-range: the system is frustrated. For $T{=}T_c$ (panel (b)), the protocol $s_0$ becomes linearly unstable around $t{=}T/2$ and the local AFM interactions around the instability point dictate the new optimal protocol structure $s_{\Delta(T)}$ (cf. Eq. \ref{['eq:stab_s_Delta']}).