Protocol Discovery for the Quantum Control of Majoranas by Differentiable Programming and Natural Evolution Strategies

Quantum control, which refers to the active manipulation of physical systems described by the laws of quantum mechanics, constitutes an essential ingredient for the development of quantum technology. Here we apply differentiable programming (DP) and natural evolution strategies (NES) to the optimal transport of Majorana zero modes in superconducting nanowires, a key element to the success of Majorana-based topological quantum computation. We formulate the motion control of Majorana zero modes as an optimization problem for which we propose a new categorization of four different regimes with respect to the critical velocity of the system and the total transport time. In addition to correctly recovering the anticipated smooth protocols in the adiabatic regime, our algorithms uncover efficient but strikingly counterintuitive motion strategies in the nonadiabatic regime. The emergent picture reveals a simple but high-fidelity strategy that makes use of pulselike jumps at the beginning and the end of the protocol with a period of constant velocity in between the jumps, which we dub the jump-move-jump protocol. We provide a transparent semianalytical picture, which uses the sudden approximation and a reformulation of the Majorana motion in a moving frame, to illuminate the key characteristics of the jump-move-jump control strategy. We verify that the jump-move-jump protocol remains robust against the presence of interactions or disorder, and corroborate its high efficacy on a realistic proximity-coupled nanowire model. Our results demonstrate that machine learning for quantum control can be applied efficiently to quantum many-body dynamical systems with performance levels that make it relevant to the realization of large-scale quantum technology.

Popular Summary

Quantum technologies exploit the laws of quantum mechanics with the objective of deepening our understanding of complex natural systems, improving artificial intelligence, and impacting the industry more broadly. A key element for their practical realization is the development of strategies to encode, manipulate, and control quantum information without causing errors. In this respect, Majorana zero modes, which are effective “quasi”particles arising in $p$-wave superconductors, offer a promising prospect due to their nonlocal nature making Majorana-based quantum devices naturally robust against errors. Here we investigate the optimal way to transport a Majorana from position $A$ to position $B$, an operation required for the manipulation of quantum information encoded in Majoranas.

We formulate this question as a game where an agent attempts different paths for the Majorana motion and gets rewarded depending on how well it reached the target state at $B$. To find the optimal strategy, we train the agent with two machine-learning algorithms; differential programming and natural evolution strategies. These algorithms uncover a counterintuitive jump-move-jump strategy to move the Majorana when the total transport time is small. In this protocol, the Majorana is “shaken” with the objective of bringing it to a stable state moving at a constant velocity before it is shaken again to bring it to rest at the end. Our work shows that advanced machine-learning algorithms can be applied to develop strategies for the control of large quantum devices and to help overcome obstacles to realizing large-scale quantum technology.

Figure 1

(a) In the Majorana game, the agent moves the left Majorana ${\gamma }_L$ from a position $x_A$ to a position $x_B$ by tuning the external potential profile $V(x,t)$. (b) The agent attempts different movement paths and tries to find the optimal one that minimizes the infidelity $\mathcal{I}(T)$. (c) Different Majorana control regimes and their corresponding strategies [boxes in (d)]: (I) critical regime ($v_{\mathrm{avg}}>v_{\mathrm{crit}}$), (II) subcritical regime ($v_{\mathrm{crit}}/2<v_{\mathrm{avg}}<v_{\mathrm{crit}}$), (III) short time $T$, low-velocity regime, and (IV) (super)adiabatic regime (long time $T$ and low-velocity $v_{\mathrm{avg}}$). (d) In regimes (I)-(II)-(III) we find the jump-move-jump (JMJ) strategy whereas in regime (IV) we recover a smooth superadiabatic protocol. The infidelities are $\mathcal{I}(T)=0.3575$ in regime (I), $\mathcal{I}(T)=0.1589$ in (II), $\mathcal{I}(T)=0.0057$ in (III) and $\mathcal{I}(T)=0.0005$ in regime (IV); the parameters for the JMJ strategies displayed in regimes (I), (II), (III) can be found in Appendix pp7 and we set $x_L(0)=x_A=5.0$ to ensure a smooth potential profile $V(x,t)$.

