Performance analysis of multi-shot shadow estimation
Shadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for $K$ times after the same unitary evolution, and repeats this procedure for...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2023-06-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2023-06-29-1044/pdf/ |
| _version_ | 1797792281296633856 |
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| author | You Zhou Qing Liu |
| author_facet | You Zhou Qing Liu |
| author_sort | You Zhou |
| collection | DOAJ |
| description | Shadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for $K$ times after the same unitary evolution, and repeats this procedure for $M$ rounds of random sampled unitary. As a result, there are $MK$ times measurements in total. Here we analyze the performance of shadow estimation in this multi-shot scenario, which is characterized by the variance of estimating the expectation value of some observable $O$. We find that in addition to the shadow-norm $\|O \|_{\mathrm{shadow}}$ introduced in \cite{huang2020predicting}, the variance is also related to another norm, and we denote it as the cross-shadow-norm $\|O \|_{\mathrm{Xshadow}}$. For both random Pauli and Clifford measurements, we analyze and show the upper bounds of $\|O \|_{\mathrm{Xshadow}}$. In particular, we figure out the exact variance formula for Pauli observable under random Pauli measurements. Our work gives theoretical guidance for the application of multi-shot shadow estimation. |
| first_indexed | 2024-03-13T02:31:58Z |
| format | Article |
| id | doaj.art-355a15c3b5c04fbc812e1e145bae8ab6 |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-03-13T02:31:58Z |
| publishDate | 2023-06-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-355a15c3b5c04fbc812e1e145bae8ab62023-06-29T14:01:52ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2023-06-017104410.22331/q-2023-06-29-104410.22331/q-2023-06-29-1044Performance analysis of multi-shot shadow estimationYou ZhouQing LiuShadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for $K$ times after the same unitary evolution, and repeats this procedure for $M$ rounds of random sampled unitary. As a result, there are $MK$ times measurements in total. Here we analyze the performance of shadow estimation in this multi-shot scenario, which is characterized by the variance of estimating the expectation value of some observable $O$. We find that in addition to the shadow-norm $\|O \|_{\mathrm{shadow}}$ introduced in \cite{huang2020predicting}, the variance is also related to another norm, and we denote it as the cross-shadow-norm $\|O \|_{\mathrm{Xshadow}}$. For both random Pauli and Clifford measurements, we analyze and show the upper bounds of $\|O \|_{\mathrm{Xshadow}}$. In particular, we figure out the exact variance formula for Pauli observable under random Pauli measurements. Our work gives theoretical guidance for the application of multi-shot shadow estimation.https://quantum-journal.org/papers/q-2023-06-29-1044/pdf/ |
| spellingShingle | You Zhou Qing Liu Performance analysis of multi-shot shadow estimation Quantum |
| title | Performance analysis of multi-shot shadow estimation |
| title_full | Performance analysis of multi-shot shadow estimation |
| title_fullStr | Performance analysis of multi-shot shadow estimation |
| title_full_unstemmed | Performance analysis of multi-shot shadow estimation |
| title_short | Performance analysis of multi-shot shadow estimation |
| title_sort | performance analysis of multi shot shadow estimation |
| url | https://quantum-journal.org/papers/q-2023-06-29-1044/pdf/ |
| work_keys_str_mv | AT youzhou performanceanalysisofmultishotshadowestimation AT qingliu performanceanalysisofmultishotshadowestimation |