Propagation of generalized Pauli erros in qudit Clifford circuits

Introduction An important question is how quantum protocols perform on noisy devices. If ρ was the output state of an ideal protocol, the noisy output is ρ'=ε(ρ) for some error channel ε.  Based on the so-called... [ view full abstract ]

Introduction

An important question is how quantum protocols perform on noisy devices. If ρ was the output state of an ideal protocol, the noisy output is ρ'=ε(ρ) for some error channel ε. 

Based on the so-called error probability tensor introduced in [1], we develop a mathematical tool for analyzing error propagation. It can be applied to find ε for the following protocols: The corresponding quantum circuit is composed out of qudit Clifford gates whose erroneous versions are modeled as ideal followed by a generalized Pauli Channel (see below).

Methods 
We consider qudits (with computational basis |0>,|1>,...,|D-1>, where D≥2 is arbitrary) as fundamental unit of quantum information. Noise is modeled by generalized Pauli error channels, i.e., CPTP maps whose Kraus operators are proportional to generalized Pauli operators [2]. The tool we have developed allows the investigation of quantum circuits which are composed out of qudit Clifford gates, i.e., gates that transform generalized Pauli operators into one another.

In this way, the error channel ε, which tells the error statistics, will be some generalized Pauli error channel - at any point of the protocol. Hence, the problem is reduced to how error probabilities of generalized Pauli errors are changed during the protocol. 

Results [4]

We have found explicit formulas for the transformation of the aforementioned error probabilities. Applying an (ideal) gate corresponds to a permutation, and collecting up errors at an error channel corresponds to a tensor equation.

We demonstrate the usefulness of our tool by generalizing [3] - the error analysis of a third generation quantum repeater protocol - from qubits  to qudits. This analysis is valid for a broad class of qudit stabilizer codes, e.g., all quantum polynomial codes encoding a single logical qudit. In contrast to other previous error analyses of qudit repeaters, we do not only compute the fidelity of the distributed state but also its full error statistics, hence ρ'=ε(ρ) itself.

Discussion

For many quantum communication protocols (e.g., entanglement distribution), only Clifford gates are needed. Hence, our results could be applied to optimize repeater settings, such as qudit dimension, repeater spacing or the employed quantum error correction code.

• [1] S. Janardan et al., Quantum Inf. Process. 3065-3079 (2016).
• [2] D. Gottesman, Chaos Solitons Fractals 10 1749-1758 (1999).
• [3] M. Epping et al.,  Appl. Phys. B 122: 54 (2016).
• [4] D. Miller et al., arXiv:1807.06030 [quant-ph] (2018).