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| quantum-gate [May 25, 2026 at 10:14] – Ivan Janevski | quantum-gate [June 13, 2026 at 03:13] (current) – external edit 127.0.0.1 |
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| # Quantum gate | # Quantum gate |
| **Quantum gates** 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. |
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| 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. |
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| Quantum gates work in a similar way. They are defined by matrices. They follow the rules of a mathematical structure called the Lie group. | 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. |
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| | 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. |
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| ## List of quantum gates | |
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| - [[i-gate]] | |
| - [[x-gate]] | |
| - [[y-gate]] | |
| - [[z-gate]] | |
| - [[r-gate]] | |
| - [[s-gate]] | |
| - [[t-gate]] | |
| - [[h-gate]] | |
| - [[p-gate]] | |
| - [[u-gate]] | |
| - [[ccnot-gate]] | |
| - [[cnot-gate]] | |
| - [[swap-gate]] | |
| - [[iswap-gate]] | |
