1. Preparing Topological States on a Quantum Computer Martin Schwarz (1), Kristan Temme (1), Frank Verstraete (1) Toby Cubitt (2), David Perez-Garcia (2) (1) University of Vienna (2) Complutense University, Madrid STV, Phys. Rev. Lett. 108, 110502 (2012) STVCP-G, (QIP 2012; paper in preparation)

2. Talk Outline Crash course on PEPS Growing PEPS in your Back Garden The Trouble with Tribbles Topological States Crash course on G-injective PEPS Growing Topological Quantum States

3. Crash Course on PEPS! Projected Entangled Pair State

4. Crash Course on PEPS! Projected Entangled Pair State Obtain PEPS by applying maps to maximally entangled pairs

5. Crash Course on PEPS! Parent Hamiltonian 2-local Hamiltonian with PEPS as ground state. Injectivity PEPS is “injective” if are left-invertible (perhaps only after blocking together sites) Uniqueness An injective PEPS is the unique ground state of its parent Hamiltonian

6. Are PEPS Physical? PEPS accurately approximate ground states of gapped local Hamiltonians. –Proven in 1D (= MPS) [Hastings 2007] –Conjectured for higher dim (analytic & numerical evidence) PEPS preparation would be an extremely powerful computational resource: –as powerful as contracting tensor networks –PP-complete (for general PEPS as classical input)  Cannot efficiently prepare all PEPS, even using a universal quantum computer (unless BQP = PP!)

7. Are PEPS Physical? Is it possible to prepare PEPS on a quantum computer (under mild conditions on PEPS)? Which subclass of PEPS are physical? [V, Wolf, P-G, Cirac 2006]

8. Talk Outline Crash course on PEPS Growing PEPS in your Back Garden The Trouble with Tribbles Topological States Crash course on G-injective PEPS Growing Topological Quantum States

9. Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS.

10. Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS. Sequence of partial PEPS |  t i are ground states of sequence of parent Hamiltonians H t :

11. Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS. Sequence of partial PEPS |  t i are ground states of sequence of parent Hamiltonians H t :

12. Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS. Sequence of partial PEPS |  t i are ground states of sequence of parent Hamiltonians H t :

13. Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS. Sequence of partial PEPS |  t i are ground states of sequence of parent Hamiltonians H t :

14. Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS. Sequence of partial PEPS |  t i are ground states of sequence of parent Hamiltonians H t :

15. Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS. Sequence of partial PEPS |  t i are ground states of sequence of parent Hamiltonians H t :

16. Growing PEPS in your Back Garden Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

17. Growing PEPS in your Back Garden Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

18. Growing PEPS in your Back Garden Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

19. Growing PEPS in your Back Garden Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

20. Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1 Growing PEPS in your Back Garden

21. Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

22. Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1 Growing PEPS in your Back Garden

23. Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

24. Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1 Growing PEPS in your Back Garden

25. Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1 Growing PEPS in your Back Garden

26. Even if we could implement this measurement, we cannot choose the outcome, so how can we deterministically project onto P 0 ?? How can we implement the measurement, when the ground state P 0 is a complex, many-body state which we don’t know how to prepare? ?? Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

27. Measuring the Ground State How can we implement the measurement ? local Hamiltonian ) Hamiltonian simulation ) measure if energy is <  or not QPE ! Use quantum phase estimation:

28. Measuring the Ground State measure if energy is <  or not Condition 1: Spectral gap  H t ) > 1/poly How can we implement the measurement ? QPE ! Use quantum phase estimation:

29. Projecting onto the Ground State How can we deterministically project from P 0 (t) to P 0 (t+1) ? ! Use Marriot-Watrous measurement rewinding trick: P 0 (t+1) = 0 0 -s c c s 0 0 P 0 (t) = 0 0 0 1 0 0 “Jordan’s lemma” (or “CS decomposition”) Start in Jordan block of P 0 (t) containing |  t i Measure {P 0 (t+1),P 0 (t+1)? } ! stay in same Jordan block  Condition 2: Unique ground state (= injective PEPS)

30. Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: How can we deterministically project from P 0 (t) to P 0 (t+1) ?

31. Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: Measure {P 0 (t+1),P 0 (t+1)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

32. Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done How can we deterministically project from P 0 (t) to P 0 (t+1) ?

33. Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? … How can we deterministically project from P 0 (t) to P 0 (t+1) ?

34. Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) rewind by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

35. Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) go back by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

36. Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) go back by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

37. Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s c c Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) go back by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

38. Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s c s s c Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) go back by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

39. Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s c s s c Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) go back by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

40. Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s c s s c s s c c Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) go back by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

41. Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s c s s c s s c c Lemma: where How can we deterministically project from P 0 (t) to P 0 (t+1) ? ) exp fast Condition 3: Condition number  A t  > 1/poly

42. Algorithm: 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1 Growing PEPS in your Back Garden

43. Algorithm: 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1.Measure {P 0 (t+1),P 0 (t+1) ? } 2.While outcome  P 0 (t) 1.Measure {P 0 (t),P 0 (t)? } 2.Measure {P 0 (t+1),P 0 (t+1) ? } 3. t = t + 1

44. Are PEPS Physical? Is it possible to prepare PEPS on a quantum computer (under mild conditions on PEPS)? Which subclass of PEPS are physical? Condition 1: Spectral gap  H t ) > 1/poly Condition 3: Condition number  A t  > 1/poly Run-time: Condition 2: Unique ground state (= injective PEPS) Rules out all topological quantum states! 

