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New approaches using subspaces generated by real-time evolution have shown efficiency in extracting eigenstate information, but the full capabilities of such approaches are still not understood. In recent work, we developed the variational quantum phase estimation (VQPE) method, a compact and efficient real-time algorithm to extract eigenvalues on quantum hardware. Here we build on that work by theoretically and numerically exploring a generalized Krylov scheme where the Krylov subspace is constructed through a parametrized real-time evolution, which applies to the VQPE algorithm as well as others. We establish an error bound that justifies the fast convergence of our spectral approximation. We also derive how the overlap with high energy eigenstates becomes suppressed from real-time subspace diagonalization and we visualize the process that shows the signature phase cancellations at specific eigenenergies. We investigate various algorithm implementations and consider performance when stochasticity is added to the target Hamiltonian in the form of spectral statistics. To demonstrate the practicality of such real-time evolution, we discuss its application to fundamental problems in quantum computation such as electronic structure predictions for strongly correlated systems.<\/jats:p>","DOI":"10.22331\/q-2023-07-25-1066","type":"journal-article","created":{"date-parts":[[2023,7,25]],"date-time":"2023-07-25T13:03:51Z","timestamp":1690290231000},"page":"1066","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":27,"title":["Real-Time Krylov Theory for Quantum Computing Algorithms"],"prefix":"10.22331","volume":"7","author":[{"given":"Yizhi","family":"Shen","sequence":"first","affiliation":[{"name":"NASA Ames Research Center, Moffett Field, CA 94035, USA"},{"name":"KBR, 601 Jefferson St., Houston, TX 77002"},{"name":"Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA"}]},{"given":"Katherine","family":"Klymko","sequence":"additional","affiliation":[{"name":"NERSC, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA"}]},{"given":"James","family":"Sud","sequence":"additional","affiliation":[{"name":"NASA Ames Research Center, Moffett Field, CA 94035, USA"},{"name":"USRA Research Institute for Advanced Computer Science, Mountain View, CA 94043, USA"}]},{"given":"David B.","family":"Williams-Young","sequence":"additional","affiliation":[{"name":"Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA"}]},{"given":"Wibe A. de","family":"Jong","sequence":"additional","affiliation":[{"name":"Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA"}]},{"given":"Norm M.","family":"Tubman","sequence":"additional","affiliation":[{"name":"NASA Ames Research Center, Moffett Field, CA 94035, USA"}]}],"member":"9598","published-online":{"date-parts":[[2023,7,25]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Bela Bauer, Dave Wecker, Andrew J. 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