Solutions to Integer Programming from Quantum Annealing

In collaboration with Chia Cheng Chang, Chih-Chieh Chen, Travis Humble and Jim OstrowskiRIKEN iTHEMS

UC Berkeley & LBNL

Presented by

Christopher Körber | Ruhr-University Bochum & UC Berkeley | Feodor Lynen Fellow

Grid Inc. & UC Berkeley ORNL & U Tennessee U Tennessee

Qubits 2020

ckoerber.com | github.com/ckoerber | inspirehep.net: c.koerber.1

See also: • Publication: arXiv: [2009.xxxxx] • Software & Data: github.com/cchang5/quantum_linear_programming • Slides: ckoerber.com/#Talks

C. Körber, CC BY-NC 4.0

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Content

Application: The Dominating Set Problem

What is a graph?Defined by vertices (V) and edges (E)G(V, E)

Dominating SetA set of vertices {v} in {V} such that{v} + nearest neighbors = {V}

Domination number = 5

x0 = argmin(c ⋅ x)Ax ≤ b

x ∈ ℤNx ≥ 0

1. Constrained Integer Problems on an Annealer 2. Systematically Improving Results 3. Interpreting Annealer through Simulations

See also: • Publication: arXiv: [2009.xxxxx] • Software & Data: github.com/cchang5/quantum_linear_programming • Slides: ckoerber.com/#Talks

C. Körber, CC BY-NC 4.0

wikipedia.org Flux Balance Analysis

3

Ax ≤ b

x ∈ ℤN

Minimize

Under constraint

For all

x ≥ 0

Integer (Linear) Programming: Why?

Definition:

Why this problem:• Is classically NP hard • Maps ideally on annealer (integers)

xi =N

∑n=0

2nqni , qni ∈ {0,1}

• Storage optimization

• Flux Balance Analysis Analyzing the flow of metabolites through a metabolic network

• …

Examples:

See also: • Quantum annealing for systems of polynomial equations

C. C. Chang, A. Gambhir, T. S. Humble, S. Sota arXiv: [1812.06917] | Sci.Rep. 9 (2019) 1, 10258

f(x) = c ⋅ x

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Introduce slack variable s

Integer (Linear) Programming: How?

Ax ≤ b

x ∈ ℤN

Minimize

Under constraint

For all

x ≥ 0Ax + s = b

x, s ∈ ℤN

Minimize

Under constraint

For all

x, s ≥ 0

Definition: Implementation:

See also: • Quantum annealing for systems of polynomial equations

C. C. Chang, A. Gambhir, T. S. Humble, S. Sota arXiv: [1812.06917] | Sci.Rep. 9 (2019) 1, 10258

f(x) = c ⋅ x f(x) = c ⋅ x

*Also , with finite precision, possible

s ∈ RN

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x0 = argminx,s (f(x, s)) = argminx,s (c ⋅ x + pP(x, s))

Simultaneously minimize objective function in x and s

P(x, s) = (Ax + b + s) ⋅ (Ax + b + s)Penalty function:

Map objective function to bit representation

f(x, s) = ψ ⋅ Q ⋅ ψ + c

Bit vector Hamiltonian or QUBO

Constant (not important)Tψ = (x

s) ψi = 0,1

"Bit transformation"

Getting the hamiltonian (QUBO)See also: • A Tutorial on Formulating and Using QUBO Models

F. Glover, G. Kochenberger, Y. Du arXiv: [1811.11538]

p > maxx

(c ⋅ x)

Can be smaller (depending on problem)

Example: Minimum Dominating Set Problem

C. Körber, CC BY-NC 4.0

Application: Dominating Set Problem

7

Need a problem which has “nice” mapping but is classically NP hard

Application: The Dominating Set Problem

What is a graph?Defined by vertices (V) and edges (E)G(V, E)

Dominating SetA set of vertices {v} in {V} such that{v} + nearest neighbors = {V}

