Improved quantum state tomography for systems with $XX+YY$ couplings and $Z$ readouts

Quantum device characterization via state tomography plays an important role in both validating quantum hardware and processing quantum information, but it needs the exponential number of the measurements. For the systems with XX$+$YY-type couplings and $Z$ readouts, such as superconducting quantum computing (SQC) systems, traditional quantum state tomography (QST) using single-qubit readout operations at least requires $3^n$ measurement settings in reconstructing an $n$-qubit state. In this paper, I propose an improved QST by adding two-qubit evolutions as the readout operations and obtain an optimal tomographic scheme using the integer programming optimization. I, respectively, apply the alternative scheme on SQC systems with the nearest-neighbor, two-dimensional, and all-to-all connectivities on qubits. It shows that this method can reduce the number of measurements by over 60% compared with the traditional QST. In addition, comparison with the traditional scheme in the experimental feasibility and robustness against errors was made by numerical simulation. It is found that the alternative scheme has good implementability and can achieve comparable or even better accuracy than the traditional scheme. It is expected that the experimentalist from related fields can directly utilize the ready-made results for reconstructing quantum states involved in their research.

Figure 1

The workflow and schematic diagram for designing a tomographic scheme and reconstructing a state in quantum platforms. The set of the experimentally measurable operators ${\mathcal{S}}_O$ is transferred to the set ${\mathcal{S}}_{\mathrm{\Lambda }}\left(M_j\right)$ under the measurement setting $M_j\in {\mathcal{S}}_M$. Usually, some overlap operators between them (labeled by the white circles) exist. Designing an optimal tomographic scheme is to minimize the number of the set ${\mathcal{S}}_M$ (denoted by $|{\mathcal{S}}_M|$) when the set ${\mathcal{S}}_{\mathrm{\Lambda }}\equiv {\cup }_j{\mathcal{S}}_{\mathrm{\Lambda }}\left(M_j\right)$ covers ${\mathcal{S}}_P$. Then, the optimized measurement set ${\mathcal{S}}_M$ can be implemented after performing circuits and before acquiring signals, such that a state $\rho$ is reconstructed with the minimum number of measurement settings.

Figure 2

The process of QST for the given measurement settings $M_i\in {\mathcal{S}}_M^{\text{new}}$ and measurable operators $O_i\in {\mathcal{S}}_O={\{I,{\sigma }_z\}}^{\otimes n}$. Different measurement settings are adopted until all unknown parameters ${\mu }_k$ in $\rho$ are measured.

Figure 3

The three common configurations on SQC systems by taking six-qubit systems as example.

Figure 4

The values of the optimal matrix $A^t$ after removing the redundant measurement settings for a three-qubit AA configuration. It is a $15\times 64$ matrix, in which the horizontal axis is the index of 64 Pauli operators ${\mathcal{P}}_i\in \{III,II{\sigma }_x,....,{\sigma }_z{\sigma }_z{\sigma }_z\}$ and the vertical axis is the index of 15 measurement settings shown in Table 4. $A_{ij}^t=\pm 1$ means that the Pauli operator ${\mathcal{P}}_j$ can be measured in the $i\mathrm{th}$ measurement setting.

Figure 5

The values of the optimal matrix $A^t$ after removing the redundant measurement settings for a three-qubit NN configuration. It is a $16\times 64$ matrix, in which the horizontal axis is the index of 64 Pauli operators ${\mathcal{P}}_i\in \{III,I{\sigma }_z,....,{\sigma }_z{\sigma }_z{\sigma }_z\}$ and the vertical axis is the index of 16 measurement settings shown in Table 5. $A_{ij}^t=\pm 1$ means that the Pauli operator ${\mathcal{P}}_j$ can be measured in the $i\mathrm{th}$ measurement setting.

Figure 6

Comparison between traditional QST (red lines) and the proposed scheme (alternative QST, cyan lines) in the robustness against the control error. (a) 3-qubit states. (b) 5-qubit states. When the residual coupling $\zeta$ is turned off to below 3%, the new scheme achieves comparable or even better accuracy than the traditional one.

Figure 7

Comparison between the traditional QST and the proposed scheme (alternative QST and integer programming) in the number of measurements settings. The red line is the scaling of traditional QST and the blue line is the scaling limit for reconstructing a general state. In principle, the tomography scheme breaking the scaling limit of $2^n$ does not exist for general states in the Hilbert space.