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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)

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Preparing Topological States on a Quantum Computer

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  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.

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