Magic state distillation explained

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.

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^{[5] [6]} and distillation routines for qubits^{[7] [8] [9]} with various advantages have been proposed.

Stabilizer formalism

See main article: Stabilizer formalism. The Clifford group consists of a set of

n -qubit operations generated by the gates (where H is Hadamard and S is \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.^[10]

Magic states

Magic states are purified from

n copies of a mixed state \rho .^[5] These states are typically provided via an ancilla to the circuit. A magic state for the \pi/ 6 rotation operator is |M\rangle = \cos(\beta/ 2 )| 0 \rangle + e ^{i (\pi)/(4)} \sin(\beta/ 2 )| 1 \rangle where \beta = \arccos\left( (1)/(\sqrt 3) \right) . By combining (copies of) magic states with Clifford gates, can be used to make a non-Clifford gate.^[10] Since Clifford gates combined with a non-Clifford gate are 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 a follows.^[10]

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

| 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.

Notes and References

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