Domination number = 5

• Subset D of V (all nodes) • All nodes in V are

adjacent to D, or in D

Dominating Set

|D | = 5

Application: The Dominating Set Problem

What is a graph?Defined by vertices (V) and edges (E)G(V, E)

Dominating SetA set of vertices {v} in {V} such that{v} + nearest neighbors = {V}

Minimal Dominating SetSet which can not be reduced by removing a vertex

Domination number = 4

Is a dominating set which cannot be reduced by removing nodes

Minimal Dominating Set

|D | = 4

Application: The Dominating Set Problem

What is a graph?Defined by vertices (V) and edges (E)G(V, E)

Dominating SetA set of vertices {v} in {V} such that{v} + nearest neighbors = {V}

Minimal Dominating SetSet which can not be reduced by removing a vertex

Minimum Dominating SetSet with smallest domination number

Domination number = 3

Is the smallest minimal dominating set

Minimum Dominating Set

|D | = 3

Application: The Dominating Set Problem

What is a graph?Defined by vertices (V) and edges (E)G(V, E)

Dominating SetA set of vertices {v} in {V} such that{v} + nearest neighbors = {V}

Minimal Dominating SetSet which can not be reduced by removing a vertex

Domination number = 4

2

C. Körber, CC BY-NC 4.0

MDS Mapping & Theoretical Scaling

8

Problem definition

min (V

∑i=1

xi)

xi ∈ {0,1}

Find

For all

∀i ∈ {1,⋯, V}xi + ∑j∈𝒩i

xj ≥ 1Under constraint

Nearest neighbours of xi

1

2 3

4

x1 + x2 + x4 ≥ 1𝒩1 = {x2, x4}

Map to slack space

∀i ∈ {1,⋯, V}

min (V

∑i=1

xi)xi − si + ∑

j∈𝒩i

xj = 1

xi ∈ {0,1}

Find

Under constraint

For all 0 ≤ si ≤ 𝒩i

C. Körber, CC BY-NC 4.0

ψx

ψx

ψs

ψs

Form of the QUBO

9

E = (Ψx Ψs) (Qxx Qxs

0 Qss) (Ψx

Ψs) + p |V |Rescaled & Tri-diagonalized

Qxs ∼ ATs

Qxx ∼ A2 + ⋯

Adjacency matrixReason for denseness

Qss ∼ TsTTs + ⋯

Block size given by (log) number of neighbours

entr

ies

Ventries

∼V∑i=

1 log2 (𝒩

i )∈ {0,1}

len(ψ) ≲ V(1 + log2(V ))

Running the Experiment

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D-Wave Scaling: G(v)

11

Study the simplest graph possible

• Results show improvement over random guess

• Hardware precision limited: • Only solving

trivially small problems

• NOT qubit limited

G(3)

G(4)

G(6)

G(5)

Analytic solution available

min( |D(v) | ) = ⌈ v3 ⌉

Degeneracy of Ground State

NGS(v) =

1 mod (v,3) = 0

2 ⌊ v3 ⌋ + 2 mod (v,3) = 1

⌊ v3 ⌋ + 2 mod (v,3) = 2

Improving Computations

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Hfinal = ∑ij

Jijσzi σz

j + ∑i

hiσzi

Limiting factors: Many-Body Localization

13

Observations (for A(s) = 0) • if h is a constant, we recover the n-dimensional Ising Model

• beyond 1-dimension we get (anti-)ferromagnetic phase transitions • If h is random, we get the spin-glass model

H(s) = A(s)Hinit + B(s)Hfinal

Hinit = ∑i

σxi

For spin-glass Hamiltonian and/or large hi, the wavefunction is localized (and exponentially decays) in space (quantum analogy of non-ergodicity)

System fails to reach thermal equilibrium and retains a memory of its initial condition Anderson & Many-Body Localization

vs. hi ψi hi ψiMBL

s = 10 < s ≪ 1

hi ψi hi ψiNo MBL

s = 10 < s ≪ 1

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Results for different offsets and graph sizesStrong delayed

