Characterization of exact two-query quantum algorithms

Shaoliang Ye, Wei Yang, Liusheng Huang

Abstract: Quantum query model is a crucial model for quantum computing, where one query to some input variable of a Boolean function f defined on {0,1}n<math><msup is="true"><mrow is="true"><mo stretchy="false" is="true">{</mo><mn is="true">0</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">}</mo></mrow><mrow is="true"><mi is="true">n</mi></mrow></msup></math> returns the variable value. The exact query complexity, denoted as QE(f)<math><msub is="true"><mrow is="true"><mi is="true">Q</mi></mrow><mrow is="true"><mi is="true">E</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo></math>, is defined to be the minimum number of queries required to determine the function value. An important problem in this area is to give a succinct characterization of a k-query exact quantum algorithm for an arbitrary k. To date, the cases k=1<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">1</mn></math> and k=n<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mi is="true">n</mi></math> are already solved and the case k=2<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">2</mn></math> remains unknown. Our result is that there are 27 nondegenerate Boolean functions up to isomorphism with QE(f)<math><msub is="true"><mrow is="true"><mi is="true">Q</mi></mrow><mrow is="true"><mi is="true">E</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">f</mi><mo stretchy="false" is="true">)</mo></math> being two, among which only two functions can be solved by a 2-query classical algorithm. The input bit number n of the above 27 functions ranges from 2 to 6, where the case n≤3<math><mi is="true">n</mi><mo is="true">≤</mo><mn is="true">3</mn></math> is already proved and the case n=4<math><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">4</mn></math> is already found by numerically solving semidefinite programming, which is a complete characterization of quantum query algorithm. Assuming the correctness of the numerical result for n=4<math><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">4</mn></math>, we prove that there are four functions in the case n=5<math><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">5</mn></math>, one in the case n=6<math><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">6</mn></math> and none in the case n≥7<math><mi is="true">n</mi><mo is="true">≥</mo><mn is="true">7</mn></math>. We further show that the 25 functions for which quantum algorithm has advantage over classical algorithm contain essentially only four different structures.