1. On the energy landscape of 3D spin Hamiltonians with topological order Sergey Bravyi (IBM Research) Jeongwan Haah (Caltech) QEC 2011 December 6, 2011 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A Phys.Rev.Lett. 107, 150504 (2011) and arXiv:1112.????

2. Main goal: Store a quantum state reliably for a macroscopic time in a presence of hardware imperfections and thermal noise without active error correction.

3. Towards topological self-correcting memories 2D toric code [Kitaev 97] Robust against small imperfections Constant threshold with active EC [Dennis et al 2001] No-go result for the thermal noise [Alicki, Fannes, Horodecki 2008] No-go result for all 2D stabilizer code [S.B. and Terhal 2008] No-go result for some 3D stabilizer codes [Yoshida 2011] ? Add one extra dimension to our space-time: [Alicki, Horodecki 3 2008] Most promising ideas: 2D + long range anyon-anyon interactions [Chesi et al 2009, Hamma et al 2009 ] 3D topological quantum spin glasses [Chamon 2005, Haah 2011, this work]

4. Encoding, storage, and decoding for memory Hamiltonians based on stabilizer codes Memory time of the 3D Cubic Code: rigorous lower bound and numerical simulation Topological quantum order, string-like logical operators, and the no-strings rule Logarithmic energy barrier for uncorrectable errors Outline

5. Qubits live at sites of a 2D or 3D lattice. O(1) qubits per site. Memory Hamiltonians based on stabilizer codes Hamiltonian = sum of local commuting Pauli stabilizers energy 0 1 2 3 [N,k,d] error correcting code Distance d≈ L Excited states with m=1,2,3… defects

6. Example: 3D Cubic Code [Haah 2011 ] ZZZI IZ ZI IZ XI IX XI IX XX 2 qubits per site, 2 stabilizers per cube II Each stabilizer acts on 8 qubits

7. Stabilizer code Hamiltonians with TQO: previous work 2D toric code and surface codes [Kitaev 97] 2D surface codes with twists [Bombin 2010] 2D topological color codes [Bombin and Martin-Delgado 2006] 3D toric code [Castelnovo, Chamon 2007] 3D topological spin glass model [Chamon 2005] 3D models with membrane condensation [Hamma,Zanardi, Wen 2004] Bombin, Martin-Delgado 2007] 4D toric code [Alicki, Horodecki 3 ] The only example of quantum self-correction

8. Storage: Markovian master equation Must be local, trace preserving, completely positive Evolution starts from a ground state of H. Lindblad operators L k act on O(1) qubits and have norm O(1). Each qubit is acted on by O(1) Lindblad operators.

9. Davies weak coupling limit Lindblad operator transfers energy from the system to the bath (quantum jump). The spectral density obeys detailed balance: Heat bath Memory system

10. Decoding Syndrome measurement: perform non-destructive eigenvalue measurement for each stabilizer G a. Error correction algorithm Measured syndrome Correcting Pauli operator The net action of the decoder: is the projector onto the subspace with syndrome s A list of all measured eigenvalues is called a syndrome.

11. Defect = spatial location of a violated stabilizer, decoder’s task is to annihilate the defects in a way which is most likely to return the memory to its original state. Defect diagrams will be used to represent syndromes. Example: 2D surface code: Z Z X X 1 3 X-error Z-error 2 4 2 4 1 3 Creates defects at squares 1,3 Creates defects at squares 2,4

12. Renormalization Group (RG) decoder* 1. Find connected defect clusters 2. For each connected cluster C Try to annihilate C by a Pauli operator acting inside b(C). Record the annihilation operator. 3 4. Increase unit of length by factor 2. 5. Go to the first step 3. Stop if no defects are left. 1 2 45 * J. Harrington, PhD thesis (2004), Duclos-Cianci and Poulin (2009) Measured syndrome Find the minimum enclosing box b(C).

