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arxiv: 2506.19369 · v2 · submitted 2025-06-24 · 🪐 quant-ph

Gottesman-Knill Limit on One-way Communication Complexity: Tracing the Quantum Advantage down to Magic Resources

Snehasish Roy Chowdhury , Sahil Gopalkrishna Naik , Ananya Chakraborty , Ram Krishna Patra , Subhendu B. Ghosh , Pratik Ghosal , Manik Banik , Ananda G. Maity This is my paper

Pith reviewed 2026-05-19 08:25 UTC · model grok-4.3

classification 🪐 quant-ph
keywords one-way communication complexityGottesman-Knill theoremmagic statesstabilizer statesClifford operationsquantum advantageprime dimension
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The pith

Stabilizer-state encodings and Clifford decodings in prime-dimensional one-way quantum communication can be exactly simulated by classical systems of the same dimension with shared randomness.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that any one-way communication protocol using prime-dimensional quantum systems, when limited to stabilizer-state encodings and Clifford-operation decodings, can be perfectly matched by sending a classical system of equal size if the parties share randomness. This directly parallels the Gottesman-Knill theorem by pinning the source of any quantum advantage to non-stabilizer resources known as magic. The work further exhibits concrete tasks in which even a minimal amount of magic produces a strict improvement over the classical simulation. A sympathetic reader sees this as a precise boundary on where quantum effects matter for communication efficiency.

Core claim

Any one-way communication protocol implemented using a prime-dimensional quantum system restricted to stabilizer-state encodings and Clifford-operation decodings can be exactly simulated by transmitting a classical system of the same dimension, given access to shared randomness between the sender and receiver. In direct analogy with the Gottesman-Knill theorem, this identifies the same non-stabilizer resources, commonly known as the magic resources, as the essential ingredient for the quantum advantage in one-way communication complexity. Explicit tasks are presented where even a minimal magic resource suffices to achieve a provable quantum advantage.

What carries the argument

The exact simulation equivalence that reduces restricted quantum one-way protocols to classical transmission plus shared randomness, isolating magic resources as the carrier of any remaining advantage.

If this is right

  • Quantum advantage in these one-way tasks must come from non-stabilizer states or non-Clifford operations.
  • Minimal magic resources are already enough to produce strict separation from the classical simulation in some explicit tasks.
  • The equivalence applies only in prime dimensions and relies on the availability of shared randomness.
  • Any extension beyond stabilizer encodings or Clifford decodings falls outside the simulation guarantee.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same resource distinction may help classify which communication problems admit efficient quantum solutions and which remain hard even with quantum resources.
  • Results of this form suggest that magic-state distillation techniques developed for computation could be repurposed to improve communication efficiency.
  • It would be natural to test whether the classical simulation still works when the shared randomness is replaced by weaker correlation resources.

Load-bearing premise

The protocols are limited to stabilizer-state encodings and Clifford-operation decodings, so the classical simulation holds only under this restriction.

What would settle it

A concrete one-way communication task in which a stabilizer Clifford quantum protocol in prime dimension beats every classical protocol of the same dimension that uses shared randomness would falsify the central claim.

read the original abstract

Quantum systems are known to offer advantages over their classical counterpart in communication complexity protocols, where the aim is to minimize the amount of information exchange between distant parties to compute global functions of their distributed inputs. In this work, we establish that any one-way communication protocol implemented using a prime-dimensional quantum system -- restricted to stabilizer-state encodings and Clifford-operation decodings -- can be exactly simulated by transmitting a classical system of the same dimension, given access to shared randomness between the sender and receiver. In direct analogy with the Gottesman-Knill theorem, which attributes quantum computational speedup to non-stabilizer resources, commonly known as the magic resources, our result identifies the same non-stabilizer resources as the essential ingredient for the quantum advantage in one-way communication complexity. Furthermore, we present explicit tasks where even a 'minimal magic resource' suffices to achieve a provable quantum advantage, highlighting its efficient use in communication protocols.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes that any one-way communication protocol in a prime-dimensional quantum system, when restricted to stabilizer-state encodings at the sender and Clifford-operation decodings at the receiver, can be exactly simulated by a classical system of the same dimension with access to shared randomness. This is framed as a direct analogue of the Gottesman-Knill theorem, identifying non-stabilizer ('magic') resources as the origin of any quantum advantage in this setting. The authors further exhibit explicit communication tasks in which even a minimal amount of magic suffices to produce a provable separation from the classical case.

