5cmquantum simulation of condensed matter in programmable...

Quantum simulation of condensed matter inprogrammable qubit lattices

Jack Raymond, D-Wave Systems

Workshop IV: New Architectures and AlgorithmsScience at Extreme Scales: Where Big Data Meets Large-Scale ComputingNovember 2018

Copyright © D-Wave Systems Inc.

How does one simulate a quantum magnet with a D-Wave processor?

Experiments on single unit cells readily indicate quantum behaviour. Showing suchbehaviour over an entire processor is much more challenging.

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Outline

I Annealing for heuristic optimizationI Annealing for equilibrium samplingI The anti-ferromagnetic, and spin-glass, phases on D-wave devicesI The Kosterlitz-Thouless phase (nuts and bolts)I The Kosterlitz-Thouless phase on D-wave devices

2 / 40

Annealing for heuristic optimization


D-Wave quantum annealing system in a clamshell

shielded room (1nT) cryostat (10mK) sample holder processor (2048 qubits)

I Implementation of transverse �eld Ising model quantum annealing

I Prepare system of superconducting currents (±1 spins) in a uniformsuperposition (�at potential)

I Evolve the physical system introducing interactions, default evolutiontime is 5 µs

I Finish in a low-energy state of the target model (complicated potential)

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Initial plan: classical minimization problems

I Work with binary variables: ±1 (Ising model)

I Energy function E : {−1,+1}n → R represents “cost” of states

I Find minimum energy state: ground state

I Near-optima often useful, depending on application

4 / 40


Hamiltonian: Transverse �eld Ising model

Annealing parameter 0 ≤ s ≤ 1

H(s) = − Γ(s)[

∑i

σxi

]︸︷︷︸

quantum�uctuations

+ J(s)[

∑i

hiσzi + ∑

ijJijσ

zi σz

j

]︸︷︷︸

classical IsingHamiltonian

Two approaches to minimize the classical Ising Hamiltonian

I Classical thermal annealing, Γ(s) = 0 (Kirkpatrick, Gelatt, Vecchi,Science, 1983)

I Quantum annealing, Γ(s)→ 0 (Kadowaki+Nishimori, PRE, 1998)

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Simulated (thermal) annealing

0 102468

1012

Annealing parameter s

Ener

gy(G

Hz)

Annealing schedule

J(s): Ising problem energy scalekBT: temperature

s = 0

ψ0 ψ1, ψ2

s < s∗

ψ0 ψ1, ψ2

s = s∗

ψ0 ψ1, ψ2

s = 1

ψ0 ψ1, ψ2

Bypass energy barriers: hop over (classical)

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Quantum annealing

0 102468

1012


Ener

gy(G

Hz)

Annealing schedule

Γ(s): quantum �uctuationsJ(s): Ising problem energy scalekBT: temperature

s = 0

|ψ0 + ψ1 + ψ2〉

s < s∗

|ψ0〉 |ψ1 + ψ2〉

s = s∗

|ψ0〉 |ψ1 + ψ2〉

s = 1

ψ0 ψ1, ψ2

Bypass energy barriers: tunnel through (quantum) or hop over (classical)

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Ising models can represent hard problems

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Quantuma annealing can yield an exponential speedup

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Some scienti�c milestones

2010 Demonstration of quantum annealing(Johnson et al., Nature)

2013 First benchmarking study, 3000x faster than CPLEX(McGeoch + Wang, ACM CF‘13)

2015 Entanglement demonstrated in D-Wave processor(Lanting et al., PRX)

2016 Multiqubit cotunneling confers scaling advantage over SA(Denchev et al., PRX)

2018 Scaling advantage over simulated annealing(Albash + Lidar, PRX)

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Annealing for equilibrium sampling


New demonstrations: Quantum simulation

Harris et al., Phase transitions in aprogrammable quantum spin glass simulator

Science 361 6398 162-165 (2018)

King et al., Observation of topological phenomenain a programmable lattice of 1,800 qubits

Nature 560 7719 (2018)

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Annealing protocols for equilibrium sampling

