Improved Separations between Quantum and Classical Communication Complexity of Total Functions

arXiv:2609.16726 · quant-ph, cs.CC · Submitted 2026-09-15 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Improved Separations between Quantum and Classical Communication Complexity of Total Functions".

Mira: This research refines previous frameworks to establish larger exponential separations between quantum and classical communication complexity for total functions, specifically achieving polylogarithmic quantum communication versus super-linear randomized communication.

Kai: First, who's behind it and why it matters.

Title and authors: Kai: Moving into the title and authors of "Improved Separations between Quantum and Classical Communication Complexity of Total Functions," I see it’s focused squarely on total functions, which are functions mapping from pairs of inputs to a single output.

Mira: That focus on total functions is what makes this paper distinct; it's not just about partial functions anymore, which means the complexity analysis has to account for the entire input space.

Lev: When we look at the authors, they seem very comfortable bridging theoretical complexity and practical communication bounds, which is always a sign that their results are robust enough to withstand scrutiny.

Kai: I think what’s important here is that they're refining Gavinsky’s framework and obtaining larger separations by moving away from older construction methods.

Mira: The real implication of the title is establishing tighter, more meaningful bounds for what quantum communication can actually achieve versus classical randomized communication in a total setting.

Lev: If the input size overhead N scales as they show, we need to consider whether that scaling is manageable when dealing with actual data structures or large AI models.

Kai: It seems the main goal is demonstrating that polylogarithmic quantum complexity can beat super-linear randomized complexity under specific conditions for total functions.

Mira: That's the punchline: they are showing that the gap between quantum and classical communication is wider than previously thought when you look at these specific function classes.

The paper's summary: Kai: So, summarizing what they’ve done in "Improved Separations between Quantum and Classical Communication Complexity of Total Functions," they are showing a total function with polylogarithmic quantum communication complexity versus a randomized complexity that is at least e (sqrt n) for two messages.

Mira: That specific result, Theorem two is quite significant because it sets the floor for how much separation we can guarantee using just two quantum messages <ref:2609.16726#pg0>.

Lev: A lower bound like e (sqrt n) means that even with a relatively small number of quantum messages, you still need a substantial classical communication cost to achieve the same result.

Kai: And they also have Theorem three which states that for any constant integer k greater than or equal to two, there's a function where the quantum complexity is polylogarithmic in n and the randomized complexity is at least e (n one-one/k) <ref:2609.16726#pg0>.

Mira: That second theorem shows they can extend these separations across different message counts, which suggests a more general principle applies to these total functions.

Lev: The technical contribution involving Theorem four is key because it provides a general conversion from quantum versus randomized communication separations for partial functions to separations for total functions by reducing input-length overhead <ref:2609.16726#pg1>.

Kai: That reduction in overhead is what allows them to extend constructions, and the main insight they developed was using a seeded linear extractor to replace the lookup table with just one compressed cell.

The paper's improvements: Mira: The primary improvement they highlight is replacing the large input-length overhead from previous constructions, which used O(ns) per party, with a much smaller overhead of O(n + s) input bits per party.

Kai: That reduction in the required input size to get those separations is what makes the results more applicable for larger problems, like those you see in distributed systems.

Lev: From an error correction perspective, if we can reduce the required input size from a linear dependence on s to something closer to n, that makes running these protocols on noisy hardware much more feasible.

Mira: The use of the seeded linear extractor, detailed in Lemma five is technically sophisticated; it guarantees that the fixed-seed maps are binary-linear and surjective, which is a very strong property for constructing certificates <ref:2609.16726#pg2,are binary-linear and surjective>.

Kai: And they show how this leads to specific applications where they achieve concrete separation exponents like one/four which then gets reduced to one/two using Proposition ten leading directly to Theorem two <ref:2609.16726#pg1>.

Lev: Applying Theorem four to functions like Shapem yields an exponent of one/four and then the subsequent reduction gives us the final result of a separation exponent of one/two which is a solid benchmark for their method <ref:2609.16726#pg1>.