Figure 2

Overview of the optimal controls in the four different Majorana motion regimes [(I)–(IV)] obtained after optimization with DP, NES, and simulated annealing (SA). In all panels we plot the domain-wall position $x_L(t)$ as a function of time starting from $x_L(0)=x_A=5.0$. The red dotted line in regime (I) has a slope equal to the critical velocity. Regimes (I)–(III) show a pulselike motion at the edges and an on average constant velocity in the bulk of the protocol. In regime (IV) the strategy is to slowly build up to constant speed and then slow down again. In these simulations we set $N=110$, $\mu =1$, $w=1$, $\mathrm{\Delta }=0.3$, $V_{\mathrm{height}}=30.1$, and $\sigma =1$. The constraint parameters $(l,T)$ are given by $\{(4.32,12),(4.95,22),(0.48,8),(2.4,40)\}$ for regimes (I), (II), (III), and (IV), respectively.

Figure 3

(a) Setup of the ML-inspired simple control strategy consisting of two edge pulses with a forward jump $\mathrm{\Delta }{x}_{\mathrm{forward}}$ and backward jump $\mathrm{\Delta }{x}_{\mathrm{back}}$ together with a constant linear motion in between the dressed jumps. (b) Infidelity $\mathcal{I}(T)$ surface plot for the scans over the free parameters $\mathrm{\Delta }{x}_{\mathrm{forward}}$ and $\mathrm{\Delta }{x}_{\mathrm{back}}$ of the model strategy with $\mathrm{\Delta }{t}_{\mathrm{back}}=0.07w^{-1}$ and $\mathrm{\Delta }{t}_{\mathrm{forward}}=0.01w^{-1}$ in regime (I) $(l=4.32,T=12)$. The red line indicates the line where the size of the forward jump $\mathrm{\Delta }{x}_{\mathrm{forward}}$ is equal to the total movement length $l$. The parameters of the Majorana control setup are the same as in Fig. 2. The optimal dressed jumps appear at a diagonal set of parameters $\mathrm{\Delta }{x}_{\mathrm{forward}}-\mathrm{\Delta }{x}_{\mathrm{back}}\sim C$, where $C\approx 0.96$. We note that the best protocol has a jump size bigger than the movement length $\mathrm{\Delta }{x}_{\mathrm{forward}}>l$, which means that it jumps over the target position $x_L(T)=x_B$ and then jumps back within the range $x_A<x_L<x_B$, as can be seen in Fig. 1 (d) [for regimes (I) and (III)].

Figure 4

The jump-move-jump protocol maximizes the product of the two functions $O_{\delta }$ [Eq. (8)] and $O_v$ [Eq. (9)]. Here $l=4.32$ and $T=12$ [regime (I)] and we use $s=2.45$ and $\beta =0.065$. The ground-state projection analysis (solid blue) captures the key features of the ML-inspired (one-wall) JMJ strategy (blue stars) for low velocities $v$ but tends to overestimate the penalty for moving at velocities near $v_{\mathrm{crit}}$.

Figure 5

Contours showing the infidelity $\mathcal{I}$ for the bare (one-wall) jump-move-jump strategy [Eq. (10)] as a function of distance $l$ and time $T$ using the same fitted parameters as Fig. 4. The red dashed line indicates when $v_{\mathrm{avg}}=l/T=v_{\mathrm{crit}}$, which can be used to distinguish the different Majorana motion regimes as defined in Fig. 1 (c). The jump-move-jump protocol allows for low infidelities even in cases where the average velocity exceeds $v_{\mathrm{crit}}$ [regime (I)].

