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Preparing Topological States on a Quantum Computer

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### Preparing Topological States on a Quantum Computer

Growing PEPS in your Back Garden

Growing PEPS in your Back Garden

Growing PEPS in your Back Garden

Growing PEPS in your Back Garden

Growing PEPS in your Back Garden

Growing PEPS in your Back Garden

Growing PEPS in your Back Garden

Growing PEPS in your Back Garden

Growing PEPS in your Back Garden

Growing PEPS in your Back Garden

Talk Outline

Talk Outline

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

- 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!

- Projected Entangled Pair State

Crash Course on PEPS!

- Projected Entangled Pair State

Obtain PEPS by applying maps to maximally entangled pairs

- 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

- 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!)

- 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 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 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

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

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 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 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:

- 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:

- 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:

- 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 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 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 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

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

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

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

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

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

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

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

- How can we implement the measurement , when the ground state P0is a complex, many-body state which we don’t know how to prepare?

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??

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 State

- How can we implement the measurement ?

! Use quantum phase estimation:

measure if energy is < or not

- Condition 1: Spectral gap (Ht) > 1/poly

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 State

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

! Use Marriot-Watrous measurement rewinding trick:

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 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 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 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 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 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)?}

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)?}

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 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)?}

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)?}

s

c

s

- )exp fast

- Lemma: where

- 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 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 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 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 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

0

1

1

P0(t) =

0

0

c1 s1

c2 s2

-s2 c2

-s1 c1

Projecting onto the Ground State0

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 State

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

Projecting onto the Ground State

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

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

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

- 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

- 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]

- 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]
- …

- 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

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

- How can we exploit this?

- 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 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 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 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)]

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 StatesAlgorithm

- 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

- 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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