Computational Bounds for f-Routing
Abstract
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∈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∈DTIME(poly(nqs)). 3. If the strategies are Clifford+T circuits using at most t magic gates, then f∈DTIME(poly(nq2t)). 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≤log 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.
Related
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BibTeX
@misc{renard2026computational,
title = {{Computational Bounds for $f$-Routing}},
author = {Oren Renard and Nicholas Spooner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40278},
eprint = {2609.40278},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40278},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Computational Bounds for f-Routing
AU - Oren Renard
AU - Nicholas Spooner
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40278
UR - https://arxiv.org/abs/2609.40278
AB - 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∈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∈DTIME(poly(nqs)). 3. If the strategies are Clifford+T circuits using at most t magic gates, then f∈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≤log 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.
ER -