Quantum algorithm for evaluating operator size with Bell measurements

Operator size growth describes the scrambling of operators in quantum dynamics and stands out as an essential physical concept for characterizing quantum chaos. Important as it is, a scheme for the direct measuring of operator size on a quantum computer is still absent. Here, we propose a quantum algorithm for the direct measuring of the operator size and its distribution based on Bell measurement. The algorithm is verified with spin chains and, meanwhile, the effects of Trotterization error and quantum noise are analyzed. It is revealed that saturation of operator size growth can be due to quantum chaos itself or be a consequence of quantum noise. Nevertheless, it is found that the error mitigation will effectively reduce the influence of noise, so as to restore the distinguishability of oscillation and saturation behaviors of operator scrambling. Our work provides a feasible protocol for investigating quantum chaos on noisy quantum computers by measuring operator size growth.

Figure 1

The schematic diagram for the quantum circuit to calculate the operator size of operator $\widehat{O}\left(t\right)$. Here, ${|0\rangle }_s$ and ${|0\rangle }_a$ represent the system qubit and the auxiliary qubit, respectively. The $\widehat{O}\left(t\right)$ operator implements only on the system qubits.

Figure 2

The operator size of $X_3\left(t\right)$ for the quantum Ising model with system site $N=5$. (a) The operator size distribution of the integrable system ($h_z=0$) with different time $t$. (b) The quantum chaotic one ($h_z=0.3$). Both of them are numerical results using a quantum computer simulator without circuit noise. (c),(d) The corresponding operator size for (a) and (b). The red lines are ED results and the dashed lines are the result of a quantum computer simulator by Bell measurements without circuit noise.

Figure 3

(a) The Trotter errors with time for both the integrable system ($h_z=0$) and quantum chaotic one ($h_z=0.3$) with the Trotter step $r=100$. (b) The Trotter errors with different Trotter steps for a total time $t=2$.

Figure 4

Noise effects and error mitigation for the operator size growth. (a),(b) The operator size of $X_3\left(t\right)$ for the integrable system and chaotic one with different depolarizing noise rates. (c),(d) Results of error mitigation for (a) and (b), respectively, by extrapolation to the zero-noise limit with different $n_c$ for a noise rate $p={10}^{-3}$.