Characterizing local noise in QAOA circuits

J.R. Johansson, P.D. Nation, and F. Nori, “QuTiP: An open-source Python framework for the dynamics of open quantum systems,” Computer Physics Communications 183

Original

2012

Earlier work this paper cites.

E. Farhi, J. Goldstone, and S. Gutmann, “A Quantum Approximate Optimization Algorithm,” arXiv:1411.4028 (2014)

Original

2014

Earlier work this paper cites.

C. Xue, Z.-Y. Chen, Y.-C. Wu, and G.-P. Guo, “Effects of Quantum Noise on Quantum Approximate Optimization Algorithm,” arXiv:1909.02196 (2019)

Original

2019

Cited alongside, same era.

S. Hadfield, Z. Wang, B. O’Gorman, E. G. Rieffel, D. Venturelli, and R. Biswas, “From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz,” Algorithms 12

2019

Cited alongside, same era.

The ‘dots’ in Eq. ( 1

Cited in the paper.

We also find the same general effect in our simulations: optimal angles in the noiseless case are approximately optimal angles when performing optimization with noise. We use the BFGS algorithm to optimize QAOA angles, using random initial angles each time. We also repeat over several initializations to help avoid being trapped in local optima. Note that optimized angles for different circuit depths in general are completely different, since we optimize over the entire set of angles, and not round by round. In our simulations the angles are in range [ 0 , 2 ​ π ] [0,2\pi]

Cited in the paper.

For now we assume no restriction on the K j K_{j} other than that ℰ p \mathcal{E}_{p} is a quantum map, and so to preserve the trace, ∑ j K j † ​ K j = M ​ 𝕀 \sum\displaylimits_{j}K_{j}^{\dagger}K_{j}=M\mathbb{I}

Cited in the paper.

For qubit-local depolarizing noise, M = 4 M=4 , with K i = σ i K_{i}=\sigma_{i} , the identity and three Pauli x , y , z x,y,z operators. Although we could move the identity term out of the sum in Eq. ( 4

Cited in the paper.

To be more formal, we can write ℰ p ( n ) = Id ⊗ ⋯ ⊗ Id ⊗ ℰ p ⊗ Id ⊗ ⋯ ⊗ Id \mathcal{E}_{p}^{(n)}=\mathrm{Id}\otimes\dots\otimes\mathrm{Id}\otimes\mathcal{E}_{p}\otimes\mathrm{Id}\otimes\dots\otimes\mathrm{Id} , with the map ℰ p \mathcal{E}_{p} in the n n -th position, and the remaining N − 1 N-1 identity Id \mathrm{Id} channels acting on the other qubits, Id ​ X = X \mathrm{Id}X=X

Cited in the paper.

K j ( n ) = 𝕀 ⊗ ⋯ ⊗ 𝕀 ⊗ K j ⊗ 𝕀 ⊗ 𝕀 K_{j}^{(n)}=\mathbb{I}\otimes\dots\otimes\mathbb{I}\otimes K_{j}\otimes\mathbb{I}\otimes\mathbb{I} , with K j K_{j} on the n n -th qubit

Cited in the paper.

In Ref. Xue et al. 2019 , the ‘ N N ’ in F 1 F_{1} is the number of noise operations applied (number of gates). In our model, for QAOA-1, N N is indeed the number of qubits, since we apply noise after each round and individually on each qubit

2019

Later among the works it cites.

In Ref. Xue et al. 2019 , if there are terms proportional to the identity a ​ 𝐈 a\mathbf{I} in H c H_{c} , so that Tr ⁡ [ H c ] ≠ 0 \mathrm{Tr}[H_{c}]\not=0 , the effect is to add a term ( 1 − ( 1 − p ) η ​ N ) ​ a (1-(1-p)^{\eta N})a to C noise C_{\mathrm{noise}} . In our analysis we do not need to do this as the equation we derive handles such a case through the freedom of parameters

2019

Later among the works it cites.