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Improved classical shadows from local symmetries in the Schur basis

Daniel Grier, Sihan Liu, Gaurav Mahajan

Abstract

We study the sample complexity of the classical shadows task: what is the fewest number of copies of an unknown state you need to measure to predict expected values with respect to some class of observables? Large joint measurements are likely required in order to minimize sample complexity, but previous joint measurement protocols only work when the unknown state is pure. We present the first joint measurement protocol for classical shadows whose sample complexity scales with the rank of the unknown state. In particular we prove $\mathcal O(\sqrt{rB}/ε^2)$ samples suffice, where $r$ is the rank of the state, $B$ is a bound on the squared Frobenius norm of the observables, and $ε$ is the target accuracy. In the low-rank regime, this is a nearly quadratic advantage over traditional approaches that use single-copy measurements. We present several intermediate results that may be of independent interest: a solution to a new formulation of classical shadows that captures functions of non-identical input states; a generalization of a ``nice'' Schur basis used for optimal qubit purification and quantum majority vote; and a measurement strategy that allows us to use local symmetries in the Schur basis to avoid intractable Weingarten calculations in the analysis.

Improved classical shadows from local symmetries in the Schur basis

Abstract

We study the sample complexity of the classical shadows task: what is the fewest number of copies of an unknown state you need to measure to predict expected values with respect to some class of observables? Large joint measurements are likely required in order to minimize sample complexity, but previous joint measurement protocols only work when the unknown state is pure. We present the first joint measurement protocol for classical shadows whose sample complexity scales with the rank of the unknown state. In particular we prove samples suffice, where is the rank of the state, is a bound on the squared Frobenius norm of the observables, and is the target accuracy. In the low-rank regime, this is a nearly quadratic advantage over traditional approaches that use single-copy measurements. We present several intermediate results that may be of independent interest: a solution to a new formulation of classical shadows that captures functions of non-identical input states; a generalization of a ``nice'' Schur basis used for optimal qubit purification and quantum majority vote; and a measurement strategy that allows us to use local symmetries in the Schur basis to avoid intractable Weingarten calculations in the analysis.
Paper Structure (30 sections, 34 theorems, 177 equations, 1 figure, 1 table, 4 algorithms)

This paper contains 30 sections, 34 theorems, 177 equations, 1 figure, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Proof overview and technical contributions
  3. Population Classical Shadows
  4. Measurements in Schur Basis
  5. Nice Schur Basis and Local Symmetries
  6. Discussion and open problems
  7. Preliminaries
  8. Representation theory
  9. Nice Schur Basis
  10. Building classical shadows
  11. Reduction to Population Classical Shadows
  12. Algorithm for Population Classical Shadow Task
  13. Step 1: Generic Pre-Processing
  14. Step 2: Row Symmetric Measurement for partition
  15. Expected value of estimator
  16. ...and 15 more sections

Key Result

Theorem 1

The multi-copy sample complexity of the classical shadows taskSee defn:classical_shadows for a precise definition of the classical shadows task. for unknown state $\chi \in \mathbbm C^{d \times d}$ of rank $r$ and observables $O_1, \ldots, O_M$ with $\left\|O_i\right\|_{\infty} \le 1$ and $\mathop{\

Figures (1)

  • Figure 1: $F \colon V \to W$ is $G$-linear iff the diagram commutes.

Theorems & Definitions (93)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Informal
  • Definition 4: Classical Shadows Task
  • Definition 5: Population Classical Shadow Task
  • Definition 6: Group Representation
  • Lemma 7: Irrep Connectivity
  • proof
  • Lemma 8: Schur-Weyl duality
  • Definition 9: $G$-linear maps and subrepresentation space isomorphism
  • ...and 83 more