Solutions to Integer Programming from Quantum Annealing€¦ · Solutions to Integer Programming...
Transcript of Solutions to Integer Programming from Quantum Annealing€¦ · Solutions to Integer Programming...
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
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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
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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
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MDS Mapping & Theoretical Scaling
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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
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ψx
ψx
ψs
ψs
Form of the QUBO
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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)
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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
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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
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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
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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
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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 [email protected]
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)