Distributed Quantum Inner Product Estimation with Structured Random Circuits

Distributed inner product estimation (DIPE) is a fundamental task in quantum information, aiming to estimate the inner product between two unknown quantum states prepared on distributed quantum platforms. Existing rigorous sample complexity analyses are limited to unitary $4$-designs, which pose significant practical challenges for near-term quantum devices. This work addresses this challenge by exploring DIPE with structured random circuits. We first establish that DIPE with an arbitrary unitary $2$-design ensemble achieves an average sample complexity of $\mathcal{O}(\sqrt{2^n})$, where $n$ is the number of qubits. We then analyze ensembles below unitary $2$-designs -- specifically, the brickwork and local unitary $2$-design ensembles -- demonstrating average sample complexities of $\mathcal{O}(\sqrt{2.18^n})$ and $\mathcal{O}(\sqrt{2.5^n})$, respectively. Furthermore, we analyze the state-dependent sample complexity. For brickwork ensembles, we develop a tensor network approach to compute the asymptotic state-dependent sample complexity, showing that it converges to $\mathcal{O}(\sqrt{2.18^n})$ as the circuit depth increases. Remarkably, we find that DIPE with the global Clifford ensemble requires $\Theta(\sqrt{2^n})$ copies, matching the performance of unitary $4$-designs. For both local and global Clifford ensembles, we find that the efficiency can be further enhanced by the nonstabilizerness of states. Additionally, for approximate unitary $4$-designs, the performance exponentially approaches that of exact unitary $4$-designs as the circuit depth increases. Our results provide theoretically guaranteed methods for implementing DIPE with experimentally feasible unitary ensembles.