In quantum computing and quantum information theory, the Clifford gates are the elements of the Clifford group, a set of mathematical transformations which normalize the n-qubit Pauli group, i.e., map tensor products of Pauli matrices to tensor products of Pauli matrices through conjugation. The notion was introduced by Daniel Gottesman and is named after the mathematician William Kingdon Clifford.[1] Quantum circuits that consist of only Clifford gates can be efficiently simulated with a classical computer due to the Gottesman–Knill theorem.
The Clifford group is generated by three gates: Hadamard, phase gate S, and CNOT.[2][3][4] This set of gates is minimal in the sense that discarding any one gate results in the inability to implement some Clifford operations; removing the Hadamard gate disallows powers of in the unitary matrix representation, removing the phase gate S disallows in the unitary matrix, and removing the CNOT gate reduces the set of implementable operations from to . Since all Pauli matrices can be constructed from the phase and Hadamard gates, each Pauli gate is also trivially an element of the Clifford group.
The gate is equal to the product of and gates. To show that a unitary is a member of the Clifford group, it suffices to show that for all that consist only of the tensor products of and , we have .
^Gottesman, Daniel (1998-01-01). "Theory of fault-tolerant quantum computation" (PDF). Physical Review A. 57 (1): 127–137. arXiv:quant-ph/9702029. Bibcode:1998PhRvA..57..127G. doi:10.1103/physreva.57.127. ISSN 1050-2947. S2CID 8391036.
^Nielsen, Michael A.; Chuang, Isaac L. (2010-12-09). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press. ISBN 978-1-107-00217-3.
^Gottesman, Daniel (1998-01-01). "Theory of fault-tolerant quantum computation". Physical Review A. 57 (1): 127–137. arXiv:quant-ph/9702029. Bibcode:1998PhRvA..57..127G. doi:10.1103/PhysRevA.57.127. ISSN 1050-2947. S2CID 8391036.
^Gottesman, Daniel (1997-05-28). Stabilizer Codes and Quantum Error Correction (PhD thesis). Caltech. arXiv:quant-ph/9705052. Bibcode:1997PhDT.......232G.
quantum computing and quantum information theory, the Cliffordgates are the elements of the Clifford group, a set of mathematical transformations which...
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circuits, like classical logic gates are for conventional digital circuits. Unlike many classical logic gates, quantum logic gates are reversible. It is possible...
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{\displaystyle \rho } , a tomographically complete set of gates U {\displaystyle U} (e.g. Cliffordgates), a set of M {\displaystyle M} observables { O i } {\displaystyle...
{\displaystyle \rho } , a tomographically complete set of gates U {\displaystyle U} (e.g Cliffordgates), a set of M {\displaystyle M} observables { O i } {\displaystyle...
qubit rotational gates and 2 qubit entangling gates. The number of repetitions of 1-qubit rotational gates and 2-qubit entangling gates is called the Depth...
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