Not directly, as far as I know, since explicitly computing large multiples of points in $E(\mathbb Q)$ is infeasible. However, people have considered lifting points from $E(\mathbb F_p)$ to $E(\mathbb Q)$ or to the $p$-adics $E(\mathbb Q_p)$ in order to devise algorithms to solve the discrete log problem in $E(\mathbb F_p)$ (although, unsuccessfully so far). Here are a few papers to get you started:

• Elliptic curve discrete logarithms and the index calculus. Advances in cryptology—ASIACRYPT'98 (Beijing), 110–125, Lecture Notes in Comput. Sci., 1514, Springer, Berlin, 1998
• The xedni calculus and the elliptic curve discrete logarithm problem. Des. Codes Cryptogr. 20 (2000), no. 1, 5–40; Analysis of the xedni calculus attack. Des. Codes Cryptogr. 20 (2000), no. 1, 41–64.

Not directly, as far as I know, since explicitly computing large multiples of points in $E(\mathbb Q)$ is infeasible. However, people have considered lifting points from $E(\mathbb F_p)$ to $E(\mathbb Q)$ or to the $p$-adics $E(\mathbb Q_p)$ in order to devise algorithms to solve the discrete log problem in $E(\mathbb F_p)$ (although, unsuccessfully so far). Here are a few papers to get you started:

• Elliptic curve discrete logarithms and the index calculus. Advances in cryptology—ASIACRYPT'98 (Beijing), 110–125, Lecture Notes in Comput. Sci., 1514, Springer, Berlin, 1998
• The xedni calculus and the elliptic curve discrete logarithm problem. Des. Codes Cryptogr. 20 (2000), no. 1, 5–40; Analysis of the xedni calculus attack. Des. Codes Cryptogr. 20 (2000), no. 1, 41–64.
• Lifting and elliptic curve discrete logarithms, Selected Areas of Cryptography (SAC 2008), Lecture Notes in Computer Science 5381, Springer-Verlag, Berlin, 2009, 82--102.