Quantum complexity theory: Difference between revisions

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One of the reasons quantum complexity theory is studied are the implications of quantum computing for the modern [[Church–Turing thesis|Church-Turing thesis]]. In short the modern Church-Turing thesis states that any computational model can be simulated in polynomial time with a [[probabilistic Turing machine]].<ref name=":02">{{Cite journal|last=Vazirani|first=Umesh V.|date=2002|title=A survey of quantum complexity theory|url=http://dx.doi.org/10.1090/psapm/058/1922899|journal=Proceedings of Symposia in Applied Mathematics|volume=58|pages=193–217|doi=10.1090/psapm/058/1922899|isbn=9780821820841|issn=2324-7088}}</ref><ref name=":32">{{Cite book|last=Nielsen, Michael A., 1974-|url=https://www.worldcat.org/oclc/665137861|title=Quantum computation and quantum information|date=2010|publisher=Cambridge University Press|others=Chuang, Isaac L., 1968-|isbn=978-1-107-00217-3|edition=10th anniversary|___location=Cambridge|oclc=665137861}}</ref> However, questions around the Church-Turing thesis arise in the context of quantum computing. It is unclear whether the Church-Turing thesis holds for the quantum computation model. There is much evidence that the thesis does not hold. It may not be possible for a probabilistic Turing machine to simulate quantum computation models in polynomial time.<ref name=":02" />

Both quantum computational complexity of functions and classical computational complexity of functions are often expressed with [[asymptotic notation]]. Some common forms of asymptotic notion of functions are <math>O(T(n))</math>, <math>\Omega(T(n))</math>, and <math>\Theta(T(n))</math>. <math>O(T(n))</math> expresses that something is bounded above by <math>cT(n)</math> where <math>c</math> is a constant such that <math>c>0</math> and <math>T(n)</math> is a function of <math>n</math>, <math>\Omega(T(n))</math> expresses that something is bounded below by <math>cT(n)</math> where <math>c</math> is a constant such that <math>c>0</math> and <math>T(n)</math> is a function of <math>n</math>, and <math>\Theta(T(n))</math> expresses both <math>O(T(n))</math> and <math>\Omega(T(n))</math>.<ref name=":12">{{Citation|last=Cleve|first=Richard|title=An Introduction to Quantum Complexity Theory|date=2000|url=http://dx.doi.org/10.1142/9789810248185_0004|work=Quantum Computation and Quantum Information Theory|volume=|pages=103–127|publisher=WORLD SCIENTIFIC|doi=10.1142/9789810248185_0004|arxiv=quant-ph/9906111|bibcode=2000qcqi.book..103C|isbn=978-981-02-4117-9|s2cid=958695|access-date=October 10, 2020}}</ref> These notations also their own names. <math>O(T(n))</math> is called [[Big O notation]], <math>\Omega(T(n))</math> is called Big Omega notation, and <math>\Theta(T(n))</math> is called Big Theta notation.

== Complexity Classes Overview ==

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=== Query Models of Directed Graphs ===

One type of problem that quantum computing can make easier to solve are graph problems. If we are to consider the amount of queries to a graph that are required to solve a given problem, let us first consider the most common types of graphs, called [[directed graph]]s, that are associated with this type of computational modelling. In brief, directed graphs are graphs where all edges between vertices are unidirectional. Directed graphs are formally defined as the graph <math>G=(N,E)</math>, where N is the set of vertices, or nodes, and E is the set of edges.<ref>{{Cite web|last=Nykamp|first=D.Q.|date=|title=Directed Graph Definition|url=https://mathinsight.org/definition/directed_graph|url-status=live|archive-url=|archive-date=|access-date=|website=}}</ref>

==== Adjacency Matrix Model ====

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==== Grover's Algorithm ====

An example depicting the power of quantum computing is [[Grover's algorithm]] for searching unstructured databases. The algorithm's quantum query complexity is <math display="inline">O{\left(\sqrt{N}\right)}</math>, a quadratic improvement over the best possible classical query complexity <math>O(N)</math>, which is a [[linear search]]. While Grover's algorithm is more optimized than the best possible classical algorithm, we know that Grover's algorithm is not one hundred percent optimal.<ref>{{Cite journal|last=Ambainis|first=Andris|date=October 28, 2005|title=Polynomial degree vs. quantum query complexity|url=https://linkinghub.elsevier.com/retrieve/pii/S0022000005000899|journal=Journal of Computer and System Sciences|language=en|volume=72|issue=2|pages=220–238|doi=10.1016/j.jcss.2005.06.006|via=}}</ref> Optimization of a query algorithm refers to how the algorithm compares to the most efficient theoretical algorithm that solves the same problem. An algorithm is said to be [[Asymptotic optimality|asymptotically optimized]] if at worst, it performs at a constant factor worse than the most efficient possible algorithm. Note that an algorithm is still considered to be optimized if it performs worse than the most efficient possible algorithm, as long as the algorithm doesn't get exponentially worse than the most efficient possible algorithm, as the number of inputs increases.

==== Deutsch-Jozsa Algorithm ====

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* {{cite book | author1-link= Michael Nielsen| author1=Nielsen, Michael |author2-link = Isaac L. Chuang |author2=Chuang, Isaac |title=Quantum Computation and Quantum Information | title-link=Quantum Computation and Quantum Information |publisher=Cambridge University Press |___location=Cambridge |year=2000 |isbn=978-0-521-63503-5 |oclc= 174527496}}

* {{cite book |last1=Arora |first1=Sanjeev|author1-link=Sanjeev Arora |last2=Barak |first2=Boaz|author2-link=Boaz Barak |title=Computational Complexity: A Modern Approach |url=https://archive.org/details/computationalcom00aror |url-access=limited |date=2016 |publisher=Cambridge University Press |isbn=978-0-521-42426-4 |pages=[https://archive.org/details/computationalcom00aror/page/n226 201]–236}}

* {{cite arXiv|eprint=0804.3401v1|author1=John Watrous|authorlinkauthor-link=John Watrous (computer scientist)|title=Quantum Computational Complexity|class=quant-ph|year=2008}}

* Watrous J. (2009) [https://link.springer.com/referencework/10.1007/978-0-387-30440-3 Quantum Computational Complexity]. In: Meyers R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY