  As small quantum computers are becoming available on different physical
platforms, a benchmarking task known as cross-platform verification has been
proposed that aims to estimate the fidelity of states prepared on two quantum
computers. This task is fundamentally distributed, as no quantum communication
can be performed between the two physical platforms due to hardware
constraints, which prohibits a joint SWAP test. In this paper we settle the
sample complexity of this task across all measurement and communication
settings. The essence of the task, which we call distributed quantum inner
product estimation, involves two players Alice and Bob who have $k$ copies of
unknown states $\rho,\sigma$ (acting on $\mathbb{C}^{d}$) respectively. Their
goal is to estimate $\mathrm{Tr}(\rho\sigma)$ up to additive error
$\varepsilon\in(0,1)$, using local quantum operations and classical
communication. In the weakest setting where only non-adaptive single-copy
measurements and simultaneous message passing are allowed, we show that
$k=O(\max\{1/\varepsilon^2,\sqrt{d}/\varepsilon\})$ copies suffice. This
achieves a savings compared to full tomography which takes $\Omega(d^3)$ copies
with single-copy measurements. Surprisingly, we also show that the sample
complexity must be at least
$\Omega(\max\{1/\varepsilon^2,\sqrt{d}/\varepsilon\})$, even in the strongest
setting where adaptive multi-copy measurements and arbitrary rounds of
communication are allowed. This shows that the success achieved by shadow
tomography, for sample-efficiently learning the properties of a single system,
cannot be generalized to the distributed setting. Furthermore, the fact that
the sample complexity remains the same with single and multi-copy measurements
contrasts with single system quantum property testing, which often demonstrate
exponential separations in sample complexity with single and multi-copy
measurements.