Figure 6

(a) Instantaneous infidelity $I(t)$ with respect to the static (lab) frame ground state (main panel) and moving-frame ground state (inset) as a function of the protocol time $t$ for various dressed jumps in regime (I), which are depicted by the different colors (red, blue, green, gray) and shown in the inset of (b). The optimal dressed jump (blue) has the minimum infidelity with the moving-frame ground state and becomes the strategy with the minimum static frame infidelity after about $t\approx 4$. (b) The time-evolved state $|\psi (t)⟩$ expanded in the single and two particle excitations $|{\psi }_v^i⟩$ of the moving-frame basis with energies $E_v^i$ at $t=T/2$ for a forward jump (red), an optimal dressed jump (blue), and nonoptimal dressed jump (green) for regime (I) as given in the inset. The occupation probabilities ($\delta$ functions) are convoluted for visualization purposes. The forward jump strategy (red) occupies the excitations close to the bottom of the band in the moving frame $(E_v^i\approx 0.08)$ more compared to the dressed jumps (blue and green). In this figure all times are in units of $1/\omega$.

Figure 7

Main panel: instantaneous infidelity $I(t)$ of the optimal JMJ protocol from the noninteracting system in regime (I) run in a system with several different finite interaction strengths $u$. The jumps are robust against interactions whereas infidelity increases during the move part of the protocol. Inset: the final infidelity values $\mathcal{I}(T)$ increase approximately linearly with interaction strength for both a linear as well as the best JMJ protocol. For all interaction strengths the JMJ protocol outperforms the linear protocol. The other parameters for these simulations are the same as in Fig. 2 and the parameters of the JMJ protocol can be found in Appendix pp7.

Figure 8

Main panel: disorder averaged instantaneous infidelity $⟨I(t)⟩$ of the optimal JMJ protocol from the clean wire in regime (I) for different disorder strengths $\lambda$. The jumps are robust against weak disorder whereas the infidelity increases during the move part of the protocol. The disorder averaging is done over 500 realizations. Inset: the final infidelity values $⟨\mathcal{I}(T)⟩$ increase approximately quadratically with disorder strength for both a linear as well as the best JMJ protocol. For all disorder strengths the JMJ protocol outperforms the linear protocol. The other parameters for these simulations are the same as in Fig. 2 and the parameters of the JMJ protocol can be found in Appendix pp7.

Figure 9

(a) Mode dispersion in the moving frame for several different velocities. (b) Localization of the Majorana zero modes in the moving frame for several different velocities. It can be seen that the gap closes for $v=v_{\mathrm{crit}}=\mathrm{\Delta }$ and simultaneously the Majoranas delocalize in the moving frame.

Figure 10

(a) Infidelity as a function of frequency $\omega$, the black dashed line indicates the resonance frequency ${\omega }_{\mathrm{res}}=\mathrm{\Delta }{k}_F=0.3$. At $\omega \mapsto \mathrm{\infty }$ the infidelity is the same as for constant linear motion with $x_L(t)=(v_{\max }/2)t$. (b) Infidelity as a function of frequency and $v_{max}$. For velocities $v_{\max }>v_{\mathrm{crit}}=0.3$ there are no low adiabatic frequencies anymore. The parameters used in these simulations are $N=140$, $\mu =1$, $w=1$, $\mathrm{\Delta }=0.3$, and $\sigma =1$.

Figure 11

(a) Comparison between the gradient of the infidelity $\mathcal{I}$ with respect to the control $x_L$ obtained with automatic differentiation (AD) and finite difference. The gradients $d\mathcal{I}/d{x}_L$ are evaluated for a linear protocol $x_L(t)=v_{\mathrm{avg}}t$ as a function of time $t$ in regime (I). The AD gradient matches up with the finite-difference gradient and we note that at the beginning and end of the protocol the magnitude of the gradient is the largest. (b),(c) Optimized parameterized bang-bang protocols in regimes (II) (b) and (IV) (c) with the simulated annealing method. The position profiles $x_L(t)$ are plotted in the main panels and the velocity profiles $v_L(t)$ in the insets. The bang-bang protocols seem to approximate the JMJ protocol in regime (II) and superadiabatic protocol in regime (IV). The parameters for these simulations are the same as in Fig. 2 in the main text.

