Quantum Bit (Qubit)

Qubit#

Definition. A quantum bit is a superposition of two states \(\ket{0}\) and \(\ket{1}\) which are normalized and orthogonal to each other.

Thus for example \(\ket{\psi} = \alpha\ket{0} + \beta\ket{1}\) is a qubit. The two special states \(\ket{0}\) and \(\ket{1}\) are known as computational basis states and form an orthonormal basis for a two-dimensional complex vector space. The prabability for a measurement of the state \(\ket{\psi}\) to fall on \(\ket{0}\) is \(|\alpha|^2\) and for \(\ket{1}\), \(|\beta|^2\). A special state, for which \(|\alpha|^2 = |\beta|^2\): \(\ket{+} = \frac{1}{\sqrt{2}} \ket{0} + \frac{1}{\sqrt{2}}\ket{1}\).

It is customary to put \(\ket{\psi} = e^{i\gamma}(\cos\frac{\theta}{2}\ket{0} + e^{i\phi}\sin\frac{\theta}{2}\ket{1})\). Since the factor \(e^{i\gamma}\) has no observable efects, effectively

\[\ket{\psi} = \cos\frac{\theta}{2} \ket{0}+ e^{i\phi}\sin\frac{\theta}{2}\ket{1}.\]

The two numbers, \(\theta\) and \(\phi\), define a point on the unit three-dimensional sphere, called the Bloch sphere. \(\phi\) is the azimuth, and \(\theta\) is the polar angle.

A single qubit gate is simply a unitary operator of dimension 2. For example,

the NOT gate: \(\ket{1}\bra{0}+\ket{0}\bra{1}\). This exchanges \(\ket{0}\) and \(\ket{1}\).
the Z gate: \(\ket{0}\bra{0} -\ket{1}\bra{1}\). This flips the sign of \(\ket{1}\) and thus gives \(-\ket{1}\).
the Hadamard gate: \(\sqrt{½}(\ket{0}\bra{0} -\ket{1}\bra{1} + \ket{1}\bra{0} + \ket{0}\ket{1})\). THis turns \(\ket{0}\) into \(\sqrt{½}(\ket{0}+\ket{1})\) and \(\ket{1}\) into \(\sqrt{½}(\ket{0}-\ket{1})\). (rotation of the Bloch sphere about the \(y\) axis by \(90^\circ\) and then about the \(x\) axis by \(180^\circ\).)

Since unitary 2-dimensional operators form the group \(SU(2)\), an arbitrary single qubit unitary gate can be decomposed as a product of rotations

\[\begin{bmatrix} \cos\frac{\gamma}{2} & -\sin\frac{\gamma}{2} \\ \sin\frac{\gamma}{2} & \cos\frac{\gamma}{2} \end{bmatrix}\]

and a rotation about the \(z\) axis \(e^{-i\beta/2}\ket{0}\bra{0} + e^{i\beta/2}\ket{1}\bra{1}\), together with a global phase shift.

Multiple qubits#

A \(n\)-qubit state is simply a superposition of \(n\)-tensor product states. A two qubit system has four computational basis states \(\ket{00},\ket{01},\ket{10},\ket{11}\). An important two qubit state is the Bell state or EPR pair:

\[\frac{\ket{00}+\ket{11}}{2}.\]

The Bell state has the following property. Upon measuring the first qubit, one obtains two possible results:

\(\ket{0}\) with probability \(½\) so that the post-measurement state becomes \(\ket{00}\).
\(\ket{1}\) with probability \(½\), leaving the post-measurement state \(\ket{11}\).

A measurement of the second qubit always gives the same result as the measurement of the first qubit. Thus the measurement on the two qubits are correlated. Interestingly, correlations still exist after applying some operations to the first/second qubit. Entanglement.