Bence Bakó, Adam Glos, Özlem Salehi, Zoltán Zimborás

Near-optimal circuit design for variational quantum optimization

"Optimization process"

Abstract formulation of the problem
(shortest route)

Mathematical formulation
(e.g. Integer linear program)

Optimization algorithm
(quantum or classical)

For quantum computing: also prepare a quantum circuits!

Problem is encoded into the Ising model

$$H_c(s) = -\sum_{i,j}J_{ij}s_is_j - \sum_i h_i s_i$$

Follows the adiabatic evolution defined through time-dependent Hamiltonian

$H(t) =\frac{1-t}{\tau} \:H_{\rm mix} + \frac{t}{\tau}\: H_c$

Quantum Annealing

Slow evolution during time $\tau$

Implemented by D-Wave quantum annealers

For the gate based model

$$|\gamma,\beta\rangle = \prod_{i=1}^p \exp(-\mathrm{i} \beta_iH_{\rm mix})\exp(-\mathrm{i} \gamma_iH_c) |+^n\rangle $$

$\gamma_i$ and $\beta_i$ optimized by external classical procedure

Quantum Approximate Optimization Algorithm

Zhou, Leo, et al. "Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices." Physical Review X 10.2 (2020): 021067.

Can be viewed as a trotterization of AQC

Get the corresponding Ising model

$H = -\frac{1}{2^n} \prod_{i=1}^n (1-s_i)$

$s_i \in \{-1,1\}$

$b_i \leftarrow \frac{1-s_i}{2}$

Implementing QAOA

Start with a
QUBO or HOBO

$H = -\prod_{i=1}^n b_i$

$b_i \in \{0,1\}$

Exponential number of CNOT gates!

Exponential number of terms !

Get the corresponding Ising model

$H = -\frac{1}{2^n} \prod_{i=1}^n (1-s_i)$

$s_i \in \{-1,1\}$

$b_i \leftarrow \frac{1-s_i}{2}$

Start with a
QUBO or HOBO

$H = -\prod_{i=1}^n b_i$

$b_i \in \{0,1\}$

Can we do something else?

Implementing QAOA

$O(n^2)$ gates on LNN!

FUNC-QAOA

This is classical AND operator!

$H = -\prod_{i=1}^n b_i$

$b_i \in \{0,1\}$

New Idea

Start with a pseudocode function!

$|f\rangle \leftarrow AND(b_i$ for $i=1$ to $n$)

$|f\rangle \leftarrow R_z(-i\theta) |f \rangle $

Uncompute $|f\rangle $

Number of gates
Number of physical qubits
Effective space size
Number of parameterized gates
Depth
Depth on LNN
Energy span

What else are important?

Quality Measures

Number of gates
Number of physical qubits
Effective space size
Number of parameterized gates
Depth
Depth on LNN
Energy span

What else are important?

Quality Measures

5 gates 1 parametrized

Number of gates
Number of physical qubits
Effective space size
Number of parameterized gates
Depth
Depth on LNN
Energy span

What else are important?

Quality Measures

Number of gates
Number of physical qubits
Effective space size
Number of parameterized gates
Depth
Depth on LNN
Energy span

What else are important?

Quality Measures

Linear Nearest Neighbour

All-to-All

Number of gates
Number of physical qubits
Effective space size
Number of parameterized gates
Depth
Depth on LNN
Energy span

What else are important?

Quality Measures

Claim: FUNC-QAOA allows near-optimal circuit designs for various problems

Given: Graph with $n$ nodes and $K$ colors

Aim: Maximize (total weight of the edges) number of edges connecting different colors

(Weighted) Max-$K$-Cut

What is the best that can be achieved?

$K$ colors $\implies \log K$ bits to encode each color in binary

Number of qubits and effective space size

$k^n$ possible assignments of colors to nodes

$\implies$ smallest eff. space size is $n \log K$

$n$ nodes $\implies n\log K$ bits in total

011000

3 nodes colored blue green red

Has impact on the evolution!

$\implies$ $n \log K$ bits are needed to encode the solutions

Bounded by the number of degrees in the problem

For Max-$K$-Cut $\implies$ number of possible edges $n^2$

Number of (parametrized) gates

Bounded by

number of gates / number of qubits

Depth (LNN)

Energy span

Minimum objective value: All nodes have different color

Maximum objective value: All nodes have the same color

$\implies$ $n^2$ possible edges

Has impact on number of measurements!

