# Quantum Machine Learning for Distributed Quantum Protocols with Local Operations and Noisy Classical Communications

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Entanglement Distillation

#### 1.3. Quantum State Discrimination

#### 1.4. Main Contributions

- As observed in Figure 1, we first introduce NA-LOCCNet as a novel PQC-based architecture for the distributed entanglement distillation (see Figure 4) that is designed with the goal of maximizing the average fidelity while accounting for the randomness caused by communication errors.
- Then, we adapt the NA-LOCCNet framework for the problem of the distributed quantum state discrimination (see Figure 9), with the goal of maximizing the average probability of successful detection for quantum state discrimination.
- The introduced NA-LOCCNet is shown via experiments to have significant advantages over existing protocols designed for noiseless communications. Furthermore, in quantum state discrimination, we make the important observation that, depending on the level of classical noise, a larger level of entanglement-breaking noise can be advantageous to facilitate successful distributed discrimination.

#### 1.5. Organization

#### 1.6. Notations and Definitions

## 2. Learning Entanglement Distillation with Noisy Classical Communication

#### 2.1. Problem Formulation

#### 2.1.1. Setting

#### 2.1.2. Performance Metrics

#### 2.2. Existing Distillation Protocols

#### 2.2.1. DEJMPS Protocol

#### 2.2.2. LOCCNet

#### 2.3. Noise Aware-LOCCNet

#### 2.3.1. Design Objective

#### 2.3.2. Architecture of the PQCs

#### 2.3.3. Optimization

#### 2.4. Experiments

## 3. Learning Quantum State Discrimination with Noisy Classical Communication

#### 3.1. Setting and Performance Metrics

#### 3.1.1. Setting

#### 3.1.2. Performance Metrics

#### Helstrom Bound

#### Positive Partial Transpose (PPT) Bound

#### 3.2. LOCCNet

#### 3.3. Noise Aware-LOCCNet

#### 3.4. Experiments

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Distributed entanglement distillation at two quantum-enabled devices (Alice and Bob) aided by a noisy classical communication channel to a third party (Charlie). Alice and Bob implement PQCs as local operations and they communicate over a noisy classical link from Alice to Bob.

**Figure 2.**Distributed quantum state discrimination at two quantum-enabled devices, Alice and Bob. Alice and Bob implement parameterized quantum circuits (PQCs) as local operations and they communicate over a noisy classical link from Alice to Bob.

**Figure 3.**LOCCNet circuit for distributed entanglement distillation of two S states [8].

**Figure 4.**Proposed Noise Aware-LOCCNet (NA-LOCCNet) circuit for distributed entanglement distillation of two S states.

**Figure 5.**Average output fidelity as a function of the bit flip probability p of the noisy classical channels from Alice and Bob to Charlie for input fidelity $F=0.6$ in (1).

**Figure 6.**Average output fidelity, conditioned on a successful distillation, as a function of the input fidelity F in (1) for bit flip probability $p=0.25$ on the noisy classical channels from Alice and Bob to Charlie. The black dashed line corresponds to the reference performance of a scheme that simply outputs the input state.

**Figure 7.**Probability of success as a function of the input fidelity F in (1) for bit flip probability $p=0.25$ on the noisy classical channels from Alice and Bob to Charlie.

**Figure 8.**Illustration of the LOCCNet protocol [8] for distributed quantum state discrimination, which operates on a single pair of qubits $(S=1)$.

**Figure 9.**The proposed NA-LOCCNet protocol for distributed quantum state discrimination that operates over $S=2$ qubit pairs and adapts to the classical and quantum noise levels p and $\gamma $.

**Figure 10.**Average success probability as a function of the bit flip probability p of the noisy classical channel from Alice to Bob for the AD channel noise parameter $\gamma =0.8$.

**Figure 11.**Average success probability as a function of the AD channel noise parameter $\gamma $ for the bit flip probability $p=0.25$ of the noisy classical channel from Alice to Bob.

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**MDPI and ACS Style**

Chittoor, H.H.S.; Simeone, O.
Quantum Machine Learning for Distributed Quantum Protocols with Local Operations and Noisy Classical Communications. *Entropy* **2023**, *25*, 352.
https://doi.org/10.3390/e25020352

**AMA Style**

Chittoor HHS, Simeone O.
Quantum Machine Learning for Distributed Quantum Protocols with Local Operations and Noisy Classical Communications. *Entropy*. 2023; 25(2):352.
https://doi.org/10.3390/e25020352

**Chicago/Turabian Style**

Chittoor, Hari Hara Suthan, and Osvaldo Simeone.
2023. "Quantum Machine Learning for Distributed Quantum Protocols with Local Operations and Noisy Classical Communications" *Entropy* 25, no. 2: 352.
https://doi.org/10.3390/e25020352