[Paper Review] Classical shadows based on locally-entangled measurements

Matteo Ippoliti|arXiv (Cornell University)|May 18, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces classical shadows based on randomized locally-entangled measurements—specifically Bell and GHZ bases—demonstrating that such protocols reduce sample complexity for estimating Pauli expectation values. By tuning entanglement in measurement bases, the method achieves a quadratic improvement in sample complexity (from ∼3ᵏ to ∼3ᵏ/²) for many operators, outperforming standard Pauli shadows and matching or surpassing shallow shadows in specific tasks.

ABSTRACT

We study classical shadows protocols based on randomized measurements in $n$-qubit entangled bases, generalizing the random Pauli measurement protocol ($n = 1$). We show that entangled measurements ($n\geq 2$) enable nontrivial and potentially advantageous trade-offs in the sample complexity of learning Pauli expectation values. This is sharply illustrated by shadows based on two-qubit Bell measurements: the scaling of sample complexity with Pauli weight $k$ improves quadratically (from $\sim 3^k$ down to $\sim 3^{k/2}$) for many operators, while others become impossible to learn. Tuning the amount of entanglement in the measurement bases defines a family of protocols that interpolate between Pauli and Bell shadows, retaining some of the benefits of both. For large $n$, we show that randomized measurements in $n$-qubit GHZ bases further improve the best scaling to $\sim (3/2)^k$, albeit on an increasingly restricted set of operators. Despite their simplicity and lower hardware requirements, these protocols can match or outperform recently-introduced "shallow shadows" in some practically-relevant Pauli estimation tasks.

Motivation & Objective

• To develop a family of classical shadow protocols based on randomized entangled measurements that improve sample complexity for estimating Pauli expectation values.
• To explore the trade-offs between measurement locality, entanglement, and sample efficiency in quantum state learning.
• To demonstrate that hardware-efficient, few-body entangled measurements can match or exceed the performance of more complex shallow shadow protocols.
• To identify structured measurement ensembles that enable improved scaling for specific classes of operators while maintaining low hardware requirements.

Proposed method

• The protocol uses randomized unitary transformations based on n-qubit entangled bases (e.g., Bell or GHZ states) to perform measurements, generalizing the standard Pauli shadow framework.
• Each measurement basis is chosen from an ensemble of unitaries that entangle qubits locally, with the degree of entanglement tunable via the choice of basis.
• Classical post-processing constructs inverted snapshots via the shadow channel, with the shadow norm derived from the inverse of the measurement channel’s eigenvalues.
• The shadow norm for a Pauli operator P of weight k is computed as ‖P‖_sh² = 3ᵏ/² for Bell measurements and (3/2)ᵏ for GHZ measurements, reflecting improved scaling.
• The method leverages Pauli invariance of the measurement ensemble to derive operator-dependent sample complexity bounds.
• The framework interpolates between Pauli shadows (n=1) and GHZ shadows (n large), with intermediate protocols offering tunable trade-offs between performance and operator coverage.

Experimental results

Research questions

• RQ1 Can randomized measurements in entangled bases reduce the sample complexity of estimating Pauli expectation values compared to standard Pauli shadows?
• RQ2 How does the degree of entanglement in measurement bases affect the trade-off between sample complexity and the set of learnable operators?
• RQ3 To what extent can locally-entangled shadows outperform shallow shadows in practical Pauli estimation tasks?
• RQ4 What is the optimal scaling of sample complexity for different classes of Pauli operators under entangled measurement protocols?

Key findings

• Bell-based classical shadows reduce the sample complexity for estimating k-weight Pauli operators from ∼3ᵏ to ∼3ᵏ/², achieving a quadratic improvement.
• GHZ-based shadows further improve the scaling to ∼(3/2)ᵏ for large n, though only for a restricted set of operators compatible with the GHZ basis structure.
• The protocol with two-qubit Bell measurements achieves a significant reduction in sample complexity while requiring only two-qubit entangling gates, making it hardware-efficient.
• For certain classes of operators—particularly those with contiguous support in 1D—locally-entangled shadows can outperform shallow shadows in terms of sample efficiency.
• The method interpolates between Pauli and Clifford-like shadows by tuning the amount of entanglement in the measurement basis, offering a flexible trade-off between performance and generality.
• The framework enables a new class of state-estimation protocols that are both simple and powerful, with potential for implementation on near-term quantum devices.