Essential Concepts in Quantum Computing
Table of Contents
1. Qubits
The unit of computation is qubit (short for quantum bits). A qubit in a quantum computer consists of two perfectly distinguishable statuses of a (usually microscopic) quantum system. We usually use a unit vector to refer to the quantum state.
Notations in Quantum Computing
In quantum computing, vectors are denoted in bracket:
- \( \ket{a} \) means a column vector \(\vec{a} = \begin{pmatrix}x\\y\end{pmatrix}\)
- \( \bra{a} \) means the adjoint of row vector \( \vec{a} \), i.e., take conjugate1 of each element \( a^{\dag} = \begin{pmatrix}x^{\star} & y^{\star}\end{pmatrix} \)
- \( \braket{a|b} \) means the dot product of these 2 vectors.
- \( \ket{a}\bra{b} \) means a 2-by-2 matrix.
There exist 2 basis vectors: ground state \( \ket{0}=\begin{pmatrix}1\\0\end{pmatrix} \) and excited state \( \ket{1}=\begin{pmatrix}0\\1\end{pmatrix} \).
A qubit can be in other states than \( \ket{0} \) and \( \ket{1} \). In general, an arbitrary qubit state is
\[ \ket{\psi} = \alpha\ket{0}+\beta\ket{1} = \begin{pmatrix}\alpha\\\beta\end{pmatrix}, \text{ where } \alpha,\beta \in \mathbb{C}, |\alpha|^{2}+|\beta|^{2}=1 \]
The coefficients \(\alpha,\beta\) are called amplitudes of \( \ket{\psi} \).
2. Measurement of Qubits
The outcome of a qubit cannot be read directly. We must perform a measurement.
- Measurement
- Each measurement can only produce one outcome \( \ket{0} \) or \( \ket{1} \). Although a single measurement can only have one outcome, different measurement may have different outcomes. But, we can statistically figure out the state of the qubit after a large amount of measurements, that is the Born’s Rule.
Born’s Rule
The probability of getting the outcome \(x\) when measuring any quantum state \( \ket{\psi} \) in the computational basis is \[ \mathbb{P}(x) = | \braket{x|\psi} |^{2}, \quad x=0,1 \]
Footnotes:
The conjugate of a complex \( z = a + bi \) is \( z^{\star} = a - bi \).