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The Scrooge ensemble in many-body quantum systems

Max McGinley, Thomas Schuster

TL;DR

This work provides a rigorous analytic framework for Scrooge ensembles, a structured generalization of Haar randomness, in many-body quantum systems. By deriving a simple, controlled approximation for the Scrooge moments and establishing exponential concentration of local subsystem properties, it separates universal nonlocal fluctuations from local observables. The results connect measurement statistics to Porter-Thomas and Wishart distributions, reveal long-range conditional mutual information, and prove lower bounds on the randomness, time, and circuit resources required to form Scrooge designs, with implications for benchmarking, sampling hardness, and complexity growth in quantum dynamics. Together, these findings offer a quantitative foundation for using Scrooge ensembles to model scrambling under conservation laws and to diagnose and certify quantum-device performance.

Abstract

In many physical settings, the statistical properties of quantum states are thought to be described by the Scrooge ensemble, a more structured generalization of the Haar ensemble. In this work, we prove several key results on the properties and complexity of Scrooge-random states in macroscopic quantum systems, and provide a general-purpose calculus for evaluating their moments. A key theme of our results is a separation between universal random fluctuations in non-local properties and exponential concentration of all local properties. Implications for device benchmarking, sampling advantages beyond random circuits, quantum complexity growth, and the physical origin of Scrooge-random states are discussed.

The Scrooge ensemble in many-body quantum systems

TL;DR

This work provides a rigorous analytic framework for Scrooge ensembles, a structured generalization of Haar randomness, in many-body quantum systems. By deriving a simple, controlled approximation for the Scrooge moments and establishing exponential concentration of local subsystem properties, it separates universal nonlocal fluctuations from local observables. The results connect measurement statistics to Porter-Thomas and Wishart distributions, reveal long-range conditional mutual information, and prove lower bounds on the randomness, time, and circuit resources required to form Scrooge designs, with implications for benchmarking, sampling hardness, and complexity growth in quantum dynamics. Together, these findings offer a quantitative foundation for using Scrooge ensembles to model scrambling under conservation laws and to diagnose and certify quantum-device performance.

Abstract

In many physical settings, the statistical properties of quantum states are thought to be described by the Scrooge ensemble, a more structured generalization of the Haar ensemble. In this work, we prove several key results on the properties and complexity of Scrooge-random states in macroscopic quantum systems, and provide a general-purpose calculus for evaluating their moments. A key theme of our results is a separation between universal random fluctuations in non-local properties and exponential concentration of all local properties. Implications for device benchmarking, sampling advantages beyond random circuits, quantum complexity growth, and the physical origin of Scrooge-random states are discussed.

Paper Structure

This paper contains 25 sections, 14 theorems, 127 equations.

Table of Contents

  1. Simple approximate formula for the moments of the Scrooge ensemble
  2. Integral representations of Scrooge moments
  3. Concentration properties of $r_\phi$
  4. Relating the exact and approximate moments
  5. Concentration of all local properties
  6. Approximation of subsystem moments
  7. Properties of entropic quantities $S^*_\infty(B)$ and $\hat{S}_\infty(B|A)$
  8. Collapse of the Scrooge ensemble under small amounts of noise
  9. Proof of Theorem \ref{['thm:subsystem full']}: Moments of the Scrooge ensemble after tracing out small subregions
  10. Proof of Corollary \ref{['cor: noise']}: Collapse of the Scrooge ensemble under small amounts of local depolarizing noise
  11. Output distributions and universal fluctuations
  12. 1-bitstring marginals and the Porter-Thomas distribution
  13. 2-bitstring marginals and the $2 \times 2$ Wishart distribution
  14. $m$-bitstring marginals and the $m \times m$ Wishart distribution
  15. Universal relation between read-out noise sensitivity and off-diagonal quantum coherence
  16. ...and 10 more sections

Key Result

Theorem 1

For any $k \leq 2^{S_\infty(\rho)/3 - 1}$. The $k$-th moment of the Scrooge ensemble is approximated by up to relative error $\delta_{\rho, k}$, where Here, $S_\infty(\rho) = -\log(\max \textup{eig}\, \rho)$ is the min-entropy of $\rho$.

Theorems & Definitions (21)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Corollary 1: Scrooge-random output distributions collapse under noise
  • Corollary 2: Scrooge-random output distributions are far from $\rho$
  • Corollary 3: Marginal output distributions of Scrooge-random states are close to $\rho$
  • Corollary 4: Output distributions of noisy Scrooge-random states are close to $\rho$
  • Theorem 3: Output distributions of Scrooge designs have quantized CMI for product $\rho$
  • ...and 11 more