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Information Theory and Applications Workshop (ITA)

On the riddle of coding equality function in the garden hose model

Oded Margalit

Published Feb 1, 2014• 3 citations
Publisher

Abstract

Recently, Harry Buhrman et al introduced a novel communication complexity model, called “garden-hose”. This model sprout from a research on using quantum properties to allow for position based cryptography. SAT is one of the fundamental NP-Complete problem - finding a satisfying assignment to a CNF (conjunctive Normal Form) formula, or proving that none exists. We will not get into the details of the quantum physics, neither we are going to explain the internal work of SAT solver. Instead we will start from the mathematical garden hose model; describe the way we used SAT solver as a tool; give some lower and upper bounds on implementing the equality function; and conclude with open questions.

DOI
10.1109/ita.2014.6804262

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Recently, Harry Buhrman et al introduced a novel communication complexity model, called “garden-hose”. This model sprout from a research on using quantum properties to allow for position based cryptography. SAT is one of the fundamental NP-Complete problem - finding a satisfying assignment to a CNF (conjunctive Normal Form) formula, or proving that none exists. We will not get into the details of the quantum physics, neither we are going to explain the internal work of SAT solver. Instead we will start from the mathematical garden hose model; describe the way we used SAT solver as a tool; give some lower and upper bounds on implementing the equality function; and conclude with open questions.
We show new results about the garden-hose model. Our main results include improved lower bounds based on non-deterministic communication complexity (leading to the previously unknown Θ(n)Θ(n)Θ(n) bounds for Inner Product mod 2 and Disjointness), as well as an O(n⋅log⁡3n)O(n\cdot \log^3 n)O(n⋅log3n) upper bound for the Distributed Majority function (previously conjectured to have quadratic complexity). We show an efficient simulation of formulae made of AND, OR, XOR gates in the garden-hose model, which implies that lower bounds on the garden-hose complexity GH(f)GH(f)GH(f) of the order Ω(n2+ε)Ω(n^{2+ε})Ω(n2+ε) will be hard to obtain for explicit functions. Furthermore we study a time-bounded variant of the model, in which even modest savings in time can lead to exponential lower bounds on the size of garden-hose protocols.
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 x {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 study the communication complexity of computing functions F:{0,1}n×{0,1}n→{0,1}F:\{0,1\}^n\times \{0,1\}^n \rightarrow \{0,1\}F:{0,1}n×{0,1}n→{0,1} in the memoryless communication model. Here, Alice is given x∈{0,1}nx\in \{0,1\}^nx∈{0,1}n, Bob is given y∈{0,1}ny\in \{0,1\}^ny∈{0,1}n and their goal is to compute F(x,y) subject to the following constraint: at every round, Alice receives a message from Bob and her reply to Bob solely depends on the message received and her input x; the same applies to Bob. The cost of computing F in this model is the maximum number of bits exchanged in any round between Alice and Bob (on the worst case input x,y). In this paper, we also consider variants of our memoryless model wherein one party is allowed to have memory, the parties are allowed to communicate quantum bits, only one player is allowed to send messages. We show that our memoryless communication model capture the garden-hose model of computation by Buhrman et al. (ITCS'13), space bounded communication complexity by Brody et al. (ITCS'13) and the overlay communication complexity by Papakonstantinou et al. (CCC'14). Thus the memoryless communication complexity model provides a unified framework to study space-bounded communication models. We establish the following: (1) We show that the memoryless communication complexity of F equals the logarithm of the size of the smallest bipartite branching program computing F (up to a factor 2); (2) We show that memoryless communication complexity equals garden-hose complexity; (3) We exhibit various exponential separations between these memoryless communication models. We end with an intriguing open question: can we find an explicit function F and universal constant c>1 for which the memoryless communication complexity is at least clog⁡nc \log nclogn? Note that c≥2+εc\geq 2+\varepsilonc≥2+ε would imply a Ω(n2+ε)Ω(n^{2+\varepsilon})Ω(n2+ε) lower bound for general formula size, improving upon the best lower bound by Nečiporuk in 1966.
The aim of the paper is to answer a long-standing open problem on the relationship between NP and BQP. The paper shows that BQP contains NP by proposing a BQP quantum algorithm for the MAX-E3-SAT problem which is a fundamental NP-hard problem. Given an E3-CNF Boolean formula, the aim of the MAX-E3-SAT problem is to find the variable assignment that maximizes the number of satisfied clauses. The proposed algorithm runs in O(m2)O(m^2)O(m2) for an E3-CNF Boolean formula with mmm clauses and in the worst case runs in O(n6)O(n^6)O(n6) for an E3-CNF Boolean formula with nnn inputs. The proposed algorithm maximizes the set of satisfied clauses using a novel iterative partial negation and partial measurement technique. The algorithm is shown to achieve an arbitrary high probability of success of 1−ε1-ε1−ε for small ε>0ε>0ε>0 using a polynomial resources. In addition to solving the MAX-E3-SAT problem, the proposed algorithm can also be used to decide if an E3-CNF Boolean formula is satisfiable or not, which is an NP-complete problem, based on the maximum number of satisfied clauses.
The Boolean Satisfiability (SAT) problem is a canonical NP-complete problem and a natural candidate for quantum acceleration via search-based algorithms. In Grover-based quantum SAT solvers, the dominant computational cost stems from the construction of a reversible oracle that evaluates the Boolean formula, rendering the choice of SAT encoding crucial for overall quantum resource efficiency. Although SAT instances are conventionally expressed in Conjunctive Normal Form (CNF), such encodings typically translate into quantum circuits with significant qubit overhead and high non-Clifford gate complexity. In this work, we investigate an Exclusive-Sum-of-Products (ESOP)-based CNF (e-CNF) representation tailored for quantum SAT solving and analyze its impact on oracle construction. We derive tighter upper bounds on qubit requirements and Clifford+TTT gate counts for Grover-based SAT solvers when e-CNF encodings are employed in place of standard CNF. In addition, we propose a scalable transformation from Boolean formulas to e-CNF and present a systematic procedure for interpreting e-CNF representations as reversible quantum circuits suitable for oracle implementation. Experimental evaluation on representative SAT benchmarks demonstrates that the proposed e-CNF-based approach yields substantial and consistent reductions in quantum resources, including qubit count, T-gate complexity, and circuit depth, when compared to CNF-based oracle constructions. These results establish e-CNF as an effective quantum-aware SAT encoding that significantly improves the practicality of oracle-based quantum SAT solving.