# Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State Preparation with QAOA

**Remark**

**1**(Inversion symmetry)

**.**

**Proposition**

**1**(Overlap invariance)

**.**

**Proof.**

**Remark**

**2.**

**Remark**

**3**(Global optimisation)

**.**

**Remark**

**4**(Layerwise training)

**.**

## 3. $\mathit{p}=\mathbf{1}$ QAOA

**Theorem**

**1.**

**Proof.**

**Remark**

**5**(Trivial solutions)

**.**

**Remark**

**6.**

**Remark**

**7.**

**Theorem**

**2.**

**Proof.**

**Remark**

**8.**

**Remark**

**9.**

## 4. Empirical Findings Missing Analytical Theory

#### 4.1. Parameter Concentration in $p\ge 2$ QAOA

#### 4.2. Last Layer Behaviour

#### 4.3. Saturation in Layerwise Training at $p=n$

#### 4.4. Removing Saturation in Layerwise Training

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Optimal angles of $p=5$ depth circuit for $n\in [6;17]$. While the first layers can be approximately described by a linear relation, the last layer fits ${\gamma}_{p}+2{\beta}_{p}=\pi $. Moreover, the values of the last layer’s parameters are evidently distinct from the previous layers.

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**MDPI and ACS Style**

Rabinovich, D.; Sengupta, R.; Campos, E.; Akshay, V.; Biamonte, J.
Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation. *Mathematics* **2022**, *10*, 2601.
https://doi.org/10.3390/math10152601

**AMA Style**

Rabinovich D, Sengupta R, Campos E, Akshay V, Biamonte J.
Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation. *Mathematics*. 2022; 10(15):2601.
https://doi.org/10.3390/math10152601

**Chicago/Turabian Style**

Rabinovich, Daniil, Richik Sengupta, Ernesto Campos, Vishwanathan Akshay, and Jacob Biamonte.
2022. "Progress towards Analytically Optimal Angles in Quantum Approximate Optimisation" *Mathematics* 10, no. 15: 2601.
https://doi.org/10.3390/math10152601