Grover Mixers for QAOA: Shifting Complexity from Mixer Design to - PowerPoint PPT Presentation

Los Alamos National Laboratory LA-UR-20-28071 Grover Mixers for QAOA: Shifting Complexity from Mixer Design to State Preparation you joint work with 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 Variants October 12, 2020 | 2

Los Alamos National Laboratory QAOA QAOA is a heuristic for combinatorial optimization. It prepares a state 𝜸, 𝜹 = 𝑉 ! 𝛾 " 𝑉 # 𝛿 " ⋯ 𝑉 ! 𝛾 $ 𝑉 # 𝛿 $ 𝑉 % 0 from which one would like to sample good solutions with high probability. QAOA is specified by • an initial state preparation unitary 𝑉 % , preparing some superposition of feasible states. • a phase separator unitary 𝑉 # (𝛿) , phasing basis states |𝑦⟩ proportional to their objective value by 𝑓 &' ( )*+(-) . • a mixer unitary 𝑉 ! 𝛾 preserving and interfering feasible solutions, • 𝑞 rounds with individual angle parameters 𝛾 $ , … , 𝛾 " , 𝛿 $ , … , 𝛿 " . October 12, 2020 | 3

Los Alamos National Laboratory Quantum Approximate Optimization Algorithm Farhi, Goldstone, Gutmann (2014) U S U P ( γ 1 ) U M ( β 1 ) U M ( β p ) | 0 i e − i β 1 X e − i β p X H γ i | 0 i e − i β 1 X e − i β p X H β , γ γ β γ | H C | β ... | 0 i e − i γ 1 H C e − i β 1 X e − i β p X H γ β , γ | 0 i e − i β 1 X e − i β p X H β h β | 0 i e − i β 1 X e − i β p X H P 1 | x i √ 2 n | {z } all x p rounds with angles γ 1 , β 1 ,..., γ p , β p October 12, 2020 | 4

Los Alamos National Laboratory Quantum Alternating Operator Ansatz Hadfield, Wang, O’Gorman, Rieffel, Venturelli, Biswas (2017) U S | 0 i γ i | 0 i γ β , γ β γ | H C | β ... | 0 i U M ( β 0 ) U P ( γ k ) U M ( β k ) β , γ γ | 0 i β h β | 0 i | {z } some feasible | x i p rounds with angles γ 1 , β 1 ,..., γ p , β p October 12, 2020 | 5

Los Alamos National Laboratory Grover Mixer QAOA U M ( β k ) = e − i β k | F ih F | | 0 i γ i | 0 i γ β , γ β γ | H C | β U † . . . | 0 i U S U P ( γ k ) U S S γ β , γ | 0 i β h β | 0 i Z � β k / π P 1 p | x i | {z } | F | x 2 F p rounds with angles γ 1 , β 1 ,..., γ p , β p P October 12, 2020 | 6

Los Alamos National Laboratory Grover Mixer QAOA II For constraint optimization problems for which there is an efficiently implementable unitary 𝑉 % to prepare a superposition of all feasible states 𝐺 , 1 𝑉 % 0 = 𝐺 ≔ 5 𝑦 , 𝐺 -∈0 we can efficiently implement a Grover Mixer 𝑉 ! 𝛾 = 𝑓 &' 1 |0⟩⟨0| = 𝐽𝑒 − (1 − 𝑓 &' 1 )|𝐺⟩⟨𝐺|. Caveat: Likely not always possible, e.g. for Maximum Independent Set. October 12, 2020 | 7

Los Alamos National Laboratory Agenda GM-QAOA Illustration • Densest 𝑙 -Subgraph • Equal Amplitudes for Equal-valued States Advantages of GM-QAOA • Permutation-based optimization problems • Teaser: Maximum 𝑙 -Vertex Cover, Portfolio Rebalancing October 12, 2020 | 8

