1 / 61

Preparing Topological States on a Quantum Computer

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)

kamran
Télécharger la présentation

Preparing Topological States on a Quantum Computer

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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. Parent Hamiltonian2-local Hamiltonian with PEPS as ground state. • InjectivityPEPS is “injective” if are left-invertible(perhaps only after blocking together sites) • UniquenessAn injective PEPS is the unique ground state of its parent Hamiltonian Crash Course on PEPS!

  6. 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!) 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) But...

  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 |ti are ground states of sequence of parent HamiltoniansHt:

  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 |ti are ground states of sequence of parent HamiltoniansHt:

  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 |ti are ground states of sequence of parent HamiltoniansHt:

  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 |ti are ground states of sequence of parent HamiltoniansHt:

  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 |ti are ground states of sequence of parent HamiltoniansHt:

  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 |ti are ground states of sequence of parent HamiltoniansHt:

  16. Growing PEPS in your Back Garden Algorithm • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Project onto ground state of Ht+1 • t = t + 1

  17. Growing PEPS in your Back Garden Algorithm • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Project onto ground state of Ht+1 • t = t + 1

  18. Growing PEPS in your Back Garden Algorithm • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Project onto ground state of Ht+1 • t = t + 1

  19. Growing PEPS in your Back Garden Algorithm • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Project onto ground state of Ht+1 • t = t + 1

  20. Growing PEPS in your Back Garden Algorithm • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Project onto ground state of Ht+1 • t = t + 1

  21. Growing PEPS in your Back Garden Algorithm • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Project onto ground state of Ht+1 • t = t + 1

  22. Growing PEPS in your Back Garden Algorithm • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Project onto ground state of Ht+1 • t = t + 1

  23. Growing PEPS in your Back Garden Algorithm • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Project onto ground state of Ht+1 • t = t + 1

  24. Growing PEPS in your Back Garden Algorithm • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Project onto ground state of Ht+1 • t = t + 1

  25. Growing PEPS in your Back Garden Algorithm • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Project onto ground state of Ht+1 • t = t + 1

  26. How can we implement the measurement , when the ground state P0is a complex, many-body state which we don’t know how to prepare? Growing PEPS in your Back Garden Algorithm • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Project onto ground state of Ht+1 • t = t + 1   ??  • Even if we could implement this measurement, we cannot choose the outcome, so how can we deterministically project onto P0??

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

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

  29. 0 0 0 0 c s 1 P0(t) = P0(t+1) = 0 -s c 0 0 0 0 “Jordan’s lemma” (or “CS decomposition”) Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: • Start in Jordan block of P0(t) containing |ti • Measure {P0(t+1),P0(t+1)?} ! stay in sameJordan block • Condition 2: Unique ground state (= injective PEPS)

  30. Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick:

  31. Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: • Measure {P0(t+1),P0(t+1)?}

  32. c Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: • Measure {P0(t+1),P0(t+1)?} • Outcome P0(t+1))done

  33. s Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: c • Measure {P0(t+1),P0(t+1)?} • Outcome P0(t+1))done • Outcome P0(t+1) ?…

  34. Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: c s • Measure {P0(t+1),P0(t+1)?} • Outcome P0(t+1))done • Outcome P0(t+1) ?)rewind by measuring {P0(t),P0(t)?}

  35. Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: c s • Measure {P0(t+1),P0(t+1)?} • Outcome P0(t+1))done • Outcome P0(t+1) ?)go back by measuring {P0(t),P0(t)?}

  36. Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: c s • Measure {P0(t+1),P0(t+1)?} • Outcome P0(t+1))done • Outcome P0(t+1) ?)go back by measuring {P0(t),P0(t)?}

  37. c c Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: c s • Measure {P0(t+1),P0(t+1)?} • Outcome P0(t+1))done • Outcome P0(t+1) ?)go back by measuring {P0(t),P0(t)?}

  38. c s Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: c c s s • Measure {P0(t+1),P0(t+1)?} • Outcome P0(t+1))done • Outcome P0(t+1) ?)go back by measuring {P0(t),P0(t)?}

  39. Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: c c s s c s • Measure {P0(t+1),P0(t+1)?} • Outcome P0(t+1))done • Outcome P0(t+1) ?)go back by measuring {P0(t),P0(t)?}

  40. c s c s Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: c c s s c s • Measure {P0(t+1),P0(t+1)?} • Outcome P0(t+1))done • Outcome P0(t+1) ?)go back by measuring {P0(t),P0(t)?}

  41. c s c s • )exp fast • Lemma: where Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: c c s s c s • Condition 3: Condition number (At ) > 1/poly

  42. Growing PEPS in your Back Garden Algorithm: • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Project onto ground state of Ht+1 • t = t + 1

  43. Growing PEPS in your Back Garden Algorithm: • t = 0 • Prepare max-entangled pairs (= ground state of H0) • Grow the PEPS vertex by vertex: • Measure {P0(t+1),P0(t+1)?} • While outcome P0(t) • Measure {P0(t),P0(t)?} • Measure {P0(t+1),P0(t+1)?} • t = t + 1

  44. Run-time: 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 (Ht) > 1/poly Condition 2: Unique ground state (= injective PEPS) Condition 3: Condition number (At ) > 1/poly 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. 0 0 1 1 P0(t) = 0 0 c1 s1 c2 s2 -s2 c2 -s1 c1 Projecting onto the Ground State 0 0 P0(t+1) = “Jordan’s lemma” (or “CS decomposition”) • State could be spread over any of the Jordan blocks of P0(t) containing |t(k)i. • Probability of measuring P0(t+1)can be 0.

  47. Projecting onto the Ground State • Probability of measuring P0(t+1)could be 0.

  48. s Projecting onto the Ground State • Probability of measuring P0(t+1)could be 0.

  49. Projecting onto the Ground State • Probability of measuring P0(t+1)could be 0. s

  50. Projecting onto the Ground State • Probability of measuring P0(t+1)could be 0.

More Related