Conceptual aspects of geometric quantum computation

Abstract

Geometric quantum computation is the idea that geometric phases can be used to implement quantum gates, i.e., the basic elements of the Boolean network that forms a quantum computer. Although originally thought to be limited to adiabatic evolution, controlled by slowly changing parameters, this form of quantum computation can as well be realized at high speed by using nonadiabatic schemes. Recent advances in quantum gate technology have allowed for experimental demonstrations of different types of geometric gates in adiabatic and nonadiabatic evolution. Here, we address some conceptual issues that arise in the realizations of geometric gates. We examine the appearance of dynamical phases in quantum evolution and point out that not all dynamical phases need to be compensated for in geometric quantum computation. We delineate the relation between Abelian and non-Abelian geometric gates and find an explicit physical example where the two types of gates coincide. We identify differences and similarities between adiabatic and nonadiabatic realizations of quantum computation based on non-Abelian geometric phases.

Appendix

We prove that the space of all dark subspaces of the tripod system is G(3; 2). We do this by demonstrating that for any $\vert {\psi }\rangle \in \text {Span} \{ \vert {0}\rangle , \vert {1}\rangle , \vert {a}\rangle \}$ there exists $\varvec{\omega }$ such that

$$\begin{aligned} P_d (\varvec{\omega }) \vert {\psi }\rangle = 0. \end{aligned}$$

(38)

By using the linear independence of the two dark states $\vert {D_0 (\varvec{\omega })}\rangle , \vert {D_1 (\varvec{\omega })}\rangle$, it follows that Eq. (38) is equivalent to

$$\begin{aligned} \langle D_j (\varvec{\omega }) \vert {\psi }\rangle = 0, \ \ \ j=0,1, \end{aligned}$$

(39)

for $\vert {\psi }\rangle = \lambda _0 \vert {0}\rangle + \lambda _1 \vert {1}\rangle + \lambda _a \vert {a}\rangle$ with arbitrary complex-valued $\lambda _0,\lambda _1,\lambda _a$ such that $|\lambda _0|^2 + |\lambda _1|^2 + |\lambda _a|^2 \ne 0$. By using the explicit form of the two dark states (parameterization taken from Ref. [43]), we find

$$\begin{aligned} \sin \phi e^{-iS_{31}} \lambda _0 - \cos \phi e^{-iS_{32}} \lambda _1= & {} 0 , \nonumber \\ \cos \theta \cos \phi e^{-iS_{31}} \lambda _0 + \cos \theta \sin \phi e^{-iS_{32}} \lambda _1- \sin \theta \lambda _a= & {} 0 , \end{aligned}$$

(40)

where $\omega _0 = \sin \theta \cos \phi e^{iS_1}$, $\omega _1 = \sin \theta \sin \phi e^{iS_2}$, $\omega _a = \cos \theta e^{iS_3}$, and $S_{kl} = S_k - S_l$.

Assume first that $\lambda _0 \ne 0$ and define $z_1=\lambda _1/\lambda _0, z_a=\lambda _a/\lambda _0$. We find

$$\begin{aligned} \tan \phi e^{-iS_{21}}= & {} z_1, \nonumber \\ \cot \theta e^{-iS_{31}}= & {} z_a \cos \phi . \end{aligned}$$

(41)

This can be solved for all $z_1,z_a$ since $\theta ,\phi ,S_{21},S_{31}$ are independent variables. Explicitly, one finds $\phi = \tan ^{-1} |z_1|$, $S_{21} = -\arg z_1$, $\theta = \cot ^{-1} \left[ |z_a|/ \sqrt{1+|z_1|^2}\right]$, and $S_{31} = -\arg z_a$. Next, we assume that $\lambda _0 = 0$ but $\lambda _1 \ne 0$, and define $\tilde{z}_a = \lambda _a/\lambda _1$. We find

$$\begin{aligned} \phi= & {} \frac{\pi }{2} , \nonumber \\ \cot \theta e^{-iS_{32}}= & {} \tilde{z}_a \end{aligned}$$

(42)

with solution $\theta = \cot ^{-1} | \tilde{z}_a|$ and $S_{32} = -\arg \tilde{z}_a$. Finally, if $\lambda _0 = \lambda _1=0$, then $\theta = 0$ solves Eq. (40).