Fast and virtually exact quantum gate generation in U(n) via iterative Lyapunov methods

ABSTRACT

A new approach called Reference Input Generation Algorithm (RIGA) is introduced for quantum gate generation. Let ${\overline{X}}^{\ell -1}\left(t\right)$ be the trajectory that is obtained in step $\ell -1$, with ${\overline{X}}^{\ell -1}\left(t\right)=I$. In step $\ell$, ${\overline{X}}^{\ell -1}$ is right-translated in order to displace ${\overline{X}}^{\ell -1}\left(T_f\right)$ to $X_{goal}$. The translated trajectory is used as a reference for a Lyapunov-based tracking control law, generating ${\overline{X}}^{\ell }\left(T\right)$, and so on. A proof of the exponential convergence of ${\overline{X}}^{\ell -1}\left(T_f\right)$ to zero is provided for $T_f$ large enough. Two examples present numerical experiments regarding N coupled qubits. The first example considers N = 3 with a known minimum time $T^{\ast }$. It presents excellent results for $T_f=T^{\ast }$. The second example is a benchmark for the comparison between RIGA and GRAPE, considering a Hadamard gate for the systems with N = 2, 3, …, 10 qubits. The runtime of RIGA could be improved, and GRAPE was implemented in a faster CPU. However, RIGA presents results that are similar to GRAPE, with faster runtime in some cases, showing that the RIGA is indeed a promising algorithm.

KEYWORDS:

• Controllability
• quantum control
• right-invariant systems
• Lyapunov stability
• quantum gates

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Gradient Ascent Pulse Engineering (GRAPE).

2 It is shown in Silveira, Pereira da Silva, and Rouchon (2014) that $\mathcal{W}$ is a positively invariant set for the time-varying closed-loop system (6(6) $\dot{\tilde{X}}\left(t\right)=\sum _{k=1}^m{\tilde{u}}_k\left(t\right){\tilde{S}}_k\left(t\right)\tilde{X}\left(t\right),\tilde{X}\left(0\right)={\tilde{X}}_0={\overline{X}}_0^{†},$(6) )–(7(7a) $\begin{array}{rl}{\tilde{u}}_k\left(t\right)& ={\tilde{U}}_k\left(\overline{X}\right(t),\tilde{X}(t\left)\right)=f_k\text{Tr}\left[Z\left(\tilde{X}\left(t\right)\right){\tilde{S}}_k\left(t\right)\right],\end{array}$(7a) ).

3 The closed-loop system is also equivalently written as (1(1) $\begin{array}{rl}\dot{X}\left(t\right)& =-\left(H_0+\sum _{k=1}^m{u}_k\left(t\right)H_k\right)X\left(t\right)\\ & =S_{0}X\left(t\right)+\sum _{k=1}^m{u}_k\left(t\right)S_{k}X\left(t\right),X\left(0\right)=I,\end{array}$(1) )–(5(5) $\dot{\overline{X}}\left(t\right)=S_0\overline{X}\left(t\right)+\sum _{k=1}^m{\overline{u}}_k\left(t\right)S_k\overline{X}\left(t\right),\overline{X}\left(0\right)={\overline{X}}_0,$(5) )–(7(7a) $\begin{array}{rl}{\tilde{u}}_k\left(t\right)& ={\tilde{U}}_k\left(\overline{X}\right(t),\tilde{X}(t\left)\right)=f_k\text{Tr}\left[Z\left(\tilde{X}\left(t\right)\right){\tilde{S}}_k\left(t\right)\right],\end{array}$(7a) ).

4 Phase 2 only works when $\mathcal{V}\left(R\right)=\text{dist}\left({\overline{X}}_{old}\right(T_f),X_{goal})$ is finite, that is, $R\in \mathcal{W}$ (see (3(3) $\mathcal{W}=\{W\in \text{U}(n)|det(I+W)\ne 0\}.$(3) )). When this is not the case, it is possible to apply the two-step procedure presented in Silveira et al. (2016) that uses the square root of R in each iteration.

5 This is the Algorithm 2 in the next subsection.

6 Recall that $\tilde{W}=U^{†}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}[\exp ({\theta }_1),\dots ,\exp ({\theta }_n\left)\right]U$, where $U\in U\left(n\right)$. For this define $\sqrt[p]{\tilde{W}}=U^{†}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}[\exp ({\theta }_1/p),\dots ,\exp ({\theta }_n/p\left)\right]U$. Since $\underset{p\to \mathrm{\infty }}{lim}\sqrt[p]{\tilde{W}}=I$, then such p always exists.

7 It is shown in Silveira et al. (2016) that ${\overline{B}}_c^{\mathcal{V}}\left(I\right)$ is indeed a compact set in $U\left(n\right)$.

8 Since $\mathcal{V}\left(\tilde{X}\right(t\left)\right)$ is non-decreasing and ${\overline{B}}_c^{\mathcal{V}}\left(I\right)$ is compact, then $\tilde{X}\left(t\right)$ is well-defined and remains in ${\overline{B}}_c^{\mathcal{V}}\left(I\right)\subset \mathcal{W}$ for $t\in [0,T_f]$.

9 Proposition A.2 of Appendix 3 implies that this is true almost surely when M is large enough.

10 This high order of brackets indicates that the control problem is rather nontrivial.

11 According to the erratum associated to Khaneja et al. (2002) (available at doi:10.113/PhysRevA.68.049903), the range of θ is $[0,2\pi ]$, and not $[0,4\pi ]$ as it was in the original paper.

12 See the supplementary material for the values of $({\mathbf{a}}_1,{\mathbf{b}}_1)$.

13 The value J=1 is considered in Leung et al. (2017) but the absolute value of inputs are bounded by $5J_0$, which is equivalent to consider $J=J_0$, $\parallel u_{x_k}\left(t\right)\parallel \le 5$ and $\parallel u_{y_k}\left(t\right)\parallel \le 5$, $k=1,\dots ,N$.

14 The data of runtime of GRAPE that is considered in Figure 3 is the one of Figure 7 of Leung et al. (2017) after correction according to their Figure 3 (since Figure 7 present the best result between GPU and CPU).

15 The value of λ may depend on the radius c.

16 It is invertible because its derivative is everywhere positive, and given by $\alpha \left(\theta \right)=(2-\cos (\theta /2\left)\right)/\left[8\right(\cos \left(\theta /2\right){)}^4]$.

17 As the Lyapunov function is nonnegative it follows that ${\tilde{M}}_a<1$.

18 See Proposition A.10 of Appendix 6 for the definition of $c_1,c_3$ and, $k_1$.

19 Recall that this means that $\mathcal{V}\left({\tilde{X}}_0\right)\le c$.

Additional information

Funding

The first author was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil, Project 305546/2016-3, and by Fundacão de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Brazil, Project 18/17463-7.