# Introduction of paper: Simulating noisy quantum circuits with matrix product density operators

In this article, I would like to introduce a research paper on noise simulator using matrix product density operator (MPDO). It shows that by multiplying the density matrix by the tensor product instead of the states to represent the noise, the noise can be reproduced with a precision that exceeds all existing methods.

Phys. Rev. Research 3, 023005 (2021) - Simulating noisy quantum circuits with matrix product density operators (aps.org)

The model in this paper divides the noise into three categories, dephasing, depolarizing, and amplitude damping, and represents them as Kraus operators, which are operators that are applied from both sides of the density matrix. This Kraus operator is represented using MPDO, which is multiplied to represent the decoherence of the system in terms of time evolution. In the Kraus representation, the noise operator is ﻿$E_k$﻿ , and

﻿$\Epsilon(\rho)=\sum_kE_k \rho E_k^\dagger$﻿

is the density operator for the system in the presence of noise. The product of this density operator and the Matrix Product Operator (MPO) in the initial state is the MPDO, and unitary operations are performed on it.

Using the MPDO, the Porter-Thomas distribution, which is realized by multiplying a random matrix by a circuit depth of 24 and iterating it a sufficient number of times, was simulated under three different types of noise. It was also confirmed that MPDO can reproduce the distribution accurately. In addition, using IBM's cloud quantum computer, the simulation and actual results agreed with high accuracy.

Although the relationship between the tensor structure and the noise remained a problem, it is significant that we were able to accurately reproduce the noise in multiple qubits.