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Los Alamos National Laboratory

Grover Mixers for QAOA: Shifting Complexity from Mixer Design to State Preparation

Andreas Bärtschi

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

joint work with Stephan Eidenbenz

LA-UR-20-28071

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

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QAOA Variants

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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".

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Quantum Approximate Optimization Algorithm

Farhi, Goldstone, Gutmann (2014)

|0i H e−iγ1HC e−iβ1X ... e−iβpX hβ β β,γ γ γ| HC |β β β,γ γ γi |0i H e−iβ1X e−iβpX |0i H e−iβ1X e−iβpX |0i H e−iβ1X e−iβpX |0i H e−iβ1X e−iβpX

1 √ 2n

P

all x

|xi US UP (γ1) UM(β1) UM(βp) | {z }

p rounds with angles γ1,β1,...,γp,βp

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Quantum Alternating Operator Ansatz

Hadfield, Wang, O’Gorman, Rieffel, Venturelli, Biswas (2017)

US |0i UM(β0) UP (γk) UM(βk) ... hβ β β,γ γ γ| HC |β β β,γ γ γi |0i |0i |0i |0i some feasible |xi | {z }

p rounds with angles γ1,β1,...,γp,βp

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Grover Mixer QAOA

UM(βk) = e−iβk|F ihF | |0i US UP (γk) U †

S

US . . . hβ β β,γ γ γ| HC |β β β,γ γ γi |0i |0i |0i |0i Zβk/π

1

p

|F |

P

x2F

|xi | {z }

p rounds with angles γ1,β1,...,γp,βp

P

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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, 𝑉% 0 = 𝐺 ≔ 1 𝐺 5

• ∈0

𝑦 , 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.

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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

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GM-QAOA Illustration

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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).

𝑟0: 𝑟1: 𝑟2: 𝑟3:

3: |0111⟩ 2: |1110⟩ |1101⟩ 1: |1011⟩ 𝑉%: Prepare superposition

• f all 4 feasible states:

𝐸5

6 =

1 4 5

• ∈ 7,$ !
• 95

|𝑦⟩ 𝑉# 𝛿 : Phase according to

• bjective value.

Equal amplitudes for equally good basis states: The Grover Mixer 𝑉! 𝛾 = 𝐽𝑒 − (1 − 𝑓&' 1)|𝐸5

6⟩⟨𝐸5 6| will also retain the same

amplitudes among all basis states of the same objective value.

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Advantages of GM-QAOA

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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. Permutation Matrix 1 1 1 1

city 1 city 2 city 3 city 4 tour position 1 tour position 2 tour position 3 tour position 4

State preparation of 𝟐

𝒐! ∑𝒚∈𝑻𝒐 |𝒚⟩

• Conceptually easier than designing a mixer that

mixes between all permutations / rows of the matrix / ...

• Grover Mixer simultaneously mixes all

permutations, no need for partial mixers / Trotterization.

• Improvement in the number of gates.

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{x | x = xn21 . . . x1x0 such that 8 0  c < n: X

j⌘c mod n

xj = 1 (col. constraints) 8 0  r < k: X

bi/nc=r

xi = 1 (row constraints) r = n 1: X

bi/nc=r

xi = n k (bitmask)}

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. Permutation Matrix 1 1 1 1

city 1 city 2 city 3 city 4 tour position 1 tour position 2 tour position 3 tour position 4

State preparation of 𝟐

𝒐! ∑𝒚∈𝑻𝒐 |𝒚⟩

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Permutation-based optimization problems

W3 W3 W3 W3 |0i W4 |0i |0i |0i |0i |0i |0i |0i |0i W2 W2 W2 |0i W3 W2 W2 |0i W2 W2 |0i W2 W2 |0i |0i |0i |0i

• 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.

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Other Applications (Teaser, see the paper)

Maximum 𝒍-Vertex Cover Grover Mixer can be used for fixed Hamming Weight subspaces (so far only 𝑌𝑍-Model Mixers known).

• Ring-Mixer (𝑌𝑍 interactions along a

ring of qubits), performance not great, 𝑃(𝑜)-depth circuit on LNN.

• Grover Mixer based on Dicke States,

𝑃(𝑜)-depth circuit on LNN, evidence for intermediate performance.

• Clique-Mixer (𝑌𝑍 interactions along a

clique of qubits), performance good, no circuit implementation known. Portfolio Rebalancing Discrete Portfolio Rebalancing with feasible subspaces given by ”bands” defined by fixed difference of

• number of long positions,
• number of short positions.
• Grover Mixer is the first

Mixer to mix between different bands,

• Initial total amplitude of

bands can be controlled.