Figure 12

(a) Energy dispersion of the Hamiltonian, Eq. (F4), in the moving frame for three different velocities. The velocity tilts the dispersion and the gap closes when $v=v_{\mathrm{crit}}$. The model parameters used for this dispersion are $B=\alpha =1.0,\mathrm{\Delta }=0.5$ and $\mu =-0.55,V=0.0$. (b) Excitation energy eigenvalues $E_n$ of the NW model on a lattice for open-boundary conditions as a function of the chemical potential $\mu$. The red dashed line indicates $\mu =-0.55$. The other parameters are the same as for (a). We get a zero mode (blue line) but also some Andreev bound states (e.g., orange line) that come down when we bring $\mu$ closer to the bottom of the band.

Figure 13

Overview of the optimal controls in the four different Majorana motion regimes (I–IV) obtained after optimization with DP, NES, and the parameterized JMJ protocols for the proximity-coupled NW model as given by Eq. (F2). In all panels we plot the domain-wall position $x_L(t)$ as a function of time starting from $x_L(0)=x_A=5.0$. The red dotted line in regime (I) has a slope equal to the critical velocity $v_{\mathrm{crit}}\approx 0.27$ for(a) and (e) and $v_{\mathrm{crit}}\approx 0.28$ for (i) and (m). The obtained strategies are similar to the strategies obtained for the Kitaev chain in the main text with pulselike motion at the beginning and end in regimes (I)-(II)-(III) being smoother motion in the middle and in regime (IV) a smooth adiabatic protocol. In the simulations for (a)–(h) we set $N=100$, $\mu =0.0$, $w=1$, $\mathrm{\Delta }=0.8$, $B=2$, $\alpha =0.8$, $V_{\mathrm{height}}=30.1$, and $\sigma =1$ and for (i)–(s) we set $N=100$, $\mu =-0.55$, $w=1$, $\mathrm{\Delta }=0.5$, $B=1$, $\alpha =1.0$, $V_{\mathrm{height}}=30.1$, and $\sigma =1$. The constraint parameters $(l,T)$ are given by $\{(4.32,12),(4.95,22),(0.48,8),(2.4,40)\}$ for regimes (I), (II), (III), and (IV), respectively.

Figure 14

(a) The DP optimized majorana motion strategy in regime (III) $(T=8,l=0.48)$ on top of the optimized (scans over the free parameters) JMJ model strategy. The optimal JMJ model strategy captures the main features (jumps) of the DP strategy well. (b) Infidelity $\mathcal{I}(T)$ obtained for 100 linearly interpolated profiles between the DP strategy and JMJ model strategy as shown in (a). We observe a convex decreasing function towards the more flexible DP result.

Figure 15

Overview of simulation results of running the optimal protocols obtained in the clean noninteracting system in a system with interactions (top row) or disorder (bottom row) in all four regimes. In all regimes the infidelity increases with disorder strength $\lambda$ and also with interaction strength $u$. The parameters for these simulations are the same as in Fig. 2 in the main text and the disorder averaging is done over 500 disorder realizations.

Figure 16

In the $p$-wave model the $O_{\delta }\sim \exp (-{\delta }^2/s^2)$ have an interesting relationship to the two key length scales. The fit parameter $s$ is reduced by a factor of about $1/2$ when the coherence length $\xi =1/(m\mathrm{\Delta })$ is made very large. On the other hand, its value tends to increase linearly in proportion to ${\lambda }_F=2\pi /k_F$. The topological gap $E_{\mathrm{gap}}\sim \mathrm{\Delta }{k}_F=2\pi /(m\xi {\lambda }_F)$ is, however, inversely proportional to both length scales.