Experimental Efficient Influence Sampling of Quantum Processes

Abstract

Characterizing quantum processes is essential for unlocking the potential of quantum devices. However, standard quantum process tomography is resource-intensive and becomes infeasible on large-scale systems. Despite alternative approaches have been successfully developed for specific scenarios, they typically rely on multi-qubit gates or extensive prior knowledge, limiting their practicability and scalability. To address these challenges and complement existing approaches, we introduce influence sampling, an efficient and scalable protocol that quantifies the influence of a quantum process on all qubit subsets using only single-qubit test gates, with sample complexity independent of system size. Using a photonic platform, we demonstrate influence sampling to identify high-influence qubits, reduce the full process to a smaller effective process, i.e., a junta approximation, and then learn it. We further confirm scalability by applying the protocol to a 24-qubit system and validate the junta approximation on a two-qubit process. These results establish influence sampling as a critical characterization technique, facilitating process learning and device assessment.

Figures (11)

• Figure 1: (a) Quantum circuit for influence sampling. The initial qubits are randomly prepared on the computational bases, and the same single-qubit test gate $U_l$, is applied to each qubit, chosen from $\{I,H,R_x\left(\frac{\pi}{2}\right)\}$. Sampling results are obtained by recording which qubits flip relative to their initial state. (b) Experimental setup. Four qubits are encoded on polarization and path DoF of a pair of photons. State preparation and implementation of $U_l$ are merged in a group of wave plates. The target quantum processes $\Phi$ are constructed from some subprocesses $\Phi_{\{i\}}$ and $\Phi_{\{i,j\}}$, as shown in (c). Measurements in the computational basis are performed through four switchable measurement groups controlled by two toggling HWPs (TH1 and TH2). FC: fiber coupler; Pol: polarizer; BD: beam displacer; K9: K9 glass; PP: phase plate;
• Figure 2: (a) Experimental influence sampling and junta process learning. The target quantum process is $\mathrm{CU_{s,\{1,2\}}}\otimes I_{\{3\}}\otimes I_{\{4\}}$. $\mathrm{CU}_{\mathrm{s},\{1,2\}}$ is a controlled-$U_s$ gate with $\ket{q_1}$ as control and $\ket{q_2}$ as target; $U_s = \sum_{j=1}^3\sigma_j/\sqrt{3}$. For each test gate $U_l$, we acquire $M \gtrsim 2.7 \times 10^5$ samples of $\mathcal{T}_l$, forming the collection $\mathcal{K}_l$. The empirical distributions $\hat{p}(\mathcal{T}_l)$ are shown on the left inset. From these we compute influence samplers $\mathbb{E}X^S_l$ for various qubit sets $S$ (middle left), and obtain lower ($\text{IL}_S$, $\text{IL}_S^{(\mathrm{II})}$) and upper bounds ($\text{IU}_S$, $\text{IU}_S^{(\mathrm{II})}$) on $\mathrm{Inf}_S[\Phi]$ (middle right). $\text{IU}_{\{i\}}$ identify the high-influence subset $T$ and its complement $T^c$. $\text{IU}_{T^c}^{(\mathrm{II})}$ yields the bound $\epsilon$ on the junta‑approximation error. The subprocess $\Phi_T$ on $T$ is then learned by QPT, giving $\widetilde{\chi}^{\Phi_T}$ (right inset) and hence the 2-junta approximation $\widetilde{\Phi}_T\otimes\mathcal{I}_{T^c}$. Red frames outside the bars indicate theoretical values. Gray bars between $\text{IL}_S$ and $\text{IU}_S$ represent theoretical influence values. (b) Measured influence samplers for three single-qubit rotation gates. (c) Influence sampling on a 24-qubit quantum circuit, where the process consists primarily of a CZ gate and a controlled phase-damping process. The dashed black lines mark the single‑qubit decision thresholds $\delta_0 = 0.006$ and $\delta_1 = 0.003$. Error bars are generated via Poisson distribution.
• Figure 3: (a) Measured influence bounds for an imperfect two-qubit identity process $\mathcal{I}^{\mathrm{(e)}}$ with $M\approx1.4\times10^5$. (b) Experimental evaluation of the junta-approximation distances and their bounds. $\widetilde{\mathcal{I}}^{\mathrm{(e)}}$ and $\widetilde{\mathcal{I}}^{\mathrm{(e)}}_{T}$ are reconstructed via two-qubit QPT, and via single-qubit QPT with the qubits in $T^c$ initialized in the maximally mixed state and subsequently traced out, respectively.
• Figure 4: State preparation and test gate implementation for the first ($q_1$) and second ($q_2$)qubits. (a) Experimental setup. (b) Configurations of wave-plates for different initial states and test gates. Identical setups and encoding approaches are used for the third ($q_3$) and fourth ($q_4$) qubits.
• Figure 5: Experimental realizations of quantum subprocesses. The order of optical element labels is from left to right, from top to bottom. Here, the schematics of $\Phi_{\{i\}}$ also denote the same optical elements acting on both path modes. For two-qubit subprocesses $\Phi_{\{i,j\}}$, the path qubit is the control and the polarization qubit is the target. K9 glass compensates for path length differences in CNOT, CZ and $\mathrm{CU_s}$. K9: K9-glass, PP: phase plate.

Theorems & Definitions (4)

• Theorem 1
• proof
• Proposition 2
• proof