Weak delayed

Time-dependence of the Hamiltonian

14

We can effectively advance or delay A(s) and B(s) qubitwise on DWave

MBL inspired hypothesis:Delaying strong external fields yields less disorder and a more delocalized system

hmid =max( |hi | ) + min( |hi | )

2

Group qubits of embedded Hamiltonian according to final external magnetic field

|hi | ≤ hmid

|hi | > hmidGroup Strong

Group Weak

Variation in schedules

Advance qubit groups with different schedules (after embedding)

Hfinal(s) = ∑ij

Bij(s)Jijσzi σz

j + ∑i

Bi(s)hiσzi

H(s) = Hinit(s) + Hfinal(s)Hinit(s) = ∑

i

Ai(s)σ xi

See also: • D-Wave: QPU-Specific Anneal Schedules

support.dwavesys.com

Bi(s)Ai(s)

Offset delay Offset delay

Simulating the Annealer

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Simulating the Annealer

16

Solving the underlying equations

Modelvon Neumann equations + Local decoherence + Full-counting statistics decoherence

∂tρ(t) =−iℏ

[H(t), ρ(t)] + ℒL(ρ(t), {hi(t)}) + ℒFC(ρ(t), H(t))

Lindblad operator

• Captures decay of a 2-level system to ground state • Relaxes to local (non-interacting) ground state • Depends on local magnetic fields • Free parameters:

• Local decoherence rate • Temperature (shared)

Local decoherence Full-counting statistics• Captures interaction with thermal environment • System relaxes to eigenstates of H(t) with a Boltzmann distrib. • Depends on energy spectrum • Free parameters:

• Full-counting decoherence rate • Temperature (shared)

• Extract D-Wave Hamiltonian after embedding • Solve equations for same anneal schedule as D-Wave

See also: • Lindblad-equation approach for the full counting

statistics of work and heat in driven quantum systems M. Silaev, T. T. Heikkilä, and P. Virtanen arXiv: [1312.3476] | Phys. Rev. E 90, 022103

C. Körber, CC BY-NC 4.0

Final state distribution vs. annealing offset

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of lowest 3 states

• Free parameters: 22.5 milliKelvin and 1 ~ 15 ns coherence time (agree with D-Wave tech report) • Result is free of any hardware unknowns (do not expect exact matching)

• Ground state offset scaling reproduced by simulation • The first non-degenerate excited state is populated with correct scaling (mainly Full-counting decoherence) • Simulation likely capturing the majority of the physics • Simulation suggests Many-Body Localization scaling hypothesis

Strong delayed Weak delayed

Strong delayed Weak delayed

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See also: • Publication: arXiv: [2009.xxxxx] • Software & Data: github.com/cchang5/quantum_linear_programming • Slides: ckoerber.com/#TalksSummary

• Mapping for ILP to Annealer provided • For MDS Line Graph G(v)

- Annealer beats random guessing - MBL inspired schedule adjustment improve results

• Simulation agrees with experiment - Model suggests improvements may caused by quantum effects

Future • Provide

- ILP Mapping Software (Python) - Simulation Software - Study data

• Quantum Horizons Grant - DOE is funding our effort to expand in this direction - Funding for additional Postdoc: contact chiachang@berkeley.edu

Special Thank You to • My collaborators :) • D-Wave Support Team

Vlad Papish, David Johnson and many others

In collaboration with Chia Cheng Chang, Chih-Chieh Chen, Travis Humble and Jim OstrowskiRIKEN iTHEMS

UC Berkeley & LBNL

Presented byChristopher Körber | Ruhr-University Bochum & UC Berkeley | Feodor Lynen Fellow

Grid Inc. & UC Berkeley ORNL & U Tennessee U Tennessee

ckoerber.com | github.com/ckoerber | inspirehep.net: c.koerber.1

(Jason)