On the sample complexity of purity and inner product estimation (2410.12712v1)

Abstract: We study the sample complexity of the prototypical tasks quantum purity estimation and quantum inner product estimation. In purity estimation, we are to estimate $tr(\rho^2)$ of an unknown quantum state $\rho$ to additive error $\epsilon$. Meanwhile, for quantum inner product estimation, Alice and Bob are to estimate $tr(\rho\sigma)$ to additive error $\epsilon$ given copies of unknown quantum state $\rho$ and $\sigma$ using classical communication and restricted quantum communication. In this paper, we show a strong connection between the sample complexity of purity estimation with bounded quantum memory and inner product estimation with bounded quantum communication and unentangled measurements. We propose a protocol that solves quantum inner product estimation with $k$-qubit one-way quantum communication and unentangled local measurements using $O(median{1/\epsilon^2,2^{n/2}/\epsilon,2^{n-k}/\epsilon^2})$ copies of $\rho$ and $\sigma$. Our protocol can be modified to estimate the purity of an unknown quantum state $\rho$ using $k$-qubit quantum memory with the same complexity. We prove that arbitrary protocols with $k$-qubit quantum memory that estimate purity to error $\epsilon$ require $\Omega(median{1/\epsilon^2,2^{n/2}/\sqrt{\epsilon},2^{n-k}/\epsilon^2})$ copies of $\rho$. This indicates the same lower bound for quantum inner product estimation with one-way $k$-qubit quantum communication and classical communication, and unentangled local measurements. For purity estimation, we further improve the lower bound to $\Omega(\max{1/\epsilon^2,2^{n/2}/\epsilon})$ for any protocols using an identical single-copy projection-valued measurement. Additionally, we investigate a decisional variant of quantum distributed inner product estimation without quantum communication for mixed state and provide a lower bound on the sample complexity.

• The paper introduces a protocol achieving an upper bound of O(med{1/ε², 2^(n/2)/ε, 2^(n−k)/ε²}) copies for inner product estimation that also adapts to purity estimation.
• The paper proves a lower bound of Ω(med{1/ε², 2^(n/2)/√ε, 2^(n−k)/ε²}) copies for purity estimation with k-qubit memory, highlighting fundamental resource limits.
• The paper reveals a crucial connection between purity and inner product estimation, suggesting that improvements in one task can inform more efficient protocols for quantum benchmarking.

Sample Complexity of Purity and Inner Product Estimation in Quantum Systems

This paper investigates the sample complexity in estimating two vital quantum metrics: quantum purity estimation and quantum inner product estimation. These estimations are critical for tasks such as quantum benchmarking and cross-platform verification. Quantum purity estimation focuses on determining the purity $\mathrm{tr}(\rho^2)$ of an unknown quantum state $\rho$, while quantum inner product estimation involves estimating $\mathrm{tr}(\rho\sigma)$ for unknown quantum states $\rho$ and $\sigma$. The paper identifies connections between these tasks in contexts of limited quantum memory and communication, and proposes a comprehensive analysis of their sample complexities.

Key Results

The paper presents the following findings:

1. Upper Bound on Sample Complexity: The protocol proposed for quantum inner product estimation utilizes $k$-qubit one-way quantum communication and unentangled measurements, requiring $$O(\mathrm{med}\{1/\epsilon^2, 2^{n/2}/\epsilon, 2^{n-k}/\epsilon^2\})$$ copies of $\rho$ and $\sigma$. This protocol can be adapted to estimate the purity of $\rho$ with the same sample complexity.
2. Lower Bound on Sample Complexity: It is proven that any protocol for purity estimation with $k$-qubit quantum memory requires $$\Omega(\mathrm{med}\{1/\epsilon^2, 2^{n/2}/\sqrt{\epsilon}, 2^{n-k}/\epsilon^2\})$$ copies. This indicates a similar lower bound for quantum inner product estimation with $k$-qubit one-way quantum communication.
3. Improved Lower Bound for Specific Measurements: For purity estimation using identical single-copy projection-valued measurements, the lower bound is elevated to $\Omega(\max\{1/\epsilon^2, 2^{n/2}/\epsilon\})$.
4. Connection Between Tasks: A crucial connection is established between the sample complexities of purity and inner product estimation, demonstrating that algorithms for one can inform solutions for the other under certain constraints.
5. Decisional Variant Investigation: A decisional variant of quantum distributed inner product estimation is also analyzed. Specifically, for mixed states without quantum communication, a lower bound on sample complexity is provided.

Implications

The findings have substantial implications in quantum information science:

• Resource Efficiency: Understanding sample complexity aids in developing efficient algorithms for quantum benchmarking and verification, crucial for the scalability of quantum technologies.
• Communication and Memory Constraints: The analysis emphasizes the impact of limited quantum memory and communication on state estimation tasks, guiding the design of protocols compatible with current quantum network capabilities.
• Theoretical Insights: The established bounds offer theoretical insights into the intrinsic difficulty of estimating quantum properties, contributing to complexity theory in quantum computing.

Future Directions

The results suggest several avenues for further research:

• Arbitrary Quantum Communication Models: Exploring sample complexity under models allowing more general quantum communication could yield tighter bounds.
• Refinement of Bound Gaps: Closing the gap between the upper and lower bounds, especially concerning the $\sqrt{\epsilon}$ factor, remains an open question.
• Broader Distributed Estimation Problems: Extending techniques to other quantum properties could provide a more general framework for distributed quantum estimation tasks.

By delineating the complex interplay between quantum memory, communication, and measurement, this paper sets the stage for future advancements in both theoretical and practical quantum computing contexts.