Optimal tradeoffs for estimating Pauli observables (2404.19105v2)

Abstract: We revisit the problem of Pauli shadow tomography: given copies of an unknown $n$-qubit quantum state $\rho$, estimate $\text{tr}(P\rho)$ for some set of Pauli operators $P$ to within additive error $\epsilon$. This has been a popular testbed for exploring the advantage of protocols with quantum memory over those without: with enough memory to measure two copies at a time, one can use Bell sampling to estimate $|\text{tr}(P\rho)|$ for all $P$ using $O(n/\epsilon^4)$ copies, but with $k\le n$ qubits of memory, $\Omega(2^{(n-k)/3})$ copies are needed. These results leave open several natural questions. How does this picture change in the physically relevant setting where one only needs to estimate a certain subset of Paulis? What is the optimal dependence on $\epsilon$? What is the optimal tradeoff between quantum memory and sample complexity? We answer all of these questions. For any subset $A$ of Paulis and any family of measurement strategies, we completely characterize the optimal sample complexity, up to $\log |A|$ factors. We show any protocol that makes $\text{poly}(n)$-copy measurements must make $\Omega(1/\epsilon^4)$ measurements. For any protocol that makes $\text{poly}(n)$-copy measurements and only has $k < n$ qubits of memory, we show that $\widetilde{\Theta}(\min{2^n/\epsilon^2, 2^{n-k}/\epsilon^4})$ copies are necessary and sufficient. The protocols we propose can also estimate the actual values $\text{tr}(P\rho)$, rather than just their absolute values as in prior work. Additionally, as a byproduct of our techniques, we establish tight bounds for the task of purity testing and show that it exhibits an intriguing phase transition not present in the memory-sample tradeoff for Pauli shadow tomography.

• The paper establishes a framework that quantifies the tradeoff between quantum memory usage and sample complexity in estimating Pauli observables.
• It derives tight bounds on sample complexity for arbitrary subsets of Pauli operators, achieving scaling that interpolates between 1/√ and 1/⁴ error regimes.
• The study extends protocols to estimate exact expectation values and sets rigorous limits for quantum state purity testing in practical quantum measurements.

Overview of "Optimal tradeoffs for estimating Pauli observables"

The paper "Optimal tradeoffs for estimating Pauli observables" by Chen, Gong, and Ye addresses the problem of Pauli shadow tomography, a quantum measurement task central to extracting key properties from unknown quantum states. Specifically, it involves estimating the expectation values of Pauli operators for a given quantum state. The paper explores the intricacies of striking an optimal balance between the resources of quantum memory and sample complexity in this context, focusing on the efficiency achievable with and without quantum memory.

Main Contributions

The authors propose a thorough investigation into the resource tradeoffs for estimating expectation values $tr(P\rho)$ given multiple copies of an unknown $n$-qubit quantum state $\rho$. Several pivotal questions are addressed:

1. Subset-dependence and Sample Complexity: They propose an efficient characterization of the sample complexity for any subset $A$ of Pauli operators. Through rigorous mathematical frameworks, they derive bounds up to factors of $\log|A|$, encapsulating the dependence of complexity on the structure of $A$.
2. Memory Constraints and $c$-copy Measurements: They derive that for single-copy measurements, an algorithm must incur an additive error scaling of $1/^2$. However, for protocols leveraging multiple copies ($c = 1$) or utilizing bounded quantum memory ($k < n$ qubits), their results provide a novel characterization of sample complexity that interpolates between the $1/^2$ scaling and $1/^4$ scaling depending on the memory and copy constraints.
3. Quantum Memory and Sample Tradeoff: They identify a crucial tradeoff between the quantum memory available and the sample complexity involved. This tradeoff is smoothly characterized through $\widetilde{\Theta}(\min\{2^n/^2, 2^{n-k}/^4\})$, giving rise to efficient quantum strategies depending solely on the number of qubits accessible.
4. Learning Protocol and Absolute Values of Observables: A significant innovation is their framework extending beyond prior absolute value estimations to enable the estimation of actual values $tr(P\rho)$. This enhancement improves the practical applicability of quantum measurement protocols when the full information about quantum states is necessary.
5. Purity Testing Paradigms: As a corollary of their techniques, the paper establishes stringent bounds for purity testing of quantum states, providing insights into scenarios where minimal state overlap results in stark phase transitions in measurement complexities.

Implications and Future Directions

The results present significant advancements in understanding the efficacy of quantum measurements under different physical and algorithmic constraints. This work lays down the theoretical foundation for developing more efficient quantum measurement protocols that can operate at the boundary of quantum resource capabilities.

The authors' numerical results and theoretical confirmations open pathways for experimental validation where quantum memory resources are a primary concern, aligning closely with near-term quantum devices. The stereo interplay between memory and sample complexity offers strategic insights critical to quantum computing applications such as quantum chemistry, where estimations of particular observables underpin critical computations.

Furthermore, the exploration on obliviousness in quantum protocols suggests intriguing implications for quantum information compression and storage, through characterizing limits on state retrievability from compressed quantum data.

Conclusion

Overall, this paper offers robust mathematical structures and hypotheses that advance both theoretical understanding and practical implementations of quantum state measurements. It deeply explores quantum measurement's frontier in terms of resource optimization, a cardinal line of inquiry governing the feasibility of future quantum technologies. The framework outlined provides a baseline for more sophisticated explorations into quantum compute-task efficiencies, imperative for sustaining quantum technology's upward trajectory. The lessons gleaned herein are essential steps toward a paradigm where optimal quantum resource utilization dictates the future of quantum innovation and practical applications.