The Quantum Alternating Operator Ansatz on Maximum k-Vertex Cover - - PowerPoint PPT Presentation

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Los Alamos National Laboratory LA-UR-20-28072 The Quantum Alternating Operator Ansatz on Maximum k-Vertex Cover you joint work with Jeremy Cook and Stephan Eidenbenz Andreas Brtschi nt CCS-3 Information Sciences baertschi@lanl.gov wo

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you nt wo

Los Alamos National Laboratory

The Quantum Alternating Operator Ansatz

  • n Maximum k-Vertex Cover

Andreas BΓ€rtschi

CCS-3 Information Sciences baertschi@lanl.gov IEEE International Conference on Quantum Computing and Engineering (QCE20) October 12 – 16, 2020

joint work with Jeremy Cook and Stephan Eidenbenz

LA-UR-20-28072

Managed by Triad National Security, LLC for the U.S. Department of Energy's NNSA

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Los Alamos National Laboratory October 12, 2020 | 2

QAOA and k-Vertex Cover

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Los Alamos National Laboratory October 12, 2020 | 3

Quantum Alternating Operator Ansatz (QAOA)

QAOA is a heuristic for constraint combinatorial optimization. Given a problem

  • ver inputs 𝑦 ∈ 𝐸 βŠ‚ 0,1 ! with objective function 𝑔 𝑦 ∢ 𝐸 β†’ ℝ, prepare a state

βƒ— 𝛾, βƒ— 𝛿 = 𝑓"#$!%"𝑓"#&!%# β‹― 𝑓"#$$%"𝑓"#&$%# πœ” from which one would like to sample good solutions with high probability. QAOA is specified by

  • an initial state πœ” in a feasible subspace given by the domain 𝐸,
  • a phase separating cost Hamiltonian 𝐼',

diagonal in the computational basis: 𝐼' 𝑦 = 𝑔(𝑦) 𝑦 ,

  • a mixing Hamiltonian 𝐼( preserving and interfering solutions in 𝐸,
  • π‘ž rounds with individual angle parameters 𝛾), … , 𝛾*, 𝛿), … , 𝛿*.
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Los Alamos National Laboratory October 12, 2020 | 4

Maximum k-Vertex Cover

Maximize the number of Edges covered by 𝑙 out of all π‘œ Vertices

  • f an input Graph (here 𝑙 = 4, π‘œ = 9).
  • Greedy: 10 edges covered
  • Maximum: 11 edges covered
  • In general (for non-constant 𝑙):

NP-hard to approximate to 1 βˆ’ πœ— , UG-hard to approximate to 0.944, classical polynomial-time 0.92-approximation Simple objective function Single Equality Constraint βˆ‘π’˜βˆˆπ‘Ύ π’šπ’˜ = 𝒍 𝑦) = 1 𝑦. = 0

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Los Alamos National Laboratory October 12, 2020 | 5

Maximum k-Vertex Cover

Maximize the number of Edges covered by 𝑙 out of all π‘œ Vertices

  • f an input Graph (here 𝑙 = 4, π‘œ = 9).

𝑔(𝑦) = ?

/,1 ∈2

𝑃𝑆(𝑦/, 𝑦1) = ?

/,1 ∈2

1 βˆ’ (1 βˆ’ 𝑦/)(1 βˆ’ 𝑦1) Simple objective function Single Equality Constraint βˆ‘π’˜βˆˆπ‘Ύ π’šπ’˜ = 𝒍 𝐼' = 1 4 ?

/,1 ∈2

3 𝐽𝑒 βˆ’ 𝜏/

3𝜏1 3 βˆ’ 𝜏/ 3 βˆ’ 𝜏1 3

𝑦! = 1 βˆ’ 𝜏!

"

2

𝑦) = 1 𝑦. = 0

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Los Alamos National Laboratory October 12, 2020 | 7

Experiments

Initial States Mixers Angle Selection Strategies Dicke States Ring Mixer Monte Carlo Random States Complete Graph Mixer Basin Hopping Interpolation We compared different QAOA approaches for Maximum 𝑙-Vertex Cover

  • n random ErdΕ‘s-RΓ©nyi 𝐻!,* graphs on π‘œ = 7,8,9,10 vertices with 𝑙 = ⌊

! .βŒ‹ and

edge probability π‘ž = 0.5: Dicke States: Equal Superposition of all Hamming-Weight 𝑙 Basis States, Random State: Randomly chosen Hamming-Weight 𝑙 Basis State. Ring Mixer: Hamming-weight preserving XY-interactions on a Ring, Complete Mixer: Hamming-weight preserving XY-interactions on a Clique.

