The garden-hose model

Abstract

Abstract. We define a new model of communication complexity, called the garden-hose model. Informally, the garden-hose complexity of a function f : {0, 1} n × {0, 1} n → {0, 1} is given by the minimal number of water pipes that need to be shared between two parties, Alice and Bob, in order for them to compute the function f as follows: Alice connects her ends of the pipes in a way that is determined solely by her input x ∈ {0, 1} n and, similarly, Bob connects his ends of the pipes in a way that is determined solely by his input y ∈ {0, 1} n . Alice turns on the water tap that she also connected to one of the pipes. Then, the water comes out on Alice's or Bob's side depending on the function value f (x, y). We prove almost-linear lower bounds on the garden-hose complexity for concrete functions like inner product, majority, and equality, and we show the existence of functions with exponential garden-hose complexity. Furthermore, we show a connection to classical complexity theory by proving that all functions computable in log-space have polynomial garden-hose complexity. We consider a randomized variant of the garden-hose complexity, where Alice and Bob hold pre-shared randomness, and a quantum variant, where Alice and Bob hold pre-shared quantum entanglement, and we show that the randomized garden-hose complexity is within a polynomial factor of the deterministic garden-hose complexity. Examples of (partial) functions are given where the quantum garden-hose complexity is logarithmic in n while the classical garden-hose complexity can be lower bounded by n c for constant c > 0. Finally, we show an interesting connection between the garden-hose model and the (in)security of a certain class of quantum position-verification schemes.

Position-Based Quantum Cryptography: Impossibility and Constructions

“…Can we show security in the bounded-quantum-storage model [21] where adversaries are limited to store, say, a linear fraction of the communicated qubits, thus restricting the amount of available entanglement? These questions do remain open in general but have already triggered quite a bit of follow-up work [3,11,46] providing partial answers (see section 1.3 for details). Proof.…”

Bounds on Instantaneous Nonlocal Quantum Computation

“…Proof. We refer to this LOBC protocol as U2E, and it is similar in spirit to the protocol of Vaidman and Buhrman et al [11], [15]. Following the terminology of Ref.…”

“…Following the terminology of Ref. [11], we refer to teleportation * as the standard teleportation protocol except with no classical communication and no Pauli correction on the receiving end. Protocol U2E: Two ebit protocol for U ∈ L • Input an arbitrary two-qubit state |ψ AB .…”

“…This is of vital importance for realistic quantum information processing in which maintaining long coherence time lengths is a formidable challenge. The time-constrained nature of LOBC processing has found cryptographic application in the task of position verification [7]- [11], and we review this connection in Section III.…”

Practical position-based quantum cryptography

“…A different class of position-based verification protocols based on the nonlocal computation of Boolean functions was introduced by Buhrman et al in [13], for which they suggested a new type of attacks based on the Gardenhose complexity of the Boolean function. They showed in particular that finding an explicit Boolean function with polynomial circuit complexity (so that the honest prover can compute it) but exponential attack complexity in the garden-hose model is at least as difficult as separating the classes of languages P and L, corresponding respectively to decision problems decidable in polynomial time or logarithmic space.…”