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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)

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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

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Crash Course on PEPS! Projected Entangled Pair State

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Crash Course on PEPS! Projected Entangled Pair State Obtain PEPS by applying maps to maximally entangled pairs

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Crash Course on PEPS! Parent Hamiltonian 2-local Hamiltonian with PEPS as ground state. Injectivity PEPS is “injective” if are left-invertible (perhaps only after blocking together sites) Uniqueness An injective PEPS is the unique ground state of its parent Hamiltonian

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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!)

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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]

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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

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Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS.

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Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS. Sequence of partial PEPS | t i are ground states of sequence of parent Hamiltonians H t :

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Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS. Sequence of partial PEPS | t i are ground states of sequence of parent Hamiltonians H t :

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Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS. Sequence of partial PEPS | t i are ground states of sequence of parent Hamiltonians H t :

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Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS. Sequence of partial PEPS | t i are ground states of sequence of parent Hamiltonians H t :

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Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS. Sequence of partial PEPS | t i are ground states of sequence of parent Hamiltonians H t :

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Growing PEPS in your Back Garden Start with maximally entangled pairs at every edge, and convert this into target PEPS. Sequence of partial PEPS | t i are ground states of sequence of parent Hamiltonians H t :

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Growing PEPS in your Back Garden Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

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Growing PEPS in your Back Garden Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

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Growing PEPS in your Back Garden Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

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Growing PEPS in your Back Garden Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

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Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1 Growing PEPS in your Back Garden

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Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

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Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1 Growing PEPS in your Back Garden

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Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

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Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1 Growing PEPS in your Back Garden

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Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1 Growing PEPS in your Back Garden

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Even if we could implement this measurement, we cannot choose the outcome, so how can we deterministically project onto P 0 ?? How can we implement the measurement, when the ground state P 0 is a complex, many-body state which we don’t know how to prepare? ?? Algorithm 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1

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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:

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Measuring the Ground State measure if energy is < or not Condition 1: Spectral gap H t ) > 1/poly How can we implement the measurement ? QPE ! Use quantum phase estimation:

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Projecting onto the Ground State How can we deterministically project from P 0 (t) to P 0 (t+1) ? ! Use Marriot-Watrous measurement rewinding trick: P 0 (t+1) = 0 0 -s c c s 0 0 P 0 (t) = 0 0 0 1 0 0 “Jordan’s lemma” (or “CS decomposition”) Start in Jordan block of P 0 (t) containing | t i Measure {P 0 (t+1),P 0 (t+1)? } ! stay in same Jordan block Condition 2: Unique ground state (= injective PEPS)

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Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: How can we deterministically project from P 0 (t) to P 0 (t+1) ?

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Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: Measure {P 0 (t+1),P 0 (t+1)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

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Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done How can we deterministically project from P 0 (t) to P 0 (t+1) ?

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Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? … How can we deterministically project from P 0 (t) to P 0 (t+1) ?

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Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) rewind by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

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Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) go back by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

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Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) go back by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

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Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s c c Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) go back by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

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Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s c s s c Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) go back by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

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Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s c s s c Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) go back by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

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Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s c s s c s s c c Measure {P 0 (t+1),P 0 (t+1)? } Outcome P 0 (t+1) ) done Outcome P 0 (t+1) ? ) go back by measuring {P 0 (t),P 0 (t)? } How can we deterministically project from P 0 (t) to P 0 (t+1) ?

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Projecting onto the Ground State ! Use Marriot-Watrous measurement rewinding trick: c s c s s c s s c c Lemma: where How can we deterministically project from P 0 (t) to P 0 (t+1) ? ) exp fast Condition 3: Condition number A t > 1/poly

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Algorithm: 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1 Growing PEPS in your Back Garden

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Algorithm: 1. t = 0 2. Prepare max-entangled pairs (= ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1.Measure {P 0 (t+1),P 0 (t+1) ? } 2.While outcome P 0 (t) 1.Measure {P 0 (t),P 0 (t)? } 2.Measure {P 0 (t+1),P 0 (t+1) ? } 3. t = t + 1

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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 H t ) > 1/poly Condition 3: Condition number A t > 1/poly Run-time: Condition 2: Unique ground state (= injective PEPS) Rules out all topological quantum states!

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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

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Projecting onto the Ground State P 0 (t+1) = 0 0 -s 1 c 1 c 1 s 1 “Jordan’s lemma” (or “CS decomposition”) State could be spread over any of the Jordan blocks of P 0 (t) containing | t (k) i. Probability of measuring P 0 (t+1) can be 0. P 0 (t) = 0 0 0 1 0 1 -s 2 c 2 c 2 s 2

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Projecting onto the Ground State Probability of measuring P 0 (t+1) could be 0.

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Projecting onto the Ground State Probability of measuring P 0 (t+1) could be 0. s

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Projecting onto the Ground State Probability of measuring P 0 (t+1) could be 0. s

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Projecting onto the Ground State Probability of measuring P 0 (t+1) could be 0.

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Projecting onto the Ground State Probability of measuring P 0 (t+1) could be 0. We can get stuck! (never make it to )

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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

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Crash Course on G-injective PEPS! [Schuch, Cirac, P-G 2010] G-injective PEPS PEPS maps left-invertible on invariant subspace of symmetry group G. G-isometric PEPS G-injective PEPS where = projector onto G-invariant subspace. Topological state Degenerate ground state of Hamiltonian whose ground states cannot be distinguished by local observables. G-injective PEPS = Topological state Parent Hamiltonian has topologically degenerate ground states (degeneracy = # “pair conjugacy classes” of G)

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Crash Course on G-injective PEPS! [Schuch, Cirac, P-G 2010] Many important topological quantum states are G-injective PEPS: Kitaev’s toric code Quantum double models Resonant valence bond states [Schuch, Poilblanc, Cirac, P-G, arXiv:1203.4816] …

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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

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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?

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Algorithm 1. t = 0 2. Prepare max-entangled pairs (ground state of H 0 ) 3. Grow the PEPS vertex by vertex: 1. Project onto ground state of H t+1 2. t = t + 1 Growing Topological Quantum States Idea: Get into the G-invariant subspace. Stay there!

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Growing Topological Quantum States Algorithm 1. t = 0 2. Prepare G-isometric PEPS (ground state of H 0 ) 3. Deform vertex by vertex to G-injective PEPS: 1. Project onto ground state of H t+1 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

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Growing Topological Quantum States Algorithm 1. t = 0 2. Prepare G-isometric PEPS (ground state of H 0 ) 3. Deform vertex by vertex to G-injective PEPS: 1. Project onto ground state of H t+1 2. t = t + 1 G-isometric PEPS = quantum double models ! algorithms known for preparing these exactly [e.g. Aguado, Vidal, PRL 100, 070404 (2008)]

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Growing Topological Quantum States Algorithm 1. t = 0 2. Prepare G-isometric PEPS (ground state of H 0 ) 3. Deform vertex by vertex to G-injective PEPS: 1. Project onto ground state of H t+1 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!

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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]

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