Quantum Bayesian Networks

November 2, 2008

Qubit Mixers

Filed under: Uncategorized — rrtucci @ 3:04 pm
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A qubit lives in a complex vector space spanned by two vectors

|0\rangle= \left[\begin{array}{c}1\\ 0 \end{array}\right]\;\;,\;\;|1\rangle = \left[\begin{array}{c}0\\ 1\end{array}\right]\;.

Define the following projection operators:

\overline{n} = P_0 = |0\rangle\langle 0|=\left[\begin{array}{c}1\;\;0\\ 0\;\;0\end{array}\right]\;\;,\;\;  n = P_1 = |1\rangle\langle 1| =\left[\begin{array}{c}0\;\;0\\ 0\;\;1\end{array}\right]\;.

n is called the number operator because n|b\rangle = b|b\rangle for b \in \{0,1\}. Note \overline{n} =  1 -n.

The Pauli matrices are defined by:
\sigma_X = \left[\begin{array}{c}0\;\;1\\1\;\;0\end{array}\right]\;\;,\;\;\sigma_Y =\left[\begin{array}{c}0\;-i\\i\;\;0\end{array}\right]\;\;,\;\;\sigma_Z = \left[\begin{array}{c}1\;\;0\\ 0\;-1\end{array}\right]\;.
One also defines \vec{\sigma} = (\sigma_X,\sigma_Y,\sigma_Z).

The lingua franca for expressing a sequence of quantum computer commands are quantum circuits with circuit elements of some basic types. Here are some circuit elements in order of increasing complexity: (in these figures, 0,1,2 are qubit labels)

Fig.1 Single-qubit rotation and CNOT

Fig.1 (Pinwheel) Single-qubit rotation and Controlled Not (CNOT)

Fig.2 (Farm Windmill) Multiply Controlled Rotation

Fig.2 (Farm Windmill) Multiply Controlled Rotation

Fig.3 (Windfarm) Multiplexor

Fig.3 (Windfarm) Multiplexor

The pinwheel, farm windmill and wind farm are my poetic analogues.
One can generalize the operators of Figs.2 and 3 to have more controls (the dark dots in Fig.2 and the half-moon nodes of Fig.3) and more complicated targets (the square nodes in Figs.2 and 3).

Let N_B be the number of bits and N_S= 2^{N_B} the number of states. The operators in Fig.1 are a “universal” set: any matrix in SU(N_S) can be constructed in terms of them. The operators in Figs. 2 and 3 can be expressed in terms of those in Fig.1.

The operators in Figs. 1 and 2 appear frequently in the quantum computing literature. The multiplexor operator of Fig.3 is much less common. Only I and a few other workers have used it in their papers. Multiplexors have some very useful properties. I’ve used them in the theory behind my quantum compiler, Qubiter ( a quantum compiler decomposes an input unitary matrix into a sequence of elementary operations (SEO), elementary operations such as single qubit rotations and CNOTs.) My next paper uses multiplexors heavily. As soon as the paper is in ArXiv, I will update this post with a link to it.

Update: The paper is now available here. It contains a whole section introducing multiplexors.

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