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Lattice-Based Quantum Advantage from Rotated Measurements

Yusuf Alnawakhtha, Atul Mantri, Carl A. Miller, Daochen Wang

TL;DR

A new technique that uses the entire range of qubit measurements from the XY-plane and shows an optimized two-round proof of quantumness whose security can be expressed directly in terms of the hardness of the LWE (learning with errors) problem.

Abstract

Trapdoor claw-free functions (TCFs) are immensely valuable in cryptographic interactions between a classical client and a quantum server. Typically, a protocol has the quantum server prepare a superposition of two-bit strings of a claw and then measure it using Pauli-$X$ or $Z$ measurements. In this paper, we demonstrate a new technique that uses the entire range of qubit measurements from the $XY$-plane. We show the advantage of this approach in two applications. First, building on (Brakerski et al. 2018, Kalai et al. 2022), we show an optimized two-round proof of quantumness whose security can be expressed directly in terms of the hardness of the LWE (learning with errors) problem. Second, we construct a one-round protocol for blind remote preparation of an arbitrary state on the $XY$-plane up to a Pauli-$Z$ correction.

Lattice-Based Quantum Advantage from Rotated Measurements

TL;DR

A new technique that uses the entire range of qubit measurements from the XY-plane and shows an optimized two-round proof of quantumness whose security can be expressed directly in terms of the hardness of the LWE (learning with errors) problem.

Abstract

Trapdoor claw-free functions (TCFs) are immensely valuable in cryptographic interactions between a classical client and a quantum server. Typically, a protocol has the quantum server prepare a superposition of two-bit strings of a claw and then measure it using Pauli- or measurements. In this paper, we demonstrate a new technique that uses the entire range of qubit measurements from the -plane. We show the advantage of this approach in two applications. First, building on (Brakerski et al. 2018, Kalai et al. 2022), we show an optimized two-round proof of quantumness whose security can be expressed directly in terms of the hardness of the LWE (learning with errors) problem. Second, we construct a one-round protocol for blind remote preparation of an arbitrary state on the -plane up to a Pauli- correction.
Paper Structure (29 sections, 23 theorems, 104 equations, 15 figures)

This paper contains 29 sections, 23 theorems, 104 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Works
  4. Proof of quantumness.
  5. Blind RSP.
  6. Technical Overview
  7. Preliminaries
  8. Model of Quantum Computation
  9. Learning With Errors
  10. Parameters and assumptions
  11. A Simple Encryption Algorithm
  12. Trapdoors for LWE matrices
  13. Rotated Measurements on Generalized GHZ States
  14. An Optimized Proof of Quantumness
  15. Completeness
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $\lambda$ denote the security parameter. Suppose that $n, m, q, \sigma, \tau$ are functions of $\lambda$ that satisfy the constraints given in fig:parameters from sec:prelim, and suppose that the $\textnormal{LWE}_{n,q,G ( \sigma, \tau )}$ problem is hard. Then, there exists a two round interact

Figures (15)

  • Figure 1: Parameters and assumptions.
  • Figure 2: The single-bit public-key encryption algorithm $K$. $pk$ is the public key, $s$ is the secret key, $b$ is the message, and $ct$ is the ciphertext.
  • Figure 3: The single-bit public-key encryption algorithm $J$.
  • Figure 4: The proof of quantumness protocol, including the behavior of the verifier (Alice).
  • Figure 5: The behavior of an honest quantum prover in Protocol Q in \ref{['fig:pfprot']}.
  • ...and 10 more figures

Theorems & Definitions (45)

  • Theorem 1.1: Informal
  • Theorem 1.2: Informal
  • Lemma 3.1: Corollary 9 of canonne2020gaussian
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • Definition 4.1: Generalized GHZ state
  • ...and 35 more