Computational Bounds for f-Routing

The f-routing protocol is a leading candidate for quantum position verification (Kent, Munro, and Spiller, 2011), but security guarantees for explicit functions remain limited. We prove unconditional resource lower bounds for uniform attackers; our new techniques bypass communication-complexity bounds central to previous works, which are inherently at most linear in the input length. We show that, for input length n and sufficiently small constant ε>0, a uniformly generated strategy using q qubits and having description length poly(q), with success probability at least 1-ε on every input, implies the following computational bounds on f: 1. If the strategies are arbitrary quantum channels, then f in QSZK(poly(nq)), where QSZK(T) is the class of languages having quantum statistical zero knowledge proofs in which the verifier runs in time T (and the simulator in time poly(T)). 2. If the strategies are explicit Pauli-sparse unitaries on q qubits that have at most s nonzero Pauli coefficients, then f in DTIME(poly(nqs)). 3. If the strategies are Clifford+T circuits using at most t magic gates, then f in DTIME(poly(nq2 t)). Time and space hierarchies then yield explicit functions secure against polynomial and even quasipolynomial qubits q under our computational restrictions. These bounds exceed the q n bound of Bluhm, Christandl, and Speelman (2022) for inner product function f= IP, at the cost of restricting adversarial computation and increasing honest evaluation complexity.