Optimal tradeoffs for estimating Pauli observables

TL;DR

For any subset of Paulis and any family of measurement strategies, the optimal sample complexity is completely characterize, up to <tex>$\log\vert A\vert$</tex> factors.

Abstract

We revisit the problem of Pauli shadow tomography: given copies of an unknown $n$-qubit quantum state $ρ$, estimate $\text{tr}(Pρ)$ for some set of Pauli operators $P$ to within additive error $ε$. 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 $|\text{tr}(Pρ)|$ for all $P$ using $O(n/ε^4)$ copies, but with $k\le n$ qubits of memory, $Ω(2^{(n-k)/3})$ 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 $ε$? 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 $Ω(1/ε^4)$ measurements. For any protocol that makes $\text{poly}(n)$-copy measurements and only has $k < n$ qubits of memory, we show that $\widetildeΘ(\min\{2^n/ε^2, 2^{n-k}/ε^4\})$ copies are necessary and sufficient. The protocols we propose can also estimate the actual values $\text{tr}(Pρ)$, 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.

Figures (7)

• Figure 1: Comparison of bounds in \ref{['thm:PauliShadowAllK']} and \ref{['thm:Purity']}, and chen2022exponential for shadow tomography and purity testing.
• Figure : (a) Learning without quantum memory
• Figure : (a) Learning without quantum memory
• Figure : (c) Learning with $c$-copy measurements
• Figure : (e) $(c, \pazocal{M})$ learning protocols

Theorems & Definitions (76)

• Theorem 1: Master theorem
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5: Informal, see Theorem \ref{['thm:ccopyLower']}
• Theorem 6
• Theorem 7
• Lemma 1: E.g., Lemma 4.10 in chen2022exponential
• Lemma 2
• Lemma 3: See, e.g., meleIntroductionHaarMeasure2023