Conclusion: Kai: To wrap up, these results from "Improved Separations between Quantum and Classical Communication Complexity of Total Functions" show that we can achieve polylogarithmic quantum communication against super-linear randomized communication with two messages and beyond.

Mira: The paper essentially provides a general mechanism—Theorem four—to translate partial function separations into total function separations by using compressed cells, which significantly reduces the input size required for these constructions <ref:2609.16726#pg1>.

Lev: For us on the hardware side, it means that while the theory shows these separations exist, we need to be careful how that input-length overhead translates into actual resource consumption when we try to implement them with quantum hardware.

Kai: The implication is that AI systems could design distributed training architectures where the communication cost for verifying data distribution scales predictably based on the complexity of the underlying function.

Mira: I think this work has big implications for how we can design verification protocols for complex models because it gives us a way to quantify exactly what quantum advantage we see when checking if an input adheres to a highly structured learned function.

Lev: If these separations hold up under the overhead reduction, then AI agents in distributed environments could leverage quantum communication for much faster consensus mechanisms under strict message limits compared to purely randomized classical methods.

Kai: It’s certainly a solid piece of complexity theory that gives us better tools to analyze the resource trade-offs in quantum computation versus classical communication.

Graduate School of Mathematics Nagoya University

quant-ph, cs.CC

Submitted: 2026-09-15

Updated: 2026-10-05

Comments: 15 pages; v2: minor edits

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 90/100

The gist: This research refines previous frameworks to establish larger exponential separations between quantum and classical communication complexity for total functions, specifically achieving

Key concepts

Total Function
A function where both inputs (Alice's and Bob's) are required to compute a single output. Unlike partial functions, total functions require both parties to contribute information for the final result.
Quantum Communication Complexity (Q(f))
Measures the minimum number of quantum bits Alice and Bob need to exchange to compute a function f. The paper shows that for certain total functions, this complexity can be polylogarithmic in the input size, which is much smaller than classical bounds.
Randomized Communication Complexity (R(f))
Measures the minimum number of bits Alice and Bob need to exchange when allowed to use randomness. The work establishes that for total functions, this complexity can be super-linear in the input size, specifically achieving bounds like $\Omega e(\sqrt{n})$ or $\Omega e(n^{1-1/k})$.
Seeded Linear Extractor
A technical tool used to replace large lookup tables with a single 'compressed' cell. This technique reduces the input overhead required by Alice and Bob from $O(ns)$ to $O(n+s)$, making the construction more efficient.

Terminology

Summary

This research refines previous frameworks to establish larger exponential separations between quantum and classical communication complexity for total functions, specifically achieving polylogarithmic quantum communication versus super-linear randomized communication. This work is significant because it improves upon earlier results by obtaining larger separation exponents, such as moving from an exponent of 1/6 to 1/2 for two-message protocols and demonstrating separations against more messages.

Key Results and Theorems

The paper introduces several key theorems that establish these improved separations:

(Theorem 2)

There is a total function with a quantum communication complexity of polylog(n) and a randomized communication complexity of at least omegae (√n).

(Theorem 3)

For every constant integer k ≥ 2, there is a total function with Q2(⌈k/2⌉+1(f) = polylog(n) and R(f) = omegae (n(1−1/k)).

Technical Contributions to the Construction

The main technical contribution is a general conversion from quantum versus randomized communication separations for partial functions to separations for total functions, encapsulated in Theorem 4. This theorem allows the extension of constructions from partial functions to total functions by reducing input-length overhead.

  1. The construction involves setting up a function F with an r-message quantum protocol of cost q ≥ 1 and a domain indicator DF(x, y) that has a Boolean circuit of size at most s, where s ≤ poly(n).

  2. Theorem 4 states that there is a total function Ftot with N = Oe(n + s) input bits per party, such that R(Ftot) = omegae (R(F)) and Qr+1(Ftot) = Oe(q).

  3. The construction uses a compressed cell represented as U + V, where Alice holds U and Bob holds V, and the candidate certificate is EL(U + V), where EL is a linear map obtained by fixing the extractor seed to L.

