Preparing Topological States on a Quantum Computer

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!
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)

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 Hamiltonians Ht

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 Hamiltonians Ht

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 Hamiltonians Ht

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 Hamiltonians Ht

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 Hamiltonians Ht

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 Hamiltonians Ht

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 Ht1
• 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 Ht1
• 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 Ht1
• 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 Ht1
• 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 Ht1
• 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 Ht1
• 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 Ht1
• 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 Ht1
• 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 Ht1
• 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 Ht1
• t t 1

26
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 Ht1
• t t 1

?
?
??
?

• Even if we could implement this measurement, we
  cannot choose the outcome, so how can we
  deterministically project onto P0??

27
Measuring the Ground State

• How can we implement the measurement
  ?

! Use quantum phase estimation
measure if energy is lt ? or not
28
Measuring the Ground State

• How can we implement the measurement
  ?

! Use quantum phase estimation
measure if energy is lt ? or not

• Condition 1 Spectral gap ?(Ht) gt 1/poly

29
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick

• Start in Jordan block of P0(t) containing ?ti
• Measure P0(t1),P0(t1)? ! stay in same Jordan
  block
• Condition 2 Unique ground state ( injective
  PEPS)

30
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick
31
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick

• Measure P0(t1),P0(t1)?

32
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick

• Measure P0(t1),P0(t1)?

• Outcome P0(t1) ) done

33
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick
c

• Measure P0(t1),P0(t1)?

• Outcome P0(t1) ) done

• Outcome P0(t1) ?

34
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick
c
s

• Measure P0(t1),P0(t1)?

• Outcome P0(t1) ) done

• Outcome P0(t1) ?) rewind by measuring
  P0(t),P0(t)?

35
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick
c
s

• Measure P0(t1),P0(t1)?

• Outcome P0(t1) ) done

• Outcome P0(t1) ?) go back by measuring
  P0(t),P0(t)?

36
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick
c
s

• Measure P0(t1),P0(t1)?

• Outcome P0(t1) ) done

• Outcome P0(t1) ?) go back by measuring
  P0(t),P0(t)?

37
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick
c
s

• Measure P0(t1),P0(t1)?

• Outcome P0(t1) ) done

• Outcome P0(t1) ?) go back by measuring
  P0(t),P0(t)?

38
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick
c
c
s
s

• Measure P0(t1),P0(t1)?

• Outcome P0(t1) ) done

• Outcome P0(t1) ?) go back by measuring
  P0(t),P0(t)?

39
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick
c
c
s
s
c
s

• Measure P0(t1),P0(t1)?

• Outcome P0(t1) ) done

• Outcome P0(t1) ?) go back by measuring
  P0(t),P0(t)?

40
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick
c
c
s
s
c
s

• Measure P0(t1),P0(t1)?

• Outcome P0(t1) ) done

• Outcome P0(t1) ?) go back by measuring
  P0(t),P0(t)?

41
Projecting onto the Ground State

• How can we deterministically project from P0(t)
  to P0(t1)?

! Use Marriot-Watrous measurement rewinding trick
c
c
s
s
c
s

• Condition 3 Condition number ?(At ) gt 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 Ht1
• 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(t1),P0(t1)?
• While outcome ? P0(t)
• Measure P0(t),P0(t)?
• Measure P0(t1),P0(t1)?
• 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 ?(Ht) gt 1/poly
Condition 2 Unique ground state ( injective
PEPS)
Condition 3 Condition number ?(At ) gt 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
Projecting onto the Ground State
0
0
P0(t1)
Jordans 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(t1) can be 0.

47
Projecting onto the Ground State

• Probability of measuring P0(t1) could be 0.

48
Projecting onto the Ground State

• Probability of measuring P0(t1) could be 0.

49
Projecting onto the Ground State

• Probability of measuring P0(t1) could be 0.

s
50
Projecting onto the Ground State

• Probability of measuring P0(t1) could be 0.

51
Projecting onto the Ground State

• Probability of measuring P0(t1) could be 0.

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
54
Crash Course on G-injective PEPS! Schuch,
Cirac, P-G 2010

• Many important topological quantum states
  areG-injective PEPS
• Kitaevs toric code
• Quantum double models
• Resonant valence bond statesSchuch, Poilblanc,
  Cirac, P-G, arXiv1203.4816

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
Growing Topological Quantum States

• Recall key Lemma relating probability c of
  successful measurement to condition number
  where

• A(t) no longer invertible (only invertible on
  G-invariant subspace) ) zero eigenvalues ) ? 1
  ) c 0 (bad!)

57
Growing Topological Quantum States

• Idea
• Get into the G-invariant subspace.
• Stay there!

• Algorithm
• t 0
• Prepare max-entangled pairs (ground state of H0)
• Grow the PEPS vertex by vertex
• Project onto ground state of Ht1
• t t 1

58
Growing Topological Quantum States

• Idea
• Get into the G-invariant subspace.
• Stay there!

• Algorithm
• t 0
• Prepare G-isometric PEPS (ground state of H0)
• Deform vertex by vertex to G-injective PEPS
• Project onto ground state of Ht1
• t t 1

For (suitable representation of) trivial group G
1,G-isometric PEPS maximally entangled
pairs! recover original algorithm
59
Growing Topological Quantum States

• Algorithm
• t 0
• Prepare G-isometric PEPS (ground state of H0)
• Deform vertex by vertex to G-injective PEPS
• Project onto ground state of Ht1
• 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
• t 0
• Prepare G-isometric PEPS (ground state of H0)
• Deform vertex by vertex to G-injective PEPS
• Project onto ground state of Ht1
• t t 1

! 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-upOzols, Roetteler, Roland, 2011