Qubits and bra-ket notation

A classical bit is either 0 or 1. A qubit has two basic states too, but it can also be in a blend of both at once. This page shows how physicists write those states down, which is the notation the rest of these pages use.

Kets

A state is written inside a ket, the symbol $\ket{\;}$ . The two basic states of a qubit are

\ket{0} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad \ket{1} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}

Under the notation, a state is just a column of numbers. $\ket{0}$ and $\ket{1}$ play the role of 0 and 1, and every qsengine circuit starts with all its qubits in $\ket{0}$ .

Superposition

Any other state of one qubit is a combination of the two basic states:

\ket{\psi} = \alpha\ket{0} + \beta\ket{1} = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}

$\alpha$ and $\beta$ are amplitudes. They are complex numbers, and they must satisfy

|\alpha|^2 + |\beta|^2 = 1

because, as the measurement page explains, the squares are probabilities and must add up to 1. A state with both amplitudes non-zero is in superposition.

Making a superposition

The Hadamard gate turns $\ket{0}$ into an equal superposition:

H\ket{0} = \frac{1}{\sqrt{2}}\ket{0} + \frac{1}{\sqrt{2}}\ket{1}

qsengine shows the two amplitudes directly. $1/\sqrt{2} \approx 0.7071$ :

import qsengine

qc = qsengine.QuantumCircuit(1)
qc.h(0)

print(qc.statevector())
print(qc.run(shots=1000, seed=3))

Output

[0.70710678+0.j 0.70710678+0.j]
{'0': 493, '1': 507}

0: 493
1: 507

Equal amplitudes, so 0 and 1 each come up about half the time.

Gates are matrices

A gate on one qubit is a 2 by 2 matrix, and applying it multiplies the state by that matrix. The X gate swaps the two amplitudes, which is why it acts as a NOT:

X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad X\ket{0} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \ket{1}

Every gate in qsengine is a matrix like this. The gates page lists them all.

Bras

The mirror image of a ket is a bra, $\bra{\psi}$ : the same numbers laid out as a row and complex-conjugated. Putting a bra against a ket gives a single number, the overlap of two states:

\braket{0|\psi} = \alpha, \qquad \braket{1|\psi} = \beta

That is how you pick an amplitude out of a state, and it is the start of the measurement rule.