Eg. CNOT is a conditional gate that performs an X-gate on the second qubit, if the state of the first qubit (control) is <math>|1\rangle</math>. <math>CNOT= \begin{bmatrix} 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
\end{bmatrix}
</math>. This matrix swaps the amplitudes of |01⟩ and |11⟩ in the statevector. <math>\text{CNOT}(x,y) = (x, x\otimes y)</math>.
<math>\text{CNOT}|0{+}\rangle = \tfrac{1}{\sqrt{2}}(|00\rangle + |11\rangle)</math>, which is ''Bell State''. Entanglement, but [https://arxiv.org/abs/quant-ph/0212023 no-communication theorem].
<math>\text{CNOT}|{-}{-}\rangle = \tfrac{1}{2}(|00\rangle - |01\rangle - |10\rangle + |11\rangle) = |{-}{-}\rangle </math>. Affects the state of the control qubit, only.
Any controlled quantum gate is <math>\text{Controlled-U} = \begin{bmatrix}I &0\\0&U\end{bmatrix}</math> and in Qiskit formalism is written in matrix as
<math>\text{Controlled-U} =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & u_{00} & 0 & u_{01} \\
0 & 0 & 1 & 0 \\
0 & u_{10} & 0 & u_{11}\\
\end{bmatrix}
</math>
Controlled-Z. Because <math>H X H = Z</math> and <math>H Z H = X</math> we can write
<syntaxhighlight>
qc = QuantumCircuit(2)
# also a controlled-Z
qc.h(t)
qc.cx(c,t)
qc.h(t)
qc.draw('mpl')
</syntaxhighlight>
Controlled-Y is
<syntaxhighlight>
qc = QuantumCircuit(2)
# a controlled-Y
qc.sdg(t)
qc.cx(c,t)
qc.s(t)
qc.draw('mpl')
</syntaxhighlight>
or Controlled-H is
<syntaxhighlight>
qc = QuantumCircuit(2)
# a controlled-H
qc.ry(pi/4,t)
qc.cx(c,t)
qc.ry(-pi/4,t)
qc.draw('mpl')
</syntaxhighlight>
Swap gate
An arbitrary controlled-controlled-U for any single-qubit rotation U. We need <math>V = \sqrt U</math> and <math>V^\dagger</math>
<syntaxhighlight>
#The controls are qubits a and b, and the target is qubit t.
#Subroutines cu1(theta,c,t) and cu1(-theta,c,t) need to be defined
from qiskit import QuantumCircuit, execute, Aer
qc = QuantumCircuit(1) # Create a quantum circuit with one qubit
initial_state = [0,1] # Define initial_state as |1>
qc.initialize(initial_state, 0) # Apply initialisation operation to the 0th qubit
qc.draw('text') # Let's view our circuit (text drawing is required for the 'Initialize' gate due to a known bug in qiskit)
backend = Aer.get_backend('statevector_simulator') # Tell Qiskit how to simulate our circuit
result = execute(qc,backend).result() # Do the simulation, returning the result
out_state = result.get_statevector()
print(out_state) # Display the output state vector
from qiskit.visualization import plot_histogram, plot_bloch_vector
qc.measure_all()
qc.draw()
result = execute(qc,backend).result()
counts = result.get_counts()
plot_histogram(counts)
Take superposition as initial state
initial_state = [1/sqrt(2), 1j/sqrt(2)] # Define state |q>
The Bloch Sphere
from qiskit_textbook.widgets import plot_bloch_vector_spherical
coords = [pi/2,0,1] # [Theta, Phi, Radius]
plot_bloch_vector_spherical(coords) # Bloch Vector with spherical coordinates
Qiskit allows measuring in the Z-basis, only.
Theory
Quantum operations are reversible, thus the reversible computing. That makes some complications to the gate design.
Three Qubits
Toffoli gate made using CNOTs.
For universal computations we need more qubits. Eg. the AND gate is not reversible, and thus we need eg. Toffoli (CCNOT) gate.
Toffoli gate performs on target qubit if both control cubits are set to state .
qc = QuantumCircuit(3)
a = 0
b = 1
t = 2
# Toffoli with control qubits a and b and target t
qc.ccx(a,b,t)
qc.draw()