Techniques for Improving Separations

The authors detail specific techniques used to achieve the improved bounds:

  1. The main insight is using a seeded linear extractor to replace the lookup table by a single compressed cell, which reduces the input-length overhead from O(ns) per party (from previous constructions) to Oe(n + s).

  2. Lemma 5 provides an explicit strong (k,ε)-seeded extractor E: F(4b/2 × F c/2 → F(b/2), where c = ⌈KE log2 b⌉ admits fixed-seed maps that are binary-linear and surjective.

  3. Theorem 4 is applied to specific functions: Shapem from [8] yields an exponent of 1/4, which is then reduced to 1/2 using Proposition 10, resulting in Theorem 2.

Application to Specific Functions

The paper demonstrates the utility of these methods by applying Theorem 4 to two specific total functions:

(Application 1: Total function from Shape)

Using the Shapem function, whose domain predicate Dm has a circuit size of O(m log2 m) (Proposition 10), Theorem 4 is applied with n = 2m and s = O(m log2 m). This yields N = Oe(m) input bits per party, leading to R(Shapem) = omegae (√n) and Q2 (Shapem) being polylogarithmic in N.

(Application 2: Total functions from k-Forrelation)

For the k-forrelation function Forrk,m, the domain indicator has a circuit size s = O(kmb + k 2m log2 m) (Proposition 11). Applying Theorem 4 results in a total function with N = Oe(k(m)) input bits per party, achieving randomized complexity R(fk,m) = omegae (k(m−1/k)) and quantum complexity Q2⌈k/2⌉+1(f) = Ok(log m).

Lower Bound Proof

The proof of Theorem 4 relies on establishing the lower bound R(Ftot) = omegae (R(F)) via Theorem 7, which is derived from a variant of [3, Theorem 6]. This involves setting a small error parameter δ = 1/(1022c) and showing that any protocol for Ftot can be reduced to a protocol for F with a cost bounded by O(c2 CC(Π)). The final contradiction in the proof of Theorem 4.

Improvements for AI systems

Here are the specific improvements to AI systems that can be derived from this research, focusing on leveraging the demonstrated separations in communication complexity:

  1. Improve Scalability of Distributed Machine Learning Training/Inference: By applying Theorem 4, which relates randomized communication complexity to total function separations, we can design distributed learning architectures (like federated learning or decentralized training) where the required input size for a local model update scales polynomially with the circuit size needed to verify the data distribution.

  2. Enhance Robustness of Data Privacy in Distributed Systems: The construction in Theorem 4 allows replacing large lookup tables (which are memory-intensive and prone to leakage) with a compressed cell representation using seeded linear extractors. This means distributed AI systems can verify complex data constraints (like domain membership for specialized data types) without needing to store an exponentially large table of certificates, significantly reducing the input-length overhead per party while maintaining strong separation guarantees between randomized and quantum communication costs.

  3. Optimize Quantum-Enhanced Distributed Consensus: Theorem 3 provides a framework for functions based on k-forrelation (which models complex relational constraints) that allows for exponential separations in communication complexity between quantum and classical protocols using a constant number of messages. This implies that AI agents operating in high-stakes, distributed environments can leverage quantum communication to achieve significantly faster or more robust consensus mechanisms under strict message limits compared to purely randomized classical methods.

  4. Develop Efficient Verification Protocols for Complex Models: The ability to convert partial function verification (checking if a data point belongs to a model's domain) into total function verification (Theorem 4) means that AI systems can rigorously verify the integrity of massive, complex models by quantifying the communication cost required for both classical and quantum verifiers. This allows researchers to precisely determine the quantum advantage in verifying whether an input adheres to a highly structured, complex learned function.

  5. Improve Query-to-Communication Lifting for Model Refinement: The paper's methodology (Theorem 4 and its proof structure) provides a general technique for lifting query complexity bounds from specialized functions to broader total functions. This can be adapted to create more efficient algorithms for model refinement where the cost of querying a black-box function is translated into an efficient communication protocol, potentially speeding up iterative optimization loops in AI training.

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