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--- quantum-gate [May 14, 2026 at 11:38] – external edit 127.0.0.1
+++ quantum-gate [June 13, 2026 at 03:13] (current) – external edit 127.0.0.1
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 # Quantum gate
-**Quantum gates** (or  **unitary quantum operators**) are the quantum computing equivalent of classical logic gates like "AND", "OR", "XOR", "NOT".
+**A quantum gate** is a unitary operation applied to one or more qubits. It is the quantum computing analog of a classical logic gate, but with two key differences: quantum gates are represented by matrices rather than truth tables, and they are always reversible — every quantum gate has an inverse.
-Classical logic gates are defined by their truth tables. They follow a mathematical structure called Boolean algebra. With the advancement of MOSFET transistor it became possible to implement these logic gates electrically.
+A quantum gate acting on $n$ qubits is a $2^n \times 2^n$ unitary matrix $U$, meaning $U^\dagger U = I$. Applying a gate to a state vector $\lvert\psi\rangle$ is a matrix-vector multiplication $U\lvert\psi\rangle$. Unitarity ensures that the total probability remains 1 after the operation. The reversibility of quantum gates follows directly from unitarity: the inverse of $U$ is $U^\dagger$, so every gate operation can be undone by applying the conjugate transpose.
+Classical logic gates like AND and OR are irreversible — given only the output, you cannot recover both inputs. Quantum gates cannot be irreversible in this way because unitary evolution is bijective. The only irreversible step in a quantum computation is **measurement**, which collapses the state vector and cannot be undone.
+Single-qubit gates are $2 \times 2$ unitary matrices. The most general single-qubit gate is the [[u-gate|U gate]] $U(\theta, \phi, \lambda)$, which subsumes all others as special cases. Two-qubit gates are $4 \times 4$ unitary matrices; the [[cx-gate|CX gate]] is the standard entangling two-qubit gate. Together, single-qubit gates and CX form a universal gate set: any $n$-qubit unitary can be approximated to arbitrary precision using only these.
-Quantum gates work in a similar way. They are defined by matrices. They follow the rules of a mathematical structure called the Lie group.