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

• 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

Crash Course on PEPS!

• 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

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)

• 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?

• 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






Growing PEPS in your Back Garden

• 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 Hamiltonians Ht :

Growing PEPS in your Back GardenAlgorithm

1. t = 0

2. Prepare max-entangled pairs (= ground state of H0)

3. Grow the PEPS vertex by vertex:

1. Project onto ground state of Ht+1

2. t = t + 1

Algorithm

1. t = 0




2. t = t + 1



1. t = 0




2. t = t + 1

Algorithm

1. t = 0




2. t = t + 1



1. t = 0




2. t = t + 1

Algorithm

1. t = 0




2. t = t + 1



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

• How can we implement the measurement , when the ground state P0 is a complex, many-body state which we don’t know how to prepare?

??

Algorithm

1. t = 0




2. t = t + 1

Measuring the Ground State

• How can we implement the measurement ?

local Hamiltonian ) Hamiltonian simulation )

measure if energy is < or not

QPE

! Use quantum phase estimation:

Measuring the Ground State

measure if energy is < or not

• Condition 1: Spectral gap Ht) > 1/poly

• How can we implement the measurement ?

QPE

! Use quantum phase estimation:

Projecting onto the Ground State

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

(t+1)?

! Use Marriot-Watrous measurement rewinding trick:

P0(t+1) =

00

-s c

c s

00

P0(t) =

00

01

00

“Jordan’s lemma” (or “CS decomposition”)• Start in Jordan block of P0

(t) containing |ti

• Measure {P0(t+1),P0

(t+1)?} ! stay in same Jordan block

Condition 2: Unique ground state (= injective PEPS)




(t+1)?




(t+1)?}


(t+1)?



c


(t+1)?}

• Outcome P0(t+1) ) done


(t+1)?



c

s


(t+1)?}


• Outcome P0(t+1) ? …


(t+1)?



c

s


(t+1)?}


• Outcome P0(t+1) ?

) rewind by measuring {P0(t),P0

(t)?}


(t+1)?



c

s


(t+1)?}



) go back by measuring {P0(t),P0

(t)?}


(t+1)?



c

s

c

c


(t+1)?}




(t)?}


(t+1)?



c

s

cs

sc


(t+1)?}




(t)?}


(t+1)?



c

s

c

s

s

c


(t+1)?}




(t)?}


(t+1)?



c

s

c

s

s

c

s

s

c

c


(t+1)?}




(t)?}


(t+1)?



c

s

c

s

s

c

s

s

c

c

• Lemma: where


(t+1)?

• ) exp fast

• Condition 3: Condition number At > 1/poly

Algorithm:1. t = 0

2. Prepare max-entangled pairs (= ground state of H0)3. Grow the PEPS vertex by vertex:


2. t = t + 1


Growing PEPS in your Back GardenAlgorithm:1. t = 0

2. Prepare max-entangled pairs (= ground state of H0)3. Grow the PEPS vertex by vertex:

1. Measure {P0(t+1),P0

(t+1)?}

2. While outcome P0(t)

1. Measure {P0(t),P0

(t)?}

2. Measure {P0(t+1),P0

(t+1)?}3. 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 3: Condition number At > 1/poly

Run-time:

Condition 2: Unique ground state (= injective PEPS)

Rules out all topological quantum states!

Talk Outline







P0(t+1) =

00

-s1 c1

c1 s1

“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.

P0(t) =

00

01

01

-s2 c2

c2 s2


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



s



We can get stuck! (never make it to )

Talk Outline






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]

• 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 Outline






Growing Topological Quantum States

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

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

where

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

• How can we exploit this?

Algorithm

1. t = 0

2. Prepare max-entangled pairs (ground state of H0)



2. t = t + 1

Growing Topological Quantum StatesIdea:• Get into the G-invariant subspace.• Stay there!

Growing Topological Quantum States

Algorithm

1. t = 0

2. Prepare G-isometric PEPS (ground state of H0)

3. Deform vertex by vertex to G-injective PEPS:


2. t = t + 1

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

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

Growing Topological Quantum StatesAlgorithm

1. t = 0




2. 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 StatesAlgorithm

1. t = 0




2. t = t + 1

Key Lemma: If initial state is already in G-invariant subspace, prob. successful measurement

is condition number restricted to G-invariant subspace

! 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-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-up

[Ozols, Roetteler, Roland, 2011]

Preparing Topological States on a Quantum Computer

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