Broken symmetry in a two-qubit quantum control landscape

We analyze the physics of optimal protocols to prepare a target state with high fidelity in a symmetrically coupled two-qubit system. By varying the protocol duration, we find a discontinuous phase transition, which is characterized by a spontaneous breaking of a ${\mathbb{Z}}_2$ symmetry in the functional form of the optimal protocol, and occurs below the quantum speed limit. We study in detail this phase and demonstrate that even though high-fidelity protocols come degenerate with respect to their fidelity, they lead to final states of different entanglement entropy shared between the qubits. Consequently, while globally both optimal protocols are equally far away from the target state, one is locally closer than the other. An approximate variational mean-field theory which captures the physics of the different phases is developed.

Figure 1

Quantum control phase diagram for the symmetrically coupled two-qubit problem (inset). As a function of the protocol duration $T$, the infidelity landscape exhibits both continuous and discontinuous phase transitions (vertical dashed lines) featuring overconstrained (red), correlated and glassy (blue and green), symmetry-broken (green), and controllable (yellow) phases, detected by the correlator $q\left(T\right)$. The fidelity $F_h\left(T\right)$ of the optimal protocol and the corresponding magnetization order parameter $m\left(T\right)$ feature distinctive behavior in the different phases with nonanalyticities at the phase boundaries emerging in the limit of vanishing protocol time-step size. Time is in units of the inverse qubit interaction strength.

Figure 2

Entanglement entropy $S_{\mathrm{ent}}\left(T\right)$ against infidelity $I_h\left(T\right)$ for the final state at three different protocol durations $T$ across the symmetry-breaking transition. Each data point corresponds to evolution following a protocol $h_{\alpha }\left(t\right)$ from the set of local infidelity minima obtained using SD. An entanglement gap between two large clusters is present in the symmetry-broken phase. The protocol step size is fixed at $\delta t=0.00125$. Time is in units of the inverse qubit interaction strength.

Figure 3

(a) Low-infidelity manifold of the optimization landscape as a function of the protocol duration (the optimal fidelity is shifted to zero for every $T$). The colors indicate the infidelity density. (b) The infidelity gap $\mathrm{\Delta }$ as a function of $T$ vanishes at the symmetry-breaking point $T_{\mathrm{sb}}$ and the optimal protocol becomes degenerate. The number of bangs in a protocol is kept fixed at $N_T=28$. Time is in units of the inverse qubit interaction strength.

Figure 4

The set of ${10}^3$ protocols (sample of local infidelity minima obtained using SD) can be divided into two disjoint subsets according to the mean entanglement entropy (see Fig. 2), leaving a total of three sets: the low-$S_{\mathrm{ent}}$ protocols (red, down), the high-$S_{\mathrm{ent}}$ protocols (blue, up), and all protocols (green, middle). Symmetry breaking in the low-infidelity manifold of the control landscape becomes evident. The time-step size $\delta t=0.00125$. Time is in units of the inverse qubit interaction strength.

Figure 5

Equal-time correlations in the low-infidelity landscape as a function of time $t$. The averaging is done over a sample of ${10}^3$ local infidelity minima. The time-step size $\delta t=0.00125$. Time is in units of the inverse qubit interaction strength.

Figure 6

Nonequal-time correlations in the low-infidelity landscape as a function of time $t$. The averaging is done over a sample of ${10}^3$ local infidelity minima. The time-step size $\delta t=0.00125$. Time is in units of the inverse qubit interaction strength.

Figure 7

(a) Entanglement entropy $S_{\mathrm{ent}}\left(T\right)$ shared between the qubits and (b) local observables after evolution with the best protocol(s) found using SD as a function of the protocol duration $T$ for $\delta t=0.00125$. A bifurcation is visible in the symmetry-broken phase where the optimal protocol is doubly degenerate. Time is in units of the inverse qubit interaction strength.

Figure 8

Variational theory for the symmetrically coupled two-qubit system. (a) Generic form of the family of variational protocols, with pulse durations ${\tau }^{\left(i\right)}$, which allow for symmetry breaking, to be chosen by optimizing the fidelity (see text). (b) Optimal pulse durations against the total protocol time $T$ feature kinks at the durations corresponding to the control phase transitions. (c) The optimal variational fidelity (dashed line) compared to the best fidelity obtained using SD. Time is in units of the inverse qubit interaction strength.

Figure 9

Variational protocols for (a) $T=0.3$, (b) $T=0.8$, and (c) $T=2.0$. At $T=2.0$, the best variational protocol breaks the ${\mathcal{Z}}_2$ symmetry: $h\left(t\right)\ne -h(T-t)$. Time is in units of the inverse qubit interaction strength.

Figure 10

Protocol time-step ($\delta t$) scaling of the quantities used to detect the control phase transitions: (a) Edwards-Anderson-like order parameter, (b) magnetization order parameter, (c) entanglement entropy of the (nearly) optimal best encountered protocol, and (d) standard deviation of of the entanglement entropy over the sample of local infidelity minima. The sample size of the local infidelity minima contains $N_{\mathrm{real}}={10}^3$ independent realizations obtained using stochastic descent (SD). The time step decreases from the topmost to the lowermost curve, and the transitions become sharper with decreasing $\delta t$. Time is in units of the inverse qubit interaction strength.

Figure 11

(a) Variational proof of controllability of the system, using a simple 3D symmetric ansatz (black dashed line). (b) The optimal protocol within this variational family reaches unit fidelity, which is marked by a vertical asymptote in the logarithmic variational fidelity. The green dotted line shows the position of the true QSL for the system. For comparison, the fidelity obtained using SD is also shown (solid blue line). Time is in units of the inverse qubit interaction strength.