Abstract
Simulating quantum circuits with classical computers requires resources growing exponentially in terms of system size. Real quantum computer with noise, however, may be simulated polynomially with various methods considering different noise models. In this work, we simulate random quantum circuits in 1D with Matrix Product Density Operators (MPDO), for different noise models such as dephasing, depolarizing, and amplitude damping. We show that the method based on Matrix Product States (MPS) fails to approximate the noisy output quantum states for any of the noise models considered, while the MPDO method approximates them well. Compared with the method of Matrix Product Operators (MPO), the MPDO method reflects a clear physical picture of noise (with inner indices taking care of the noise simulation) and quantum entanglement (with bond indices taking care of two-qubit gate simulation). Consequently, in case of weak system noise, the resource cost of MPDO will be significantly less than that of the MPO due to a relatively small inner dimension needed for the simulation. In case of strong system noise, a relatively small bond dimension may be sufficient to simulate the noisy circuits, indicating a regime that the noise is large enough for an `easy' classical simulation. Moreover, we propose a more effective tensor updates scheme with optimal truncations for both the inner and the bond dimensions, performed after each layer of the circuit, which enjoys a canonical form of the MPDO for improving simulation accuracy. With truncated inner dimension to a maximum value $\kappa$ and bond dimension to a maximum value $\chi$, the cost of our simulation scales as $\sim ND\kappa^3\chi^3$, for an $N$-qubit circuit with depth $D$.
Highlights
The quantum computer has the potential to outperform the best possible classical computers in many tasks such as factoring large numbers
We show that the method based on matrix product states (MPSs) fails to approximate the noisy output quantum states for any of the noise models considered, while the matrix product density operators (MPDOs) method approximates them well
In case of weak system noise, the resource cost of the MPDO will be significantly less than that of the matrix product operators (MPOs) due to a relatively small inner dimension needed for the simulation
Summary
The quantum computer has the potential to outperform the best possible classical computers in many tasks such as factoring large numbers. We propose a more effective canonical tensor update scheme, performed after each layer of the circuit, which would truncate the inner dimension to some maximum value κ and the bond dimension to some maximum value χ with a canonicalization of the MPDO for improving simulation accuracy The complexity of this scheme is only proportional to DN for an N-qubit circuit with depth D. We demonstrate that the MPDO approximates the noisy output quantum states well, while the method based on matrix product states fails to approximate the noisy output quantum states for any of the noise models considered This indicates that the bond dimension truncation method of the MPS simulation might not represent any local noise model in real physical systems.
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