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Noisy decoding by shallow circuits with parities: classical and quantum

Jop Briët, Harry Buhrman, Davi Castro-Silva, Niels M. P. Neumann

TL;DR

The paper proves a sharp dichotomy for decoding corrupted ECCs with shallow circuits: classical NC$^0[\oplus]$ decoders fail to recover a non-negligible fraction of messages under any positive noise, for any code, while a constant-depth quantum circuit can efficiently decode the Hadamard code with probability $\Omega(\varepsilon^2)$ under adversarial corruption. The classical hardness is established via a polynomial-equidistribution analysis under biased inputs and a novel analytic-rank framework that separates high-rank (pseudorandom) from low-rank (structured) polynomial maps. On the quantum side, the Hadamard decoder builds a BV-inspired, non-local-game–like strategy using GHZ states and a quantum fan-out gadget to achieve constant-depth decoding, plus OR-reduction techniques to improve circuit size. The results yield a clear quantum-versus-classical separation for a natural, practically motivated decoding task and motivate further exploration of shallow-quantum advantages for other low-degree codes and biased-noise regimes. The work combines higher-order Fourier-analytic methods with distributed quantum circuit design to push the boundary of what constant-depth quantum circuits can achieve in information-theoretic tasks.

Abstract

We consider the problem of decoding corrupted error correcting codes with NC$^0[\oplus]$ circuits in the classical and quantum settings. We show that any such classical circuit can correctly recover only a vanishingly small fraction of messages, if the codewords are sent over a noisy channel with positive error rate. Previously this was known only for linear codes with large dual distance, whereas our result applies to any code. By contrast, we give a simple quantum circuit that correctly decodes the Hadamard code with probability $Ω(\varepsilon^2)$ even if a $(1/2 - \varepsilon)$-fraction of a codeword is adversarially corrupted. Our classical hardness result is based on an equidistribution phenomenon for multivariate polynomials over a finite field under biased input-distributions. This is proved using a structure-versus-randomness strategy based on a new notion of rank for high-dimensional polynomial maps that may be of independent interest. Our quantum circuit is inspired by a non-local version of the Bernstein-Vazirani problem, a technique to generate ``poor man's cat states'' by Watts et al., and a constant-depth quantum circuit for the OR function by Takahashi and Tani.

Noisy decoding by shallow circuits with parities: classical and quantum

TL;DR

The paper proves a sharp dichotomy for decoding corrupted ECCs with shallow circuits: classical NC decoders fail to recover a non-negligible fraction of messages under any positive noise, for any code, while a constant-depth quantum circuit can efficiently decode the Hadamard code with probability under adversarial corruption. The classical hardness is established via a polynomial-equidistribution analysis under biased inputs and a novel analytic-rank framework that separates high-rank (pseudorandom) from low-rank (structured) polynomial maps. On the quantum side, the Hadamard decoder builds a BV-inspired, non-local-game–like strategy using GHZ states and a quantum fan-out gadget to achieve constant-depth decoding, plus OR-reduction techniques to improve circuit size. The results yield a clear quantum-versus-classical separation for a natural, practically motivated decoding task and motivate further exploration of shallow-quantum advantages for other low-degree codes and biased-noise regimes. The work combines higher-order Fourier-analytic methods with distributed quantum circuit design to push the boundary of what constant-depth quantum circuits can achieve in information-theoretic tasks.

Abstract

We consider the problem of decoding corrupted error correcting codes with NC circuits in the classical and quantum settings. We show that any such classical circuit can correctly recover only a vanishingly small fraction of messages, if the codewords are sent over a noisy channel with positive error rate. Previously this was known only for linear codes with large dual distance, whereas our result applies to any code. By contrast, we give a simple quantum circuit that correctly decodes the Hadamard code with probability even if a -fraction of a codeword is adversarially corrupted. Our classical hardness result is based on an equidistribution phenomenon for multivariate polynomials over a finite field under biased input-distributions. This is proved using a structure-versus-randomness strategy based on a new notion of rank for high-dimensional polynomial maps that may be of independent interest. Our quantum circuit is inspired by a non-local version of the Bernstein-Vazirani problem, a technique to generate ``poor man's cat states'' by Watts et al., and a constant-depth quantum circuit for the OR function by Takahashi and Tani.
Paper Structure (30 sections, 21 theorems, 65 equations, 2 figures)

This paper contains 30 sections, 21 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Error models
  3. Circuits
  4. Quantum advantage
  5. The Hadamard code
  6. Our results
  7. Classical setting
  8. Quantum setting
  9. Related results and discussion
  10. Future directions
  11. Techniques
  12. Polynomial equidistribution
  13. Building the quantum circuit
  14. Warm-up: The linear case and a non-local game
  15. Impossibility of decoding for linear maps
  16. ...and 15 more sections

Key Result

Theorem 2.1

For any $\rho\in [0,1)$, $d\in \mathbb{N}$ and $\varepsilon\in (0,1]$, there is a $k_0 = k_0(d, \rho, \varepsilon)\in \mathbb{N}$ such that the following holds. Let $k \geq k_0$ and $n$ be positive integers, $E:\mathbb{F}_2^k\to \mathbb{F}_2^n$ be any map and $\phi:\mathbb{F}_2^n\to\mathbb{F}_2^k$ b

Figures (2)

  • Figure 1: The quantum circuit to generate a 3-qubit GHZ state. First we obtain a poor man's cat state $\frac{1}{\sqrt{2}}(\ket{z}+\ket{\bar{z}})$ with each $z\in\mathbb{F}_2^3$ equally likely to be found. The parity gates compute a prefix sum on the measurement results $d_1$ and $d_2$ and determine if a qubit has to be flipped to obtain the GHZ state.
  • Figure 2: Implementation of a quantum fan-out gate with one control qubit $\ket{\phi}$ and two target qubits $\ket{x_1}$ and $\ket{x_2}$. Only single and two-qubit gates and classical parity gates are used. The bottom three qubits are in the GHZ$_3$ state.

Theorems & Definitions (28)

  • Theorem 2.1: Impossibility of decoding by NC$^0[\oplus]$
  • Theorem 2.2: Decoding Hadamard with QNC$^0[\oplus]$
  • Corollary 2.2
  • Remark 2.3
  • Theorem 2.4: Quantum-vs-classical separation
  • Theorem 2.5: MAJORITY from list-Hadamard
  • Corollary 2.6: Hardness of list-Hadamard
  • Theorem 3.1: Impossibility of decoding by polynomial maps
  • Theorem 5.1: Bias implies low rank
  • Definition 5.2: Analytic rank
  • ...and 18 more