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Agenda

Initial State and Mixer Choice

  • Dicke States vs. Random States
  • Ring Mixer vs. Complete Graph Mixer

Angle Selection Strategies

  • Angle Correlations
  • Strategy evaluations

Summary

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Los Alamos National Laboratory October 12, 2020 | 9

Initial State and Mixer Choice

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Hamming Weight 𝒍 Subspace

Problem is constraint to solutions 𝑦 of Hamming Weight 𝑙. We need to start in this subspace and stay in it: Initial State

  • Random Basis State |π‘¦βŸ© of

Hamming Weight x = 𝑙,

  • r
  • Dicke State

𝐸4

! =

1 π‘œ 𝑙 ?

5 64

|π‘¦βŸ© Mixer Choice for 𝒇"π’‹πœΈπ‘°π‘΅ 𝜏)

5𝜏. 5 + 𝜏) :𝜏. : =

1 1

  • 𝐼;#!<

= βˆ‘#6=

!

𝜏#

5𝜏#>) 5

+ 𝜏#

:𝜏#>) :

  • 𝐼?@#A/B = βˆ‘#CD 𝜏#

5𝜏 D 5 + 𝜏# :𝜏 D :

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Los Alamos National Laboratory October 12, 2020 | 11

Complete Graph vs. Ring Mixer and Dicke vs. k-state

  • Complete Graph mixer finds better solutions at lower round counts
  • Dicke starting state outperforms randomly selected k-state starting state

Complete Graph Mixer Ring Mixer

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Los Alamos National Laboratory October 12, 2020 | 12

Complete Graph Mixer outperforms Ring Mixer

  • Complete Graph Mixer

consistently outperforms Ring Mixer (by a ratio > 1 even w/ confidence intervals)

  • Ratio decreases for higher

rounds, when both Mixers are close to the optimum.

(Plot data taken over 100 graphs of size 7)

Approximation ratio of the complete mixer 𝒔𝑳 compared to the approximation ratio of the ring mixer 𝒔𝑺 for several rounds:

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Angle Selection Strategies

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Monte Carlo Angle Sampling does not work well

  • Monte Carlo Angle Sampling: take p random

angles for gamma and beta

  • Plot (right) shows a declining approximation

ratio if a total of 1000 sample angles are distributed across π‘ž rounds

(Data on 10 random graphs; Complete Graph mixer; Dicke states)

  • Plot (right) shows number of Monte Carlo

samples required to get an increase in best approximation ratios found (π‘ž wrt π‘ž βˆ’ 1)

  • Exponential scaling in 𝒒 makes Monte Carlo

not seem promising as angle selection strategy

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Los Alamos National Laboratory October 12, 2020 | 15

Angle Selection: Are angles good for many graphs?

Heat plots show expectation values for 1-round QAOA averaged over 100 graphs. Ø Hope for generally good values. Complete Graph Mixer Ring Mixer

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Los Alamos National Laboratory October 12, 2020 | 16

Are good angles correlated across rounds and graphs?

Beta Plot shows best 𝛾 and 𝛿 values for π‘ž = 6-round QAOAs of typical graphs. Red line: Average over all tested graphs (graph size 7 – 10). Gamma

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Good angles for increasing rounds p?

𝛾 Correlation holds across different values of the number of rounds π‘ž. Ø Interpolation strategy for angle selection Ξ³ π‘ž = 5 π‘ž = 6 π‘ž = 7

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Los Alamos National Laboratory October 12, 2020 | 18

Comparison of Angle Selection Strategies

Strategies

  • Monte Carlo: β€œrandom angles”
  • Basin Hopping: β€œdetect and

escape local minima”

  • Interpolation: β€œlearn angles

from other graphs”

  • Optimal: β€œfull angle exploration

at 0.01𝜌 resolution” Experimental Setup

  • Same number of samples for all

three strategies and for all p Findings: only interpolation manages to increase performance up to 4 rounds, Basin hopping and Monte Carlo do not profit from > 𝟐 round.

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Summary

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Conclusion k-Vertex Cover QAOA

Main Findings

  • Dicke States outperform Random 𝑙-states as initial state
  • Complete Graph Mixer outperforms Ring Mixer
  • Interpolation method finds angles that are good across graphs and rounds

Additional Findings

  • Monte Carlo angle sampling works poorly
  • Ring mixer not periodic, no circuit implementation of Complete Graph Mixer

Future work & Open questions

  • Do results hold for larger graphs? Conjecture: performance gaps increase
  • How well does the Grover Mixer perform? (see other talk in this session)
  • Do findings generalize to other optimization problems?