Entanglement in Adiabatic Computation Andrew G. Green Tanja Duric Chris Hooley Philip Crowley Jonathan Keeling Walter Vinci Steve H Simon Paul Warburton Vadim Oganesyan
Adiabatic quantum computation provides a direct overlap between condensed matter approaches and quantum information. Ideas and analytical methods from quantum phase transitions, many-body localization, out-of-equilibrium dynamics and the dynamics of decoherence may all be used to assess its power and limitations. I will review some of the basic ideas of adiabatic computation – from the performance of an ideal computation, to the limiting effects of the environment and how one might hope to mitigate them.
Outline: Introduction to Adiabatic Quantum Computation Entanglement as a Resource in AQC Environmental Restriction of Entanglement Towards Q Adiabatic Error Correction Conclusions
Introduction to Adiabatic Computation Classical Adiabatic Transport Adiabatic transport well-known in classical systems • Very useful – but not in computation • Low connectivity of classical state space •
Introduction to Adiabatic Computation Classical Adiabatic Transport Non-adiabatic Adiabatic Adiabatic transport well-known in classical systems • Very useful – but not in computation • Low connectivity of classical state space •
Introduction to Adiabatic Computation Classical Adiabatic Transport Non-adiabatic Adiabatic Quantum mechanics increases the connectivity of state space • unsolved solved Add Q
Introduction to Adiabatic Computation Quantum Adiabatic Computation State space of Quantum Mechanics is fully connected • Quantum Adiabatic Theorem [Born+Fock (1928)] : A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenstate and the rest of the Hamiltonian’s spectrum. Quantum Adiabatic Algorithm • H ( t ) = s ( t ) ˆ ˆ H start + [1 − s ( t )] ˆ H end Caveats: determine the limitations of Adiabatic Q Computation • E E ~ ν ( t ) ∆ Computation time > max t ∆ 2 ( t ) t ∝ e − π ∆ 2 2 ν ~ t ~ ν ( t ) = h 1 | ∂ t H| 0 i
Power of Ideal Adiabatic Q Computation Limitations of AQC Slow down when gap smallest • [van Dam, Mosca, and Vazirani arXiv:quant-ph/0206003 (2002)] [Farhi, Goldstone, and Gutmann arXiv:quant-ph/0208135 (2002)] [Caneva, et al, PRA84, 012312 (2011)] [Farhi, Goldstone, and Gutmann,,JQIC 11, 181, (2011)] Map to fixed gap • [Hastings PRL103, 050502 (2009); Hastings + Freedman arXiv:1302.5733] Quantum critical • [Caneva, Fazio and Santoro, PRB76, 144427 (2007)] Localized and Many-body localized states • [Altshuler, Krovi and Roland PNAS107, 12446 (2010)] [Laumann, Moessner, Scardicchio and Sondhi Phys.Rev.Lett.109, 030502 (2012), EPJST 224, 75, (2015)]
Power of Ideal Adiabatic Q Computation Realizations Adiabatic State Preparation NMR and atomic condensates • [Bloch PR70, 460 (1946), Bloch, Hansen, Packard, PR70, 474, (1946)] Quantum Magnet • [Brooke, Bitko, Rosenbaum & Aeppli, Science 284, 779(1999)] Dwave • [Johnson et al. Nature 473, 194 (2011)] Simulated Q Annealing/Adiabatic • Classical [Kirkpatrick, Gelatt & Vecchi. Science 220, 671 (1983)] [Metropolis, Rosenbluth, Rosenbluth, Teller & Teller, J Chem Phys21, 1087 (1953)] Quantum [Kadowaki & Nishimori,. PRE 58, 5355 (1998)] [Martonak, Santoro & Tosatti,PRB66, 094203 (2002)] [Santoro, Martonak, Tosatti, Car, Science 295, 2427 (2002)] [Farhi, Goldstone, Gutmann, Lapan, Lundgren, Preda Science 292, 472 (2001)]
Introduction to Adiabatic Computation Quantum Adiabatic Computation State space of Quantum Mechanics is fully connected • Quantum Adiabatic Theorem [Born+Fock (1928)] : A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenstate and the rest of the Hamiltonian’s spectrum. Quantum Adiabatic Algorithm • H ( t ) = s ( t ) ˆ ˆ H start + [1 − s ( t )] ˆ H end Q. Can we quantify the role of entanglement in Adiabatic Quantum Computation?
Outline: Introduction to Adiabatic Quantum Computation Entanglement as a Resource in AQC [Crowley et al PRA90, 042317 (2014)] Environmental Restriction of Entanglement Towards Q Adiabatic Error Correction Conclusions
Entanglement as a resource in AQC Quantifying Entanglement unsolved solved Add Q Superposition reconnects state space of single particle/spin • Entanglement reconnects many-body state space • Quantify entanglement ~ bond order D of tensor network • Q1. What can we do with a given entanglement resource? Q2. How does the environment constrain entanglement?
