Benchmarking quantum state transfer on quantum devices

Quantum state transfer (QST) provides a method to send arbitrary quantum states from one system to another. Such a concept is crucial for transmitting quantum information into the quantum memory, quantum processor, and quantum network. The standard benchmark of QST is the average fidelity between the prepared and received states. In this work we provide a new benchmark which reveals the nonclassicality of QST based on spatiotemporal steering (STS). More specifically, we show that the local-hidden-state (LHS) model in STS can be viewed as the classical strategy of state transfer. Therefore, we can quantify the nonclassicality of the QST process by measuring the spatiotemporal steerability. We then apply the spatiotemporal steerability measurement technique to benchmark quantum devices including the IBM quantum experience and QuTech quantum inspire under QST tasks. The experimental results show that the spatiotemporal steerability decreases as the circuit depth increases, and the reduction agrees with the noise model, which refers to the accumulation of errors during the QST process. Moreover, we provide a quantity to estimate the signaling effect which could result from gate errors or intrinsic non-Markovian effect of the devices.

Figure 1

Circuit implementation of QST. Here we prepare the initial states by implementing the $u_3$ operation in qubit $Q_1$. The states can be transferred to the $Q_2$ by applying the $U_{1,2}\left(\theta \right)$ operation decomposed with a CRX followed by a CNOT operation. We then iterate the $U_{l,l+1}\left(\theta \right)$ operation on the $n$-qubit chain. Finally, the states will be transferred to $Q_n$, which can then be obtained by the procedure of quantum state tomography.

Figure 2

Schematic illustration of the perfect $n$-qubit QST process ${\mathrm{\Lambda }}_t$, which consists of several iterations of quantum operations. For the $l\mathrm{th}$ iteration, the state is transferred from $Q_l$ to $Q_{l+1}$ by turning on the qubit-qubit interaction [depicted in Eqs. (11) and (12)] for a period of time $\mathrm{\Delta }t=\pi /\left(2J\right)$, where $J$ is $1/2$. Therefore, to transfer Alice's prepared states from $Q_1$ to $Q_n$, the iteration has to be performed $n-1$ times.

Figure 3

Circuit decomposition of ${\mathcal{V}}_{l,l+1}\left(\theta \right)$ and $U_{l,l+1}\left(\theta \right)$. $R_x\left(\theta \right)$ is the rotation $X$ operation with rotating angle $\theta$.

Figure 4

Value of $\mathcal{STSR}$ with respect to different angle $\theta$ for different qubit number $n$. Here the initial assemblage for $Q_1$ is ${\{{\varrho }_{a|x}=P_{\text{A}}\left(a\right|x){\tilde{\varrho }}_{a|x}\}}_{a,x}$ where $P_{\text{A}}\left(a\right|x)=1/2$ for all $a,x$ and ${\tilde{\varrho }}_{a|x}$ are the eigenstates of Pauli operators.

Figure 5

The experimental values of the $\mathcal{STSR}$ under the QST. The evolution operators transfer the state from $Q_1$ to $Q_n$. The experimental results of (a) and (c) are implemented in Mar 2020 after maintenance, whereas those for (b) and (d) were complete in Jan 2020 before the maintenance. In (a) and (b) we show the spatiotemporal steerability as a quantification of the QST process. When $\theta =\pi$, the ideal evolution circuit, corresponding to the perfect QST process, provides the maximal spatiotemporal steerability. Although the experimental $\mathcal{STSR}$ cannot reach the theoretical prediction, the trend of the experimental results functions similarly with the ideal case. Moreover, one can observe that the $\mathcal{STSR}$ well fitted the noisy model for (a) and (b). One can see that the signaling effect in (d) actually dominates $\mathcal{STSR}$ in (b) by Eq. (6).