Simulating noisy quantum circuits with matrix product density operators

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 one dimension with matrix product density operators (MPDOs), for different noise models such as dephasing, depolarizing, and amplitude damping. 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 MPDO method approximates them well. Compared with the method of matrix product operators (MPOs), 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 the 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, which is further supported by a comparison with the experimental results on an IBM cloud device. 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 $D$.

Figure 1

The structure of the matrix product density operators (MPDO). Instead of directly using a tensor $M$ to form a matrix product operator, we choose $M$ to be composed by the red tensor $T^{\left[r\right]}$ and the blue tensor $T^{\left[b\right]}$, where $T^{\left[b\right]}={\left(T^{\left[r\right]}\right)}^{†}$. By so, the MPDO will automatically satisfy the hermiticity of a density matrix, although $T\mathrm{s}$ are general fourth-order tensors. In this way, the parameters of the model can be fully used on the one hand, and the inner indices between the $T^{\left[b\right]}$ and $T^{\left[r\right]}$ have emerged on the other hand. Those inner indices can be used to describe the classic entropy of the system. At the same time, when the classic entropy is small, the model's computational cost will become significantly smaller.

Figure 2

(a) Applying a single-qubit gate on the corresponding tensor of the MPDO. (b) Applying a two-qubit gate on two corresponding tensors of the MPDO. The new tensors would be obtained by the singular value decomposition (SVD). The SVD would increase the bond dimension between those two tensors, while we could find the global optimal truncation of tensors based on the singular value of SVD. (c) Applying noise models on the corresponding tensor is equivalent to applying each noise operator on tensor separately and then direct sum them on the inner indices. The increased dimensions of inner indices can also be truncated using SVD.

Figure 3

Sketch of the 1D random circuit with qubits number $N=7$ and depth $D=5$. The gray dotted line outlines the structure of one layer. The circuit is interleaved by layers of single-qubit gates (colored boxes) and two-qubit gates (the dots connected to a blank box). The single-qubit gates are randomly sampled from the set of universal single-qubit quantum gates. The two-qubit gates are either control-not or control-z with equal probability.

Figure 4

Fidelity comparison for ${\rho }_0, {\rho }_e, {\rho }_d$, and ${\rho }_s$, where $r=\mathcal{F}({\rho }_0,{\rho }_s)=\mathcal{F}({\rho }_0,{\rho }_e)$. (a) The dephasing noise; (b) the depolarizing noise; (c) the amplitude damping noise.

Figure 5

Exact noisy simulation of cumulated $p$ distributions for 1D random circuit with 15 qubits and 24 layers, under three different error models with different noise rate. (a) The dephasing noise; (b) the depolarizing noise; (c) the amplitude damping noise. As the system noise increases, the cumulated $p$ distribution gradually deviates from the Porter-Thomas distribution.

Figure 6

Approximate results of the cumulated $p$ distribution for different noise simulators. The system is a 1D random circuit with 15 qubits and 24 layers, under three different error models. (a) The dephasing noise; (b) the depolarizing noise; (c) the amplitude damping noise. The green dashed line refers to the result of exact simulation. As a signal that MPS cannot faithfully be simulated with noise, when $\chi$ increase, the distribution of MPS simulation will only get closer and closer to the Porter-Thomas distribution, while MPDO can gradually approach the correct cumulated $p$ distribution.

Figure 7

Analysis result of the deviation from Porter-Thomas distribution under the depolarizing noise

Figure 8

$\mathcal{F}({\rho }_e,{\rho }_d)$ for different noise rates and different MPDO with maximum bond dimension $\chi$ and maximum inner dimension $\kappa =2\chi$. (a) The dephasing noise; (b) the depolarizing noise; (c) the amplitude damping noise.

Figure 9

$\mathcal{F}({\rho }_e,{\rho }_d)$ for different noise rates and different MPDO with inner dimension $\kappa$. (a) The dephasing noise; (b) the depolarizing noise; (c) the amplitude damping noise.

Figure 10

Structure of ${\mathrm{ibmq}}_{-}{16}_{-}\text{melbourne}$.

Figure 11

K-L divergence between simulations and experiments on the $\text{ibmq}\_16\_\text{melbourne}$ device. (a) K-L divergence $H$ between experimental results and simulation results of MPS with maximum bond dimension $\chi$ for circuits with $D$ layers. (b) K-L divergence $H$ between experimental results and simulation results of MPDO with maximum bond dimension $\chi$ and maximum inner dimension $\kappa =2\chi$ for circuits with $D$ layers. (Inset) K-L divergence $H$ between experimental results and simulation results of MPDO with maximum inner dimension $\kappa$ and maximum bond dimension $\chi =32$ for circuits with $D$ layers.

Figure 12

Encoder of [5,1,3] code.

Figure 13

Recovered fidelity versus gate noise $\epsilon$ for [5,1,3] perfect code. The blue line is the fidelity without quantum error correction. (a) The dephasing gate noise; (b) the depolarizing gate noise; (c) the amplitude damping gate noise; (d) the collective dephasing gate noise.