Entanglement in Adiabatic Computation Andrew G. Green Tanja Duric - - PowerPoint PPT Presentation

Entanglement in Adiabatic Computation

Andrew G. Green Tanja Duric Philip Crowley Walter Vinci Paul Warburton Chris Hooley Jonathan Keeling Steve H Simon 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

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

• Adiabatic transport well-known in classical systems
• Very useful – but not in computation
• Low connectivity of classical state space

Introduction to Adiabatic Computation

Adiabatic Non-adiabatic

Classical Adiabatic Transport

• Quantum mechanics increases the connectivity of state space

Introduction to Adiabatic Computation

Add Q unsolved solved

Adiabatic Non-adiabatic

Introduction to Adiabatic Computation Quantum Adiabatic Computation

• State space of Quantum Mechanics is fully connected
• Quantum Adiabatic Algorithm
• Caveats: determine the limitations of Adiabatic Q Computation

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.

ˆ H(t) = s(t) ˆ Hstart + [1 − s(t)] ˆ Hend

E t

Computation time

> maxt ~ν(t) ∆2(t)

∆

∝ e− π∆2

2ν~

t

E

~ν(t) = h1|∂tH|0i

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)]

Power of Ideal Adiabatic Q Computation

Introduction to Adiabatic Computation Quantum Adiabatic Computation

• State space of Quantum Mechanics is fully connected
• Quantum Adiabatic Algorithm

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.

ˆ H(t) = s(t) ˆ Hstart + [1 − s(t)] ˆ Hend

• 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

Add Q unsolved solved

Quantifying Entanglement

• 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

• Q1. What can we do with a given entanglement resource?
• Q2. How does the environment constrain entanglement?

Quantifying Entanglement

• Superposition reconnects state space of single particle/spin
• Entanglement reconnects many-body state space
• Quantify entanglement ~ bond order D of tensor network

|φi = X

{σ}

Aσ1

i Aσ2 ij Aσ3 jkAσ4 kl ...|σ1, σ2, σ3, σ4, ...i

Aσ1 Aσ2 Aσ3 Aσ

i Aσ2 ij σ3 jk

Entanglement as a resource in AQC Entanglement vs Tunneling

• Sum of classical/product states
• Each i,j,k,… corresponds to product state
• Transfer of weight between => tunneling

|φi = X

{σ}

Aσ1

i Aσ2 ij Aσ3 jkAσ4 kl ...|σ1, σ2, σ3, σ4, ...i

Aσ1 Aσ2 Aσ3 Aσ

i Aσ2 ij σ3 jk

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

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)]

|φi = X

{σ}

Aσ1

i Aσ2 ij Aσ3 jkAσ4 kl ...|σ1, σ2, σ3, σ4, ...i

Aσ1 Aσ2 Aσ3 Aσ

i Aσ2 ij σ3 jk

Entanglement as a resource in AQC

• Q1. What can we do with a given entanglement resource?
• Q2. How does the environment constrain entanglement?

Quantifying Entanglement

• Superposition reconnects state space of single particle/spin
• Entanglement reconnects many-body state space
• Quantify entanglement ~ bond order D of tensor network

|φi = X

{σ}

Aσ1

i Aσ2 ij Aσ3 jkAσ4 kl ...|σ1, σ2, σ3, σ4, ...i

Aσ1 Aσ2 Aσ3 Aσ

i Aσ2 ij σ3 jk

Entanglement as a resource in AQC Q1’. How much entanglement required to solve a given problem adiabatically? Quantifying Entanglement

• Environment restricts useable entanglement resources
• (I will discuss Q2. how? shortly)
• Capture with fixed D tensor network
• Refine question . . .

|φi = X

{σ}

Aσ1

i Aσ2 ij Aσ3 jkAσ4 kl ...|σ1, σ2, σ3, σ4, ...i

Aσ1 Aσ2 Aσ3 Aσ

i Aσ2 ij σ3 jk

Entanglement as a resource in AQC Projected Dynamics

• Continually project onto variational manifold
• Time-dependent variational principle (TDVP) [Haegeman et al PRL 2011]

Hilbert Space Variational Sub-manifold (e.g. fixed D MPS)

idt|ψi = i|∂Aβψi ˙ Aβ ⇡ H|ψi

• 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

• Continually project onto variational manifold
• Time-dependent variational principle (TDVP) [Haegeman et al PRL 2011]

Hilbert Space Variational Sub-manifold (e.g. fixed D MPS)

• Q1. How much entanglement required to solve

a given problem adiabatically?