0 0.2 0.4 0.6 0.8 10

5

10

15

20

s = t/ta

Ener

gysc

ale,

GH

z

Standard

0 0.2 0.4 0.6 0.8 1

s = t/ta

Quench

0 0.2 0.4 0.6 0.8 1

s = t/ta

Pause and Quench

I Pause beyond the equilibration time, then quench faster than physical dynamicsI Allows access to non-classical distributions, where superposition is non-trivial

Success requires both equilibration up to s, and fast quenchWe access quantum distributions projected on the classical space

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0 5 10 15 200

0.2

0.4

0.6

0.8

1

Annealing time t (µs)

Ann

ealin

gpa

ram

eters

reverse anneal protocol

I We can also start from a classical initial condition (reverse annealing)I We can check escape from controlled subspacesI and potentially avoid dynamical obstacles

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0 5 10 15 200

0.2

0.4

0.6

0.8

1


Ann

ealin

gpa

ram

eters


0 5 10 15 200

0.2

0.4

0.6

0.8

1


Ann

ealin

gpa

ram

eters


I We can also start from a classical initial condition (reverse annealing)I We can check escape from controlled subspacesI and potentially avoid dynamical obstaclesI and processes can be daisy-chained together to access long time scales

Success requires that the quench does not set us back to zero knowledge

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What is the alternative?

I Quantum equilibria and dynamics require exponential resources to simulateI Spin glass and KT phases are relatively resilient to approximation methodsI However, quantum equilibria of the transverse �eld Ising model allow a

path-integral representation

I Sampling by Markov chain Monte Carlo methods (QMC, PIMC)I Allows a check on results at scaleI Local path dynamics are not equivalent to physical dynamics of wavefunctions

14 / 40


What is the alternative?

I Quantum equilibria and dynamics require exponential resources to simulateI Spin glass and KT phases are relatively resilient to approximation methodsI However, quantum equilibria of the transverse �eld Ising model allow a

path-integral representationI Sampling by Markov chain Monte Carlo methods (QMC, PIMC)I Allows a check on results at scaleI Local path dynamics are not equivalent to physical dynamics of wavefunctions

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The anti-ferromagnetic, and spin-glass,phases on D-wave devices


The 3D AFM lattice (Harris et al.)I First large-scale quantum simulation result on a QA processorI Simulate quantum phase transition of doped AFM latticeI Parameters T/J, Γ/J, doping probability p.

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Magnetization measurement

Anneal Hold Quench

s

0

1

time

time

Transverse Field Control

Longitudinal Field Control

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28−1

−0.5

0

0.5

1

AFM

ordering

p=0.100, instance=81, AFM Magnetization versus s

s

mean magnetization=0

mAFM

Count

0

20

40

60

80

100

120

140

160

180

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Susceptibility measurement

Anneal Hold Quench

s

0

1

time

time

Transverse Field Control

Longitudinal Field Control +

- 0.16 0.18 0.2 0.220

0.5

1

1.5

2

2.5

s

χAFM×

105(Φ

−1

0)

p = 0p = 0.05p = 0.1p = 0.15p = 0.2

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The Phase diagram

Sketching out the phase diagram

I Binder cumulant crossings give pc

I Susceptibility peak gives Γc

I Deviations consistent with �nite temperature/size e�ects

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The Kosterlitz-Thouless phase (nuts andbolts)


The Fully Frustrated Square-octagonal lattice (King et al.)I First physical lattice demonstration the transverse �eld Ising model KT phaseI Simulation of the KT phase transitionI Parameters T/J and Γ/J

.51

QA s = 0.20 QA s = 0.26

QMC s = 0.20 QMC s = 0.26

Com

plex

orde

rpa

ram

eterψ=meiθ

0 0.5 1 1.5 20

0.1

0.2

0.3

s = 0.30

s = 0.25

s = 0.20

KT

Ordered

PM

Γ/J

T/J

QA @ 8.4 mKUpper transition

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The Kosterlitz-Thouless phase transition

2016 Nobel for theoretical discoveries of topological phase transitionsand topological phases of matter

Vadim Berezinskii J. Michael Kosterlitz David Thouless

I Most easily described in 2D XY model.I Finite-temperature phase transition, but does not exist in 2D

Ising model without quantum �uctuations.

