Clifford gates

Short description: Definition of quantum circuits

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 1/2 in the unitary matrix representation, removing the phase gate S disallows i in the unitary matrix, and removing the CNOT gate reduces the set of implementable operations from 𝐂n to 𝐂1n. 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 Y gate is equal to the product of X and Z gates. To show that a unitary U is a member of the Clifford group, it suffices to show that for all P∈𝐏n that consist only of the tensor products of X and Z, we have UPU†∈𝐏n.

## Contents

* 1 Common generating gates
* 1.1 Hadamard gate
* 1.2 S gate
* 1.3 CNOT gate
* 2 Building a universal set of quantum gates
* 3 See also
* 4 References

## Common generating gates

### Hadamard gate

The Hadamard gate

H=12[111−1]

is a member of the Clifford group as HXH†=Z and HZH†=X.

### S gate

The Phase gate

S=[100eiπ2]=[100i]=Z

is a Clifford gate as SXS†=Y and SZS†=Z.

### CNOT gate

The CNOT gate applies to two qubits. It is a (C)ontrolled NOT gate, where a NOT gate is performed on qubit 2 if and only if qubit 1 is in the 1 state.

CNOT=[1000010000010010].

Between X and Z there are four options:

CNOT combinations  P | CNOT P CNOT †

|

X⊗I | X⊗X
I⊗X | I⊗X
Z⊗I | Z⊗I
I⊗Z | Z⊗Z

## Building a universal set of quantum gates

The Clifford gates do not form a universal set of quantum gates as some gates outside the Clifford group cannot be arbitrarily approximated with a finite set of operations. An example is the phase shift gate (historically known as the π/8 gate):

T=[100eiπ4]=S=Z4.

The following shows that the T gate does not map the Pauli-X gate to another Pauli matrix:

TXT†=[100eiπ4][0110][100e−iπ4]=[0e−iπ4eiπ40]∉𝐏1

However, the Clifford group, when augmented with the T gate, forms a universal quantum gate set for quantum computation.[5] Moreover, exact, optimal circuit implementations of the single-qubit Z-angle rotations are known.[6][7]

## See also

* Magic state distillation
* Clifford algebra

## References

1. ↑ Gottesman, Daniel (1998-01-01). "Theory of fault-tolerant quantum computation". Physical Review A 57 (1): 127–137. doi:10.1103/physreva.57.127. ISSN 1050-2947. Bibcode: 1998PhRvA..57..127G. https://authors.library.caltech.edu/3850/1/GOTpra98.pdf.
2. ↑ Nielsen, Michael A.; Chuang, Isaac L. (2010-12-09) (in en). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press. ISBN 978-1-107-00217-3. https://books.google.com/books?id=j2ULnwEACAAJ.
3. ↑ Gottesman, Daniel (1998-01-01). "Theory of fault-tolerant quantum computation" (in en). Physical Review A 57 (1): 127–137. doi:10.1103/PhysRevA.57.127. ISSN 1050-2947. Bibcode: 1998PhRvA..57..127G. https://link.aps.org/doi/10.1103/PhysRevA.57.127.
4. ↑ Gottesman, Daniel (1997-05-28). Stabilizer Codes and Quantum Error Correction (PhD thesis). Caltech. arXiv:quant-ph/9705052. Bibcode:1997PhDT.......232G.
5. ↑ Forest, Simon; Gosset, David; Kliuchnikov, Vadym; McKinnon, David. "Exact Synthesis of Single-Qubit Unitaries Over Clifford-Cyclotomic Gate Sets". Journal of Mathematical Physics.
6. ↑ Ross, Neil J.; Selinger, Peter (2014). Optimal ancilla-free Clifford+ T approximation of z-rotations.
7. ↑ Kliuchnikov, Vadym; Maslov, Dmitri; Mosca, Michele (2013). "Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates". Quantum Information and Computation 13 (7–8): 607–630. doi:10.26421/QIC13.7-8-4.

* v
* t
* e

Quantum information science

General|

* Quantum computing
* Qubit
* physical vs. logical
* DiVincenzo's criteria
* Quantum information
* Quantum programming
* Quantum processors
* Cloud-based quantum computing
* Timeline of quantum computing

Theorems|

* Bell's theorem
* Gleason's theorem
* Gottesman–Knill theorem
* Holevo's theorem
* Margolus–Levitin theorem
* No-broadcast theorem
* No-cloning theorem
* No-communication theorem
* No-deleting theorem
* No-hiding theorem
* No-teleportation theorem
* PBR theorem
* Quantum threshold theorem

Quantum
communication|

* Quantum capacity
* Classical capacity
* Entanglement-assisted classical capacity
* Quantum channel
* Quantum network
* Quantum cryptography
* Quantum key distribution
* BB84
* SARG04
* Three-stage quantum cryptography protocol
* Quantum teleportation
* Superdense coding
* LOCC
* Entanglement distillation

Quantum algorithms|

* Universal quantum simulator
* Deutsch–Jozsa algorithm
* Grover's algorithm
* Quantum Fourier transform
* Shor's algorithm
* Simon's problem
* Quantum phase estimation algorithm
* Quantum counting algorithm
* Quantum annealing
* Quantum algorithm for linear systems of equations
* Amplitude amplification

Quantum
complexity theory|

* Quantum Turing machine
* EQP
* BQP
* QMA
* PostBQP
* QIP

Quantum
computing models|

* Quantum circuit
* Quantum logic gate
* One-way quantum computer
* cluster state
* Adiabatic quantum computation
* Topological quantum computer

Quantum
error correction|

* Stabilizer codes
* Entanglement-Assisted Quantum Error Correction
* Shor code
* Steane code
* CSS code
* Quantum convolutional codes
* Toric code

Physical
implementations|

| Quantum optics|

* Cavity QED
* Circuit QED
* Linear optical quantum computing
* KLM protocol
* Boson sampling

|

Ultracold atoms|

* Trapped ion quantum computer
* Optical lattice

Spin-based|

* Nuclear magnetic resonance QC
* Kane QC
* Loss–DiVincenzo QC
* Nitrogen-vacancy center

Superconducting
quantum computing|

* Charge qubit
* Flux qubit
* Phase qubit
* Transmon

Software|

* libquantum
* OpenQASM
* Q#
* Qiskit
* IBM Q Experience

0.00

(0 votes)

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Clifford_gates&oldid=3907599"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template