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Quantum Pseudorandomness Cannot Be Shrunk In a Black-Box Way

Samuel Bouaziz--Ermann, Garazi Muguruza

TL;DR

The paper investigates whether PRSs can be shrunk from polynomial to logarithmic qubit length without losing pseudorandomness. It builds on Kretschmer's quantum oracle to establish a relativized separation: PRSs exist but short-PRSs do not, showing PRSs are strictly weaker than short-PRSs in this setting. The argument hinges on pseudodeterministic variants and their implications for promise-class separations ($\mathsf{PromiseBQP}$ vs $\mathsf{PromiseQMA}$) via a PD-OWF-based construction. This result informs the landscape of quantum cryptographic primitives by clarifying the limits of output-size shrinking and the role of pseudodeterminism, while leaving open whether the separation extends beyond quantum oracles to the classical setting.

Abstract

Pseudorandom Quantum States (PRS) were introduced by Ji, Liu and Song as quantum analogous to Pseudorandom Generators. They are an ensemble of states efficiently computable but computationally indistinguishable from Haar random states. Subsequent works have shown that some cryptographic primitives can be constructed from PRSs. Moreover, recent classical and quantum oracle separations of PRS from One-Way Functions strengthen the interest in a purely quantum alternative building block for quantum cryptography, potentially weaker than OWFs. However, our lack of knowledge of extending or shrinking the number of qubits of the PRS output still makes it difficult to reproduce some of the classical proof techniques and results. Short-PRSs, that is PRSs with logarithmic size output, have been introduced in the literature along with cryptographic applications, but we still do not know how they relate to PRSs. Here we answer half of the question, by showing that it is not possible to shrink the output of a PRS from polynomial to logarithmic qubit length while still preserving the pseudorandomness property, in a relativized way. More precisely, we show that relative to Kretschmer's quantum oracle (TQC 2021) short-PRSs cannot exist (while PRSs exist, as shown by Kretschmer's work).

Quantum Pseudorandomness Cannot Be Shrunk In a Black-Box Way

TL;DR

The paper investigates whether PRSs can be shrunk from polynomial to logarithmic qubit length without losing pseudorandomness. It builds on Kretschmer's quantum oracle to establish a relativized separation: PRSs exist but short-PRSs do not, showing PRSs are strictly weaker than short-PRSs in this setting. The argument hinges on pseudodeterministic variants and their implications for promise-class separations ( vs ) via a PD-OWF-based construction. This result informs the landscape of quantum cryptographic primitives by clarifying the limits of output-size shrinking and the role of pseudodeterminism, while leaving open whether the separation extends beyond quantum oracles to the classical setting.

Abstract

Pseudorandom Quantum States (PRS) were introduced by Ji, Liu and Song as quantum analogous to Pseudorandom Generators. They are an ensemble of states efficiently computable but computationally indistinguishable from Haar random states. Subsequent works have shown that some cryptographic primitives can be constructed from PRSs. Moreover, recent classical and quantum oracle separations of PRS from One-Way Functions strengthen the interest in a purely quantum alternative building block for quantum cryptography, potentially weaker than OWFs. However, our lack of knowledge of extending or shrinking the number of qubits of the PRS output still makes it difficult to reproduce some of the classical proof techniques and results. Short-PRSs, that is PRSs with logarithmic size output, have been introduced in the literature along with cryptographic applications, but we still do not know how they relate to PRSs. Here we answer half of the question, by showing that it is not possible to shrink the output of a PRS from polynomial to logarithmic qubit length while still preserving the pseudorandomness property, in a relativized way. More precisely, we show that relative to Kretschmer's quantum oracle (TQC 2021) short-PRSs cannot exist (while PRSs exist, as shown by Kretschmer's work).
Paper Structure (5 sections, 7 theorems, 12 equations, 1 figure)

This paper contains 5 sections, 7 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Open questions.
  4. Preliminaries
  5. Separating PRSs from short-PRSs

Key Result

theorem 1

There exists a quantum oracle $\mathcal{O}$ relative to which PRSs exist but short-PRSs do not.

Figures (1)

  • Figure 1: Relation between different primitives. Proven, expected and our result.

Theorems & Definitions (14)

  • theorem 1: \ref{['thm:main']}, informal
  • definition 1: Pseudorandom quantum states C:JiLiuSon18
  • definition 2: Quantum Pseudo-deterministic One-Way Functions BS23
  • theorem 2: Adapted from BS23
  • definition 3
  • definition 4
  • theorem 3: Kre21
  • theorem 4
  • proposition 1
  • proof
  • ...and 4 more