Linear gate bounds against natural functions for position-verification
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Abstract
A quantum position-verification scheme attempts to verify the spatial location of a prover. The prover is issued a challenge with quantum and classical inputs and must respond with appropriate timings. We consider two well-studied position-verification schemes known as f-routing and f-BB84. Both schemes require an honest prover to locally compute a classical function f of inputs of length n, and manipulate O(1) size quantum systems. We prove the number of quantum gates plus single qubit measurements needed to implement a function f is lower bounded linearly by the communication complexity of f in the simultaneous message passing model with shared entanglement. Taking f(x,y)=∑ixiyi to be the inner product function, we obtain a Ω(n) lower bound on quantum gates plus single qubit measurements. The scheme is feasible for a prover with linear classical resources and O(1) quantum resources, and secure against sub-linear quantum resources.