# Depolarizing Channel Mismatch and Estimation Protocols for Quantum Turbo Codes

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## Abstract

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## 1. Introduction

## 2. Preliminaries: The Quantum Depolarizing Channel and Quantum Turbo Codes

#### 2.1. Quantum Depolarizing Channel

#### 2.2. Quantum Turbo Codes

## 3. Quantum Turbo Decoder Performance with Depolarizing Probability Mismatch

#### Interleaver Impact

- S-random interleaver with parameter $S=25$ and
- JPL interleaver.

## 4. Estimating the Depolarizing Probability

#### 4.1. Off-Line Estimation Framework

#### 4.1.1. Quantum Channel Estimation

- Number of channel invocations: For our encoding-decoding system in Figure 1, sequential channel invocations are not considered. The reason is that once a quantum state goes through the operation of the depolarization channel, it cannot be sent again through the channel. Therefore, the number of channel invocations will be set to $m=1$.
- Unitary ${\widehat{U}}_{1}$ transformation: The goal of the unitary transformation, ${\widehat{U}}_{1}$, applied to the n input quantum probes, $\sigma $, is to introduce correlations among the quantum probes. In the particular case where the transformation is diagonal, it results in independent instances of the quantum probes, i.e., in an independent channel use protocol. Figure 6 shows such an estimation protocol.Furthermore, if n is set to one, the above scheme reduces to the simplest estimation protocol, called single-qubit, single-channel (SQSC). If we denote by ${J}_{1}\left(p\right)$ the Fisher information of this SQSC estimation protocol, then for any n greater than one, it can be shown [22] that the corresponding overall Fisher information ${J}_{n}\left(p\right)$ is given by ${J}_{n}\left(p\right)=n{J}_{1}\left(p\right)$. Therefore, for n channel invocations, the quantum Cramér–Rao bound is:$$\mathrm{var}\left(\widehat{p}\right)\ge \frac{1}{n{J}_{1}\left(p\right)},$$
- Input probe $\sigma $: Two different state probes $\sigma $ are considered (We consider noiseless probes that are only affected by the depolarizing channel. Research about constructing robust quantum probe states to face such adverse noise has been addressed in [25,26].).
- Unentangled pure states: The Fisher information for unentangled pure states as probes has been calculated in [27] to be:$${J}_{1}\left(p\right)=\frac{9}{8p(3-2p)}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{J}_{n}\left(p\right)=n\frac{9}{8p(3-2p)}.$$
- Maximally entangled pure states: When entanglement is available, maximally entangled pure states or EPR pairs $|{\mathsf{\Phi}}^{+}\rangle =\left(\right|00\rangle +|11\rangle )/\sqrt{2}$ can be used as probes for the depolarizing channel. It can be shown that if just one of the qubits of $|{\mathsf{\Phi}}^{+}\rangle =\left(\right|00\rangle +|11\rangle )/\sqrt{2}$ is transformed by the depolarizing channel (i.e., the EPR pair goes through an extended channel ${\mathcal{N}}_{D}\otimes \mathrm{id}$), then the corresponding Fisher information (Note that the expressions in [24,27] are given for the depolarizing channel defined as (1). Here, we use the relationship $p=\frac{3}{4}\u03f5$ to adapt such expressions for the depolarizing channel defined as (2).) is [24]:$${J}_{1}\left(p\right)=\frac{9}{16p(1-p)}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{J}_{n}\left(p\right)=n\frac{9}{16p(1-p)}.$$Note that this type of protocol requires that one of the entangled qubits is not affected by noise. This is not an issue in our scenario, since the codes we consider are entanglement assisted (there is pre-shared entanglement between the coder and the decoder), and thus, this protocol is suitable for the estimation of the depolarizing probability. It can be shown that the Fisher information value in (9), higher than in (8) due to entanglement, is the largest value that can be achieved by SQSC estimation protocols for the depolarizing channel [22].

#### 4.1.2. Computation of the Average Word Error Rate

#### 4.2. On-Line Estimation Framework

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QECC | Quantum error correction code |

QCC | Quantum convolutional code |

QTC | Quantum turbo code |

QLDPC | Quantum low density parity check |

QIRCC | Quantum irregular convolutional code |

QURC | Quantum unity rate code |

EXIT | Extrinsic information transfer |

EPR | Einstein–Podolsky–Rosen |

SISO | Soft-input soft-output |

JPL | Jet Propulsion Laboratory |

WER | Word error rate |

SQSC | Single-qubit single-channel |

SLD | Symmetric logarithmic derivative |

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**Figure 1.**Schematic of the QTC. Note that the c pre-shared EPR pairs ${|\mathsf{\Phi}\rangle}^{+}$ are needed for the inner encoder to be both recursive and non-catastrophic [4]. ${\mathrm{P}}_{i}^{a}(.)$ and ${\mathrm{P}}_{i}^{e}(.)$ denote the a priori and extrinsic probabilities related to each of the SISO decoders used for turbo decoding.

