$N$-Party Hadamard Test for Distributed Quantum Computation

Quantum computers promise computational advantages over classical computers, but hardware-imposed limitations remain a major obstacle. The Hadamard test mitigates these limitations by estimating expectation values associated with resource-intensive quantum operations using simple quantum circuits at the cost of additional classical sampling, and therefore underlies many quantum algorithms. However, in distributed quantum computing (DQC), which offers a promising route to scalability, its use is hindered by the need for nonlocal controlled operations. Here we introduce an $N$-party Hadamard test for DQC that estimates the same expectation values as the standard Hadamard test without implementing nonlocal controlled operations. The protocol instead uses pre-shared entanglement together with local operations and classical communication, which are standard resources in DQC settings. To demonstrate its utility, we apply it to unitary operations for clustered Hamiltonian simulation and to projectors for stabilizer-state preparation, showing lower sampling overheads than previous approaches by exploiting pre-shared entangled ancilla states. Moreover, we numerically demonstrate Bell-state preparation from Werner states to show favorable sampling efficiency and noise robustness relative to conventional purification, circuit knitting/cutting, and probabilistic error cancellation. Our work provides a general strategy for bringing Hadamard-test-based algorithms to DQC, facilitating practical and flexible quantum computation.