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<title>Hidden shift problem</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Hidden shift problem</span></span>

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<p>

In <a href="Quantum_computing" title="Quantum computing">quantum computing</a>, the <b>hidden shift problem</b> is a type of <a href="Oracle_machine" title="Oracle machine">oracle</a>-based problem. Various versions of this problem have quantum algorithms which can run much more quickly than known non-quantum methods for the same problem. In its general form, it is equivalent to the <a href="Hidden_subgroup_problem" title="Hidden subgroup problem">hidden subgroup problem</a> for the <a href="Dihedral_group" title="Dihedral group">dihedral group</a>.<sup id="cite_ref-CvD05_1-0" class="reference"><a href="#cite_note-CvD05-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It is a major open problem to understand how well quantum algorithms can perform for this task, as it can be applied to break <a href="Lattice-based_cryptography" title="Lattice-based cryptography">lattice-based cryptography</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Problem_statement">Problem statement</h2></div>

<p>The hidden shift problem states: Given an <a href="Oracle_machine" title="Oracle machine">oracle</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}">

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<annotation encoding="application/x-tex">{\displaystyle O}</annotation>

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</math></span><img src="./9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}" loading="lazy"></span> that encodes two functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>, there is an <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-bit string <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math></span><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span> for which <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=f(x+s)}">

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<annotation encoding="application/x-tex">{\displaystyle g(x)=f(x+s)}</annotation>

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</math></span><img src="./cdfeb1a40d437a17fc81c5736e5bccdb5fba4bb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.702ex; height:2.843ex;" alt="{\displaystyle g(x)=f(x+s)}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>. Find <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math></span><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span>.<sup id="cite_ref-WH02_4-0" class="reference"><a href="#cite_note-WH02-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>Functions such as the <a href="Legendre_symbol" title="Legendre symbol">Legendre symbol</a> and <a href="Bent_function" title="Bent function">bent functions</a>, satisfy these constraints.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Algorithms">Algorithms</h2></div>

<p>With a <a href="Quantum_algorithm" title="Quantum algorithm">quantum algorithm</a> that is defined as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |s\rangle =H^{\otimes n}O_{f}H^{\otimes n}O_{\hat {g}}H^{\otimes n}|0^{n}\rangle }">

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle |s\rangle =H^{\otimes n}O_{f}H^{\otimes n}O_{\hat {g}}H^{\otimes n}|0^{n}\rangle }</annotation>

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</math></span><img src="./c47cedb73f7e86a8755ece9281f02325a58046c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:29.279ex; height:3.509ex;" alt="{\displaystyle |s\rangle =H^{\otimes n}O_{f}H^{\otimes n}O_{\hat {g}}H^{\otimes n}|0^{n}\rangle }" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> is the <a href="Hadamard_gate" class="mw-redirect" title="Hadamard gate">Hadamard gate</a> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {g}}}">

<mover>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {g}}}</annotation>

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</math></span><img src="./9351259e90225fb86e6d884862bd7ccc579db8ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.232ex; height:2.509ex;" alt="{\displaystyle {\hat {g}}}" loading="lazy"></span> is the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>, certain instantiations of this problem can be solved in a polynomial number of queries to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle O}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFChildsvan_Dam2007" class="citation cs2">Childs, Andrew M.; van Dam, Wim (2007), <a rel="nofollow" class="external text" href="https://dl.acm.org/citation.cfm?id=1283383.1283515">"Quantum algorithm for a generalized hidden shift problem"</a>, in Bansal, Nikhil; Pruhs, Kirk; Stein, Clifford (eds.), <i>Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, Louisiana, USA, January 7-9, 2007</i>, SIAM, pp. <span class="nowrap">1225–</span>1232, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0507190">quant-ph/0507190</a></span></cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFLomont2004" class="citation cs2">Lomont, Chris (November 4, 2004), <i>The Hidden Subgroup Problem - Review and Open Problems</i>, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0411037">quant-ph/0411037</a></span></cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFRegev2004" class="citation journal cs1">Regev, Oded (January 2004). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://epubs.siam.org/doi/10.1137/S0097539703440678">"Quantum Computation and Lattice Problems"</a></span>. <i>SIAM Journal on Computing</i>. <b>33</b> (3): <span class="nowrap">738–</span>760. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0097539703440678">10.1137/S0097539703440678</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0097-5397">0097-5397</a>.</cite></span>

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<cite id="CITEREFDamHallgrenIp2002" class="citation journal cs1">Dam, Wim van; Hallgren, Sean; Ip, Lawrence (2002). "Quantum Algorithms for some Hidden Shift Problems". <i><a href="SIAM_Journal_on_Computing" title="SIAM Journal on Computing">SIAM Journal on Computing</a></i>. <b>36</b> (3): <span class="nowrap">763–</span>778. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0211140">quant-ph/0211140</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS009753970343141X">10.1137/S009753970343141X</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11122780">11122780</a>.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFRötteler2008" class="citation book cs1">Rötteler, Martin (2008). "Quantum algorithms for highly non-linear Boolean functions". <i>Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms</i>. Vol. 402. <a href="Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>. pp. <span class="nowrap">448–</span>457. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0811.3208">0811.3208</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9781611973075.37">10.1137/1.9781611973075.37</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-89871-701-3</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9615826">9615826</a>.</cite></span>

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