# how to use HOMOTOpic maps to improve Quantum optimization Approximation Algorithms

Institute of Theoretical and Applied Informatics, Polish Academy of Sciences
Ludmila Botelho

# Quantum computing devices

Noisy

Intermediate-

Scale

Quantum computing

Where is $$\omega$$?

N

Prime factors of

?

# What can we do?

• Quantum Approximate Optimization Algorithm (QAOA)

### Ground state = optimal solution

H_{\text{obj}} = - \sum_{i > j} J_{ij}Z_i Z_j - \sum _i h_iZ_i,
H_{\text{mix}} = \sum _i X_i,
|\vec{\gamma}, \vec{\beta}\rangle=\prod_{i=1}^{L}\exp\left(-\mathrm{i}\beta_{j}H_{\mathrm{mix}}\right)\exp\left(-\mathrm{i}\gamma_{j}H_{\mathrm{obj}}\right)|+\rangle^{\otimes N},
E(\vec{\gamma},\vec{\beta})\,=\,\langle\vec{\gamma},\vec{\beta}|H_{\text{obj}}|\vec{\gamma},\vec{\beta}\rangle.

# What can we do?

• QAOA based on Adiabatic Quantum Computing
T\gg\Delta^{-2}\max_{s\in[0,1]}\left\|\left[\frac{\partial H(s)}{\partial t}\right]^2\right\|

H(s) = (1-s)H_{\text{mix}} + sH_{\text{obj}},

Interpolation between Hamiltonians

# Energy landscape

C+\sum A_{i,j}\cos(\theta(E_i-E_j)+B_{i,j}),

Text

# Homotopic map

H_{\text{mix}}
g_2
{\mathcal{F}}(\alpha,x)=g_{1}(\alpha)f_{\mathrm{targ}}(x)+g_{2}(\alpha)f_{\mathrm{init}}(x)
\begin{array}{llllllllll} {{g_1(0)=0,}} & {{g_2(0)=1,}} \\ {{g_1(1)=1,}} & {{g_2(1)=0.}} \end{array}
g_1
H_{\text{obj}}

# Hamiltonian-Oriented Homotopy QAOA

H_{\text{mix}}
H_{\text{obj}}
E_{\alpha}(\vec{\gamma},\vec{\beta})\,=\,\langle\vec{\gamma},\vec{\beta}|H(\alpha)|\vec{\gamma},\vec{\beta}\rangle,
g_{1}(\alpha) = 1 -\alpha, \quad g_{2}(\alpha) = \alpha,

# Conclusions

•  The method does not increase the impact of the noise
• We numerically confirm that our method outperforms state-of-the-art approaches
• Our approach is well-motivated for nonlinear optimized functions

By ludmilaasb

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