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==== Deutsch-Jozsa algorithm ====
The [[Deutsch-Jozsa algorithm]] is a quantum algorithm designed to solve a toy problem with a smaller query complexity than is possible with a classical algorithm. The toy problem asks whether a function <math>f:\{0,1\}^n\rightarrow\{0,1\}</math> is constant or balanced, those being the only two possibilities.<ref name=":32"/> The only way to evaluate the function <math>f</math> is to consult a [[black box]] or [[Oracle machine|oracle]]. A classical [[deterministic algorithm]] will have to check more than half of the possible inputs to be sure of whether or not the function is constant or balanced. With <math>2^n</math> possible inputs, the query complexity of the most efficient classical deterministic algorithm is <math>2^{n-1}+1</math>.<ref name=":32" /> The Deutsch-Jozsa algorithm takes advantage of quantum parallelism to check all of the elements of the ___domain at once and only needs to query the oracle once, making its query complexity <math>1</math>.<ref name=":32" />
==Other theories of quantum physics==
It has been speculated that further advances in physics could lead to even faster computers. For instance, it has been shown that a non-local hidden variable quantum computer based on [[De Broglie–Bohm theory|Bohmian Mechanics]] could implement a search of an {{mvar|N}}-item database in at most <math>O(\sqrt[3]{N})</math> steps, a slight speedup over [[Grover's algorithm]], which runs in <math>O(\sqrt{N})</math> steps. Note, however, that neither search method would allow quantum computers to solve [[NP-complete]] problems in polynomial time.<ref name="auto">{{cite web |title=Quantum Computing and Hidden Variables |last=Aaronson |first=Scott |url=http://www.scottaaronson.com/papers/qchvpra.pdf}}</ref> Theories of [[quantum gravity]], such as [[M-theory]] and [[loop quantum gravity]], may allow even faster computers to be built. However, defining computation in these theories is an open problem due to the [[problem of time]]; that is, within these physical theories there is currently no obvious way to describe what it means for an observer to submit input to a computer at one point in time and then receive output at a later point in time.<ref>{{Cite journal |first=Scott |last=Aaronson |title=NP-complete Problems and Physical Reality |journal=ACM SIGACT News |volume=2005 |arxiv=quant-ph/0502072 |year=2005 |bibcode=2005quant.ph..2072A |author-link=Scott Aaronson}} See section 7 "Quantum Gravity": "[...] to anyone who wants a test or benchmark for a favorite quantum gravity theory,[author's footnote: That is, one without all the bother of making numerical predictions and comparing them to observation] let me humbly propose the following: ''can you define Quantum Gravity Polynomial-Time?'' [...] until we can say what it means for a 'user' to specify an 'input' and 'later' receive an 'output'—''there is no such thing as computation, not even theoretically.''" (emphasis in original)</ref><ref>{{cite web |url=http://www.dwavesys.com/en/pressreleases.html#lm_2011 |title=D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation |access-date=30 May 2011 |date=25 May 2011 |publisher=D-Wave |archive-date=22 December 2020 |archive-url=https://web.archive.org/web/20201222041457/https://www.dwavesys.com/en/pressreleases.html#lm_2011 |url-status=dead }}</ref>
==See also==
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