Preparing topological states on a quantum computer
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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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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)

STVCP-G, (QIP 2012; paper in preparation)


Talk outline

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


Crash course on peps

Crash Course on PEPS!

  • Projected Entangled Pair State


Crash course on peps1

Crash Course on PEPS!

  • Projected Entangled Pair State

Obtain PEPS by applying maps to maximally entangled pairs


Crash course on peps2

  • 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!


Are peps physical

  • 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...


Are peps physical1

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]


Talk outline1

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


Growing peps in your back garden

Growing PEPS in your Back Garden

  • Start with maximally entangled pairs at every edge, and convert this into target PEPS.


Growing peps in your back garden1

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:


Growing peps in your back garden2

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:


Growing peps in your back garden3

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:


Growing peps in your back garden4

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:


Growing peps in your back garden5

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:


Growing peps in your back garden6

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:


Growing peps in your back garden7

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


Growing peps in your back garden8

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


Growing peps in your back garden9

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


Growing peps in your back garden10

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


Growing peps in your back garden11

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


Growing peps in your back garden12

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


Growing peps in your back garden13

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


Growing peps in your back garden14

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


Growing peps in your back garden15

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


Growing peps in your back garden16

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


Growing peps in your back garden17

  • 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??


Measuring the ground state

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


Measuring the ground state1

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


Projecting onto the ground state

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)


Projecting onto the ground state1

Projecting onto the Ground State

  • How can we deterministically project from P0(t) to P0(t+1)?

! Use Marriot-Watrous measurement rewinding trick:


Projecting onto the ground state2

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)?}


Projecting onto the ground state3

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


Projecting onto the ground state4

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) ?…


Projecting onto the ground state5

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)?}


Projecting onto the ground state6

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)?}


Projecting onto the ground state7

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)?}


Projecting onto the ground state8

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)?}


Projecting onto the ground state9

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)?}


Projecting onto the ground state10

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)?}


Projecting onto the ground state11

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)?}


Projecting onto the ground state12

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


Growing peps in your back garden18

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


Growing peps in your back garden19

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


Are peps physical2

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! 


Talk outline2

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


Projecting onto the ground state13

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.


Projecting onto the ground state14

Projecting onto the Ground State

  • Probability of measuring P0(t+1)could be 0.


Projecting onto the ground state15

s

Projecting onto the Ground State

  • Probability of measuring P0(t+1)could be 0.


Projecting onto the ground state16

Projecting onto the Ground State

  • Probability of measuring P0(t+1)could be 0.

s


Projecting onto the ground state17

Projecting onto the Ground State

  • Probability of measuring P0(t+1)could be 0.


Projecting onto the ground state18

  • We can get stuck! (never make it to )

Projecting onto the Ground State

  • Probability of measuring P0(t+1)could be 0.


Talk outline3

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


Crash course on g injective peps schuch cirac p g 2010

  • G-injective PEPSPEPS maps left-invertible on invariant subspace of symmetry group G.

  • G-isometric PEPSG-injective PEPS where = projector onto G-invariant subspace.

  • Topological stateDegenerate ground state of Hamiltonian whose ground states cannot be distinguished by local observables.

  • G-injective PEPS = Topological stateParent Hamiltonian has topologically degenerate ground states (degeneracy = # “pair conjugacy classes” of G)

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


Crash course on g injective peps schuch cirac p g 20101

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

  • Many important topological quantum states areG-injective PEPS:

    • Kitaev’s toric code

    • Quantum double models

    • Resonant valence bond states[Schuch, Poilblanc, Cirac, P-G, arXiv:1203.4816]


Talk outline4

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


Growing topological quantum states

  • However, G-injectivity ) restriction of A(t) to G-invariant subspace is invertible.

  • How can we exploit this?

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!)


Growing topological quantum states1

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 Ht+1

    • t = t + 1


Growing topological quantum states2

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 Ht+1

    • t = t + 1

For (suitable representation of) trivial group G = 1,G-isometric PEPS = maximally entangled pairs!recover original algorithm


Growing topological quantum states3

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 Ht+1

    • t = t + 1

G-isometric PEPS = quantum double models! algorithms known for preparing these exactly[e.g. Aguado, Vidal, PRL 100, 070404 (2008)]


Growing topological quantum states4

Key Lemma:If initial state is already in G-invariant subspace, prob. successful measurement is condition number restricted to G-invariant subspace

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 Ht+1

    • t = t + 1

! Marriot-Watrous measurement rewinding trick works!


Conclusions

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-injectivePEPS 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]


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