quantum network designers

Abstract: Understanding the entanglement cost of non-local quantum computation (NLQC) is relevant to complexity theory, cryptography, quantum gravity, and related areas. A central special case is $f$-routing, motivated in part by quantum position verification. Proving lower bounds on its entanglement cost in the fully robust setting has been a major open problem in NLQC. Motivated by this problem, we establish two related lower bounds. First, we study the shared-randomness cost of robust conditional disclosure of secrets (CDS). The connection between CDS and $f$-routing established by Allerstorfer et al. (Quantum 2024) makes understanding the randomness complexity of robust CDS a natural step toward lower bounds for the fully robust routing problem. We show that the shared-randomness cost of robust CDS is lower bounded by the logarithm of deterministic SMP communication complexity, even when communication and private randomness are unrestricted. Our lower bound is tight for the equality function. Second, we consider one-sided-perfect $f$-routing, in which the protocol is exact on one input class and has constant error on the other. By exploiting the positivity of the low-rank matrix arising in the method of Asadi, Culf, and May (ITCS 2025), we derive a general lower bound on the entanglement cost in terms of sign rank. In particular, this yields a linear lower bound on the entanglement cost of routing for the inner-product function in both one-sided-perfect settings, matching the known upper bound.

Abstract: We study a resource-efficient approach for implementing logical fanout operations in fault-tolerant distributed quantum computing using transversal operations on quantum error-correcting code blocks. Logical fanout, comprising multiple controlled-NOT operations from a common control qubit to target qubits located at remote nodes, is an important primitive for distributed quantum computation but can require substantial non-local communication when implemented directly between encoded blocks. We exploit the structure of encoded blocks and the availability of transversal logical operations to construct distributed fanout circuits that reduce the required non-local operations while preserving the logical action of the fanout operation. The construction is developed for encoded quantum information and illustrated using Bivariate Bicycle (BB)-code blocks. We analyze the resulting physical gate, entanglement, circuit-depth, and ancilla requirements. The approach provides a systematic method for implementing large logical fanout operations across distributed error-corrected quantum processors. Also, we study a distributed implementation of the global gate GCZ involving logical qubits (encoded using BB-code blocks), exploiting the concurrency in transversal distributed fanouts.