[Crowley et al PRA (2014), Bauer et al, arXiv:1501.06914]

| i | i ⇡ H| i ih∂Aαψ|∂Aβψi ˙ Aβ = h∂Aαψ|H|ψi

Entanglement as a resource in AQC

Accessible Region of Hilbert Space Time of Adiabatic Sweep Unsolved

Classical Quantum Poly[N] Exp[N]

Poly[N] Exp[N]

Easy Hard

I II

Full Hilbert Space Accessible

Solved

Success and Failure of AQC

• 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

Success and Failure of AQC

• 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

Hilbert Space Variational Sub-manifold (e.g. fixed D MPS)

| i | i ⇡ H| i ih∂Aαψ|∂Aβψi ˙ Aβ = h∂Aαψ|H|ψi

Entanglement as a resource in AQC

Accessible Region of Hilbert Space Temperature

a. b. c.

Reintroducing the Environment

• Thermal fluctuations – performance with T isnt monotonic
• [Crowley et al PRA90, 042317 (2014), Bauer et al ArXiv:1501.06914]

Entanglement as a resource in AQC Reintroducing the Environment

• Thermal activation
• Assisted tunneling (thermal activation on higher D manifold)
• Dissipative bias
• Decoherence (reduction of entanglement resources)

Hilbert Space Variational Sub-manifold (e.g. fixed D MPS)

+noise+dissipation

| i | i ⇡ H| i ih∂Aαψ|∂Aβψi ˙ Aβ = h∂Aαψ|H|ψi

Entanglement as a resource in AQC Reintroducing the Environment

• Other schemes can use state space
• Thermal anneal on enhanced state space?
• Balance of decoherence and thermal exploration.
• What is the best scheme for a given resource?

Hilbert Space Variational Sub-manifold (e.g. fixed D MPS)

+noise+dissipation

| i | i ⇡ H| i ih∂Aαψ|∂Aβψi ˙ Aβ = h∂Aαψ|H|ψi

Outline:

Introduction to Adiabatic Quantum Computation Entanglement as a Resource in AQC Environmental Restriction of Entanglement

[Crowley, Oganesyan, Green in preparation]

Towards Q Adiabatic Error Correction Conclusions

Environmental Restriction of Entanglement Quantum Langevin Equation for Entangled States

• Couple to bosonic bath via
• Derive from Keldysh field theory over MPS states

[Green et al, ArXiv:1607 .01778]

• r otherwise . . .

Hilbert Space Variational Sub-manifold (e.g. fixed D MPS)

hHintispins = ˆ XF(A)

h | i h |H| i ih∂Aαψ|∂Aβψi ˙ Aβ = h∂Aαψ| ˆ H|ψi + γ ⇤ ∂AαFdtF + η∂AαF

i γ = 1 ω h ˆ X ˆ XiR hηηi = i 2h ˆ X ˆ XiK ⇡ 2γT

Environmental Restriction of Entanglement Quantum Langevin Equation for Entangled States

• Couple to bosonic bath via
• Derive from Keldysh field theory over MPS states

[Green et al, ArXiv:1607 .01778]

• r otherwise . . .

Hilbert Space Variational Sub-manifold (e.g. fixed D MPS)

hHintispins = ˆ XF(A)

Fluctuation Dissipation TDVP

h | i h |H| i ih∂Aαψ|∂Aβψi ˙ Aβ = h∂Aαψ| ˆ H|ψi + γ ⇤ ∂AαFdtF + η∂AαF

i γ = 1 ω h ˆ X ˆ XiR hηηi = i 2h ˆ X ˆ XiK ⇡ 2γT

Environmental Restriction of Entanglement Quantum Langevin Equation for Entangled States

• Couple to bosonic bath via
• Derive from Keldysh field theory over MPS states

Hilbert Space Variational Sub-manifold (e.g. fixed D MPS)

hHintispins = ˆ XF(A)

h | i h |H| i ih∂Aαψ|∂Aβψi ˙ Aβ = h∂Aαψ| ˆ H|ψi + γ ⇤ ∂AαFdtF + η∂AαF

• Q2. How does the environment constrain entanglement?

i γ = 1 ω h ˆ X ˆ XiR hηηi = i 2h ˆ X ˆ XiK ⇡ 2γT

Examples

• 1 spin:

Landau-Lifshitz-Gilbert Eqns Dissipative bias Decoherence

• 2 spins:

Dissipative bias Decoherence => coupled Landau-Lifshitz-Gilbert

• Many Spins:

??

