Dirac notation and operators
Quantum computing is a complex field that requires linear algebra and quantum mechanics to understand. How to write quantum states in a way that's easy to understand and work with? A handy notation to write quantum states is the Dirac bra-ket notation.
What's Dirac bra-ket notation?
Dirac bra-ket notation, or Dirac notation for short, is a shorthand notation that eases writing quantum states and computing linear algebra. In this notation, the possible states of quantum system are described by using symbols called kets, which look like this $| \rangle$.
For example, $|0\rangle$ and $|1\rangle$ represents the 0 and 1 states of a qubit, respectively.
A qubit in the state $|\psi\rangle = |0\rangle$ means that the probability of observing 0 when you measure the qubit is 100 %. Similarly, if you measure a qubit in the state $|\psi\rangle =|1\rangle$, you always get 1.
For example, a qubit in superposition can be written as $|\psi\rangle = \frac1{\sqrt2} |0\rangle + \frac1{\sqrt2} |1\rangle$. This state is a superposition of the $|0\rangle$ and $|1\rangle$ states. The probability of measuring 0 is $\frac12$ and the probability of measuring 1 is also $\frac12$.
What are quantum operators?
Quantum computing is about manipulating quantum states to perform computations. A quantum operator is a function that acts on a state of a quantum system and transforms it to another state. For example, you can transform a $|0\rangle$ state into a $|1\rangle$ state, by applying the X operator.
$$X |0\rangle = |1\rangle$$
The X operator is also called the Pauli-X gate. It's a fundamental quantum operation that flips the state of a qubit. There're three Pauli gates: X, Y, and Z. Each gate or operator has a specific effect on the qubit state.
| Operator | Effect on $\ket{0}$ | Effect on $\ket{1}$ |
|---|---|---|
| X | $X \ket{0} = \ket{1}$ | $X\ket{1} = \ket{0}$ |
| Y | $Y\ket{0}=i\ket{1}$ | $Y\ket{1}=-i\ket{0}$ |
| Z | $Z\ket{0}=\ket{0}$ | $Z\ket{1}=-\ket{1}$ |
Note
Sometimes you may read or hear the term quantum gates instead of quantum operations. The term quantum gate is an analogy to classical logic gates. It's rooted in the early days of quantum computing when quantum algorithms were visualized as diagrams similar to circuit diagrams in classical computing.
You can use an operator to put a qubit in superposition. The Hadamard operator, H, puts a qubit that's in the state $|0\rangle$ into superposition of $|0\rangle$ and $|1\rangle$ states. Mathematically, this equation is
$$ H |0\rangle = \frac1{\sqrt2} |0\rangle + \frac1{\sqrt2} |1\rangle.$$
In this case, the probability of measuring each state is $P(0)=\left|\frac1{\sqrt{2}}\right|^2=\frac12$ and $P(1)=\left|\frac1{\sqrt{2}}\right|^2=\frac12$. Each state has a 50% probability of being measured. You also can check that $\frac12 + \frac12 = 1$.
What does it mean to do a measurement?
There are many interpretations of the concept of measurement in quantum mechanics, but the details are beyond the scope of this module. For quantum computing, you don't have to worry about it.
In this module, you understand measurement to be the informal idea of "observing" a qubit, which immediately collapses the quantum superposition to one of the two basis states that correspond to 0 and 1. For example, if you measure a qubit in the state $|\psi\rangle = \frac1{\sqrt2} |0\rangle + \frac1{\sqrt2} |1\rangle$, it means that you force the qubit to take one of the two possible states, and you'll observe either 0 or 1 with equal probability.
To learn more about measurement in the context of quantum mechanics and its historical discussion, see the Wikipedia article about the Measurement problem.