At least one cost depends siginificantly on K

State-of-the art

X-QAOA

Mixer: X-mixer
Initial state: Product of equal superposition
Encoding: One-hot

3 nodes colored
blue green red

001

010

100

010-100-001

b_{0,0}b_{0,1}b_{0,2}-b_{1,0}b_{1,1}b_{1,2}-b_{2,0}b_{2,1}b_{2,2}

State-of-the art

XY-QAOA

Mixer: XY-mixer
Initial state: Product of W-state
Encoding: One-hot

Each $b_i$ starts in an equal superspsition of one-hot states

\frac{1}{\sqrt{3}} (|001\rangle + |010\rangle + |100\rangle)

State-of-the art

HOBO

Mixer: X-mixer
Initial state: Product of equal superposition
Encoding: Binary

$|11\rangle $ not valid if there are only 3 colors

State-of-the art

Fuchs-QAOA

Mixer: X-mixer
Initial state: Product of equal superposition
Encoding: Binary

$|10\rangle $ and $|11\rangle $ represent same color if there are only 3 colors

Fuchs, Franz G., et al. "Efficient Encoding of the Weighted MAX-k-CUT on a Quantum Computer Using QAOA."

State-of-the art

Fuchs-QAOA

Very bad when $K=2^k+1$

Penalize states representing same colors

FUNC-QAOA for Max-$K$-Cut

FUNC-QAOA

Mixer: Grover-mixer
Initial state: Product of equal superposition of valid colors
Encoding: Binary

\frac{1}{\sqrt{3}} (|00\rangle + |01\rangle + |10\rangle)

Each $c_i$ starts in an equal superspsition of valid colors

Dependency on $K$ disappears

FUNC-QAOA for Max-$K$-Cut

FUNC-QAOA

We reached optimal quality measures
We essentially dropped the dependency on $K$
Fits very well NISQ requirements!
Very important problem!

We just took Fuchs idea and have simply chosen better initial state...

Let's go with a harder one
Travelling Salesman Problem!

YES

Was it interesting?

Given: Graph with $n$ nodes and cost between them
Aim: Find a tour with minimum total cost that visits each node exactly once

Travelling Salesman Problem

What is the best that can be achieved?

$n$ cities $\implies \log n$ bits to encode each city in binary

Number of qubits and effective space size

$n!$ permutations

$\implies$ eff. space size is $\log n! \approx n \log n$

$\implies$ $n\log n$ bits are needed to encode the solutions

00 01 10

$n$ time points $\implies n\log n$ bits in total

011000 Cities are visited in order 2-3-1

Bounded by the number of degrees in the problem

For TSP $\implies$ $n^2$ entries in the cost matrix

Number of (parametrized) gates

Bounded by

number of gates / number of qubits

Depth (LNN)

$n\log n / n^2 \implies O(n/\log n) $

Assuming each city is represented by a qubit and that it should interact with $n-1$ other qubits, it is safe to assume $O(n)$

Energy span

Minimum objective value: $n \cdot$ min cost edge

Maximum objective value: $n \cdot$ max cost edge

State-of-the art

None of the encodings matches the "natural optimal" value

State-of-the art

Initial State:
Mixer:
Encoding:

X-QAOA XY-QAOA
Equal sup. One-hot
X-mixer XY-mixer
One-hot One-hot

GM-QAOA
Permutation
Grover-mixer
One-hot

HOBO
Equal sup.
X-mixer
Binary

State-of-the art

$b_t$ encodes city visited at time $t$

Initial State:
Mixer:
Encoding:

X-QAOA XY-QAOA
Equal sup. One-hot
X-mixer XY-mixer
One-hot One-hot

GM-QAOA
Permutation
Grover-mixer
One-hot

HOBO
Equal sup.
X-mixer
Binary

State-of-the art

$b_t$ encodes city visited at time $t$

Initial State:
Mixer:
Encoding:

X-QAOA XY-QAOA
Equal sup. One-hot
X-mixer XY-mixer
One-hot One-hot

GM-QAOA
Permutation
Grover-mixer
One-hot

HOBO
Equal sup.
X-mixer
Binary

State-of-the art

$b_t$ encodes city visited at time $t$

Initial State:
Mixer:
Encoding:

X-QAOA XY-QAOA
Equal sup. One-hot
X-mixer XY-mixer
One-hot One-hot

GM-QAOA
Permutation
Grover-mixer
One-hot

HOBO
Equal sup.
X-mixer
Binary

State-of-the art

$b_t$ encodes city visited at time $t$

Initial State:
Mixer:
Encoding:

