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Improved quantum data analysis
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abstract
We provide more sample-efficient versions of some basic routines in quantum data analysis, along with simpler proofs. Particularly, we give a quantum "Threshold Search" algorithm that requires only $O((\log^2 m)/\epsilon^2)$ samples of a $d$-dimensional state $\rho$. That is, given observables $0 \le A_1, A_2, ..., A_m \le 1$ such that $\mathrm{tr}(\rho A_i) \ge 1/2$ for at least one $i$, the algorithm finds $j$ with $\mathrm{tr}(\rho A_j) \ge 1/2-\epsilon$. As a consequence, we obtain a Shadow Tomography algorithm requiring only $\tilde{O}((\log^2 m)(\log d)/\epsilon^4)$ samples, which simultaneously achieves the best known dependence on each parameter $m$, $d$, $\epsilon$. This yields the same sample complexity for quantum Hypothesis Selection among $m$ states; we also give an alternative Hypothesis Selection method using $\tilde{O}((\log^3 m)/\epsilon^2)$ samples.
Forward citations
Cited by 3 Pith papers
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Dimension-Free Polylogarithmic Quantum Shadow Tomography from Sequential Pretty-Good Measurements
New sequential pretty-good measurement protocol achieves dimension-free shadow tomography with sample complexity O(1/eps^2 * (log(M/delta))^4 / (log log(M/delta))^3).
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High-rate qLDPC processors
Non-abelian "mitten" qLDPC codes achieve 20% encoding rate with distances 10-24 on 150-975 qubits, and simulations indicate fault-tolerant processors sustaining ~10^10 logical operations at 0.1% physical error rate.
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Information-Computation Gaps in Quantum Learning via Low-Degree Likelihood
A quantum extension of the low-degree method shows that state designs imply computational hardness for many single-copy quantum measurement strategies, yielding new information-computation gaps.
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