Quantum Computing 101

Superposition

Superposition means several basis states carry amplitude at once, and those amplitudes can reinforce or cancel each other.

In this lesson you will learn to

Tell a superposition apart from a random classical bit.
Use H to create equal superpositions on one or many qubits.
Explain interference using the fact that applying H twice does nothing.
Recognize relative phase in the states |+⟩ and |−⟩.

More Than a Coin Flip

A qubit is in superposition when more than one basis state has a nonzero amplitude. The state |+⟩ = (|0⟩ + |1⟩)/√2 is the classic example: measuring it gives 0 or 1, each half the time.

So far that sounds like a fair coin. The difference appears when you act on the state before measuring. A coin flipped twice is still random, but a qubit's amplitudes can add up or cancel out. This effect, interference, is the engine behind every quantum algorithm.

A coin flipped twice is still 50/50. A qubit sent through H twice comes back to |0⟩ every time: the two routes into |1⟩ have amplitudes +½ and −½ and cancel, while the routes into |0⟩ add up to 1.

Worked example: H twice returns |0⟩

Apply H to |+⟩ one basis state at a time, then collect matching terms.

H\left(\frac{|0\rangle+|1\rangle}{\sqrt{2}}\right)=\frac{1}{\sqrt{2}}\left(H|0\rangle+H|1\rangle\right)

=\frac{1}{2}\left(|0\rangle+|1\rangle\right)+\frac{1}{2}\left(|0\rangle-|1\rangle\right)

=\underbrace{\left(\tfrac12+\tfrac12\right)}_{1}|0\rangle+\underbrace{\left(\tfrac12-\tfrac12\right)}_{0}|1\rangle=|0\rangle

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

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

print(Statevector.from_instruction(qc).probabilities_dict(decimals=3))
# {'0': 1.0}

Hadamard Gates

The Hadamard gate H is the standard way to create a superposition. It maps |0⟩ to |+⟩ and |1⟩ to |−⟩ = (|0⟩ − |1⟩)/√2.

Applying H to every qubit of a register spreads the amplitude evenly over every basis state. With n qubits there are 2ⁿ basis states, so each gets amplitude 1/√(2ⁿ) and probability 1/2ⁿ.

Applying H to every qubit of |0…0⟩ spreads the probability evenly over all basis states. Each extra qubit doubles their number: 2, 4, 8, … 2ⁿ.

Worked example: two qubits

\begin{aligned}(H\otimes H)|00\rangle&=\left(\frac{|0\rangle+|1\rangle}{\sqrt{2}}\right)\otimes\left(\frac{|0\rangle+|1\rangle}{\sqrt{2}}\right)\\[6pt]&=\frac{|00\rangle+|01\rangle+|10\rangle+|11\rangle}{2}\end{aligned}

\text{Each probability}=\left|\frac{1}{2}\right|^2=\frac{1}{4}=25\%\qquad 4\cdot\frac{1}{4}=1

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

qc = QuantumCircuit(2)
qc.h([0, 1])

print(Statevector.from_instruction(qc).probabilities_dict(decimals=3))
# {'00': 0.25, '01': 0.25, '10': 0.25, '11': 0.25}

›Check yourself: You apply H to each of 5 qubits, starting from |00000⟩. How many basis states have nonzero amplitude, and what is the probability of each?

2⁵ = 32 basis states, each with amplitude 1/√32 and probability 1/32, about 3.1%.

Relative Phase

|+⟩ and |−⟩ give exactly the same measurement statistics: 50% each. The only difference is the minus sign on |1⟩, which is called a relative phase.

Relative phase is invisible to a direct measurement but not to the next gate. Applying H turns |+⟩ into |0⟩ and |−⟩ into |1⟩, so the hidden sign becomes a certain, measurable result.

Worked example: same odds, different futures

|+⟩ and |−⟩ measure identically, but one more H separates them completely.

|+\rangle=\frac{|0\rangle+|1\rangle}{\sqrt{2}}\qquad |-\rangle=\frac{|0\rangle-|1\rangle}{\sqrt{2}}

P_{|+\rangle}(0)=P_{|+\rangle}(1)=P_{|-\rangle}(0)=P_{|-\rangle}(1)=\frac{1}{2}

H|+\rangle=|0\rangle\qquad H|-\rangle=|1\rangle

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

plus = QuantumCircuit(1)
plus.h(0)                 # |+>
minus = QuantumCircuit(1)
minus.x(0)
minus.h(0)                # |->

for name, qc in [("plus", plus), ("minus", minus)]:
    print(name, Statevector.from_instruction(qc).probabilities_dict(decimals=3))
    qc.h(0)
    print(name, "after H", Statevector.from_instruction(qc).probabilities_dict(decimals=3))
# plus {'0': 0.5, '1': 0.5}
# plus after H {'0': 1.0}
# minus {'0': 0.5, '1': 0.5}
# minus after H {'1': 1.0}

Key Takeaways

A superposition gives amplitude to several basis states at once.
Unlike a coin, amplitudes interfere: routes to the same outcome add, signs included.
H on every qubit of |0…0⟩ spreads amplitude evenly over all 2ⁿ basis states.
|+⟩ and |−⟩ have the same probabilities but different relative phase, and H reveals it.

What To Practice

Practice creating superpositions, predicting every amplitude, and steering phases.