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2D XY model

Classical 2D spin:

XY-HamiltonianH = −JXY ∑i,j

~Si ·~Sj = −JXY ∑i,j

cos (θi − θj)

Ground state:all spins aligned

Continuous rotational symmetry: O(2) or U(1)

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Topological excitations

vortex antivortex

Defects appear in vortex/antivortex pairs (Stokes’ Theorem)

But when are these pairs tightly bound?Below the KT phase transition

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Experimental Observation of KT Phase Transition

As observed physically in. . .

I super�uid 4He �lms, 1978I thin �lm superconductors, 1979I trapped atoms, 2006I graphene-tin hybrid JJ arrays,

2014

As theorized/simulated in. . .

I Triangular AFM transverse �eld Ising model

2LL

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KT phase transition in TAFM TFIM

Theoretical predictions / Monte Carlo

I Blankschtein, Ma, Berker, Grest & Soukoulis, PRB 29, 5250 (1984)I Jalabert & Sachdev, PRB 44, 686 (1991)I Moessner & Sondhi, PRB 63, 224401 (2001)I Isakov & Moessner, PRB 68, 104409 (2003)I Wenzel, Coletta, Korshunov & Mila, PRL 109, 187202(2012)

No experimental demonstration to date

Simulation or observation? Both!Programmable superconducting qubits ≈ Ising model

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AFM triangle: Order by disorder (transverse �eld Γ)Hamiltonian

H = ∑i<j

Jijσzi σz

j − Γ ∑i

σxi

6-degenerate frustrated ground stateClassical EGS = −JQuantum EGS = −J− Γ

2 3

1

+ −

+

2 3

1

+ −

−

2 3

1

+ −

+/−

E = −J E = −J E = −J

− Γ

Perturbative pictureFloppy spins (no net e�ective �eld) align with transverse �eld

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H = ∑i<j

Jijσzi σz

j − Γ ∑i

σxi


2 3

1

+ −

+

2 3

1

+ −

−

2 3

1

|↑〉 |↓〉

|↑〉+|↓〉2

E = −J E = −J E = −J− Γ


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H = ∑i<j

Jijσzi σz

j − Γ ∑i

σxi


2 3

1

+ −

+

2 3

1

+ −

−

2 3

1

|↑〉 |↓〉

|→〉

E = −J E = −J E = −J− Γ


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H = ∑i<j

Jijσzi σz

j − Γ ∑i

σxi


2 3

1

+ −

+

2 3

1

+ −

−

2 3

1

|↑〉 |↓〉

|→〉

↑pseudospin

E = −J E = −J E = −J− Γ


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Pseudospin = linear combination of 3 basis vectors

2 3

1

|↑〉 |↓〉

|→〉

1eiθ1

2eiθ2

3eiθ3

θ1 = 0

θ2 = 2π/3

θ3 = 4π/3

26 / 40


Pseudospin = linear combination of 3 basis vectors

2 3

1

|↑〉 |↓〉

|→〉

↑pseudospin

〈σz1〉 = 0

〈σz2〉 = 1

〈σz3〉 = −1

pseudospin

|→↑↓〉

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Pseudospin⇒ 6 clock states (in perturbative picture)

spin pseudospin

1 2 3

↑

↑

↑

↑

↑

↑|↑↓↓〉|↓↑↑〉

|↑↑↓〉|↓↑↓〉

|↑↓↑〉|↓↓↑〉

|↓↓↓〉|↑↑↑〉

|↑→↓〉

|↑↓→〉

|↓↑→〉

|↓→↑〉

|→↑↓〉

|→↓↑〉

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Pseudospin phase = XY modelSpin alignment⇒ sublattice ordering⇒ |ψ| = 1

2 3 1 2 3

2 3 1 2 3

3 1 2 3 1 2

::

::

::

::

::

::

::

::

::

Order parameter ψ = average pseudospinReal order parameter m = |ψ|

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2 3 1 2 3

2 3 1 2 3

3 1 2 3 1 2

::

::

::

::

:

:

:

:

:

:

:

:

:

:

Order parameter ψ = average pseudospinReal order parameter m = |ψ|

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2 3 1 2 3

2 3 1 2 3

3 1 2 3 1 2

::

::

::

::

:

:

:

:

:

:

:

:

:

:

Twisting pseudospin phase⇒ triangles with no �oppy qubitExactly like the XY model! Except Tc depends on Γ.