Entanglement as a resource in AQC Quantifying Entanglement X | φ i = A σ 1 i A σ 2 ij A σ 3 jk A σ 4 kl ... | σ 1 , σ 2 , σ 3 , σ 4 , ... i i A σ 2 A σ σ 3 { σ } ij jk A σ 1 A σ 2 A σ 3 Superposition reconnects state space of single particle/spin • Entanglement reconnects many-body state space • Quantify entanglement ~ bond order D of tensor network • Q1. What can we do with a given entanglement resource? Q2. How does the environment constrain entanglement?
Entanglement as a resource in AQC Entanglement vs Tunneling X | φ i = A σ 1 i A σ 2 ij A σ 3 jk A σ 4 kl ... | σ 1 , σ 2 , σ 3 , σ 4 , ... i i A σ 2 A σ σ 3 { σ } ij jk A σ 1 A σ 2 A σ 3 Sum of classical/product states • Each i,j,k,… corresponds to product state • Transfer of weight between => tunneling • Entanglement � connectivity � tunneling structure of state space trajectories [Jiang et al arXiv:1603.01293],[Smelyanskiy et al arXiv:1511.02581]
Entanglement as a resource in AQC View from Different Bases X | φ i = A σ 1 i A σ 2 ij A σ 3 jk A σ 4 kl ... | σ 1 , σ 2 , σ 3 , σ 4 , ... i i A σ 2 A σ σ 3 { σ } ij jk A σ 1 A σ 2 A σ 3 Adiabatic basis: entanglement structure => AQC [Farhi,et al A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem Science 292, 472-475 (2001)] Computational basis: tunneling between states => Q Anneal [Ray, Chakrabarti & Chakrabarti Phys. Rev. B 39, 11828(1989)] [Finnila et al Quantum annealing: A new method for minimizing Multidimensional functions. Chemical Physics Letters 219, 343 (1994)]
Entanglement as a resource in AQC Quantifying Entanglement X | φ i = A σ 1 i A σ 2 ij A σ 3 jk A σ 4 kl ... | σ 1 , σ 2 , σ 3 , σ 4 , ... i i A σ 2 A σ σ 3 { σ } ij jk A σ 1 A σ 2 A σ 3 Superposition reconnects state space of single particle/spin • Entanglement reconnects many-body state space • Quantify entanglement ~ bond order D of tensor network • Q1. What can we do with a given entanglement resource? Q2. How does the environment constrain entanglement?
Entanglement as a resource in AQC Quantifying Entanglement X | φ i = A σ 1 i A σ 2 ij A σ 3 jk A σ 4 kl ... | σ 1 , σ 2 , σ 3 , σ 4 , ... i i A σ 2 A σ σ 3 { σ } ij jk A σ 1 A σ 2 A σ 3 Environment restricts useable entanglement resources • (I will discuss Q2. how? shortly) • Capture with fixed D tensor network • Refine question . . . • Q1’. How much entanglement required to solve a given problem adiabatically?
Entanglement as a resource in AQC Projected Dynamics Variational Sub-manifold Hilbert Space (e.g. fixed D MPS) � id t | ψ i = � i | ∂ A β ψ i ˙ A β ⇡ H| ψ i Continually project onto variational manifold • Time-dependent variational principle (TDVP) [Haegeman et al PRL 2011] • Q1. How much entanglement required to solve a given problem adiabatically? [Crowley et al PRA (2014), Bauer et al, arXiv:1501.06914]
Entanglement as a resource in AQC Projected Dynamics Variational Sub-manifold Hilbert Space (e.g. fixed D MPS) | i � | i ⇡ H| i � i h ∂ A α ψ | ∂ A β ψ i ˙ A β = h ∂ A α ψ |H| ψ i Continually project onto variational manifold • Time-dependent variational principle (TDVP) [Haegeman et al PRL 2011] • Q1. How much entanglement required to solve a given problem adiabatically? [Crowley et al PRA (2014), Bauer et al, arXiv:1501.06914]
Entanglement as a resource in AQC Success and Failure of AQC Full Hilbert Space Accessible Accessible Region of Classical Quantum Poly[N] Exp[N] Solved Hilbert Space II I Unsolved Poly[N] Exp[N] Easy Hard Time of Adiabatic Sweep Problem solved or unsolved with given time and entanglement • Curve must form convex hull - the adiabatic success hull • I classically soluble, II quantum soluble – classically not • Interesting changes in dynamics between soluble and insoluble •
Entanglement as a resource in AQC Success and Failure of AQC Variational Sub-manifold Hilbert Space (e.g. fixed D MPS) | i � | i ⇡ H| i � i h ∂ A α ψ | ∂ A β ψ i ˙ A β = h ∂ A α ψ |H| ψ i At least 2 ways to fail: • i. Disconnected path • ii. Bifurcation – direction ill-defined at certain points • Projected dynamics is non-linear => Chaos • Projected dynamics is (semi-)classical => Chaos •
Entanglement as a resource in AQC Reintroducing the Environment Accessible Region of Hilbert Space c. b. a. Temperature Thermal fluctuations – performance with T isnt monotonic • [Crowley et al PRA90, 042317 (2014), Bauer et al ArXiv:1501.06914] •
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