**Figure 2.**Word error rate (WER) performance curves for the $1/9$-QTCs in Table 1 when different interleavers are used. Perfect channel knowledge is assumed.

**Figure 3.**WER variation for QTCs using an EXIT-random interleaver as a function of the mismatched depolarizing probability $\widehat{p}$. Each one of the curves corresponds to a value of the actual channel depolarizing probability. The QTCs considered here have an entanglement consumption rate of $6/9$.

**Figure 4.**WER variation for the QTCs with three different interleavers as a function of the mismatched depolarizing probability $\widehat{p}$. The true depolarizing probabilities selected are $p=0.33$ and $p=0.30$.

**Figure 5.**General channel identification method for a p parameter dependent quantum channel $\mathrm{\Gamma}\left(p\right)$ [22]. For the scenario of this paper, $\mathrm{\Gamma}\left(p\right)={\mathcal{N}}_{D}$. This general scenario uses n quantum probes $\sigma $ and m channel invocations. ${U}_{1},\cdots ,{U}_{m+1}$ are parameter independent transformations that can be effectively selected to estimate p. ${p}_{est}$ is an estimator function that uses the classical information $({x}_{1},\cdots ,{x}_{n})$ obtained from measuring the output quantum state ${\sigma}_{f}\left(p\right)$ in order to get the estimate $\widehat{p}$.

**Figure 6.**Estimation protocol used in this paper. n independent channel invocations are performed in order to estimate the value of p.

**Figure 7.**WER degradation for the QTCs as a function of the number of probes, n, used for the initial depolarizing probability estimation. Note that the number of probes used applies in the asymptotic setting. (

**a**) Estimation using pure quantum probes. (

**b**) Estimation using maximally-entangled quantum probes (EPR pairs).

**Figure 8.**Modified QTC decoder to perform on-line estimation of the depolarizing probability. The figure only presents the inner SISO part of the decoder, as the rest of the turbo decoder remains unchanged.

**Figure 9.**WER performance for the QTCs when the proposed on-line estimation procedure for the depolarizing probability is utilized with initial value ${\widehat{p}}^{\left(1\right)}={p}^{*}=0.3779$. For comparison purposes, the figure also shows the WER achieved when perfect channel information is available.

**Figure 10.**WER variation for the QTCs as a function of the initial value of the depolarizing probability, ${\widehat{p}}^{\left(1\right)}$. The continuous lines represent the sensitivity of the original turbo decoder [1] (no estimation of the depolarizing probability is performed, and the decoder always utilizes ${\widehat{p}}^{\left(1\right)}$), while the dashed lines represent the sensitivity of the modified decoder using the proposed on-line estimation method.

Config. | Encoder | R | E | m | Seed Transformation $\mathcal{U}$ |
---|---|---|---|---|---|

EXIT- optimized | Outer | $1/3$ | 0 | 3 | $\{1048,3872,3485,2054,$ $983,3164,3145,1824,$ ${987,3282,2505,1984\}}_{10}$ |

Inner | $1/3$ | $2/3$ | 3 | $\{4091,3736,2097,1336,$ $1601,279,3093,502,$ ${1792,3020,226,1100\}}_{10}$ |

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**MDPI and ACS Style**

Etxezarreta Martinez, J.; Crespo, P.M.; Garcia-Frías, J.
Depolarizing Channel Mismatch and Estimation Protocols for Quantum Turbo Codes. *Entropy* **2019**, *21*, 1133.
https://doi.org/10.3390/e21121133

**AMA Style**

Etxezarreta Martinez J, Crespo PM, Garcia-Frías J.
Depolarizing Channel Mismatch and Estimation Protocols for Quantum Turbo Codes. *Entropy*. 2019; 21(12):1133.
https://doi.org/10.3390/e21121133

**Chicago/Turabian Style**

Etxezarreta Martinez, Josu, Pedro M. Crespo, and Javier Garcia-Frías.
2019. "Depolarizing Channel Mismatch and Estimation Protocols for Quantum Turbo Codes" *Entropy* 21, no. 12: 1133.
https://doi.org/10.3390/e21121133