Environmental Restriction of Entanglement

Environmental Restriction of Entanglement 1 spin : Landau-Lifshitz-Gilbert Equation

Hilbert Space Variational Sub-manifold

h - Obvious inter e |ψi = Q⌦

i |nii.

• Bi = ∂liH

li

Isotropic bath Anisotropic bath

˙ li + li ⇥ h ∂liH + ηi γ ˙ li i = 0 h i ⇥ h H

• i

˙ li + li ⇥ h ∂liH + zηi γz(z · ˙ li) i = 0

• ⇥

⇥ · hηα(t)ηβ(t0)i ⇡ 2γTδ(t t0)δαβ

Environmental Restriction of Entanglement 1 spin : Landau-Lifshitz-Gilbert Equation

Hilbert Space Variational Sub-manifold

h - Obvious inter e |ψi = Q⌦

i |nii.

li ⇥ ˙ li + ∂liH γ˙ li + ηi = 0

• Bi = ∂liH

li

Isotropic bath Anisotropic bath

TDVP Dissipation Fluctuation

⇥ H li ⇥ ˙ li + ∂liH γli ⇥ (li ⇥ z)(z · ˙ li) + zηi = 0

• ⇥

⇥ · hηα(t)ηβ(t0)i ⇡ 2γTδ(t t0)δαβ

Environmental Restriction of Entanglement 1 spin : Landau-Lifshitz-Gilbert Equation

• Bi = ∂liH

li

Isotropic bath Anisotropic bath

li ⇥ ˙ li + ∂liH γ˙ li + ηi = 0

⇥ H li ⇥ ˙ li + ∂liH γli ⇥ (li ⇥ z)(z · ˙ li) + zηi = 0

• ⇥

⇥ · hηα(t)ηβ(t0)i ⇡ 2γTδ(t t0)δαβ

Decoherence:

• Take => noise only
• Random field => random axial angle
• In many spin case => reduction of

effective manifold?

k heiφi = ehφ2i/2 = ehη2i/2 = eKT t H Bi k z

Environmental Restriction of Entanglement 1 spin : Landau-Lifshitz-Gilbert Equation

• Bi = ∂liH

li

Isotropic bath Anisotropic bath

li ⇥ ˙ li + ∂liH γ˙ li + ηi = 0

⇥ H li ⇥ ˙ li + ∂liH γli ⇥ (li ⇥ z)(z · ˙ li) + zηi = 0

• ⇥

⇥ · hηα(t)ηβ(t0)i ⇡ 2γTδ(t t0)δαβ

Non-Markovian

[Crowley, Green,arXiv:1503.00651]

Dissipative Bias

• Dissipation is anisotropic in spin space
• Depends upon angle between B, z and n

Markovian

Environmental Restriction of Entanglement 2 spins: Quantum Langevin Equations

✓ ◆ nzli ⇥ ˙ li + ∂liH φi lz

i

l?

i

∂βH γnzli ⇥ (li ⇥ ˙ li) + nzηi = 0 H ˙ β + X

i=1,2

lz

i ˙

φi + ∂nzH + 2γ ˙ nz + X

i=1,2

li · ηi = 0 ˙ nz + ∂βH = 0 X

• Q. When do dynamics reduce to LLG?
• Independent, local, isotropic baths
• Expect classical physics to emerge

BUT

• Apply local unitaries

⇒ Cannot change entanglement!!??

i | i | i |n1|2 + |n2|2 = 1 ✓ ◆

˙ β +

nzl

li

lz

i ˙

φi

φi θi | | | | n = (n⇤

1, n⇤ 2)σ

✓ n1 n2 ◆

|ψi = n1|l1, l2i + n2| l1, l2i

Environmental Restriction of Entanglement 2 spins: Quantum Langevin Equations

✓ ◆ nzli ⇥ ˙ li + ∂liH φi lz

i

l?

i

∂βH γnzli ⇥ (li ⇥ ˙ li) + nzηi = 0 H ˙ β + X

i=1,2

lz

i ˙

φi + ∂nzH + 2γ ˙ nz + X

i=1,2

li · ηi = 0 ˙ nz + ∂βH = 0 X

• Q. When do dynamics reduce to LLG?
• Spin fields couple to one bath
• Entanglement field coupled to both

