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

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 Bärtschi nt CCS-3 Information Sciences baertschi@lanl.gov wo IEEE International Conference on Quantum Computing and Engineering (QCE20) October 12 – 16, 2020 Managed by Triad National Security, LLC for the U.S. Department of Energy's NNSA

Los Alamos National Laboratory QAOA and k-Vertex Cover October 12, 2020 | 2

Los Alamos National Laboratory Quantum Alternating Operator Ansatz (QAOA) QAOA is a heuristic for constraint combinatorial optimization. Given a problem over inputs 𝑦 ∈ 𝐸 ⊂ 0,1 ! with objective function 𝑔 𝑦 ∶ 𝐸 → ℝ , prepare a state 𝛿 = 𝑓 "#$ ! % " 𝑓 "#& ! % # ⋯ 𝑓 "#$ $ % " 𝑓 "#& $ % # 𝜔 ⃗ 𝛾, ⃗ from which one would like to sample good solutions with high probability. QAOA is specified by • an initial state 𝜔 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 , • 𝑞 rounds with individual angle parameters 𝛾 ) , … , 𝛾 * , 𝛿 ) , … , 𝛿 * . October 12, 2020 | 3

Los Alamos National Laboratory Maximum k-Vertex Cover Maximize the number of Edges covered by 𝑙 out of all 𝑜 Vertices of an input Graph (here 𝑙 = 4 , 𝑜 = 9 ) . 𝑦 . = 0 • Greedy : 10 edges covered • Maximum : 11 edges covered 𝑦 ) = 1 • In general (for non-constant 𝑙 ): NP-hard to approximate to 1 − 𝜗 , UG-hard to approximate to 0.944, classical polynomial-time 0.92-approximation Single Equality Constraint ∑ 𝒘∈𝑾 𝒚 𝒘 = 𝒍 Simple objective function October 12, 2020 | 4

Los Alamos National Laboratory Maximum k-Vertex Cover Maximize the number of Edges covered by 𝑙 out of all 𝑜 Vertices of an input Graph (here 𝑙 = 4 , 𝑜 = 9 ) . 𝑔(𝑦) = ? 𝑃𝑆(𝑦 / , 𝑦 1 ) 𝑦 . = 0 /,1 ∈2 𝑦 ) = 1 " 1 − 𝜏 ! = ? 1 − (1 − 𝑦 / )(1 − 𝑦 1 ) 𝑦 ! = 2 /,1 ∈2 𝐼 ' = 1 3 − 𝜏 / 3 − 𝜏 1 3 𝜏 1 3 4 ? 3 𝐽𝑒 − 𝜏 / /,1 ∈2 Single Equality Constraint ∑ 𝒘∈𝑾 𝒚 𝒘 = 𝒍 Simple objective function October 12, 2020 | 5

Los Alamos National Laboratory Experiments We compared different QAOA approaches for Maximum 𝑙 -Vertex Cover ! on random Erdős-Rényi 𝐻 !,* graphs on 𝑜 = 7,8,9,10 vertices with 𝑙 = ⌊ . ⌋ and edge probability 𝑞 = 0.5 : Initial States Mixers Angle Selection Strategies Dicke States Ring Mixer Monte Carlo Random States Complete Graph Mixer Basin Hopping Interpolation Equal Superposition of all Hamming-Weight 𝑙 Basis States, Dicke States : Randomly chosen Hamming-Weight 𝑙 Basis State. Random State : Ring Mixer : Hamming-weight preserving XY-interactions on a Ring, Complete Mixer : Hamming-weight preserving XY-interactions on a Clique. October 12, 2020 | 7

Los Alamos National Laboratory 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 October 12, 2020 | 8

Los Alamos National Laboratory Initial State and Mixer Choice October 12, 2020 | 9

Los Alamos National Laboratory 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 Mixer Choice for 𝒇 "𝒋𝜸𝑰 𝑵 Random Basis State |𝑦⟩ of 0 0 0 0 • 0 0 1 0 : = Hamming Weight x = 𝑙 , 5 + 𝜏 ) : 𝜏 . 5 𝜏 . 𝜏 ) 0 1 0 0 or 0 0 0 0 • Dicke State : 𝜏 #>) : ! 5 𝜏 #>) 5 𝐼 ;#!< = ∑ #6= 𝜏 # + 𝜏 # • 1 ! = 𝐸 4 ? |𝑦⟩ 𝑜 5 + 𝜏 # : 𝜏 : 5 𝜏 𝐼 ?@#A/B = ∑ #CD 𝜏 # 5 64 • 𝑙 D D October 12, 2020 | 10

Los Alamos National Laboratory Complete Graph vs. Ring Mixer and Dicke vs. k-state Complete Graph Mixer Ring Mixer • Complete Graph mixer finds better solutions at lower round counts • Dicke starting state outperforms randomly selected k-state starting state October 12, 2020 | 11

Los Alamos National Laboratory Complete Graph Mixer outperforms Ring Mixer Approximation ratio of the complete mixer 𝒔 𝑳 compared to the approximation ratio of the ring mixer 𝒔 𝑺 for several rounds: • 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) October 12, 2020 | 12

Los Alamos National Laboratory Angle Selection Strategies October 12, 2020 | 13

Los Alamos National Laboratory 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 𝑞 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 ( 𝑞 wrt 𝑞 − 1 ) Exponential scaling in 𝒒 makes Monte Carlo • not seem promising as angle selection strategy October 12, 2020 | 14

Los Alamos National Laboratory Angle Selection: Are angles good for many graphs? Complete Graph Mixer Ring Mixer Heat plots show expectation values for 1-round QAOA averaged over 100 graphs. Ø Hope for generally good values. October 12, 2020 | 15

Los Alamos National Laboratory Are good angles correlated across rounds and graphs? Beta Gamma Plot shows best 𝛾 and 𝛿 values for 𝑞 = 6 -round QAOAs of typical graphs. Red line: Average over all tested graphs (graph size 7 – 10) . October 12, 2020 | 16

Los Alamos National Laboratory Good angles for increasing rounds p ? 𝑞 = 6 𝑞 = 7 𝑞 = 5 𝛾 γ Correlation holds across different values of the number of rounds 𝑞 . Ø Interpolation strategy for angle selection October 12, 2020 | 17

Los Alamos National Laboratory Comparison of Angle Selection Strategies Strategies • Monte Carlo : “random angles” • Basin Hopping : “detect and escape local minima” • Interpolation : “learn angles from other graphs” • Optimal : “full 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. October 12, 2020 | 18

Los Alamos National Laboratory Summary October 12, 2020 | 19

Los Alamos National Laboratory 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? October 12, 2020 | 20