{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T07:02:37Z","timestamp":1771830157813,"version":"3.50.1"},"reference-count":26,"publisher":"Walter de Gruyter GmbH","issue":"4","license":[{"start":{"date-parts":[[2025,8,1]],"date-time":"2025-08-01T00:00:00Z","timestamp":1754006400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,8,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In this tutorial, which contains some original results, we bridge the fields of quantum computing algorithms, conservation laws, and many-body quantum systems by examining three algorithms for searching an unordered database of size\n                    <jats:italic>N<\/jats:italic>\n                    using a continuous-time quantum walk, which is the quantum analogue of a continuous-time random walk. The first algorithm uses a linear quantum walk, and we apply elementary calculus to show that the success probability of the algorithm reaches 1 when the jumping rate of the walk takes some critical value. We show that the expected value of its Hamiltonian\n                    <jats:italic>H<\/jats:italic>\n                    <jats:sub>0<\/jats:sub>\n                    is conserved. The second algorithm uses a nonlinear quantum walk with effective Hamiltonian\n                    <jats:italic>H<\/jats:italic>\n                    (\n                    <jats:italic>t<\/jats:italic>\n                    ) =\n                    <jats:italic>H<\/jats:italic>\n                    <jats:sub>0<\/jats:sub>\n                    +\n                    <jats:italic>\u03bb<\/jats:italic>\n                    |\n                    <jats:italic>\u03c8<\/jats:italic>\n                    |\n                    <jats:sup>2<\/jats:sup>\n                    , which arises in the Gross-Pitaevskii equation describing Bose-Einstein condensates. When the interactions between the bosons are repulsive,\n                    <jats:italic>\u03bb<\/jats:italic>\n                    &gt; 0, and there exists a range of fixed jumping rates such that the success probability reaches 1 with the same asymptotic runtime of the linear algorithm, but with a larger multiplicative constant. Rather than the effective Hamiltonian, we show that the expected value of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_qic-2025-0017_ieq_001.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                          <m:msub>\n                            <m:mi>H<\/m:mi>\n                            <m:mn>0<\/m:mn>\n                          <\/m:msub>\n                          <m:mo>+<\/m:mo>\n                          <m:mfrac>\n                            <m:mn>1<\/m:mn>\n                            <m:mn>2<\/m:mn>\n                          <\/m:mfrac>\n                          <m:mi>\u03bb<\/m:mi>\n                          <m:mo>|<\/m:mo>\n                          <m:mi>\u03c8<\/m:mi>\n                          <m:msup>\n                            <m:mo>|<\/m:mo>\n                            <m:mn>2<\/m:mn>\n                          <\/m:msup>\n                        <\/m:math>\n                        <jats:tex-math>{H_0} + {1 \\over 2}\\lambda |\\psi {|^2}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is conserved. The third algorithm utilizes attractive interactions, corresponding to\n                    <jats:italic>\u03bb<\/jats:italic>\n                    &lt; 0. In this case, there is a time-varying critical function for the jumping rate\n                    <jats:italic>\n                      \u03b3\n                      <jats:sub>c<\/jats:sub>\n                    <\/jats:italic>\n                    (\n                    <jats:italic>t<\/jats:italic>\n                    ) that causes the success probability to reach 1 more quickly than in the other two algorithms, and we show that the expected value of\n                    <jats:italic>H<\/jats:italic>\n                    (\n                    <jats:italic>t<\/jats:italic>\n                    )\/[\n                    <jats:italic>\n                      \u03b3\n                      <jats:sub>c<\/jats:sub>\n                    <\/jats:italic>\n                    (\n                    <jats:italic>t<\/jats:italic>\n                    )\n                    <jats:italic>N<\/jats:italic>\n                    ] is conserved.\n                  <\/jats:p>","DOI":"10.2478\/qic-2025-0017","type":"journal-article","created":{"date-parts":[[2025,8,22]],"date-time":"2025-08-22T13:59:50Z","timestamp":1755871190000},"page":"315-328","source":"Crossref","is-referenced-by-count":0,"title":["Conserved Quantities in Linear and Nonlinear Quantum Search"],"prefix":"10.2478","volume":"25","author":[{"given":"David A.","family":"Meyer","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of California , San Diego , 9500 Gilman Drive , La Jolla , CA , USA"}]},{"given":"Thomas G.","family":"Wong","sequence":"additional","affiliation":[{"name":"Department of Physics, Creighton University , 2500 California Plaza , Omaha , NE , USA"}]}],"member":"374","published-online":{"date-parts":[[2025,8,22]]},"reference":[{"key":"2026022306090191800_j_qic-2025-0017_ref_001","doi-asserted-by":"crossref","unstructured":"D.J. 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