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Topological quantum memory

### Abstract

We analyze*surface codes*, the topological quantum error-correcting codes introduced by Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of nontrivial topology, and encoded quantum operations are associated with nontrivial homology cycles of the surface. We formulate protocols for error recovery, and study the efficacy of these protocols. An order-disorder phase transition occurs in this system at a nonzero critical value of the error rate; if the error rate is below the critical value (the *accuracy threshold*), encoded information can be protected arbitrarily well in the limit of a large code block. This phase transition can be accurately modeled by a three-dimensional lattice gauge theory with quenched disorder. We estimate the accuracy threshold, assuming that all quantum gates are *local*, that qubits can be measured rapidly, and that polynomial-size classical computations can be executed instantaneously. We also devise a robust recovery procedure that does not require measurement or fast classical processing; however, for this procedure the quantum gates are local only if the qubits are arranged in *four* or more spatial dimensions. We discuss procedures for encoding, measurement, and performing fault-tolerant universal quantum computation with surface codes, and argue that these codes provide a promising framework for quantum computing architectures.

© 2002 American Institute of Physics

Received 25 October 2001
Accepted 16 May 2002
Published online 20 August 2002

/content/aip/journal/jmp/43/9/10.1063/1.1499754

1.

1.P. W. Shor, Phys. Rev. A 52, 2493 (1995).

2.

2.A. Steane, Phys. Rev. Lett. 77, 793 (1996).

3.

3.P. W. Shor, “Fault-tolerant quantum computation,” in Proceedings, 37th Annual Symposium on Foundations of Computer Science (IEEE, Los Alamitos, CA, 1996), pp. 56–65;

4.

4.A. Yu. Kitaev, “Quantum error correction with imperfect gates,” in Proceedings of the Third International Conference on Quantum Communication and Measurement, edited by O. Hirota, A. S. Holevo, and C. M. Caves (Plenum, New York, 1997).

5.

5.A. Yu. Kitaev, “

Fault-tolerant quantum computation by anyons,”

quant-ph/9707021.

6.

6.E. Knill, R. Laflamme, and W. H. Zurek, “Resilient quantum computation: error models and thresholds,” Proc. R. Soc. London, Ser. A 454, 365 (1998);

7.

7.D. Aharonov and M. Ben-Or, “Fault-tolerant quantum computation with constant error,” in Proceedings of the 29th Annual ACM Symposium on Theory of Computing (ACM, New York, 1998), p. 176;

7.D. Aharonov and

M. Ben-Or, “

Fault-tolerant quantum computation with constant error rate,”

quant-ph/9906129.

8.

8.A. Yu. Kitaev, Russ. Math. Surveys 52, 1191 (1997).

9.

9.J. Preskill, Proc. R. Soc. London, Ser. A 454, 385 (1998);

10.

10.D. Gottesman, “Stabilizer codes and quantum error correction,” Caltech Ph.D. thesis 1997;

11.

11.D. Gottesman and J. Preskill, unpublished.

12.

12.D. Gottesman, “

Fault tolerant quantum computation with local gates,”

quant-ph/9903099.

13.

13.S. B. Bravyi and

A. Yu. Kitaev, “

Quantum codes on a lattice with boundary,”

quant-ph/9810052.

14.

14.M. H. Freedman and

D. A. Meyer, “

Projective plane and planar quantum codes,”

quant-ph/9810055.

15.

15.P. Gaćs, J. Comput. Syst. Sci. 32, 15 (1986).

16.

16.D. Aharonov,

M. Ben-Or,

R. Impagliazzo, and

N. Nisan, “

Limitations of noisy reversible computation,”

quant-ph/9611028.

17.

17.A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Phys. Rev. Lett. 78, 405 (1997);

17.A. R. Calderbank,

E. M. Rains,

P. W. Shor, and

N. J. A. Sloane,

quant-ph/9605005.

18.

18.D. Gottesman, Phys. Rev. A 54, 1862 (1996);

19.

19.T. Einarsson, Phys. Rev. Lett. 64, 1995 (1990).

20.

20.X. G. Wen and Q. Niu, Phys. Rev. B 41, 9377 (1990).

21.

21.H. Nishimori, Prog. Theor. Phys. 66, 1169 (1981).

22.

22.I. A. Gruzberg, N. Read, and A. W. W. Ludwig, “Random-bond Ising model in two dimensions, the Nishimori line, and supersymmetry,” Phys. Rev. B 63, 104422 (2001);

23.

23.M. G. Alford, K.-M. Lee, J. March-Russell, and J. Preskill, Nucl. Phys. B 384, 251 (1992);

23.M. G. Alford,

K.-M. Lee,

J. March-Russell, and

J. Preskill,

hep-th/9112038.

24.

24.A. Honecker,

M. Picco, and

P. Pujol, “

Nishimori point in the 2D random-bond Ising model,”

cond-mat/00010143.

25.

25.F. Merz and

J. T. Chalker, “

The two-dimensional random-bond Ising model, free fermions and the network model,”

cond-mat/0106023.

26.

26.A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,” Phys. Rev. A 54, 1098 (1996);

27.

27.A. Steane, “Multiple particle interference and quantum error correction,” Proc. R. Soc. London, Ser. A 452, 2551 (1996);

28.

28.H. Nishimori, “Geometry-induced phase transition in the Ising model,” J. Phys. Soc. Jpn. 55, 3305 (1986).

29.

29.H. Kitatani, “The verticality of the ferromagnetic-spin glass phase boundary of the Ising Model in the plane,” J. Phys. Soc. Jpn. 61, 4049 (1992).

30.

30.J. Edmonds, “Paths, trees and flowers,” Can. J. Math. 17, 449 (1965).

31.

31.F. Barahona, R. Maynard, R. Rammal, and J. P. Uhry, J. Phys. A 15, 673 (1982).

32.

32.N. Kawashima and H. Rieger, Europhys. Lett. 39, 85 (1997);

33.

33.C. Vanderzande, Lattice Models of Polymers (Cambridge University Press, Cambridge, UK, 1998).

34.

34.N. Madras and G. Slade, The Self-Avoiding Walk (Birkhäuser, Boston, 1996).

35.

35.P.-G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, NY, 1970).

36.

36.A. Steane, Phys. Rev. Lett. 78, 2252 (1997);

37.

37.D. Gottesman, “

The Heisenberg representation of quantum computers,”

quant-ph/9807006.

38.

38.D. Gottesman, Phys. Rev. A 57, 127 (1998);

39.

39.A. Yu. Kitaev, unpublished.

40.

40.E. Dennis, Phys. Rev. A 63, 052314 (2001);

41.

41.W. Ogburn and J. Preskill, “Topological quantum computation,” Lect. Notes Comput. Sci. 1509, 341 (1999).

42.

42.M. H. Freedman,

A. Kitaev,

M. J. Larsen, and

Z. Wang, “

Topological quantum computation,”

quant-ph/0101025.

43.

43.A. L. Toom, “Stable and attractive trajectories in multicomponent systems,” in Advances in Probability 6, edited by R. L. Dobrushin (Dekke, New York, 1980), pp. 549–575.

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