Superposition

Classically, a bit can either be \(0\) or \(1\), never both simultaneously. Quantum bits (qubits), exist in a state of superposition, meaning they can represent both \(0\) and \(1\) simultaneously. This is described mathematically as:

\[\ket{\psi} = \alpha\ket{0} + \beta\ket{1}\]

where \(\alpha\) and \(\beta\) are complex coefficients.

1. IBM Quantum Composer

2. Qiskit Implementation

IBM Quantum Composer: https://quantum.ibm.com/composer/

1. IBM Quantum Composer

IBM Quantum Composer allows one to build, visualize, and run quantum circuits on quantum hardware.

• Classical bits are represented as c\(n\), where \(n\) is the number of bits.

• Qubits are labeled \(q[0], q[1], \ldots, q[n]\) from top to bottom, representing the initial state \(\ket{q[n]q[n-1] \cdots q[0]}\) in Dirac notation.

  • Example: \(q[0] = 1\), \(q[1] = 1\), \(q[2] = 0 \rightarrow \ket{011}\)

• Gate operations are placed left to right on the qubits, applying operations in order.

• The probability distribution and Q-sphere diagram represent the possible states of the qubit when measured.

New circuits start in the \(\ket{0000}\) state with 4 classical bits, as shown below. You have the ability to add or delete qubits and classical bits by clicking on the labels.

1.1 One Qubit

First, begin with the \(\ket{0}\) state. Then, apply a \(H\) gate on the first qubit.

Notice how applying the \(H\) gate creates a superposition where measuring the state results in \(\frac{1}{2}\) probability of being \(0\) and \(\frac{1}{2}\) probability of being \(1\), using only one qubit.

1.2 Two Qubits

Going further, add a second qubit and apply another \(H\) gate to the second qubit.

With two qubits, there are \(4\) possible states that can be measured with equal probability (\(\frac{1}{4}\) each).

2. Qiskit Implementation

Make sure Qiskit is installed on your machine. Run the cell below to install it.

%pip install qiskit

2.1 One Qubit

We initialize a Quantum Circuit with \(1\) qubit and then apply a \(H\) gate on the qubit.

from qiskit import QuantumCircuit
# Initialize 1 qubit circuit
circuit = QuantumCircuit(1)
# Apply Hadamard on first qubit
circuit.h(0)

circuit.draw(output="mpl")

2.2 Two Qubits

For two qubits, we initialize a Quantum Circuit with \(2\) qubits and apply \(H\) gates to each qubit.

# Initialize 2 qubit circuit
circuit = QuantumCircuit(2)
# Apply Hadamard on first qubit and second qubit
circuit.h(0)
circuit.h(1)

circuit.draw(output="mpl")