Tc = JXYπ

2= Γ

π

12(2−√

3)Only applies in perturbative regime.

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Triangular AFM phase diagram

T/J

Γ/J

KT phase

ordered

paramagnetic

Ordered KT PM

Moessner & Sondhi, 2001 Isakov & Moessner, 2003

29 / 40



T/J

Γ/J

KT phase

ordered

paramagneticOrdered KT PM

Moessner & Sondhi, 2001 Isakov & Moessner, 2003

29 / 40



T/J

Γ/J

KT phase

ordered


Moessner & Sondhi, 2001

Isakov & Moessner, 2003

29 / 40



T/J

Γ/J

KT phase

ordered


Moessner & Sondhi, 2001 Isakov & Moessner, 200329 / 40

The Kosterlitz-Thouless phase on D-wavedevices


Geometrically frustrated latticesFully-frustrated square-octagonal ≈ triangular AFM

I Same theoretical understanding in one perturbative limit (T → 0, Γ/J → 0)I Di�erent elsewhere, including non-universal phase transition propertiesI Di�ers from classical XY, and standard incoherent quantum (rotor) models

2LL

I AFM couplers have Jij = 1 ; FM couplers have Jij = −1.8

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Demonstration in D-Wave 2000Q

5.5 mm

L=15

31 / 40


Sampling with Reverse anneal, pause and quench

0 0.2 0.4 0.6 0.8 10

10

20

30

40


Ener

gysc

ale

(uni

tless

)

Γ(s)/T

J(s)/T

0.2 0.25 0.34

6

8

s

Ene

rgy

0 0.5 1 1.5 20

0.1

0.2

0.3

s = 0.30

s = 0.25

s = 0.20

KT

Ordered

PM

Γ/JT/J

QA @ 8.4 mKUpper transition

0 5 10 15 200

0.2

0.4

0.6

0.8

1


Ann

ealin

gpa

ram

eters


I QA schedule: Sequence of Hamiltonians, annealing parameter sI Pause allows long relaxation at �xed HamiltoniansI Quench allows “projective” readoutI Reverse anneal allows initialization in classical state at s = 1

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Quantum evolution Monte Carlo

0 25 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

measurementburn-in

Quantum evolution steps

Ord

erpa

ram

eter

〈m〉

Ordered initial stateRandom initial state

Input classical state (s = 1)

Quantum evolution (s = 0.26)

Read out classical state (s = 1)

0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

PM critical PM

Annealing parameter sO

rder

para

met

er〈m

〉

QMCQA

.51

QA s = 0.20 QA s = 0.26

QMC s = 0.20 QMC s = 0.26

Com

plex

orde

rpa

ram

eterψ=meiθ

I Reverse annealing “Markov” chainI Start from random and ordered state.

Bound 〈m〉 above/below

I Peak in 〈m〉 near KT phaseI U(1) symmetry in complex order parameterI Agreement with QMC

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Onset of power-law correlation decay

3 6 12

0.1

1 b = 0.369(11)

b = 0.366(21)

s = 0.26

Distance xij

Pha

seco

rrel

atio

nC

ij

QMC 8.4mKQMC 21.4mKQA 8.4mKQA 21.4mK

I HOT PM region: Exponential decayI COLD KT region: Power-law decay

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Onset of power-law correlation decay

3 6 12

0.1

1 b = 0.369(11)

b = 0.366(21)

s = 0.26

Distance xij

Pha

seco

rrel

atio

nC

ij


3 6 12

0.3

0.4

0.5

0.6

0.7

0.80.9

b/2 = 0.147(4)b/2 = 0.168(11)

s = 0.26

Lattice width L

Ord

erpa

ram

eter

〈m〉


I HOT PM region: Exponential decayI COLD KT region: Power-law decay

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Quantum simulation with D-Wave

New features, new possibilities

I Anneal features allow previously unreachableexperiments

Phase transitions and critical phenomena

I Kosterlitz-Thouless, ferromagnetic and spin glasstransitions studied

Programmable magnetic material

I Feynman’s vision for quantum computing. . . to apoint: Simulate a quantum system with aprogrammable quantum system.

