Charge qubit

In

If ${\displaystyle n}$ denotes the number of excess Cooper pairs in the island (i.e. its net charge is ${\displaystyle -2ne}$), then the Hamiltonian is:[4]

${\displaystyle H=\sum _{n}{\big [}E_{\rm {C}}(n-n_{\rm {g}})^{2}|n\rangle \langle n|-{\frac {1}{2}}E_{\rm {J}}(|n\rangle \langle n+1|+|n+1\rangle \langle n|){\big ]},}$

where ${\displaystyle n_{\rm {g}}=C_{\rm {g}}V_{\rm {g}}/(2e)}$ is a control parameter known as effective offset charge (${\displaystyle V_{\rm {g}}}$ is the gate voltage), and ${\displaystyle E_{\rm {J}}}$ the Josephson energy of the tunneling junction.

At low temperature and low gate voltage, one can limit the analysis to only the lowest ${\displaystyle n=0}$ and ${\displaystyle n=1}$ states, and therefore obtain a two-level quantum system (a.k.a. qubit).

Note that some recent papers^[8][9] adopt a different notation, and define the charging energy as that of one electron:

${\displaystyle E_{\rm {C}}=e^{2}/2(C_{\rm {g}}+C_{\rm {J}}),}$

and then the corresponding Hamiltonian is:

${\displaystyle H=\sum _{n}{\big [}4E_{\rm {C}}(n-n_{\rm {g}})^{2}|n\rangle \langle n|-{\frac {1}{2}}E_{\rm {J}}(|n\rangle \langle n+1|+|n+1\rangle \langle n|){\big ]}.}$

Benefits

To-date, the realizations of qubits that have had the most success are ion traps and NMR, with Shor's algorithm even being implemented using NMR.^[10] However, it is hard to see these two methods being scaled to the hundreds, thousands, or millions of qubits necessary to create a quantum computer. Solid-state representations of qubits are much more easily scalable, but they themselves have their own problem: decoherence. Superconductors, however, have the advantage of being more easily scaled, and they are more coherent than normal solid-state systems.^[10]

Experimental progresses

The implementation of Superconducting charge qubits have been progressing quickly since 1996. Design was theoretically described in 1997 by Shnirman,

References