Bence Bakó, Adam Glos, Özlem Salehi, Zoltán Zimborás
Simultaneous resources reduction for quantum optimization - FUNC-QAOA
Quantum annealing
- Used mostly (and currently) for combinatorial optimization
- Natively for quantum annealers
- encodes the problem into the Ising model
H(s)=−i,j∑Jijsisj−i∑hisi
- Follows the adiabatic evolution defined through time-dependent Hamiltonian
H(t)=(1−t/T)Hmix+t/TH
T very large
very slow evolution
Variational Quantum Eigensolver
- Used for general Hamiltonian
- For gate-based model
- Optimizes predefined ansatz according to ⟨ψ(θ)∣H∣ψ(θ)⟩
- θ optimized by external classical procedure
- ansatz in principle has no information about the 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 parameterized gates
- depth
- depth on LNN
- energy span
Max-K-Cut
- Graph as an input
- K colors
- maximize number of edges connecting different colors
No. qubits and effective space size
XY-QAOA for TSP
mixer: XiXj+YiYj
- n2 qubits but
- only one hot states are present (for example ∣001⟩∣010⟩∣001⟩)
- There is only nn of them
- effective space space size is log(nn)=nlogn
- both lower bouned by log of the number of solutions
∣p,r⟩=i=1∏kexp(−ipiHmix)exp(−iriH)∣+n⟩
No. (parameterized) gates
- Z⊗Z⊗Z applied
- 5 gates, but 1 parameterized gates
no. gates bounded by the number of degrees of freedom in the problem (for example number of possible edges in Max-K-Cut)
H=−∑i,jwi,jZiZj−∑iwiZi
Depth (on LNN)
lowerbounded by number of gates over number of qubits
Linear Nearest Neighbour
All-to-All
Energy span - no. measurements
Better: From Hoeffding Theorem
State-of-the-art from VQE for H=−∑i,jwi,jZiZj−∑iwiZI
∣ψ⟩↦011…0↦ solution value
difference between max-min energies
M≥2ε2−Δ2log(p/2)
M \geq \frac{-\Delta^2\log(p/2)}{2 \varepsilon^2}
M≈ε2∥ω∥12
M \approx \frac{\|\omega \|_1^2}{\varepsilon^2}
Minimal example
- Hamiltonian H=−∏i=1nbi
- Corresponding Ising model: H=−2n1∏i=1n(1−si) exponential number of terms O(2n)!
O(n2) gates on LNN!
bi←21−si
FUNC-QAOA
This is AND operations!
Max-K-Cut
- Graph as an input
- K colors
- maximize number of edges connecting different colors
- X-QAOA - one-hot states, X mixer, standard QUBO,
- XY-QAOA - one-hot states, XY mixer, standard QUBO
- HOBO - binary encoding, X-mixer, higher-order terms
- Fuchs-QAOA - as in Fuchs, Franz G., et al. "Efficient Encoding of the Weighted MAX-k-CUT on a Quantum Computer Using QAOA."
At least one cost depends siginificantly on K
Max-K-Cut
Fuchs-QAOA
Very bad when K=2k+1
- binary encoding (∣0100⟩↦∣102⟩)
- All colors have meaning - last color is multiplied
- Fix incorrectly assumed different colors
Max-K-Cut - SIM-QAOA
- superposition of only valid colors (in binary)
- a quantum version of the classical pseudocode
- Dependency on K basically disappeared
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
At∑(v∑btv−1)2+Av∑(t∑btv−1)2+Bt∑v,w∑Wv,wbt,vbt+1,w
Travelling Salesman Problem
Time 2
City 3
We need to include cost matrix n times
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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 an exceptions 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
It almost matches!
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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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.
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Wang, Zhihui, et al. "X y 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).
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Peruzzo, Alberto, et al. "A variational eigenvalue solver on a photonic quantum processor." Nature communications 5.1 (2014): 1-7.
Bence Bakó, Adam Glos , Özlem Salehi, Zoltán Zimborás Simultaneous resources reduction for quantum optimization - FUNC-QAOA
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By Adam Glos
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