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Nuts and Bits

For the uninitiated, the building block of the quantum computer is the quantum-bit or qubit. A qubit is the superposition of the states (0 and 1) of the traditional bit with a probabilistic state instead of an absolute one. The possible states of a single qubit are written as follows.

[Graphics:../Images/index_gr_2.gif]

The superposition [Graphics:../Images/index_gr_3.gif] where [Graphics:../Images/index_gr_4.gif] and [Graphics:../Images/index_gr_5.gif] are complex numbers that declare the probability that [Graphics:../Images/index_gr_6.gif] will be in the state 0 or 1, [Graphics:../Images/index_gr_7.gif] and [Graphics:../Images/index_gr_8.gif] respectively, is written as follows.

[Graphics:../Images/index_gr_9.gif]

In vector form, we can think of the state of qubits as being represented by the following.

[Graphics:../Images/index_gr_10.gif]

A collection of two qubits can be written as follows.

[Graphics:../Images/index_gr_12.gif]

[Graphics:../Images/index_gr_13.gif] generalized qubits can be written as follows.

[Graphics:../Images/index_gr_14.gif]

As with things quantum, a set of qubits are in every possible state and when we choose to observe them they will be in some state with a given probability. Notationally, we typically see the following, where [Graphics:../Images/index_gr_15.gif] and [Graphics:../Images/index_gr_16.gif] are complex numbers with the probability of the device being in the state [Graphics:../Images/index_gr_17.gif] with [Graphics:../Images/index_gr_18.gif] and in the state [Graphics:../Images/index_gr_19.gif] with [Graphics:../Images/index_gr_20.gif], etc.

[Graphics:../Images/index_gr_21.gif]

One interesting aspect of this representation is that we can take advantage of the fact that the phase information of the coefficients is actually meaningful. Using it, we can examine the interference between states, which as DiVincenzo points out can be useful from a computational standpoint, as we will see below.


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