Infeasible space reduction for QAOA through encoding change
Institute of Theoretical and Applied Informatics, Polish Academy of Sciences;
Ludmila Botelho, Adam Glos, Akash Kundu, Jarosław Adam Miszczak, Özlem Salehi, and Zoltan Zimboras
Quantum computing devices
Noisy
Intermediate-
Scale
Quantum computing
Where is ω?
N
What are prime factors of
?
What can we do? - Quantum annealing
H=−i>j∑JijZiZj−i∑hiZi,
2-Local Ising model
Pauli Z Gates on i-th qubit
Ground state = optimal solution
NP-hard!
Initialize the annelear: ∣+n⟩
HQA(t)=g(t/τ)Hmix+h(t/τ)H
Adiabatic evolution
Ground state of H (hopefully)
∣0110…010⟩
Ground state of:
Hmix=i∑Xi
What can we do? - QAOA
g(t/τ)Hmix+h(t/τ)H
Trotter Formula
Quantum Approximate Optimization Algorithm
∣p,r⟩=i=1∏kexp(−ipiHmix)exp(−iriH)∣+n⟩
Traveling salesperson problem
Traveling salesperson problem
At∑(v∑btv−1)2+Av∑(t∑btv−1)2+Bt,v,w∑Wv,wbt,vbt+1,w
QUBO formulation for n cities
How many qubits?
- Required number of qubits →n2
- Number of routes →n!
Overlap with feasible space can be reduced!
- Optimal solution encoding →log2(n!)∼nlog2n
n2→n!/2n2≈2−n2
nlog2n→ n!/2nlog2n≈2−Θ(n)
VS
Number of qubits can be reduced!
Glos, Adam, Aleksandra Krawiec, and Zoltán Zimborás. "Space-efficient binary optimization for variational computing." arXiv preprint arXiv:2009.07309 (2020).
Encodings?
Cost of route:
π:{0,⋯,n−1}→{0,⋯,n−1}
∑t=0n−1Wπ(t),π(t+1)=∑t,v,wWv,wδ(π(t),v)δ(π(t+1),w)
city visited at time t
0
1
2
3
4
W1,2
0
1
2
3
4
W1,2
time t
0
1
2
3
4
00001
10000
000
100
binary
one-hot
00010
00100
01000
001
010
011
QUBO
HOBO
EC-QAOA
HOBO
QUBO
HOBO
EC-QAOA
∣0⟩
{
∣0⟩
{
∣0⟩
{
∣0⟩
{
∣0⟩
}
∣0⟩
}
∣0⟩
}
∣0⟩
}
V
V†
QUBO
V
V
V
V†
V†
V†
Hmix
Hmix
Hmix
Hmix
Success probability of measuring the quantum state on the feasible space (left). The area spans the mean energy ± standard deviation over 40 instances of TSP. Simulation were done for TSP instances with four cites.
EC-QAOA
Initialize the quantum state
∣0⟩
{
∣0⟩
{
∣0⟩
{
∣0⟩
{
∣0⟩
}
∣0⟩
}
∣0⟩
}
∣0⟩
}
V
V†
QUBO
V
V
V
V†
V†
V†
Hmix
Hmix
Hmix
Hmix
Conclusion
- Mid-circuit error mitigation
- EC-QAOA is efficient to implement
- EC-QAOA works in a smaller space
}
\}
Available for Quantum Alternating Operator Ansatz
EC-QAOA
V
V†
QUBO
V
V
V
V†
V†
V†
Hmix
Hmix
Hmix
Hmix
Post-Selection
EC-QAOA
V
V†
QUBO
V
V
V
V†
V†
V†
Hmix
Hmix
Hmix
Hmix
V
V†
QUBO
V
V
V
V†
V†
V†
Mid-Circuit Measurement
PS
PS
PS
PS
APPLICABLE FOR OTHER QAOA SCHEMES!
Results - ERROR MITIGATION
Comparison of various error-mitigation methods. The red dashed line corresponds to "do nothing" approach. The blue solid line describes the effect with the post-selection at the end only. Finally, the green dotted line includes post-selection for both final measurement and mid-circuit measurement. Areas span the minimum and maximum value obtained. Top row shows the deviation in the energy, relative to the scenario proposed in the paper. Bottom row shows the probability of accepting circuit run.
Eours−EpureE−Epure
Generalization To VQE
Hubbard model:
Ansatz:
Possible explanation
We detect transfer between infeasible and feasible space
Conclusion
- New concept: Mid-circuit error mitigation
- Number of qubits and depth comparable to other QAOA classes
- EC-QAOA works effectively in a smaller subspace
thank you!
Infeasible space reduction for QAOA through encoding change Institute of Theoretical and Applied Informatics, Polish Academy of Sciences ; Ludmila Botelho, Adam Glos , Akash Kundu, Jarosław Adam Miszczak, Özlem Salehi, and Zoltan Zimboras
Copy of QScience Days 2021
By Adam Glos
Copy of QScience Days 2021
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