Magic state distillation

Magic state distillation is a method for creating more accurate quantum states from multiple noisy ones, which is important^[1] for building fault tolerant quantum computers. It has also been linked^[2] to quantum contextuality, a concept thought to contribute to quantum computers' power.^[3]

The technique was first proposed by Emanuel Knill in 2004,^[4] and further analyzed by Sergey Bravyi and Alexei Kitaev the same year.^[5]

Thanks to the Gottesman–Knill theorem, it is known that some quantum operations (operations in the Clifford group) can be perfectly simulated in polynomial time on a classical computer. In order to achieve universal quantum computation, a quantum computer must be able to perform operations outside this set. Magic state distillation achieves this, in principle, by concentrating the usefulness of imperfect resources, represented by mixed states, into states that are conducive for performing operations that are difficult to simulate classically.

A variety of qubit magic state distillation routines^[6][7] and distillation routines for qubits^[8][9][10] with various advantages have been proposed.

Stabilizer formalism

Main article: Stabilizer formalism

The Clifford group consists of a set of ${\displaystyle n}$-qubit operations generated by the gates {H, S, CNOT} (where H is Hadamard and S is ${\displaystyle {\begin{bmatrix}1&0\\0&i\end{bmatrix}}}$) called Clifford gates. The Clifford group generates stabilizer states which can be efficiently simulated classically, as shown by the Gottesman–Knill theorem. This set of gates with a non-Clifford operation is universal for quantum computation.^[5]

Magic states

Magic states are purified from ${\displaystyle n}$ copies of a mixed state ${\displaystyle \rho }$.^[6] These states are typically provided via an ancilla to the circuit. A magic state for the ${\displaystyle \pi /6}$ rotation operator is ${\displaystyle |M\rangle =\cos(\beta /2)|0\rangle +e^{i{\frac {\pi }{4}}}\sin(\beta /2)|1\rangle }$ where ${\displaystyle \beta =\arccos \left({\frac {1}{\sqrt {3}}}\right)}$. A non-Clifford gate can be generated by combining (copies of) magic states with Clifford gates.^[5] Since a set of Clifford gates combined with a non-Clifford gate is universal for quantum computation, magic states combined with Clifford gates are also universal.

Purification algorithm for distilling |M〉

The first magic state distillation algorithm, invented by Sergey Bravyi and Alexei Kitaev, is as follows.^[5]

Input: Prepare 5 imperfect states.

Output: An almost pure state having a small error probability.

repeat

Apply the decoding operation of the five-qubit error correcting code and measure the syndrome.

If the measured syndrome is ${\displaystyle |0000\rangle }$, the distillation attempt is successful.

else Get rid of the resulting state and restart the algorithm.

until The states have been distilled to the desired purity.

References

1. Campbell, Earl T.; Terhal, Barbara M.; Vuillot, Christophe (14 September 2017). "Roads towards fault-tolerant universal quantum computation" (PDF). Nature. 549 (7671): 172–179. arXiv:1612.07330. Bibcode:2017Natur.549..172C. doi:10.1038/nature23460. PMID 28905902. S2CID 4446310.
2. Howard, Mark; Wallman, Joel; Veitch, Victor; Emerson, Joseph (11 June 2014). "Contextuality supplies the 'magic' for quantum computation". Nature. 510 (7505): 351–355. arXiv:1401.4174. Bibcode:2014Natur.510..351H. doi:10.1038/nature13460. PMID 24919152. S2CID 4463585.
3. Bartlett, Stephen D. (11 June 2014). "Powered by magic". Nature. 510 (7505): 345–347. doi:10.1038/nature13504. PMID 24919151.
4. Knill, E. (2004). "Fault-Tolerant Postselected Quantum Computation: Schemes". arXiv:quant-ph/0402171. Bibcode:2004quant.ph..2171K. {{cite journal}}: Cite journal requires |journal= (help)
5. Bravyi, Sergey; Kitaev, Alexei (2005). "Universal quantum computation with ideal Clifford gates and noisy ancillas". Physical Review A. 71 (2): 022316. arXiv:quant-ph/0403025. Bibcode:2005PhRvA..71b2316B. doi:10.1103/PhysRevA.71.022316. S2CID 17504370.
6. Bravyi, Sergey; Haah, Jeongwan (2012). "Magic state distillation with low overhead". Physical Review A. 86 (5): 052329. arXiv:1209.2426. Bibcode:2012PhRvA..86e2329B. doi:10.1103/PhysRevA.86.052329. S2CID 4399674.
7. Meier, Adam; Eastin, Bryan; Knill, Emanuel (2013). "Magic-state distillation with the four-qubit code". Quantum Information & Computation. 13 (3–4): 195–209. arXiv:1204.4221. doi:10.26421/QIC13.3-4-2. S2CID 27799877.
8. Campbell, Earl T.; Anwar, Hussain; Browne, Dan E. (27 December 2012). "Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes". Physical Review X. 2 (4): 041021. arXiv:1205.3104. Bibcode:2012PhRvX...2d1021C. doi:10.1103/PhysRevX.2.041021.
9. Campbell, Earl T. (3 December 2014). "Enhanced Fault-Tolerant Quantum Computing in d -Level Systems". Physical Review Letters. 113 (23): 230501. arXiv:1406.3055. Bibcode:2014PhRvL.113w0501C. doi:10.1103/PhysRevLett.113.230501. PMID 25526106. S2CID 24978175.
10. Prakash, Shiroman (September 2020). "Magic state distillation with the ternary Golay code". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 476 (2241): 20200187. arXiv:2003.02717. Bibcode:2020RSPSA.47600187P. doi:10.1098/rspa.2020.0187. PMC 7544352. PMID 33071576.