Quantum gate generation for systems with drift in U(n) using Lyapunov–LaSalle techniques

ABSTRACT

This paper considers right-invariant and controllable quantum systems with m inputs u = (u₁,… , u_m) and state X(t) evolving on the unitary Lie group U(n). The ε-steering problem is introduced and solved for systems with drift: given any initial condition X₀ at the initial time instant t₀ ≥ 0, any goal state $X_{\text{goal}}\in$ U(n) and ε > 0, find a control law such that $\text{dist}(X(t_0+\bar{t}),X_{\text{goal}})\le ϵ$, where $\bar{t}>0$ is big enough and dist(X₁, X₂) is a convenient right-invariant notion of distance between two elements X₁, X₂ ∈ U(n). The purpose is to approximately generate arbitrary quantum gates corresponding to X_goal. This is achieved by solving a tracking problem for a special kind of reference trajectories $\bar{W}$: [t₀, ∞) → U(n), which are here called c-universal reference trajectories. It is shown that, for this special kind of trajectories, the tracking problem can be solved up to an error ε for any reference trajectory $\bar{X}\left(t\right)=\bar{W}\left(t\right)R$ which is a right-translation of $\bar{W}\left(t\right)$, at least when $\text{dist}(X\left(t_0\right),\bar{X}\left(t_0\right))$ is finite. Furthermore, it is shown that $\text{dist}(X\left(t\right),\bar{X}\left(t\right))$ converges uniformly exponentially to zero in the sense that the rate of convergence is independent of t₀, R and X₀. The approach considered here for showing such convergence is a generalisation of the results of a previous paper of the authors, which is mainly based on the central ideas of Jurdjevic and Quinn and Coron's return method. Taking a right-translation R such that $\bar{X}(t_0+\bar{t})=X_{\text{goal}}$, one may solve the ε-steering problem by solving the tracking problem for the reference trajectory $\bar{X}\left(t\right)$, at least when $\text{dist}(X\left(t_0\right),\bar{X}\left(t_0\right))\le c$. When $\text{dist}(X\left(t_0\right),\bar{X}\left(t_0\right))\ge c$, it is shown that the ε-steering problem can be globally solved in a two-iteration procedure. The underlying algorithmic complexity to get the steering control is essentially equivalent to the numerical integration of the Cauchy problem governing X(t). A numerical example considering a Toffoli quantum gate on U(8) for a chain of three coupled qubits that are controlled only locally is presented.

Keywords:

• Controllable right-invariant systems
• Coron's return method
• LaSalle's invariance theorem
• quantum control
• unitary group

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Recall that a n-square complex matrix X belongs to U(n) if and only if X^†X = I, and S is in $\mathfrak{u}\left(n\right)$ if and only if S^† = −S (skew-Hermitian), where S^† is the conjugate transpose of S.

2. A singularity appears if ${\bar{W}}^{†}\left(t_f\right)X_{goal}$ has at least one eigenvalue equal to −1.

3. Since the Frobenius norm is invariant with respect to left- and right-multiplication by a unitary matrix, it is clear that ‖X − Y‖ = ‖X^†(X − Y)‖ = ‖I − X^†Y‖. The equivalence of these different notions of distance is established by Lemma C.2 in Appendix C.

4. Computer simulations have shown that a better compromise between the maximum (Euclidean) norm of the input u = (u₁,… , u_m) and the final time t_f is attained by using the two-iteration procedure.

5. As ${\bar{t}}_h$ is the initial condition of the second iteration of Algorithm F.1 in Appendix 6, then one will assume that ${\bar{t}}_h$ is also an integer multiple of T in the rest of the paper. Note also that ${\bar{t}}_{ϵ}$ is not assumed to be an integer multiple of T.

6. It will be shown that the reference controls in (6) generate a c-universal reference trajectory for almost all $(\mathbf{a},\mathbf{b})\in I{R}^{mM}\times I{R}^{mM}$, and that an associated c-universal control law is given by (7c(7c) $${\tilde{U}}_k(\bar{X},\tilde{X})\triangleq f_{k}Tr\left[\tilde{Z}\left(\tilde{X}\right){\bar{X}}^{†}{S}_k\bar{X}\right],$$(7c) ).

7. Note that this implies that $\tilde{X}\left(t\right)={\bar{X}}^{†}\left(t\right)X\left(t\right)$ always remains in ${\bar{B}}_c\left(I\right)$, and so the control law is always well-defined.

8. The independence of the rate of the convergence with respect to the choice of $\bar{X}\left(\ell T\right)\in \text{U}\left(n\right)$ is equivalent to the independence of the rate with respect to the choice R ∈ U(n).

9. Property (C3(C3) $$\tilde{Z}\left(\tilde{X}\right)=U^{†}\text{diag}[\iota \alpha \left({\theta }_1\right),...,\iota \alpha \left({\theta }_n\right)]U,$$(C3) ) in Appendix C ensures that $\tilde{Z}\left(\tilde{X}\right)\in \mathcal{S}$ for all $\tilde{X}\in \mathcal{W}$.

10. Note that subsystem (17a(17a) $$\begin{array}{ccc}\hfill \dot{\bar{X}}\left(t\right)& =& S_0\bar{X}\left(t\right)+\sum _{k=1}^m{\bar{u}}_k\left(t\right)S_k\bar{X}\left(t\right),\hfill \\ \hfill \bar{X}\left(0\right)& =& {\bar{X}}_0\in \text{U}\left(n\right),\hfill \end{array}$$(17a) ) is nothing but system (1(1) $$\begin{array}{ccc}\hfill \dot{X}\left(t\right)& =& -\iota \left(H_0+\sum _{k=1}^m{u}_k\left(t\right)H_k\right)X\left(t\right)\hfill \\ & =& S_{0}X\left(t\right)+\sum _{k=1}^m{u}_k\left(t\right)S_{k}X\left(t\right),X\left(t_0\right)=X_0,\hfill \end{array}$$(1) ).

11. Not necessarily periodic.

12. The constants L and γ in (9(9) $$\begin{array}{ccc}& & \text{dist}(X\left(t\right),\bar{X}\left(t\right))\le Lexp(-\gamma t)\text{dist}(X\left(\ell T\right),\bar{X}\left(\ell T\right)),\hfill \\ & & \text{for}t\ge t_0=\ell T\hfill \end{array}$$(9) ) may depend on T > 0, on (a, b) and on c > 0.

13. For an elementary proof of this fact, see e.g. Caron and Traynor (2005).

14. After restricting to an adequate subsequence.

Additional information

Funding

The second author was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNP_q), Fundacão de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and USP-COFECUB. The third author was partially supported by Agence Nationale de la Recherche (ANR), Projet Blanc EMAQS number ANR-2011-BS01-017-01 and USP-COFECUB.