45. Talk Outline Crash course on PEPS Growing PEPS in your Back Garden The Trouble with Tribbles Topological States Crash course on G-injective PEPS Growing Topological Quantum States

46. Projecting onto the Ground State P 0 (t+1) = 0 0 -s 1 c 1 c 1 s 1 “Jordan’s lemma” (or “CS decomposition”) State could be spread over any of the Jordan blocks of P 0 (t) containing |  t (k) i. Probability of measuring P 0 (t+1) can be 0. P 0 (t) = 0 0 0 1 0 1 -s 2 c 2 c 2 s 2

47. Projecting onto the Ground State Probability of measuring P 0 (t+1) could be 0.

48. Projecting onto the Ground State Probability of measuring P 0 (t+1) could be 0. s

49. Projecting onto the Ground State Probability of measuring P 0 (t+1) could be 0. s

50. Projecting onto the Ground State Probability of measuring P 0 (t+1) could be 0.

51. Projecting onto the Ground State Probability of measuring P 0 (t+1) could be 0.  We can get stuck! (never make it to )

52. Talk Outline Crash course on PEPS Growing PEPS in your Back Garden The Trouble with Tribbles Topological States Crash course on G-injective PEPS Growing Topological Quantum States

53. Crash Course on G-injective PEPS! [Schuch, Cirac, P-G 2010] G-injective PEPS PEPS maps left-invertible on invariant subspace of symmetry group G. G-isometric PEPS G-injective PEPS where = projector onto G-invariant subspace. Topological state Degenerate ground state of Hamiltonian whose ground states cannot be distinguished by local observables. G-injective PEPS = Topological state Parent Hamiltonian has topologically degenerate ground states (degeneracy = # “pair conjugacy classes” of G)

54. Crash Course on G-injective PEPS! [Schuch, Cirac, P-G 2010] Many important topological quantum states are G-injective PEPS: Kitaev’s toric code Quantum double models Resonant valence bond states [Schuch, Poilblanc, Cirac, P-G, arXiv:1203.4816] …

55. Talk Outline Crash course on PEPS Growing PEPS in your Back Garden The Trouble with Tribbles Topological States Crash course on G-injective PEPS Growing Topological Quantum States

56. A (t) no longer invertible (only invertible on G-invariant subspace) ) zero eigenvalues )  = 1 ) c = 0 (bad!) Recall key Lemma relating probability c of successful measurement to condition number: where However, G-injectivity ) restriction of A (t) to G-invariant subspace is invertible. How can we exploit this?

57. Algorithm 1. t = 0 2. Prepare max-entangled pairs (ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1 Growing Topological Quantum States Idea: Get into the G-invariant subspace. Stay there!

58. Growing Topological Quantum States Algorithm 1. t = 0 2. Prepare G-isometric PEPS (ground state of H 0 ) 3. Deform vertex by vertex to G-injective PEPS: 1. Project onto ground state of H t+1 2. t = t + 1 Idea: Get into the G-invariant subspace. Stay there! For (suitable representation of) trivial group G = 1, G-isometric PEPS = maximally entangled pairs ! recover original algorithm

59. Growing Topological Quantum States Algorithm 1. t = 0 2. Prepare G-isometric PEPS (ground state of H 0 ) 3. Deform vertex by vertex to G-injective PEPS: 1. Project onto ground state of H t+1 2. t = t + 1 G-isometric PEPS = quantum double models ! algorithms known for preparing these exactly [e.g. Aguado, Vidal, PRL 100, 070404 (2008)]

60. Growing Topological Quantum States Algorithm 1. t = 0 2. Prepare G-isometric PEPS (ground state of H 0 ) 3. Deform vertex by vertex to G-injective PEPS: 1. Project onto ground state of H t+1 2. t = t + 1 Key Lemma: If initial state is already in G-invariant subspace, prob. successful measurement  is condition number restricted to G-invariant subspace ! Marriot-Watrous measurement rewinding trick works!

61. Conclusions Injective PEPS can be prepared efficiently on a quantum computer, under the following conditions: –Sequence of parent Hamiltonians is gapped –PEPS maps A (v) are well-conditioned G-injective PEPS can be prepared efficiently under similar conditions  includes many important topological states Alternatives to Marriot-Watrous trick: –Jagged adiabatic thm? [Aharonov, Ta-Shma, 2007] (Worse run-time, may not work for G-injective case) –Quantum rejection sampling ! quadratic speed-up [Ozols, Roetteler, Roland, 2011]