arXiv:2404.07606

Lifting with Inner Functions of Polynomial Discrepancy

Yahel Manor, Or Meir

Published Apr 11, 2024• 0 citations

Abstract

Lifting theorems are theorems that bound the communication complexity of a composed function $f\circ g^{n}$ 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.

DOI: 10.48550/arxiv.2404.07606

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arXiv:2404.07606

Lifting with Inner Functions of Polynomial Discrepancy

Yahel Manor, Or Meir

Published Apr 11, 2024• 0 citations

arXiv HTML PDF

Abstract

Lifting theorems are theorems that bound the communication complexity of a composed function

f\circ g^{n}

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.

DOI: 10.48550/arxiv.2404.07606

Related papers

Lifting with Inner Functions of Polynomial Discrepancy

Yahel Manor, Or Meir

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.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.

Leibniz international proceedings in informaticsJan 1, 20220 citations

A Lifting Theorem for Hybrid Classical-Quantum Communication Complexity

Xudong Wu, Guangxu Yang, Penghui Yao

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 $f\circ G^n$ , where $f:\{0,1\}^n\to\{\pm1\}$ and $G$ is an inner product function of $Θ(\log n)$ 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 $c$ classical bits followed by $q$ qubits to compute $f\circ G^n$ must satisfy $c+q^2=Ω\big(\max\{\mathrm{deg}(f),\mathrm{bs}(f)\}\cdot\log n\big)$ , where $\mathrm{deg}(f)$ is the degree of $f$ and $\mathrm{bs}(f)$ is the block sensitivity of $f$ . For read-once formula $f$ , this yields an almost tight trade-off: either they have to exchange $Θ\big(n\cdot\log n\big)$ classical bits or $\widetildeΘ\big(\sqrt n\cdot\log n\big)$ 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 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 $f\circ G^n$, where $f:\{0,1\}^n\to\{\pm1\}$ and $G$ is an inner product function of $Θ(\log n)$ 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 $c$ classical bits followed by $q$ qubits to compute $f\circ G^n$ must satisfy $c+q^2=Ω\big(\max\{\mathrm{deg}(f),\mathrm{bs}(f)\}\cdot\log n\big)$, where $\mathrm{deg}(f)$ is the degree of $f$ and $\mathrm{bs}(f)$ is the block sensitivity of $f$. For read-once formula $f$, this yields an almost tight trade-off: either they have to exchange $Θ\big(n\cdot\log n\big)$ classical bits or $\widetildeΘ\big(\sqrt n\cdot\log n\big)$ 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.

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Query-to-Communication Lifting Using Low-Discrepancy Gadgets

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Troy Lee, Adi Shraibman, Robert Špalek

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Anurag Anshu, Shalev Ben-David, Srijita Kundu

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 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.

Dec 7, 20201 citation

On Query-To-Communication Lifting for Adversary Bounds

Anshu, Anurag, Shalev Ben-David, Srijita Kundu

arXiv:2012.03415Leibniz international proceedings in informaticsDec 7, 20200 citations

Lifting theorems are theorems that bound the communication complexity of a composed function

f\circ g^{n}

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.

f\circ G^n

, where

f:\{0,1\}^n\to\{\pm1\}

and

G

is an inner product function of

Θ(\log n)

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

c

classical bits followed by

q

qubits to compute

f\circ G^n

must satisfy

c+q^2=Ω\big(\max\{\mathrm{deg}(f),\mathrm{bs}(f)\}\cdot\log n\big)

, where

\mathrm{deg}(f)

is the degree of

f

and

\mathrm{bs}(f)

is the block sensitivity of

f

. For read-once formula

f

, this yields an almost tight trade-off: either they have to exchange

Θ\big(n\cdot\log n\big)

classical bits or

\widetildeΘ\big(\sqrt n\cdot\log n\big)

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.

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.