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Unfortunately, this tradeoff still limits the size of tractable problems since the increased depth is often not realizable before noise dominates. Here, we develop <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">qubit-efficient<\/mml:mtext><\/mml:mrow><\/mml:math> quantum algorithms for entanglement spectroscopy which avoid this tradeoff. In particular, we develop algorithms for computing the trace of the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><\/mml:math>-th power of the density operator of a quantum system, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>T<\/mml:mi><mml:mi>r<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mi>\u03c1<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, (related to the R\u00e9nyi entropy of order <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><\/mml:math>) that use fewer qubits than any previous efficient algorithm while achieving similar performance in the presence of noise, thus enabling spectroscopy of larger quantum systems on NISQ devices. Our algorithms, which require a number of qubits independent of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><\/mml:math>, are variants of previous algorithms with width proportional to <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><\/mml:math>, an asymptotic difference. The crucial ingredient in these new algorithms is the ability to measure and reinitialize subsets of qubits in the course of the computation, allowing us to reuse qubits and increase the circuit depth without suffering the usual noisy consequences. We also introduce the notion of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">effective circuit depth<\/mml:mtext><\/mml:mrow><\/mml:math> as a generalization of standard circuit depth suitable for circuits with qubit resets. This tool helps explain the noise-resilience of our qubit-efficient algorithms and should aid in designing future algorithms. We perform numerical simulations to compare our algorithms to the original variants and show they perform similarly when subjected to noise. Additionally, we experimentally implement one of our qubit-efficient algorithms on the Honeywell System Model H0, estimating <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>T<\/mml:mi><mml:mi>r<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mi>\u03c1<\/mml:mi><mml:mi>n<\/mml:mi><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> for larger n than possible with previous algorithms.<\/jats:p>","DOI":"10.22331\/q-2021-09-02-535","type":"journal-article","created":{"date-parts":[[2021,9,2]],"date-time":"2021-09-02T13:36:53Z","timestamp":1630589813000},"page":"535","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":25,"title":["Qubit-efficient entanglement spectroscopy using qubit resets"],"prefix":"10.22331","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6173-2465","authenticated-orcid":false,"given":"Justin","family":"Yirka","sequence":"first","affiliation":[{"name":"Department of Computer Science, The University of Texas at Austin, Austin, TX 78712, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1167-6527","authenticated-orcid":false,"given":"Yi\u011fit","family":"Suba\u015f\u0131","sequence":"additional","affiliation":[{"name":"Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]}],"member":"9598","published-online":{"date-parts":[[2021,9,2]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"N. Abdessaied, R. Wille, M. Soeken, and R. Drechsler. Reducing the depth of quantum circuits using additional circuit lines. In G.W. Dueck and D.M. Miller, editors, Reversible Computation. RC 2013., page 221\u2013233, Berlin, Heidelberg, 2013. Springer-Verlag. 10.1007\/978-3-642-38986-3_18.","DOI":"10.1007\/978-3-642-38986-3_18"},{"key":"1","unstructured":"H\u00e9ctor Abraham et al. Qiskit: An open-source framework for quantum computing, 2019. URL https:\/\/github.com\/Qiskit."},{"key":"2","doi-asserted-by":"publisher","unstructured":"Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. Entanglement in many-body systems. Rev. Mod. Phys., 80 (2): 517, 2008. 10.1103\/RevModPhys.80.517.","DOI":"10.1103\/RevModPhys.80.517"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Galit Anikeeva, Isaac H Kim, and Patrick Hayden. Recycling qubits in near-term quantum computers. 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