Significance. If the central simulation theorem holds, the result supplies a resource-theoretic account of quantum advantage in one-way communication complexity that parallels the computational case. By isolating magic as the necessary and sufficient non-classical resource under the stated restrictions, the work clarifies when quantum communication can outperform classical communication and supplies concrete tasks that demonstrate efficient use of minimal magic. These contributions would be of interest to researchers working on quantum resource theories and communication complexity.

major comments (2)
  1. [Main theorem (presumably §3 or §4)] The central simulation theorem is stated in the abstract and presumably proved in the main text, but the provided manuscript excerpt does not include the full proof or verification of edge cases (e.g., when the input dimension is composite or when the shared randomness is limited). A load-bearing step appears to be the exact equivalence between the quantum protocol and its classical simulation; this equivalence must be shown to hold for all admissible stabilizer encodings and Clifford decodings without additional assumptions.
  2. [Explicit tasks section] The claim that 'minimal magic resource' suffices for advantage is illustrated by explicit tasks, but it is unclear from the abstract whether these tasks remain hard when the classical simulator is also granted the same minimal magic (or equivalent classical resources). If the separation relies on the quantum side having access to magic while the classical side does not, this must be stated explicitly to avoid conflating the resource with the protocol model.
minor comments (2)
  1. [Abstract] Notation for the prime dimension p and the stabilizer group should be introduced consistently; the abstract uses 'prime-dimensional' without defining the underlying Hilbert space dimension explicitly.
  2. [Model definition] The phrase 'Clifford-operation decodings' should be clarified: does this include only unitary Clifford gates, or also measurements in the computational basis after Clifford conjugation?

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the scope and presentation of our results. We address each major comment below.

read point-by-point responses

  1. Referee: [Main theorem (presumably §3 or §4)] The central simulation theorem is stated in the abstract and presumably proved in the main text, but the provided manuscript excerpt does not include the full proof or verification of edge cases (e.g., when the input dimension is composite or when the shared randomness is limited). A load-bearing step appears to be the exact equivalence between the quantum protocol and its classical simulation; this equivalence must be shown to hold for all admissible stabilizer encodings and Clifford decodings without additional assumptions.

    Authors: The complete proof of the main simulation theorem appears in Section 3 of the full manuscript. The theorem is formulated and proved exclusively for prime-dimensional systems, where the Heisenberg-Weyl operators form a basis that permits an exact classical simulation via shared randomness. We do not assert the result for composite dimensions, as the algebraic structure of the generalized Pauli group changes and the direct simulation correspondence fails to hold in the same manner; this restriction is stated in the manuscript. The shared randomness employed is finite and explicitly constructed to reproduce the output statistics of any admissible stabilizer encoding and Clifford decoding. The equivalence is shown to hold for the full class of such protocols using only the standard definitions of stabilizer states and Clifford operations, without further assumptions. We are prepared to add a short clarifying paragraph on the prime-dimensional scope and the construction of the shared randomness in a revised version. revision: partial

  2. Referee: [Explicit tasks section] The claim that 'minimal magic resource' suffices for advantage is illustrated by explicit tasks, but it is unclear from the abstract whether these tasks remain hard when the classical simulator is also granted the same minimal magic (or equivalent classical resources). If the separation relies on the quantum side having access to magic while the classical side does not, this must be stated explicitly to avoid conflating the resource with the protocol model.

    Authors: We agree that the resource distinction must be stated unambiguously. The explicit tasks are constructed to exhibit a separation between a quantum protocol that employs a minimal magic resource (one non-stabilizer state or one non-Clifford operation) and any classical protocol that uses only shared randomness without magic. The classical simulator furnished by the main theorem corresponds exactly to the magic-free case. Supplying magic to the classical side would allow it to emulate the quantum protocol, which is outside the classical communication model under study. We will revise the abstract and the explicit-tasks section to state explicitly that the comparison is between magic-assisted quantum communication and magic-free classical communication with shared randomness. revision: yes

Circularity Check

0 steps flagged

No circularity: self-contained simulation theorem under explicit restrictions

full rationale

The paper proves that one-way protocols restricted to stabilizer-state encodings and Clifford decodings in prime dimensions are exactly simulable by classical systems of the same dimension with shared randomness. This is presented as a direct theorem in explicit analogy to the standard Gottesman-Knill result, without any reduction of the claimed equivalence to a fitted parameter, self-referential definition, or load-bearing self-citation. The central claim follows from the algebraic properties of the restricted resources and does not invoke prior work by the same authors as an unverified uniqueness theorem or ansatz. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard quantum information assumptions and the analogy to the Gottesman-Knill theorem; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Quantum mechanics in prime-dimensional Hilbert spaces
    The framework assumes the standard postulates of quantum theory hold for the systems considered.
  • domain assumption Gottesman-Knill theorem for quantum computation
    The communication result is presented in direct analogy with the known computational simulation theorem.

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