Gates & Circuits

$U^\dagger U = U U^\dagger = I,$

where $U^\dagger$ is the conjugate transpose. Unitarity has three consequences. It preserves length, so a normalized state stays normalized and the Born probabilities keep summing to one. It is reversible: every gate has an inverse $U^{-1} = U^\dagger$, so no information is destroyed, the opposite of a classical AND gate, which maps two bits to one and forgets the input. And it is linear: a gate applied to a superposition acts on each component and the results add, which is precisely what lets amplitudes interfere. The one non-unitary, irreversible step in the whole model is measurement.

Single-qubit gates

The essential single-qubit gates are the Pauli gates and the Hadamard. Written in the computational basis:

$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad H = \tfrac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.$

The $X$ gate is the quantum bit flip: $X|0\rangle = |1\rangle$ and $X|1\rangle = |0\rangle$. The $Z$ gate is a phase flip: it leaves $|0\rangle$ alone and sends $|1\rangle \to -|1\rangle$, touching phase without touching measurement probabilities in the computational basis. The Hadamard $H$ is the workhorse that manufactures superposition:

$H|0\rangle = \tfrac{1}{\sqrt{2}}\bigl(|0\rangle + |1\rangle\bigr) = |+\rangle, \qquad H|1\rangle = \tfrac{1}{\sqrt{2}}\bigl(|0\rangle - |1\rangle\bigr) = |-\rangle.$

$H$ is its own inverse ($H^2 = I$), a fact we will use in a moment. Rounding out the toolkit, the phase gates $S$ and $T$ add a fixed relative phase to $|1\rangle$ (a quarter and an eighth turn), and together with $H$ and $\text{CNOT}$ they form a universal set, enough to approximate any quantum computation.

Two-qubit gates and entanglement: CNOT

To compute anything interesting you need gates that couple qubits. The controlled-NOT (CNOT, or CX) acts on two qubits, a control and a target, and flips the target only when the control is in state $|1\rangle$. In the basis ordered $|00\rangle, |01\rangle, |10\rangle, |11\rangle$ (control first),

$\text{CNOT} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix},$

so it maps $|00\rangle \to |00\rangle$, $|01\rangle \to |01\rangle$, $|10\rangle \to |11\rangle$, and $|11\rangle \to |10\rangle$. On classical basis inputs it looks mundane. Its power appears when the control is itself in superposition, then CNOT produces a state that cannot be factored into two independent qubits, which is entanglement.

Worked example: building a Bell state

Start both qubits in $|0\rangle$, i.e. the two-qubit state $|00\rangle$. Apply $H$ to qubit 0 (the control):

$H_0\,|00\rangle = \Bigl(\tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\Bigr)\otimes|0\rangle = \tfrac{1}{\sqrt{2}}\bigl(|00\rangle + |10\rangle\bigr).$

Now apply CNOT with control 0 and target 1. It leaves the $|00\rangle$ term untouched and flips the target of the $|10\rangle$ term to give $|11\rangle$:

$\text{CNOT}\cdot\tfrac{1}{\sqrt{2}}\bigl(|00\rangle + |10\rangle\bigr) = \tfrac{1}{\sqrt{2}}\bigl(|00\rangle + |11\rangle\bigr).$

This is the Bell state $|\Phi^+\rangle$, a maximally entangled pair. Measure it and you get $00$ or $11$, each with probability one half, but never $01$ or $10$: the two outcomes are perfectly correlated no matter how far apart the qubits are carried. Crucially, this correlation cannot be written as any product $|\psi\rangle_0 \otimes |\phi\rangle_1$ of separate single-qubit states, that non-factorability is the mathematical definition of entanglement. And it does not allow faster-than-light signaling: each qubit measured alone looks like a fair random coin, and only comparing results later reveals the correlation (the no-communication theorem).

1. 11. InitializeBoth qubits start in the zero state, giving the joint state where both read zero.
2. 22. Hadamard on the controlQubit 0 becomes an equal superposition, so the pair is now an equal blend of both-zero and control-one-target-zero.
3. 33. CNOT couples themThe target flips only in the branch where the control is one, linking the two qubits.
4. 44. Result: a Bell pairAn equal superposition of both-zero and both-one. Measuring gives perfectly correlated outcomes and nothing in between.

from qiskit import QuantumCircuit

qc = QuantumCircuit(2, 2)      # 2 qubits, 2 classical bits
qc.h(0)                        # Hadamard on qubit 0 -> superposition
qc.cx(0, 1)                    # CNOT: control 0, target 1 -> entangle
qc.measure([0, 1], [0, 1])     # measure both qubits into the classical bits

# The statevector just before measurement is (|00> + |11>)/sqrt(2).
# Sampling many shots yields roughly 50% '00' and 50% '11',
# and essentially never '01' or '10'.

Interference is the real engine

Here is the fact that separates quantum computing from a classical coin-flipping machine. Because amplitudes are signed (indeed complex), they can cancel. Apply a Hadamard twice to $|0\rangle$:

$H|0\rangle = \tfrac{1}{\sqrt{2}}\bigl(|0\rangle + |1\rangle\bigr), \qquad H\Bigl(\tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\Bigr) = \tfrac{1}{2}\bigl(|0\rangle + |1\rangle\bigr) + \tfrac{1}{2}\bigl(|0\rangle - |1\rangle\bigr) = |0\rangle.$

The $|1\rangle$ contributions have opposite signs and destructively interfere to zero, while the $|0\rangle$ contributions reinforce, you recover $|0\rangle$ with certainty. A classical process that flips a fair coin twice never returns to a definite state; a quantum one can, because the intermediate "paths" carry amplitudes that cancel. Every real quantum algorithm, Deutsch–Jozsa, Grover, Shor, is an arrangement of gates that steers destructive interference onto wrong answers and constructive interference onto right ones before the final measurement. This is why "it tries all inputs at once" is the wrong mental model: the machine does not read a giant table, it sculpts amplitudes so the answer you want survives.

Explaining entanglement three ways

Further reading

• Nielsen & Chuang, Quantum Computation and Quantum Information, Chapters 1–4 cover gates, universality, and entanglement rigorously.
• IBM Quantum Learning, interactive lessons on single- and multi-qubit gates and Bell states.
• Qiskit documentation, API reference and guides for building and running the circuits shown here.
• John Preskill, Caltech Ph219 lecture notes, deeper treatment of unitarity, entanglement, and the no-cloning and no-communication theorems.