Ivan's wiki

Single-qubit gates

Ivan Janevski (iv@undisclosed.example.com) — 2026-08-26T18:16:40+00:00

Single-qubit gates Single-qubit gates are unitary operations that act on one qubit. They are represented by $2 \times 2$ unitary matrices and form the Lie group $\mathrm{SU}(2)$. Every quantum computation can be decomposed into single-qubit rotations and two-qubit entangling gates (e.g., CNOT), making single-qubit gates the fundamental building blocks of quantum circuits.$|\psi\rangle$$U|\psi\rangle$$S^4 = I$$T^8 = I$$\theta$$\mathrm{SU}(2)$$R_X$$R_Y$$R_Z$$U(\phi, \theta, \lambda)$$2 \times 2$$…

Two-qubit gates

Ivan Janevski (iv@undisclosed.example.com) — 2026-08-26T18:03:32+00:00

Two-qubit gates Two-qubit gates are unitary operations that act on a pair of qubits. They are represented by $4 \times 4$ unitary matrices and form the Lie group $\mathrm{SU}(4)$. Unlike single-qubit gates, which only rotate individual qubits on their own Bloch spheres, two-qubit gates can create entanglement between qubits, which is what makes a quantum computer more than a collection of independent classical bits.$|\psi\rangle \in \mathbb{C}^4$$U|\psi\rangle$$n$$A \otimes B$$A, B$$\text{XX}$$…

Quantum gate

Ivan Janevski (iv@undisclosed.example.com) — 2026-08-26T18:02:48+00:00

Quantum gate Quantum gates are unitary operations that transform quantum states. Like classical logic gates (AND, OR, NOT), quantum gates manipulate qubits -- but they preserve superposition and enable entanglement. Every quantum gate is reversible (unitary matrix $U$$U^\dagger U = I$$n$$2^n \times 2^n$$\mathrm{SU}(2^n)$$|\psi\rangle$$U|\psi\rangle$

Three-qubit gates - removed

Ivan Janevski (iv@undisclosed.example.com) — 2026-08-26T17:46:37+00:00

Three-qubit gates Three-qubit gates are unitary operations that act on a triple of qubits. They are represented by $8 \times 8$ unitary matrices and form the Lie group $\mathrm{SU}(8)$. Beyond what two-qubit gates offer, three-qubit gates expose genuinely tripartite entanglement, correlation that can't be reduced to any pair of qubits, and several of them (CCX, CSWAP) are universal for classical reversible computation on their own.$|\psi\rangle \in \mathbb{C}^8$$U|\psi\rangle$$\mathrm{SU}(8)$$\…

CX gate (Controlled-NOT)

Ivan Janevski (iv@undisclosed.example.com) — 2026-08-26T17:28:45+00:00

CX gate (Controlled-NOT) CX gate (or CNOT gate, Controlled-NOT gate) is the most common two-qubit gate: flips the target qubit if the control qubit is $|1\rangle$. Essential for creating entanglement. Matrix (control on qubit 0, target on qubit 1): $$\text{CX} = \text{CNOT} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$$ Action on basis states:$$|00\rangle \to |00\rangle, \quad |01\rangle \to |01\rangle, \quad |10\rangle \to |11\rangle, \qua…

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