Bence Bakó, Adam Glos, Özlem Salehi, Zoltán Zimborás
Optimal QAOA design for the Traveling Salesman Problem
Quantum Approximate Optimization Algorithm
- A mixture of quantum annealing and VQE
- For gate-based model and combinatorial optimization
- encodes the problem into the circuit
∣p,r⟩=i=1∏kexp(−ipiHmix)exp(−iriH)∣+n⟩
- pi and ri optimized by external classical procedure
- Since the problem is encoded into the circuit - how to minimize resources needed?
H(t)=(1−t/T)Hmix+t/TH
Quality measures
number of physical qubits
effective space size
number of gates
number of parm. gates
depth
depth on LNN
energy span
log(# solutions)
log(# solutions)
# degrees of freedom
# degrees of freedom
# degrees of freedom/log(# solutions)
# degrees of freedom/log(# solutions)
max(obj. value) - min(obj. value)
measures
lower bounds
Max-K-Cut
Minimal example
- −∏i=1nbi
- Corresponding Ising model: −2n1∏i=1n(1−si)
- exponential number of terms O(2n)!
O(n2) gates on LNN!
bi←21−si
SIM-QAOA
This is AND operation!
Was it interesting?
- we reached optimal quality measures
- we essentially dropped the dependency on K
- fits very well NISQ requirements!
- very important problem!
- we essentially took Fuchs idea and have simply chosen better initial state...
Let's go with a harder problem - Travelling Salesman Problem!
YES
NO
Travelling Salesman Problem
wij
TSP - redundancy
Time 2
City 3
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cost
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Travelling Salesman Problem
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None of the encodings matches the "natural optimal" value, ...
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..., but none can! We repeated O(n2) degrees of freedom n times - O(n3) gates
MTZ ILP is an exception, as variables are not "time to city" but "city to city"
SIM-QAOA for TSP
- We start in the superposition of valid cities for each time-point
- we choose Grover Mixer for our purpose (different ones can be used)
- Hamiltonian implemented through a quantum version of the classical pseudocode (with some parallelization + swapping strategy)
TSP - SIM-QAOA
TSP - SIM-QAOA
TSP - numerics
GM-QAOA clearly better!
...but at a price
Generalization
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!
Thank you!
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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.
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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.
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Glos, Adam, Aleksandra Krawiec, and Zoltán Zimborás. "Space-efficient binary optimization for variational quantum computing." npj Quantum Information 8.1 (2022): 1-8.
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Wang, Zhihui, et al. "XY mixers: Analytical and numerical results for the quantum alternating operator ansatz." Physical Review A 101.1 (2020): 012320.
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Lucas, Andrew. "Ising formulations of many NP problems." Frontiers in physics (2014): 5.
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Farhi, Edward, Jeffrey Goldstone, and Sam Gutmann. "A quantum approximate optimization algorithm." arXiv preprint arXiv:1411.4028 (2014).
Bence Bakó, Adam Glos , Özlem Salehi, Zoltán Zimborás Optimal QAOA design for the Traveling Salesman Problem
simultaneous reduction - short
By Adam Glos
simultaneous reduction - short
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