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Quantum $X$-Secure $B$-Byzantine $T$-Colluding Private Information Retrieval

Mohamed Nomeir, Alptug Aytekin, Sennur Ulukus

TL;DR

The paper defines QXBTPIR, a quantum variant of X-secure, B-Byzantine, T-colluding PIR, where N entangled databases store K messages and up to B servers may perform arbitrary single-qudit quantum operations. It develops a CSA-based quantum coding scheme within the N-sum box framework to retain a superdense coding gain and ensure decodability under Byzantine interference, with explicit rate characterizations across regimes and extensions to eavesdroppers. A two-instance quantum CSA construction, dual QCSA matrices, and a stability analysis against interference enable reliable retrieval while identifying Byzantine behavior via consistency checks. The generalized error analysis shows any single-qudit operation can be decomposed into generalized Pauli operators, and through Kraus discretization the scheme remains robust to arbitrary Byzantine errors, highlighting strong resilience and potential practical impact for secure quantum distributed retrieval. Overall, the work advances quantum private information retrieval by quantifying Byzantine capabilities, providing concrete achievability schemes, and leveraging quantum resources to achieve doubling gains in capacity under suitable conditions.

Abstract

We consider the problems arising from the presence of Byzantine servers in a quantum private information retrieval (QPIR) setting. This is the first work to precisely define what the capabilities of Byzantine servers could be in a QPIR context. We show that quantum Byzantine servers have more capabilities than their classical counterparts due to the possibilities created by quantum encoding procedures. We focus on quantum Byzantine servers that can apply any reversible operation on their individual qudits. In this case, Byzantine servers can generate any error, i.e., this covers \emph{all} possible single qudit operations that can be applied by Byzantine servers on their qudits. We design a scheme based on cross-subspace alignment (CSA) and we show that this scheme achieves superdense coding gain in some cases.

Quantum $X$-Secure $B$-Byzantine $T$-Colluding Private Information Retrieval

TL;DR

The paper defines QXBTPIR, a quantum variant of X-secure, B-Byzantine, T-colluding PIR, where N entangled databases store K messages and up to B servers may perform arbitrary single-qudit quantum operations. It develops a CSA-based quantum coding scheme within the N-sum box framework to retain a superdense coding gain and ensure decodability under Byzantine interference, with explicit rate characterizations across regimes and extensions to eavesdroppers. A two-instance quantum CSA construction, dual QCSA matrices, and a stability analysis against interference enable reliable retrieval while identifying Byzantine behavior via consistency checks. The generalized error analysis shows any single-qudit operation can be decomposed into generalized Pauli operators, and through Kraus discretization the scheme remains robust to arbitrary Byzantine errors, highlighting strong resilience and potential practical impact for secure quantum distributed retrieval. Overall, the work advances quantum private information retrieval by quantifying Byzantine capabilities, providing concrete achievability schemes, and leveraging quantum resources to achieve doubling gains in capacity under suitable conditions.

Abstract

We consider the problems arising from the presence of Byzantine servers in a quantum private information retrieval (QPIR) setting. This is the first work to precisely define what the capabilities of Byzantine servers could be in a QPIR context. We show that quantum Byzantine servers have more capabilities than their classical counterparts due to the possibilities created by quantum encoding procedures. We focus on quantum Byzantine servers that can apply any reversible operation on their individual qudits. In this case, Byzantine servers can generate any error, i.e., this covers \emph{all} possible single qudit operations that can be applied by Byzantine servers on their qudits. We design a scheme based on cross-subspace alignment (CSA) and we show that this scheme achieves superdense coding gain in some cases.
Paper Structure (5 sections, 9 theorems, 36 equations)

This paper contains 5 sections, 9 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Results
  4. Achievable Scheme
  5. Generalization to Arbitrary Errors

Key Result

Theorem 1

For the quantum $X$-secure, $B$-Byzantine, $T$-colluding PIR (QXBTPIR), where the $B$ Byzantine servers can manipulate encoders with any single qudit quantum operation, with $N$ databases, which are allowed to share entanglement and have quantum channels for answer strings, with $K$ messages, the fo when $X+T \geq N/2$, when $X+T < \frac{N}{2} \leq N-2B$, and when $X+T < N-2B < \frac{N}{2}$.

Theorems & Definitions (15)

  • Definition 1: Quantum operation
  • Definition 2: Kraus representation
  • Theorem 1
  • Corollary 1
  • Remark 1
  • Definition 3: Dual QCSA matrices
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Corollary 2
  • ...and 5 more