Créer une présentation
Télécharger la présentation

Télécharger la présentation
## Preparing Topological States on a Quantum Computer

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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**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 Crash Course on PEPS!**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? • 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:**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:**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**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**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**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**How can we implement the measurement , when**the ground state P0is a complex, many-body state which we don’t know how to prepare? 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 ?? • 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 ) Measuring the Ground State • How can we implement the measurement ? ! Use quantum phase estimation: measure if energy is < or not**QPE**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 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)?}**c**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**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 • 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**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)?}**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 • 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)?}**c**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)?}**c**s c s • )exp fast • Lemma: where 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 • 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**Run-time:**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**0 1 1 P0(t) = 0 0 c1 s1 c2 s2 -s2 c2 -s1 c1 Projecting onto the Ground State 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.**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. s**Projecting onto the Ground State**• Probability of measuring P0(t+1)could be 0.