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Leibniz international proceedings in informatics

A Lifting Theorem with Applications to Symmetric Functions

Arkadev Chattopadhyay, Nikhil S. Mande

Published Jan 1, 2018• 2 citations
Publisher

Abstract

We use a technique of “lifting” functions introduced by Krause and Pudlak [Theor. Comput. Sci., 1997], to amplify degree-hardness measures of a function to corresponding monomial-hardness properties of the lifted function. We then show that any symmetric function F projects onto a “lift” of another suitable symmetric function f . These two key results enable us to prove several results on the complexity of symmetric functions in various models, as given below: 1. We provide a characterization of the approximate spectral norm of symmetric functions in terms of the spectrum of the underlying predicate, affirming a conjecture of Ada et al. [APPROX-RANDOM, 2012] which has several consequences. 2. We characterize symmetric functions computable by quasi-polynomial sized Threshold of Parity circuits. 3. We show that the approximate spectral norm of a symmetric function f characterizes the (quantum and classical) bounded error communication complexity of f o XOR. 4. Finally, we characterize the weakly-unbounded error communication complexity of symmetric XOR functions, resolving a weak form of a conjecture by Shi and Zhang [Quantum Information & Computation, 2009]

DOI
10.4230/lipics.fsttcs.2017.23

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We use a technique of “lifting” functions introduced by Krause and Pudlak [Theor. Comput. Sci., 1997], to amplify degree-hardness measures of a function to corresponding monomial-hardness properties of the lifted function. We then show that any symmetric function F projects onto a “lift” of another suitable symmetric function f . These two key results enable us to prove several results on the complexity of symmetric functions in various models, as given below: 1. We provide a characterization of the approximate spectral norm of symmetric functions in terms of the spectrum of the underlying predicate, affirming a conjecture of Ada et al. [APPROX-RANDOM, 2012] which has several consequences. 2. We characterize symmetric functions computable by quasi-polynomial sized Threshold of Parity circuits. 3. We show that the approximate spectral norm of a symmetric function f characterizes the (quantum and classical) bounded error communication complexity of f o XOR. 4. Finally, we characterize the weakly-unbounded error communication complexity of symmetric XOR functions, resolving a weak form of a conjecture by Shi and Zhang [Quantum Information & Computation, 2009]
We prove an essentially tight lower bound on the unbounded-error communication complexity of every symmetric function, i.e., f(x,y)=D(|x∧y|), where D: {0,1,…,n}→{0,1} is a given predicate and x,y range over {0,1} n . Specifically, we show that the communication complexity of f is between Θ(k/log5 n) and Θ(k logn), where k is the number of value changes of D in {0,1,…, n}. Prior to this work, the problem was solved only for the parity predicate D (Forster 2001). Our proof is built around two new ideas. First, we show that a predicate D gives rise to a rapidly mixing random walk on ℤ 2 , which allows us to reduce the problem to communication lower bounds for “typical” predicates. Second, we use Paturi’s approximation lower bounds (1992), suitably generalized here to clusters of real nodes in [0,n] and interpreted in their dual form, to prove that a typical predicate behaves analogous to the parity predicate with respect to a smooth distribution on the inputs.
We investigates a model of hybrid classical-quantum communication complexity, in which two parties first exchange classical messages and subsequently communicate using quantum messages. We study the trade-off between the classical and quantum communication for composed functions of the form f∘Gnf\circ G^nf∘Gn, where f:{0,1}n→{±1}f:\{0,1\}^n\to\{\pm1\}f:{0,1}n→{±1} and GGG is an inner product function of Θ(log⁡n)Θ(\log n)Θ(logn) bits. To prove the trade-off, we establish a novel lifting theorem for hybrid communication complexity. This theorem unifies two previously separate lifting paradigms: the query-to-communication lifting framework for classical communication complexity and the approximate-degree-to-generalized-discrepancy lifting methods for quantum communication complexity. Our hybrid lifting theorem therefore offers a new framework for proving lower bounds in hybrid classical-quantum communication models. As a corollary, we show that any hybrid protocol communicating ccc classical bits followed by qqq qubits to compute f∘Gnf\circ G^nf∘Gn must satisfy c+q2=Ω(max⁡{deg(f),bs(f)}⋅log⁡n)c+q^2=Ω\big(\max\{\mathrm{deg}(f),\mathrm{bs}(f)\}\cdot\log n\big)c+q2=Ω(max{deg(f),bs(f)}⋅logn), where deg(f)\mathrm{deg}(f)deg(f) is the degree of fff and bs(f)\mathrm{bs}(f)bs(f) is the block sensitivity of fff. For read-once formula fff, this yields an almost tight trade-off: either they have to exchange Θ(n⋅log⁡n)Θ\big(n\cdot\log n\big)Θ(n⋅logn) classical bits or Θ~(n⋅log⁡n)\widetildeΘ\big(\sqrt n\cdot\log n\big)Θ(n​⋅logn) qubits, showing that classical pre-processing cannot significantly reduce the quantum communication required. To the best of our knowledge, this is the first non-trivial trade-off between classical and quantum communication in hybrid two-way communication complexity.
We investigate query-to-communication lifting theorems for models related to the quantum adversary bounds. Our results are as follows: 1) We show that the classical adversary bound lifts to a lower bound on randomized communication complexity with a constant-sized gadget. We also show that the classical adversary bound is a strictly stronger lower bound technique than the previously-lifted measure known as critical block sensitivity, making our lifting theorem one of the strongest lifting theorems for randomized communication complexity using a constant-sized gadget. 2) Turning to quantum models, we show a connection between lifting theorems for quantum adversary bounds and secure 2-party quantum computation in a certain "honest-but-curious" model. Under the assumption that such secure 2-party computation is impossible, we show that a simplified version of the positive-weight adversary bound lifts to a quantum communication lower bound using a constant-sized gadget. We also give an unconditional lifting theorem which lower bounds bounded-round quantum communication protocols. 3) Finally, we give some new results in query complexity. We show that the classical adversary and the positive-weight quantum adversary are quadratically related. We also show that the positive-weight quantum adversary is never larger than the square of the approximate degree. Both relations hold even for partial functions.
A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form F(x,y)=f(x1∧y1,…,xn∧yn), F(x,y) = f(x_1 \land y_1, \ldots, x_n \land y_n), F(x,y)=f(x1​∧y1​,…,xn​∧yn​), when the outer function fff is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function fff, the bounded-error quantum and classical deterministic communication complexities of the function f∘AND2f \circ \mathrm{AND}_2f∘AND2​ are polynomially related, up to polylogarithmic factors in nnn. We prove this by showing that both are characterized--up to polynomial loss--by the logarithm of the De Morgan sparsity of fff. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.