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Near-Term Pseudorandom and Pseudoresource Quantum States

Andrew Tanggara, Mile Gu, Kishor Bharti

TL;DR

This work generalizes quantum pseudorandomness to observers with subpolynomial computational power by introducing obreak $ extbf{T}$-pseudorandom quantum states ($ extbf{T}$-PRS) and a framework for indistinguishability that scales with a class of functions $ extbf{T}$. It provides two explicit constructions of $ extbf{T}$-PRS based on quantum-secure pseudorandom phase functions and permutations, with careful analysis of how copy number and resource requirements depend on $ extbf{T}$. The authors then define $ extbf{T}$-pseudoresources and derive quantitative gaps in coherence, entanglement, and magic between high- and low-resource ensembles, showing that weaker observers see larger gaps. The results connect computational power to perceptible quantum resources and point to a broader program of computational-resource theories for quantum states and operations. Overall, the work offers a principled path to realize and analyze pseudorandomness and pseudoresources at near-term quantum scales, with implications for cryptography, learning theory, and quantum foundations.

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.

Near-Term Pseudorandom and Pseudoresource Quantum States

TL;DR

This work generalizes quantum pseudorandomness to observers with subpolynomial computational power by introducing obreak -pseudorandom quantum states (-PRS) and a framework for indistinguishability that scales with a class of functions . It provides two explicit constructions of -PRS based on quantum-secure pseudorandom phase functions and permutations, with careful analysis of how copy number and resource requirements depend on . The authors then define -pseudoresources and derive quantitative gaps in coherence, entanglement, and magic between high- and low-resource ensembles, showing that weaker observers see larger gaps. The results connect computational power to perceptible quantum resources and point to a broader program of computational-resource theories for quantum states and operations. Overall, the work offers a principled path to realize and analyze pseudorandomness and pseudoresources at near-term quantum scales, with implications for cryptography, learning theory, and quantum foundations.

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 -PRS which is indistinguishable to quantum algorithms with runtime that grows slower than polynomials in the input size . We give a set of reasonable conditions that a -PRS must satisfy and give two constructions by using quantum-secure pseudorandom functions and pseudorandom functions. For being linearithmic, linear, polylogarithmic, and logarithmic function, we characterize the amount of quantum resources a -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 , giving a general necessary condition of whether a low-resource ensemble can mimic a high-resource ensemble, forming a -pseudoresource pair. We demonstate how the necessary amount of resource decreases as the observer's computational power is more restricted, giving a -pseudoresource pair with larger resource gap for more computationally limited observers.

Paper Structure

This paper contains 11 sections, 11 theorems, 71 equations, 1 figure, 3 tables.

Table of Contents

  1. Computational Pseudorandomness and Indistinguishability
  2. Negligible distinguishability
  3. T-Pseudorandom Quantum States
  4. T-Pseudorandom Quantum State Constructions
  5. T-pseudorandom subset phase states
  6. T-pseudorandom subset states
  7. T-Pseudoresources
  8. Coherence resource gap
  9. Entanglement resource gap
  10. Magic resource gap
  11. Discussion

Key Result

Proposition 8

The set of $f(n)$-negligible functions ${\mathrm{negl}}_{f(n)}$ for any non-decreasing, non-constant function $f:{\mathbb{N}}\rightarrow{\mathbb{N}}$ satisfy the closure properties with respect to the set of repeat functions $\mathbf{K}$, where $\mathbf{K}$ is the set of constant functions. Moreover

Figures (1)

  • Figure 1: In this illustration, we consider $\mathbf{T}$-indistinguishable $n$-qubit pair of ensembles $\{|\psi_k\rangle\}_k$ and $\{|\varphi_k\rangle\}_k$ (as defined in Section \ref{['sec:T-pseudorandomness']}). A quantum algorithm $\mathcal{A}$ is given input of either $t$ copies of $|\psi\rangle$ randomly sampled from $\{|\psi_k\rangle\}_k$ or $t$-copies of $|\varphi\rangle$ randomly sampled from $\{|\varphi_k\rangle\}_k$ such that it outputs $\mathcal{A}({\mathbb{E}}_k[|\psi_k\rangle])\in\{0,1\}$ or $\mathcal{A}({\mathbb{E}}_k[|\psi_k\rangle])\in\{0,1\}$ indicating which ensemble the input state belongs to. If the runtime of $\mathcal{A}$ given $t$ copies of $n$-qubit state input is given by $s(n)\in O(\mathbf{T}(n))$, then $\mathcal{A}$ cannot guess which ensemble the input belongs to, expect for a negligible probability (as defined in Section \ref{['sec:negligible_distinguishability']}).

Theorems & Definitions (33)

  • Definition 1: Closure properties
  • Remark 2
  • Definition 3: $\mathbf{T}$-negligible functions
  • Remark 4: Polynomial-time negligible functions
  • Example 5: Linearithmic-time negligible functions
  • Example 6: Polylog-time negligible functions
  • Definition 7: Repetition consistency
  • Proposition 8
  • proof
  • Proposition 9
  • ...and 23 more