X-QAOA XY-QAOA
Equal sup. One-hot
X-mixer XY-mixer
One-hot One-hot

GM-QAOA
Permutation
Grover-mixer
One-hot

HOBO
Equal sup.
X-mixer
Binary

State-of-the art

Time-to-city representation

Initial State:
Mixer:
Encoding:

X-QAOA XY-QAOA
Equal sup. One-hot
X-mixer XY-mixer
One-hot One-hot

GM-QAOA
Permutation
Grover-mixer
One-hot

HOBO
Equal sup.
X-mixer
Binary

We need to include cost matrix $n$ times $\implies O(n^3) gates $

Time-to-city representation

City-to-city representation

This is used in MTZ-ILP, but with the cost of additional constraints to avoid subtours

City 4

City 2

$W_{1}$

$W_{2}$

$W_{3}$

$W_{4}$

$W_{5}$

$W_{6}$

We need to include each row of the cost matrix once

$\implies O(n^2) gates $

FUNC-QAOA for TSP

FUNC-QAOA

Mixer: Grover-mixer
Initial state: Product of equal superposition of valid cities
Encoding: Binary

\frac{1}{\sqrt{3}} (|00\rangle + |01\rangle + |10\rangle)

Each $b_t$ is an equal supersposition of valid cities

Permutation checking

Idea:

Count occurrence of each city in modulo 2
If they are not all 1, penalize

Cost of the route

Idea:

Switch from time-to-city to city-to city representation

FUNC-QAOA for TSP

GM-QAOA clearly better!

...but at a price

Numerics for TSP

The idea can be generalized, so far we managed to use it for

Set Cover problem
Integer Linear Problem (trade-off)
Graph Isomorphism (trade-off in general, optimal for Euclidean graphs)

Very difficult for general graph!

Generalization

Bärtschi, Andreas, and Stephan Eidenbenz. "Grover mixers for QAOA: Shifting complexity from mixer design to state preparation." 2020 IEEE International Conference on Quantum Computing and Engineering (QCE). IEEE, 2020.
Fuchs, Franz G., et al. "Efficient Encoding of the Weighted MAX-$K$-CUT on a Quantum Computer Using QAOA." SN Computer Science 2.2 (2021): 1-14.
Glos, Adam, Aleksandra Krawiec, and Zoltán Zimborás. "Space-efficient binary optimization for variational quantum computing." npj Quantum Information 8.1 (2022): 1-8.
Tabi, Zsolt, et al. "Quantum optimization for the graph coloring problem with space-efficient embedding." 2020 IEEE International Conference on Quantum Computing and Engineering (QCE). IEEE, 2020.
Wang, Zhihui, et al. "X y mixers: Analytical and numerical results for the quantum alternating operator ansatz." Physical Review A 101.1 (2020): 012320.
Lucas, Andrew. "Ising formulations of many NP problems." Frontiers in physics (2014): 5.
Farhi, Edward, Jeffrey Goldstone, and Sam Gutmann. "A quantum approximate optimization algorithm." arXiv preprint arXiv:1411.4028 (2014).
Peruzzo, Alberto, et al. "A variational eigenvalue solver on a photonic quantum processor." Nature communications 5.1 (2014): 1-7.

Thank you

I would like to thank Ozlem Salehi for sharing with me with this presentation

Near-optimal circuit design for variational quantum optimization

"Optimization process"

Quantum Annealing

Quantum Approximate Optimization Algorithm

Implementing QAOA

Implementing QAOA

New Idea

What else are important?

What else are important?

What else are important?

What else are important?

What else are important?

(Weighted) Max-\(K\)-Cut

Number of qubits and effective space size

Number of (parametrized) gates

Depth (LNN)

Energy span

At least one cost depends siginificantly on K

State-of-the art

State-of-the art

State-of-the art

State-of-the art

State-of-the art

State-of-the art

Very bad when \(K=2^k+1\)

FUNC-QAOA for Max-\(K\)-Cut

FUNC-QAOA for Max-\(K\)-Cut

Was it interesting?

Travelling Salesman Problem

Number of qubits and effective space size

Number of (parametrized) gates

Depth (LNN)

Energy span

State-of-the art

State-of-the art

State-of-the art

State-of-the art

State-of-the art

State-of-the art

State-of-the art

Time-to-city representation

City-to-city representation

City 4

City 2

FUNC-QAOA for TSP

Permutation checking

Cost of the route

FUNC-QAOA for TSP

Numerics for TSP

Generalization

Thank you