13. 1 2 RG decoder 1. Find connected defect clusters 2. For each connected cluster C Try to annihilate C by a Pauli operator acting inside b(C). Record the annihilation operator. 3. Stop if no defects are left. Find the minimum enclosing box b(C). Syndrome after the 1 st iteration

14. RG decoder Failure 1: decoder has reached the maximum unit of length, but some defects are left. The decoder stops whenever all defects have been annihilated, or when the unit of length reached the lattice size. The correcting operator is chosen as the product of all recorded annihilation operators. Failure 2: all defects have been annihilated but the correcting operator does not return the system to the original state. RG decoder can be implemented in time poly(L)

15. Main goal for this talk: Derive an upper bound on the worst-case storage error: Initial ground state Lindblad evolution RG decoder

16. Theorem 1 However, the lattice size cannot be too large: If we are willing to tolerate error ε then the memory time is at least Optimal memory time at a fixed temperature is exponential in β 2 The storage error of the 3D Cubic Code decays polynomially with the lattice size L. Degree of the polynomial is proportional to β :

17. The theorem only provides a lower bound on the memory time. Is this bound tight ? We observed the exponential decay: Numerical estimate the memory time: Monte-Carlo simulation probability of the successful decoding on the time-evolved state at time t.

18. Each data point = 400 Monte Carlo samples with fixed L and β β=5.25 β=5.1 β=4.9 β=4.7 β=4.5 β=4.3 Optimal lattice size: log(L*) as function of β Exponent in the power law as function of β log(memory time) vs linear lattice size for the 3D Cubic Code 1,000 CPU-days on Blue Gene P

19. Numerical test of the scaling

20. Main theorem: sketch of the proof

21. An error path implementing a Pauli operator P is a finite sequence of single-qubit Pauli errors whose combined action coincides with P. Energy cost = maximum number of defects along the path. vacuum P1P1 P2P2 PtPt Energy barrier of a Pauli operator P is the smallest integer m such that P can be implemented by an error path with energy cost m Some terminology

22. Errors with high energy barrier can potentially confuse the decoder. However, such errors are not likely to appear. The thermal noise is likely to generate only errors with a small energy barrier. Decoder must be able to correct them. Basic intuition behind self-correction: Lemma (storage error) Suppose the decoder corrects all errors whose energy barrier is smaller than m. Then for any constant 0<a<1 one has Boltzmann factor Entropy factor = # physical qubits = # logical qubits

23. Suppose we choose Then the entropy factor can be neglected: and In order to have a non-trivial bound, we need at least logarithmic energy barrier for all uncorrectable errors:

24. More terminology [Haah 2011] A logical string segment is a Pauli operator whose action on the vacuum creates two well-separated clusters of defects. vacuum The smallest cubic boxes enclosing the two clusters of defects are called anchors

25. More terminology A logical string segment is trivial iff its action on the vacuum can be reproduced by operators localized near the anchors: vacuum

26. No-strings rule: There exist a constant α such that any logical string segment with aspect ratio > α is trivial. Aspect ratio = Distance between the anchors Size of the acnhors 3D Cubic Code obeys the no-strings rule with α=15 [Haah 2011] No 2D stabilizer code obeys the no-strings rule [S.B., Terhal 09]

27. Theorem 2 Consider any topological stabilizer code Hamiltonian on a D- dimensional lattice of linear size L. Suppose the code has TQO and obeys the no-strings rule with some constant α. Then the RG decoder corrects any error with the energy barrier at most c log(L). The constant c depends only on α and D. Haah’s 3D Cubic Code: α=15. Recall that errors with energy barrier >clog(L) are exponentially suppressed due to the Boltzmann factor. We have shown that

28. Sketch of the proof: logarithmic lower bound on the energy barrier of logical operators

29. Idea 1: No-strings rule implies `localization’ of errors S E1E1 S1S1 E2E2 S2S2 E3E3 S’ E 100 A stream of single-qubit errors: Suppose however that all intermediate syndromes are sparse: the distance between any pair of defects is >>α. Accumulated error: E= E 1 E 2 · · · E 100 could be very non-local A stream of local errors cannot move isolated topologically charged defects more than distance α away (the no-strings rule). Localization: E=E loc · S where S is a stabilizer and E loc is supported on the α-neighborhood of S and S’ · · ·