Los Alamos National Laboratory GM-QAOA Illustration October 12, 2020 | 9

Los Alamos National Laboratory Densest k-Subgraph Given an 𝑜 -vertex graph 𝐻, maximize the number of Edges in a subgraph induced by 𝑙 vertices (here 𝑙 = 3 , 𝑜 = 4 ). 𝑉 % : Prepare superposition |0111⟩ 3: 𝑟0: of all 4 feasible states: |1110⟩ 1 2: 𝑟1: 6 = 𝐸 5 5 |𝑦⟩ 4 -∈ 7,$ ! |1101⟩ 𝑟2: - 95 |1011⟩ 𝑉 # 𝛿 : Phase according to 1: 𝑟3: objective value. Equal amplitudes for equally good basis states: The Grover Mixer 𝑉 ! 𝛾 = 𝐽𝑒 − (1 − 𝑓 &' 1 )|𝐸 5 6 ⟩⟨𝐸 5 6 | will also retain the same amplitudes among all basis states of the same objective value. October 12, 2020 | 10

Los Alamos National Laboratory Advantages of GM-QAOA October 12, 2020 | 11

Los Alamos National Laboratory Permutation-based optimization problems Optimization problems such as the Traveling Salesperson Problem, the Quadratic Assignment Problem or Maximum Common Edge Subgraph ask for a maximum/minimum-valued permutation (or bijection) of 𝑜 elements. State preparation of 𝟐 𝒐! ∑ 𝒚∈𝑻 𝒐 |𝒚⟩ Permutation Matrix 0 0 1 0 city 1 1 0 0 0 • Conceptually easier than designing a mixer that city 2 0 0 0 1 city 3 mixes between all permutations / rows of the 0 1 0 0 city 4 matrix / ... • Grover Mixer simultaneously mixes all tour position 1 tour position 2 tour position 3 tour position 4 permutations, no need for partial mixers / Trotterization. • Improvement in the number of gates. October 12, 2020 | 12

Los Alamos National Laboratory Permutation-based optimization problems Optimization problems such as the Traveling Salesperson Problem, the Quadratic Assignment Problem or Maximum Common Edge Subgraph ask for a maximum/minimum-valued permutation (or bijection) of 𝑜 elements. State preparation of 𝟐 𝒐! ∑ 𝒚∈𝑻 𝒐 |𝒚⟩ Permutation Matrix 0 0 1 0 city 1 { x | x = x n 2 � 1 . . . x 1 x 0 such that 1 0 0 0 city 2 0 0 0 1 X 8 0  c < n : x j = 1 (col. constraints) city 3 0 1 0 0 city 4 j ⌘ c mod n tour position 1 tour position 2 tour position 3 tour position 4 X 8 0  r < k : x i = 1 (row constraints) b i/n c = r X r = n � 1: x i = n � k (bitmask) } b i/n c = r October 12, 2020 | 14

Los Alamos National Laboratory Permutation-based optimization problems | 0 i | 0 i W 4 | 0 i | 0 i | 0 i | 0 i | 0 i | 0 i | 0 i W 2 W 2 W 2 | 0 i W 3 W 3 W 3 W 3 W 2 W 2 | 0 i W 3 W 2 W 2 | 0 i W 2 W 2 | 0 i | 0 i | 0 i | 0 i Fig. 5. State Preparation : (left) Initialization of the first row in a -state and in the bitmask in the last row, followed by a bitmask update. October 12, 2020 | 15

Los Alamos National Laboratory Other Applications (Teaser, see the paper) Maximum 𝒍 -Vertex Cover Portfolio Rebalancing Grover Mixer can be used for fixed Discrete Portfolio Rebalancing Hamming Weight subspaces (so far only with feasible subspaces given 𝑌𝑍 -Model Mixers known). by ”bands” defined by fixed difference of Ring-Mixer ( 𝑌𝑍 interactions along a • • number of long positions, ring of qubits), performance not great, • number of short positions. 𝑃(𝑜) -depth circuit on LNN. • Grover Mixer based on Dicke States, • Grover Mixer is the first 𝑃(𝑜) -depth circuit on LNN, evidence Mixer to mix between for intermediate performance. different bands, Clique-Mixer ( 𝑌𝑍 interactions along a • • Initial total amplitude of clique of qubits), performance good, bands can be controlled. no circuit implementation known. October 12, 2020 | 16