⇒ Decohered more rapidly ⇒ Average over intermediate timescale ⇒ If initially, does not grow ⇒ LLG equations

nzli

i | i | i |n1|2 + |n2|2 = 1 ✓ ◆

nzl

li

lz

i ˙

φi

φi θi | | | | n = (n⇤

1, n⇤ 2)σ

✓ n1 n2 ◆

n

nz = 1

˙ β +

|ψi = n1|l1, l2i + n2| l1, l2i

Environmental Restriction of Entanglement Iff projected dynamics are Landau-Lifshitz-Gilbert

nz = 1

2 spins: Quantum Langevin Equations

1|l1i 2|l2i

˙ β

t t*

Environmental Restriction of Entanglement 2 spins: Quantum Langevin Equations

• When , start with
• Then
• Same as single spin with anisotropic dissipation

i | i | i |n1|2 + |n2|2 = 1 ✓ ◆

nzl

li

lz

i ˙

φi

φi θi | | | | n = (n⇤

1, n⇤ 2)σ

✓ n1 n2 ◆ ˙ β +

• | "#i

p

| "#i 1 p 2(| "#i | #"i)

˙ n + n ⇥ [2Jx + z(η1 + η2) 2γz(z · ˙ n)] = 0

· H = Jσ1 · σ2

l1 = l2 · B = 2Jx

• | "#i

p

| #"i

2γ ⌧ 1

⌧ 2γ 1

| "#i 1 p 2(| "#i | #"i)

| #"i

• | "#i

p

Entanglement Dynamics

|ψi = n1|l1, l2i + n2| l1, l2i

Environmental Restriction of Entanglement Many spins: ? ? ?

Reference state

| "i| "i| "i| "i | "i| "i| "i| "i| "i| "i| "i

Spins coupled to local harmonic baths decohered

| "i U(t)

|ψi =

Outline:

Introduction to Adiabatic Quantum Computation Entanglement as a Resource in AQC Environmental Restriction of Entanglement Towards Q Adiabatic Error Correction Conclusions

Error Correction in Adiabatic Q Comp. General Thoughts

Open system => No Q scaling without error correction

• Remove entropy from system
• Drive entanglement into system
• Cannot compute with equilibrium states

Repetition Schemes

• Repeat problem several times on chip
• ferromagnetic interaction between copies
• Stabiliser code for bit flip errors
• More robust to local noise
• Works experimentally
• Cannot change scaling . .

[Rønnow et al, Science 345, 420 (2014)] [Pudenz, et al, Nature Comms 5, 3243 (2014)]

Error Correction in Adiabatic Q Comp.

Dynamical Decoupling

• Uses idea of spin echo and refocusing from NMR
• Reverse time–evolution over short pulses
• r Quantum Zeno effect
• removes effects of noise in certain channel
• Relies on relatively slow noise dynamics

Computing with Out-of-Equilibrium Steady States?

• Q error correction
• continually remove entropy or add entanglement
• requires supply of fresh qubits
• Out-of-equilibrium steady state
• similar entropy flux

[Viola, Lloyd, PRA58, 2733 (1998)] [Viola, Knill, Lloyd, PRL82, 2417 (1999), PRL83, 4888 (1999)] [Quiroz, Lidar, Phys. Rev. A86, 042333 (2012)] [Tanaka and Miyamoto, PRL98,160407 (2007)] [Tanaka and Nemoto, PRA81, 022320 (2010)] [Schneider and Saenz, PRA85, 050304(R) (2012)] [Quiroz, Lidar, Phys. Rev. A86, 042333 (2012)]

Error Correction in Adiabatic Q Comp.

Threshold Theorems for AQC?

• An apparent contradiction
• AQC is analogue computation
• (and variational sub-manifolds are classical)
• Analogue computation is sensitive to noise
• How to reconcile with threshold theorems?

Number of components Number of noise channels

Hilbert Space Variational Sub-manifold (e.g. fixed D MPS)

H / exp[N] / / Poly[N]

Conclusions:

Entanglement is a resource in AQC

• Project to manifold of bounded entanglement
• Reduces to (semi-) classical dynamics
• Various algorithms over manifold of bounded entanglement

Entanglement Constrained by Environment

• Dissipation biases trajectories
• Noise decoheres – shown for 2 spins
• recovered restricted dynamics

Towards Error Correction for AQC

• Compute with out-of-equilibrium steady states
• Scaling of noise channels => threshold theorems