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We prove that this belief is overly conservative by constructing a constant quantum-depth circuit for the task, inspired by the method of Shor error correction. Furthermore, our circuit demands only local gates in a two dimensional circuit \u2013 we show how to implement it in a highly parallelized way on an architecture similar to that of Google's S<\/mml:mi>y<\/mml:mi>c<\/mml:mi>a<\/mml:mi>m<\/mml:mi>o<\/mml:mi>r<\/mml:mi>e<\/mml:mi><\/mml:math> processor. With these features, our algorithm brings the central task of multivariate trace estimation closer to the capabilities of near-term quantum processors. We instantiate the latter application with a theorem on estimating nonlinear functions of quantum states with \"well-behaved\" polynomial approximations.<\/jats:p>","DOI":"10.22331\/q-2024-01-10-1220","type":"journal-article","created":{"date-parts":[[2024,1,10]],"date-time":"2024-01-10T13:03:23Z","timestamp":1704891803000},"page":"1220","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":30,"title":["Multivariate trace estimation in constant quantum depth"],"prefix":"10.22331","volume":"8","author":[{"given":"Yihui","family":"Quek","sequence":"first","affiliation":[{"name":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139"},{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"},{"name":"Information Systems Laboratory, Stanford University, Palo Alto, CA 94305, USA"}]},{"given":"Eneet","family":"Kaur","sequence":"additional","affiliation":[{"name":"Cisco Quantum Lab, Los Angeles, USA"},{"name":"Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"}]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[{"name":"School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14850, USA"},{"name":"Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA"}]}],"member":"9598","published-online":{"date-parts":[[2024,1,10]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Artur K. 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