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arXiv:1904.13056 · SIAM Journal on Computing

Query-to-Communication Lifting Using Low-Discrepancy Gadgets

Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, Toniann Pitassi

Published Jan 1, 2021• 12 citations
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Abstract

Lifting theorems are theorems that relate the query complexity of a function $f:\{0,1\}^{n}\to \{0,1\}$ to the communication complexity of the composed function $f\circ g^{n}$ for some “gadget” $g:\{ 0,1\}^{b}\times \{0,1\}^{b}\to \{0,1\}$. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget $g$. We prove a new lifting theorem that works for all gadgets $g$ that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we significantly increase the range of gadgets for which such lifting theorems hold. Our result has two main motivations: first, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications of the theorem. Second, our result can be seen as a strong generalization of a direct-sum theorem for functions with low discrepancy.

DOI
10.1137/19m1310153

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Lifting theorems are theorems that relate the query complexity of a function f:{0,1}n→{0,1}f:\{0,1\}^{n}\to \{0,1\}f:{0,1}n→{0,1} to the communication complexity of the composed function f∘gnf\circ g^{n}f∘gn for some “gadget” g:{0,1}b×{0,1}b→{0,1}g:\{ 0,1\}^{b}\times \{0,1\}^{b}\to \{0,1\}g:{0,1}b×{0,1}b→{0,1}. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget ggg. We prove a new lifting theorem that works for all gadgets ggg that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we significantly increase the range of gadgets for which such lifting theorems hold. Our result has two main motivations: first, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications of the theorem. Second, our result can be seen as a strong generalization of a direct-sum theorem for functions with low discrepancy.
Lifting theorems are theorems that bound the communication complexity of a composed function f∘gnf\circ g^{n}f∘gn in terms of the query complexity of fff and the communication complexity of ggg. Such theorems constitute a powerful generalization of direct-sum theorems for ggg, and have seen numerous applications in recent years. We prove a new lifting theorem that works for every two functions f,gf,gf,g such that the discrepancy of ggg is at most inverse polynomial in the input length of fff. Our result is a significant generalization of the known direct-sum theorem for discrepancy, and extends the range of inner functions ggg for which lifting theorems hold.
Lifting theorems are theorems that bound the communication complexity of a composed function f∘gⁿ in terms of the query complexity of f and the communication complexity of g. Such theorems constitute a powerful generalization of direct-sum theorems for g, and have seen numerous applications in recent years. We prove a new lifting theorem that works for every two functions f,g such that the discrepancy of g is at most inverse polynomial in the input length of f. Our result is a significant generalization of the known direct-sum theorem for discrepancy, and extends the range of inner functions g for which lifting theorems hold.
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.
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.
We prove a new query-to-communication lifting for randomized protocols, with inner product as gadget. This allows us to use a much smaller gadget, leading to a more efficient lifting. Prior to this work, such a theorem was known only for deterministic protocols, due to Chattopadhyay et al. [Arkadev Chattopadhyay et al., 2017] and Wu et al. [Xiaodi Wu et al., 2017]. The only query-to-communication lifting result for randomized protocols, due to Göös, Pitassi and Watson [Mika Göös et al., 2017], used the much larger indexing gadget. Our proof also provides a unified treatment of randomized and deterministic lifting. Most existing proofs of deterministic lifting theorems use a measure of information known as thickness. In contrast, Göös, Pitassi and Watson [Mika Göös et al., 2017] used blockwise min-entropy as a measure of information. Our proof uses the blockwise min-entropy framework to prove lifting theorems in both settings in a unified way.