30. Idea 1: No-strings rule implies `localization’ of errors S E1E1 S1S1 E2E2 S2S2 E3E3 S’ E 100 A stream of single-qubit errors: Accumulated error: E= E 1 E 2 · · · E 100 could be very non-local Localization: E=E loc · S where S is a stabilizer and E loc is supported on the α-neighborhood of S and S’ · · · In order for the accumulated error to have a large weight at least one of the intermediate syndromes must be non-sparse (dense)

31. Idea 2: scale invariance and RG methods A stream of local errors cannot move an isolated charged cluster of defects of size R by distance more than αR away. In order for the accumulated error to have a large weight at least one of the intermediate syndromes must be non-sparse (dense) 1.Define sparseness and denseness at different spatial scales. 2.Show that in order for the accumulated error to have a REALLY large weight (of order L), at least one intermediate syndrome must be dense at roughly log(L) spatial scales. 3.Show that a syndrome which is dense at all spatial scales must contain at least clog(L) defects.

32. Definition: a syndrome S is called sparse at level p if it can be partitioned into disjoint clusters of defects such that 1.Each cluster has diameter at most r(p)=(10 α) p, 2.Any pair of clusters merged together has diameter greater than r(p+1) Otherwise, a syndrome is called dense at level p.

33. Lemma (Dense syndromes are expensive) Suppose a syndrome S is dense at all levels 0,…,p. Then S contains at least p+2 defects. p e 0123 e e e e 4 sparse

34. Renormalization group method 0 = vacuum, S = sparse syndromes, D= dense syndromes time RG level Level-0 syndrome history. Consecutive syndromes are related by single-qubit errors. Some syndromes are sparse (S), some syndromes are dense (D). We are given an error path implementing a logical operator P which maps a ground state to an orthogonal ground state. Record intermediate syndrome after each step in the path. It defines level-0 syndrome history:

35. Renormalization group method 0 = vacuum, S = sparse syndromes, D= dense syndromes time RG level Level-1 syndrome history includes only dense syndromes at level-0. Level-1 errors connect consecutive syndromes at level-0.

36. Renormalization group method 0 = vacuum, S = sparse syndromes, D= dense syndromes time RG level Level-1 syndrome history includes only dense syndromes at level-0. Level-1 errors connect consecutive syndromes at level-0. Use level-1 sparsity to label level-1 syndromes as sparse and dense.

37. Renormalization group method 0 = vacuum, S = sparse states, D= dense states time RG level Level-2 syndrome history includes only dense syndromes at level-1. Level-2 errors connect consecutive syndromes at level-1.

38. Renormalization group method 0 = vacuum, S = sparse states, D= dense states time RG level Level-2 syndrome history includes only dense excited syndromes at level-1. Level-2 errors connect consecutive syndromes at level-1. Use level-2 sparsity to label level-2 syndromes as sparse and dense.

39. Renormalization group method 0 = vacuum, S = sparse syndromes, D= dense syndromes time RG level At the highest RG level the syndrome history has no intermediate syndromes. A single error at the level p max implements a logical operator p max

40. Key technical result: Localization of level-p errors time RG level No-strings rule can be used to `localize’ level-p errors by multiplying them by stabilizers. Localized level-p errors connecting syndromes S and S’ act on r(p)-neighborhood of S and S’. p max

41. Localization of level-p errors time RG level TQO implies that r(p max ) > L since any logical operator must be very non-local. Therefore p max is at least log(L). At least one syndrome must be dense at all levels. Such syndrome must contain at least log(L) defects. p max

42. Conclusions The 3D Cubic Code Hamiltonian provides the first example of a (partially) quantum self-correcting memory. Memory time of the encoded qubit(s) grows polynomially with the lattice size. The degree of the polynomial is proportional to the inverse temperature β. The lattice size cannot be too big: L< L* ≈ exp(β). For a fixed temperature the optimal memory time is roughly exp(β 2 )