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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)

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

- 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

- Projected Entangled Pair State

- 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...

- 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]

- 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

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

- 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:

- 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:

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

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??

QPE

local Hamiltonian ) Hamiltonian simulation )

- How can we implement the measurement ?

! Use quantum phase estimation:

measure if energy is < or not

QPE

- 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

0

c s

1

P0(t) =

P0(t+1) =

0

-s c

0

0

0

0

“Jordan’s lemma” (or “CS decomposition”)

- 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)

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

! Use Marriot-Watrous measurement rewinding trick:

- 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)?}

c

- 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

s

- 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) ?…

- 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)?}

- 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)?}

- 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

c

- 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

s

- 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)?}

- 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)?}

c

s

c

s

- 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)?}

c

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

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:
- 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

Run-time:

- 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!

- 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

0

1

1

P0(t) =

0

0

c1 s1

c2 s2

-s2 c2

-s1 c1

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.

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

s

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

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

s

- 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)

- 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!)

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

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

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

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!

- 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]