Thanks for your attention

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Extra material


Typical Pseudospin �eld at equilibrium

Typical state (L=6, Γ/J=0.842 , T/J=0.146 [s=0.26, T=8.4mK]).

37 / 40


More-ordered pseudospin �eld

Similar to perturbative ground state (L=6, Γ/J=0.842 , T/J=0.146 [s=0.26, T=8.4mK])

38 / 40


Less-ordered pseudospin �eld

Dissimilar to perturbative ground state (L=6, Γ/J=0.842 , T/J=0.146 [s=0.26,T=8.4mK]).

39 / 40


Extras: Calibration re�nement

−1 −0.5 0 0.5 10

0.1

0.2

0.3

Average qubit magnetization

Freq

uenc

y

Without shimWith shim

−1 −0.5 0 0.5 10

0.1

0.2

0.3

Average AFM spin-spin correlation

Freq

uenc

y

Without shimWith shim

a b

Use lattice symmetries to re�ne calibration

I Each qubit has average magnetization 0.I Coupler frustration probabilities obey rotational symmetry.

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Extras: “Markov” chain convergence

0 25 500.1

0.2

0.3

0.4

0.5

0.6

0.7s = 0.20, T = 8.4mK


Ord

erpa

ram

eter

〈m〉

Ordered init.Random init.

0 25 500.1

0.2

0.3

0.4

0.5

0.6

0.7s = 0.26, T = 8.4mK


Ord

erpa

ram

eter

〈m〉


0 25 500.1

0.2

0.3

0.4

0.5

0.6

0.7s = 0.30, T = 8.4mK


Ord

erpa

ram

eter

〈m〉


0 25 500.1

0.2

0.3

0.4

0.5

0.6

0.7s = 0.26, T = 15.8mK


Ord

erpa

ram

eter

〈m〉


0 25 500.1

0.2

0.3

0.4

0.5

0.6

0.7s = 0.26, T = 18.2mK


Ord

erpa

ram

eter

〈m〉


0 25 500.1

0.2

0.3

0.4

0.5

0.6

0.7s = 0.26, T = 21.4mK


Ord

erpa

ram

eter

〈m〉


39 / 40


Extras: Phase diagram

10−2 103 108 1013

10−2

10−1

100

101 Γ/J = 1.20

eat−1/2

/L

χL−c

L = 3

L = 6

L = 9

L = 12

L = 15

L = 18

L = 21

10−1 100 101 102 1030.98

1

1.02

1.04

Γ/J = 1.20

eat−1/2

/L

mLb

L = 15

L = 18

L = 21

6 9 12 15 18 21

0.7

0.8

0.9

η/2 = 0.1250 ± 0.0004

η/2 = 0.0556 ± 0.0004

T2 = 0.1824 ± 0.0012

T1 = 0.0850 ± 0.0006

Γ/J = 1.20 log-log

L

m

1.6 1.7 1.8 1.9 2 2.10

0.2

0.4

0.6

0.8

1

Γ/J

Bin

der

cum

ulan

tU

L = 3

L = 6

L = 9

L = 12

L = 15

−2 0 2 40

0.2

0.4

0.6

0.8

1

L1/ν(Γ − Γc)/Γc

U

0 0.5 1 1.5 20

0.1

0.2

Γ/JT/J

T2 from collapseT2 from η

T1 from collapseT1 from η

QCP

a b c

d e

39 / 40


Extras: Quench

0 .05 .10 .15 .20 .250

0.1

0.2

0.3

Residual classical energy per spin

Freq

uenc

yQAQMC projected

0 .05 .10 .15 .20 .250

0.1

0.2

0.3

Residual classical energy per spin

Freq

uenc

y

QAQMC quenched

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

|ψ|, QMC projected

|ψ|,

QM

Cqu

ench

ed

a b c

QA evolves during 1 µs quench.

I Huge di�erence between QMC and QA classical energies.I Classical quench erases the di�erenceI ψ mostly unchanged.

39 / 40


Extras: Embedding into qubit lattice

a b

c

40 / 40


Extras: Next-gen Pegasus topology

Planned Pegasus topology, shown at P4 scale.40 / 40

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