{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T20:37:48Z","timestamp":1778013468844,"version":"3.51.4"},"reference-count":68,"publisher":"American Association for the Advancement of Science (AAAS)","content-domain":{"domain":["spj.science.org"],"crossmark-restriction":true},"short-container-title":["Intell Comput"],"published-print":{"date-parts":[[2023,1]]},"abstract":"<jats:p>\n            The dynamic mode decomposition (DMD) algorithm is a widely used factorization and dimensionality reduction technique in time series analysis. When analyzing high-dimensional time series, the DMD algorithm requires extremely large amounts of computational power. To accelerate the DMD algorithm, we propose a quantum-classical hybrid algorithm that we call the quantum dynamic mode decomposition (QDMD) algorithm. Given a time series\n            <jats:italic>X<\/jats:italic>\n            \u00a0\u2208\u00a0\n            <jats:italic>R<\/jats:italic>\n            <jats:sup>\n              <jats:italic>n<\/jats:italic>\n              \u00a0\u00d7\u00a0(\n              <jats:italic>m<\/jats:italic>\n              \u00a0+\u00a01)\n            <\/jats:sup>\n            with\n            <jats:italic>n<\/jats:italic>\n            \u00a0\u226b\u00a0\n            <jats:italic>m<\/jats:italic>\n            , the QDMD algorithm first executes quantum singular value decomposition on a matrix related to\n            <jats:italic>X<\/jats:italic>\n            and obtains a quantum state containing the main singular values and singular vectors of the decomposed matrix, then performs a low-sampling-frequency process on the obtained quantum state and computes the low-dimensional projection of the DMD operator through the sampling results. Finally, the algorithm computes the DMD eigenvalues and prepares the amplitude-encoding states of the DMD modes using the obtained classical information and\n            <jats:italic>X<\/jats:italic>\n            . Considering the main variables, the complexity of the QDMD algorithm is\n            <jats:inline-formula>\n              <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                <mml:mover accent=\"true\">\n                  <mml:mi>O<\/mml:mi>\n                  <mml:mo stretchy=\"true\">~<\/mml:mo>\n                <\/mml:mover>\n                <mml:mfenced close=\")\" open=\"(\">\n                  <mml:mrow>\n                    <mml:mi>M<\/mml:mi>\n                    <mml:msqrt>\n                      <mml:mi>m<\/mml:mi>\n                    <\/mml:msqrt>\n                    <mml:mtext>polylog<\/mml:mtext>\n                    <mml:mfenced close=\")\" open=\"(\">\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:mfenced>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mi>\u03f5<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:mfenced>\n              <\/mml:math>\n            <\/jats:inline-formula>\n            , where\n            <jats:inline-formula>\n              <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                <mml:mi>M<\/mml:mi>\n                <mml:mo>=<\/mml:mo>\n                <mml:mover accent=\"true\">\n                  <mml:mi>O<\/mml:mi>\n                  <mml:mo stretchy=\"true\">~<\/mml:mo>\n                <\/mml:mover>\n                <mml:mfenced close=\")\" open=\"(\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mi>m<\/mml:mi>\n                      <mml:mn>3<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>\u03f5<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:mfenced>\n              <\/mml:math>\n            <\/jats:inline-formula>\n            denotes the number of samples. Compared with the classical DMD algorithm, which has complexity\n            <jats:inline-formula>\n              <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                <mml:mover accent=\"true\">\n                  <mml:mi>O<\/mml:mi>\n                  <mml:mo stretchy=\"true\">~<\/mml:mo>\n                <\/mml:mover>\n                <mml:mfenced close=\")\" open=\"(\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:msup>\n                      <mml:mi>m<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mtext>log<\/mml:mtext>\n                    <mml:mfenced close=\")\" open=\"(\">\n                      <mml:mrow>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mo>\/<\/mml:mo>\n                        <mml:mi>\u03f5<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:mfenced>\n                  <\/mml:mrow>\n                <\/mml:mfenced>\n              <\/mml:math>\n            <\/jats:inline-formula>\n            , the QDMD algorithm provides an exponential acceleration of\n            <jats:italic>n<\/jats:italic>\n            , at the cost of greater dependence on\n            <jats:italic>M<\/jats:italic>\n            and\n            <jats:italic>\u03f5<\/jats:italic>\n            . We test the effects of\n            <jats:italic>M<\/jats:italic>\n            on the QDMD algorithm in the specific application scenarios of data denoising, scene background extraction, and fluid dynamics analysis. We determined that the QDMD algorithm requires only a small number of samples\n            <jats:italic>M<\/jats:italic>\n            in specific applications, further demonstrating the quantum advantage of the QDMD algorithm in high-dimensional data analysis.\n          <\/jats:p>","DOI":"10.34133\/icomputing.0045","type":"journal-article","created":{"date-parts":[[2023,7,4]],"date-time":"2023-07-04T16:28:02Z","timestamp":1688488082000},"update-policy":"https:\/\/doi.org\/10.34133\/aaas_crossmark_01","source":"Crossref","is-referenced-by-count":5,"title":["Quantum Dynamic Mode Decomposition Algorithm for High-Dimensional Time Series Analysis"],"prefix":"10.34133","volume":"2","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2207-9998","authenticated-orcid":true,"given":"Cheng","family":"Xue","sequence":"first","affiliation":[{"name":"Institute of Artificial Intelligence, Hefei Comprehensive National Science Center, Hefei, Anhui 230026, P. R. China."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhao-Yun","family":"Chen","sequence":"additional","affiliation":[{"name":"Institute of Artificial Intelligence, Hefei Comprehensive National Science Center, Hefei, Anhui 230026, P. R. China."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tai-Ping","family":"Sun","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China."},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China."},{"name":"Hefei National Laboratory, Hefei, Anhui 230088, P. R. China."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiao-Fan","family":"Xu","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China."},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China."},{"name":"Hefei National Laboratory, Hefei, Anhui 230088, P. R. China."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Si-Ming","family":"Chen","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China."},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China."},{"name":"Hefei National Laboratory, Hefei, Anhui 230088, P. R. China."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huan-Yu","family":"Liu","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China."},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China."},{"name":"Hefei National Laboratory, Hefei, Anhui 230088, P. R. China."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xi-Ning","family":"Zhuang","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China."},{"name":"CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China."},{"name":"Hefei National Laboratory, Hefei, Anhui 230088, P. R. 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