Entanglement entropy scaling of noisy random quantum circuits in two dimensions

Whether noisy quantum devices without error correction can provide quantum advantage over classical computers is a critical issue of current quantum computation. In this work, the random quantum circuits, which are used as the paradigm model to demonstrate quantum advantage, are simulated with depolarizing noise on experiment-relevant two-dimensional architecture. With comprehensive numerical simulation and theoretical analysis, we find that the maximum achievable operator entanglement entropy, which indicates maximal simulation cost, has area law scaling with the system size for constant noise rate. On the other hand, we also find that the maximum achievable operator entanglement entropy has power-law scaling with the noise rate for fixed system size, and the volume law scaling can be obtained only if the noise rate decreases when system size increase.

Figure 1

(a) Qubits layout on a square lattice, two-qubit gates (blue links) are applied only between the nearest-neighbor sites. (b) MPO represents qubits array in two dimensions. (c) Vectorized form of the density operator. (d) Applying a two-qubit noisy gate to an MPO, first contract the applied tensors with the two-qubit noisy gate, then decompose the resulting tensor by SVD and truncate the inner bond dimension to keep $\chi$ largest singular values. (e) Swap operation of two adjacent sites of an MPO, first contract the two tensors, then swap their physical indices by SVD and keep the largest $\chi$ singular values.

Figure 2

Operator entanglement entropy as a function of circuit depth $t$ on (a) $4\times 4$ systems with $0.09\le p\le 0.14$ and (b) $6\times 6$ systems with $0.16\le p\le 0.22$.

Figure 3

Operator entanglement entropy as a function of circuit depth $t$ on $4\times L$ systems for (a) $p=0.1$ and (b) $p=0.12$. Operator entanglement entropy as a function of circuit depth $t$ on $L\times L$ systems for (c) $p=0.16$ and (d) $p=0.18$.

Figure 4

(a) Maximum achievable operator entanglement entropy as a function of $L$ on $4\times L$ systems for various noise rate $0.1\le p\le 0.2$. (b) Maximum achievable operator entanglement entropy as a function of $L$ on $L\times L$ systems for various noise rate $0.16\le p\le 0.2$.

Figure 5

Maximum achievable operator entanglement entropy as a function of noise rate $p$ for (a) $4\times 4$ systems when $0.09\le p\le 0.2$ and (b) $6\times 6$ systems when $0.16\le p\le 0.24$. Blue lines represent power function fitting.

Figure 6

The architecture of random quantum circuits. The black circles represent qudits and blue links represent two-qubit gates. The layer is repeated in the sequence of ABCDEFGH.

Figure 7

(a) $\text{Tr}\left(\rho \right)$ and fidelity $\mathcal{F}(\rho ,\sigma )$ as a function of bond dimension $\chi$ for 2D $3\times 3$ noisy quantum circuit with noise rate $p=0.08$, and the circuit depth is 24. (b) The operator EE as functions of the circuit depth $t$ for $6\times 6$ system with $p=0.16$ calculated by MPOs with different $\chi$.

Figure 8

The spectrum of singular values in the descending order for (a) $4\times 4$ systems with $p=0.09$ and (b) $6\times 6$ systems with $p=0.16$ at peak circuit depth. Bipartition is chosen at the middle of the systems. The instances with largest and smallest operator EE are chosen among 40 different random circuits.

Figure 9

The second Rényi entropy as function of depth for different system sizes with noise rate $p=0.16$, which are obtained by (a) numerical calculation and (b) Eq. (D1), respectively.