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Transcript
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 entangled pairs to maximally Crash Course on PEPS! • Parent Hamiltonian 2-local Hamiltonian ground state. • Injectivity PEPS is “injective” if with PEPS as are left-invertible (perhaps only after blocking together sites) • Uniqueness An 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 • 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 Hamiltonians Ht : 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 Hamiltonians Ht : 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 Hamiltonians Ht : 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 Hamiltonians Ht : 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 Hamiltonians Ht : 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 Hamiltonians Ht : Growing PEPS in your Back Garden Algorithm 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 Growing PEPS in your Back Garden Algorithm 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 Growing PEPS in your Back Garden Algorithm 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 Growing PEPS in your Back Garden Algorithm 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 Growing PEPS in your Back Garden Algorithm 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 Growing PEPS in your Back Garden Algorithm 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 Growing PEPS in your Back Garden Algorithm 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 Growing PEPS in your Back Garden Algorithm 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 Growing PEPS in your Back Garden Algorithm 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 Growing PEPS in your Back Garden Algorithm 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 Growing PEPS in your Back Garden Algorithm 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 • 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? • Even if we could implement this measurement, we cannot choose the outcome, so how can we deterministically project onto P0?? Measuring the Ground State • How can we implement the measurement ? ! Use quantum phase estimation: QPE measure if energy is < or not local Hamiltonian ) Hamiltonian simulation ) Measuring the Ground State • How can we implement the measurement ? ! Use quantum phase estimation: QPE measure if energy is < or not • Condition 1: Spectral gap (Ht) > 1/poly Projecting onto the Ground State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: 0 0 0 0 P0(t) = 1 P0 0 (t+1) = c -s s c 0 0 0 0 “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) 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: c • 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 s • 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)?} 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 c • 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)?} 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 State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: c c s c s s c 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 State • How can we deterministically project from P0(t) to P0(t+1)? ! Use Marriot-Watrous measurement rewinding trick: c c s c s s c s c s • • Lemma: ) exp fast where • Condition 3: Condition number (At ) > 1/poly Growing PEPS in your Back Garden Algorithm: 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 Growing PEPS in your Back Garden Algorithm: 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 2: Unique ground state (= injective PEPS) Condition 3: Condition number (At ) > 1/poly Run-time: 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 Projecting onto the Ground State 0 0 0 0 (t) P0 = 1 P0 0 (t+1) = 1 0 c1 s 1 -s1 c1 c2 s2 -s2 c2 “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. s Projecting onto the Ground State • Probability of measuring P0(t+1) could be 0. s 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 ) 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 PEPS PEPS maps left-invertible on invariant subspace of symmetry group G. • G-isometric PEPS G-injective PEPS where subspace. = projector onto G-invariant • 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) 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] • … 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 • 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!) • However, G-injectivity ) restriction of A(t) to G-invariant subspace is invertible. • How can we exploit this? Growing Topological Quantum States Idea: • Get into the G-invariant subspace. • Stay there! Algorithm 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 Growing Topological Quantum States Idea: • Get into the G-invariant subspace. • Stay there! Algorithm 1. t = 0 2. Prepare G-isometric PEPS (ground state of H0) 3. Deform vertex by vertex to G-injective PEPS: 1. Project onto ground state of Ht+1 2. 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 1. t = 0 2. Prepare G-isometric PEPS (ground state of H0) 3. Deform vertex by vertex to G-injective PEPS: 1. Project onto ground state of Ht+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)] 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: 1. Project onto ground state of Ht+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! 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]