Quantum algorithms for learning hidden strings with applications to matroid problems

Xiaowei Huang, Shihao Zhang, Lvzhou Li

Abstract: In this paper, we explore quantum speedups for the problem, inspired by matroid theory, of learning a pair of n-bit binary strings that are promised to have the same number of 1s and differ in exactly two bits, by using the max inner product oracle and the sub-set oracle. More specifically, given two string s,s′∈{0,1}n<math><mi is="true">s</mi><mo is="true">,</mo><msup is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup><mo is="true">∈</mo><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> satisfying the above constraints, for any x∈{0,1}n<math><mi is="true">x</mi><mo is="true">∈</mo><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> the max inner product oracle Omax(x)<math><msub is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mi is="true">m</mi><mi is="true">a</mi><mi is="true">x</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">x</mi><mo stretchy="false" is="true">)</mo></math> returns the max value between s⋅x<math><mi is="true">s</mi><mo is="true">⋅</mo><mi is="true">x</mi></math> and s′⋅x<math><msup is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup><mo is="true">⋅</mo><mi is="true">x</mi></math>, and the sub-set oracle Osub(x)<math><msub is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mi is="true">s</mi><mi is="true">u</mi><mi is="true">b</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">x</mi><mo stretchy="false" is="true">)</mo></math> indicates whether the index set of the 1s in x is a subset of that in s or s′<math><msup is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup></math>. We present an exact quantum algorithm consuming O(1)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></math> queries to the max inner product oracle for identifying the pair {s,s′}<math><mo stretchy="false" is="true">{</mo><mi is="true">s</mi><mo is="true">,</mo><msup is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup><mo stretchy="false" is="true">}</mo></math>, and prove that any randomized algorithm requires Ω(n/log2⁡n)<math><mi mathvariant="normal" is="true">Ω</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo stretchy="false" is="true">/</mo><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">log</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mo is="true">⁡</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> queries. Also, we present a quantum algorithm consuming n2+O(n)<math><mfrac is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msqrt is="true"><mrow is="true"><mi is="true">n</mi></mrow></msqrt><mo stretchy="false" is="true">)</mo></math> queries to the subset oracle, and prove that any randomized algorithm requires at least n−3+Ω(log2⁡n)<math><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">3</mn><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi mathvariant="normal" is="true">Ω</mi><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">log</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mo is="true">⁡</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> queries. Therefore, quantum speedups are revealed in the two oracle models. Furthermore, the above results are applied to the problem in matroid theory of finding all the bases of a 2-bases matroid, where a matroid is called k-bases if it has k bases.