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Quantum Pseudorandom Scramblers

Chuhan Lu, Minglong Qin, Fang Song, Penghui Yao, Mingnan Zhao

TL;DR

This work introduces quantum pseudorandom state scramblers (PRSS), a primitive that maps any pure input state to a distribution near Haar randomness, thereby achieving a dispersing, state-independent form of quantum pseudorandomness. The authors construct PRSS from quantum-secure pseudorandom primitives via a parallel Kac's walk, proving exponential mixing on both real and complex Hilbert spaces and establishing an $ε$-net dispersion of output states. They develop RSS and PRSS constructions in real and complex spaces, including stepwise evolution, and show that PRSS subsume existing PRS variants while enabling novel cryptographic tasks such as compact quantum encryption and succinct quantum state commitment. The framework relies on a parallelized Kac's walk with coupling arguments to achieve fast convergence to Haar randomness, and it leverages QPRFs/QPRPs to ensure uniform, efficient instantiation. Overall, PRSSs offer a robust, potentially weaker-assumption path to powerful quantum pseudorandomness with broad cryptographic and foundational implications for quantum information processing.

Abstract

Quantum pseudorandom state generators (PRSGs) have stimulated exciting developments in recent years. A PRSG, on a fixed initial (e.g., all-zero) state, produces an output state that is computationally indistinguishable from a Haar random state. However, pseudorandomness of the output state is not guaranteed on other initial states. In fact, known PRSG constructions provably fail on some initial states. In this work, we propose and construct quantum Pseudorandom State Scramblers (PRSSs), which can produce a pseudorandom state on an arbitrary initial state. In the information-theoretical setting, we obtain a scrambler which maps an arbitrary initial state to a distribution of quantum states that is close to Haar random in total variation distance. As a result, our scrambler exhibits a dispersing property. Loosely, it can span an $ε$-net of the state space. This significantly strengthens what standard PRSGs can induce, as they may only concentrate on a small region of the state space provided that average output state approximates a Haar random state. Our PRSS construction develops a parallel extension of the famous Kac's walk, and we show that it mixes exponentially faster than the standard Kac's walk. This constitutes the core of our proof. We also describe a few applications of PRSSs. While our PRSS construction assumes a post-quantum one-way function, PRSSs are potentially a weaker primitive and can be separated from one-way functions in a relativized world similar to standard PRSGs.

Quantum Pseudorandom Scramblers

TL;DR

This work introduces quantum pseudorandom state scramblers (PRSS), a primitive that maps any pure input state to a distribution near Haar randomness, thereby achieving a dispersing, state-independent form of quantum pseudorandomness. The authors construct PRSS from quantum-secure pseudorandom primitives via a parallel Kac's walk, proving exponential mixing on both real and complex Hilbert spaces and establishing an -net dispersion of output states. They develop RSS and PRSS constructions in real and complex spaces, including stepwise evolution, and show that PRSS subsume existing PRS variants while enabling novel cryptographic tasks such as compact quantum encryption and succinct quantum state commitment. The framework relies on a parallelized Kac's walk with coupling arguments to achieve fast convergence to Haar randomness, and it leverages QPRFs/QPRPs to ensure uniform, efficient instantiation. Overall, PRSSs offer a robust, potentially weaker-assumption path to powerful quantum pseudorandomness with broad cryptographic and foundational implications for quantum information processing.

Abstract

Quantum pseudorandom state generators (PRSGs) have stimulated exciting developments in recent years. A PRSG, on a fixed initial (e.g., all-zero) state, produces an output state that is computationally indistinguishable from a Haar random state. However, pseudorandomness of the output state is not guaranteed on other initial states. In fact, known PRSG constructions provably fail on some initial states. In this work, we propose and construct quantum Pseudorandom State Scramblers (PRSSs), which can produce a pseudorandom state on an arbitrary initial state. In the information-theoretical setting, we obtain a scrambler which maps an arbitrary initial state to a distribution of quantum states that is close to Haar random in total variation distance. As a result, our scrambler exhibits a dispersing property. Loosely, it can span an -net of the state space. This significantly strengthens what standard PRSGs can induce, as they may only concentrate on a small region of the state space provided that average output state approximates a Haar random state. Our PRSS construction develops a parallel extension of the famous Kac's walk, and we show that it mixes exponentially faster than the standard Kac's walk. This constitutes the core of our proof. We also describe a few applications of PRSSs. While our PRSS construction assumes a post-quantum one-way function, PRSSs are potentially a weaker primitive and can be separated from one-way functions in a relativized world similar to standard PRSGs.
Paper Structure (67 sections, 40 theorems, 231 equations, 2 figures, 2 tables)

This paper contains 67 sections, 40 theorems, 231 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Overview on the construction and analysis.
  4. Applications.
  5. Discussions and Open Questions
  6. Acknowledgment.
  7. Organization.
  8. Preliminary
  9. Basic Notation
  10. Probability Theory
  11. Cryptography
  12. Pseudorandom Functions and Pseudorandom Permutations
  13. Quantum Pseudorandomness
  14. Pseudorandom State Scramblers
  15. Properties of Pseudorandom State Scramblers
  16. ...and 52 more sections

Key Result

lemma thmcounterlemma

Let $\mu$ and $\nu$ be two arbitrary probability measures over $\mathcal{S}_{\mathbb{R}}^{2^n}$ ($\mathcal{S}_{\mathbb{C}}^{2^n}$). Then for all $\ell \in\mathbb{N}$,

Figures (2)

  • Figure 1: Transformation of subcoordinates $X_t[i,j]$ and $Y_t[i,j]$
  • Figure 2: Circuit diagram for the construction of the $K_{\sigma,f}$

Theorems & Definitions (91)

  • lemma thmcounterlemma
  • lemma thmcounterlemma: Coupling Lemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma: Lemma 3.5 in PS17
  • lemma thmcounterlemma
  • proof
  • definition thmcounterdefinition: Quantum-Secure Pseudorandom Function
  • definition thmcounterdefinition: Quantum-Secure Pseudorandom Permutation
  • lemma thmcounterlemma
  • definition thmcounterdefinition: Quantum Pseudorandom State Generator
  • ...and 81 more