Near-Term Pseudorandom and Pseudoresource Quantum States

2025-04-24

AI Summary

Preparing Haar-random states on NISQ devices remains infeasible due to hardware constraints. Method: We introduce the framework of *T-pseudorandom states* (T-PRS), defined relative to observers whose quantum algorithms run in subpolynomial time T(n) (e.g., log n, n, n log n); such states are computationally indistinguishable from Haar-random states for all T-bounded adversaries. We integrate quantum-secure pseudorandom functions (QPRFs), classical PRFs, and quantum resource theories to derive rigorous resource characterizations across multiple T(n) regimes. Contribution/Results: We formally define T-PRS and *T-pseudoresource pairs*, establishing quantitative lower bounds on quantum resources—coherence, entanglement, and magic—in terms of T(n). Crucially, we prove that stricter computational limits (i.e., smaller T(n)) induce larger resource gaps between low-resource simulable states and high-resource target states. Our results provide both theoretical foundations and practically implementable protocols for lightweight quantum cryptography and learning in the NISQ era.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

A pseudorandom quantum state (PRS) is an ensemble of quantum states indistinguishable from Haar-random states to observers with efficient quantum computers. It allows one to substitute the costly Haar-random state with efficiently preparable PRS as a resource for cryptographic protocols, while also finding applications in quantum learning theory, black hole physics, many-body thermalization, quantum foundations, and quantum chaos. All existing constructions of PRS equate the notion of efficiency to quantum computers which runtime is bounded by a polynomial in its input size. In this work, we relax the notion of efficiency for PRS with respect to observers with near-term quantum computers implementing algorithms with runtime that scales slower than polynomial-time. We introduce the $mathbf{T}$-PRS which is indistinguishable to quantum algorithms with runtime $mathbf{T}(n)$ that grows slower than polynomials in the input size $n$. We give a set of reasonable conditions that a $mathbf{T}$-PRS must satisfy and give two constructions by using quantum-secure pseudorandom functions and pseudorandom functions. For $mathbf{T}(n)$ being linearithmic, linear, polylogarithmic, and logarithmic function, we characterize the amount of quantum resources a $mathbf{T}$-PRS must possess, particularly on its coherence, entanglement, and magic. Our quantum resource characterization applies generally to any two state ensembles that are indistinguishable to observers with computational power $mathbf{T}(n)$, giving a general necessary condition of whether a low-resource ensemble can mimic a high-resource ensemble, forming a $mathbf{T}$-pseudoresource pair. We demonstate how the necessary amount of resource decreases as the observer's computational power is more restricted, giving a $mathbf{T}$-pseudoresource pair with larger resource gap for more computationally limited observers.

Problem

• Relax efficiency for PRS with near-term quantum computers.
• Introduce T-PRS for slower-than-polynomial runtime observers.
• Characterize quantum resources for T-PRS coherence and entanglement.

Innovation

• Introduces T-PRS for near-term quantum computers
• Uses quantum-secure pseudorandom functions
• Characterizes quantum resources for T-PRS