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On the Protection Against Noise for Measurement-Based Quantum Computation

Valentin Vankov Iliev

TL;DR

This paper studies how to protect pairs of commuting single-qubit gates against crosstalk in measurement-based quantum computation by encoding them as self-adjoint operators with spectrum $\{1,-1\}$ and analyzing their joint statistics on $\mathcal{H}^{\otimes 2}$ using Bell-state inputs. It derives the joint probability distribution $p_{k,\ell}$ for outcomes, connects informational independence (zero crosstalk) to the condition $p_{k,k}=\tfrac{1}{4}$ via an entropy-based noise measure, and provides explicit, geometry-driven criteria for independence when the gate directions lie in the coordinate planes $x=0$, $y=0$, or $z=0$ with concrete relations among polar and azimuthal angles and the Bell input states. The results yield exact criteria for when two commuting observables can be measured without information flow between them, enabling experimental verification and design of crosstalk-free gate pairs. Overall, the work links quantum measurement correlations, symmetry under group actions, and angle-parameterized gate representations to a practical noise-insensitivity condition in a measurement-based setting.

Abstract

Here we establish conditions for some pairs of quantum logic gates which operate on one qubit to be protected against crosstalk.

On the Protection Against Noise for Measurement-Based Quantum Computation

TL;DR

This paper studies how to protect pairs of commuting single-qubit gates against crosstalk in measurement-based quantum computation by encoding them as self-adjoint operators with spectrum and analyzing their joint statistics on using Bell-state inputs. It derives the joint probability distribution for outcomes, connects informational independence (zero crosstalk) to the condition via an entropy-based noise measure, and provides explicit, geometry-driven criteria for independence when the gate directions lie in the coordinate planes , , or with concrete relations among polar and azimuthal angles and the Bell input states. The results yield exact criteria for when two commuting observables can be measured without information flow between them, enabling experimental verification and design of crosstalk-free gate pairs. Overall, the work links quantum measurement correlations, symmetry under group actions, and angle-parameterized gate representations to a practical noise-insensitivity condition in a measurement-based setting.

Abstract

Here we establish conditions for some pairs of quantum logic gates which operate on one qubit to be protected against crosstalk.

Paper Structure

This paper contains 11 sections, 3 theorems, 21 equations.

Table of Contents

  1. Introduction, Notation
  2. Notation
  3. Introduction
  4. Two Commuting Operators on $\mathcal{H}^{\otimes 2}$
  5. A Sample Space and Two Random Variables
  6. The Probability Assignment of the Sample Space $S(\psi;\mathcal{A},\mathcal{B})$
  7. The Noise
  8. Special Types of Self-Adjoined Operators with Spectre $\{1,-1\}$
  9. Operators with Direction in Coordinate Plane $x=0$
  10. Operators with Direction in Coordinate Plane $y=0$
  11. Operators with Direction in Coordinate Plane $z=0$

Key Result

Proposition 2.1.1

The following two statements are equivalent: ( i) One has $p_{0,0}=p_{1,1}$ and $p_{0,1}=p_{1,0}$. ( ii) For all $k,\ell\in\mathbb{Z}_2$ one has $\operatorname{pr}(\mathcal{A}_{\mu,\eta}=\lambda_k)=\operatorname{pr}(\mathcal{B}_{\nu,\zeta}=\lambda_\ell)$.

Theorems & Definitions (4)

  • Proposition 2.1.1
  • Theorem 2.1.2
  • Theorem 2.2.1
  • Remark 4.2.1