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Quantum complexity theory is the subfield of
computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved ...
that deals with complexity classes defined using quantum computers, a computational model based on
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. It studies the hardness of computational problems in relation to these complexity classes, as well as the relationship between quantum complexity classes and classical (i.e., non-quantum) complexity classes. Two important quantum complexity classes are BQP and
QMA In computational complexity theory, QMA, which stands for Quantum Merlin Arthur, is the set of languages for which, when a string is in the language, there is a polynomial-size quantum proof (a quantum state) that convinces a polynomial time qua ...
.


Background

A
complexity class In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms o ...
is a collection of computational problems that can be solved by a computational model under certain resource constraints. For instance, the complexity class P is defined as the set of problems solvable by a
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer alg ...
in
polynomial time In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
. Similarly, quantum complexity classes may be defined using quantum models of computation, such as the quantum circuit model or the equivalent
quantum Turing machine A quantum Turing machine (QTM) or universal quantum computer is an abstract machine used to model the effects of a quantum computer. It provides a simple model that captures all of the power of quantum computation—that is, any quantum algori ...
. One of the main aims of quantum complexity theory is to find out how these classes relate to classical complexity classes such as P, NP, BPP, and PSPACE. One of the reasons quantum complexity theory is studied are the implications of quantum computing for the modern 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. 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. 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 O(T(n)), \Omega(T(n)), and \Theta(T(n)). O(T(n)) expresses that something is bounded above by cT(n) where c is a constant such that c>0 and T(n) is a function of n, \Omega(T(n)) expresses that something is bounded below by cT(n) where c is a constant such that c>0 and T(n) is a function of n, and \Theta(T(n)) expresses both O(T(n)) and \Omega(T(n)). These notations also their own names. O(T(n)) is called
Big O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund L ...
, \Omega(T(n)) is called Big Omega notation, and \Theta(T(n)) is called Big Theta notation.


Overview of complexity classes

Some important complexity classes are P, BPP, BQP, PP, and P-Space. To define these we first define a promise problem. A promise problem is a decision problem which has an input assumed to be selected from the set of all possible input strings. A promise problem is a pair A=(A_\text,A_\text). A_\text is the set of yes instances, A_\text is the set of no instances, and the intersection of these sets is such that A_\text \cap A_\text = \varnothing. All of the previous complexity classes contain promise problems.


BQP

The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP ("bounded error, quantum, polynomial time"). More formally, BQP is the class of problems that can be solved by a polynomial-time
quantum Turing machine A quantum Turing machine (QTM) or universal quantum computer is an abstract machine used to model the effects of a quantum computer. It provides a simple model that captures all of the power of quantum computation—that is, any quantum algori ...
with error probability of at most 1/3. As a class of probabilistic problems, BQP is the quantum counterpart to BPP ("bounded error, probabilistic, polynomial time"), the class of problems that can be efficiently solved by probabilistic Turing machines with bounded error. It is known that \mathsf and widely suspected, but not proven, that \mathsf, which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity. BQP is a subset of PP. The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that \mathsf; that is, the class of problems that can be efficiently solved by quantum computers includes all problems that can be efficiently solved by deterministic classical computers but does not include any problems that cannot be solved by classical computers with polynomial space resources. It is further suspected that BQP is a strict superset of P, meaning there are problems that are efficiently solvable by quantum computers that are not efficiently solvable by deterministic classical computers. For instance,
integer factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are s ...
and the discrete logarithm problem are known to be in BQP and are suspected to be outside of P. On the relationship of BQP to NP, little is known beyond the fact that some NP problems are in BQP (integer factorization and the discrete logarithm problem are both in NP, for example). It is suspected that \mathsf; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. As a direct consequence of this belief, it is also suspected that BQP is disjoint from the class of NP-complete problems (if any NP-complete problem were in BQP, then it follows from
NP-hard In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...
ness that all problems in NP are in BQP). The relationship of BQP to the essential classical complexity classes can be summarized as: :\mathsf It is also known that BQP is contained in the complexity class (or more precisely in the associated class of decision problems ), which is a subset of PSPACE.


Simulation of quantum circuits

There is no known way to efficiently simulate a quantum computational model with a classical computer. This means that a classical computer cannot simulate a quantum computational model in polynomial time. However, a quantum circuit of S(n) qubits with T(n) quantum gates can be simulated by a classical circuit with O(2^T(n)^3) classical gates. This number of classical gates is obtained by determining how many bit operations are necessary to simulate the quantum circuit. In order to do this, first the amplitudes associated with the S(n) qubits must be accounted for. Each of the states of the S(n) qubits can be described by a two-dimensional complex vector, or a state vector. These state vectors can also be described a linear combination of its component vectors with coefficients called amplitudes. These amplitudes are complex numbers which are normalized to one, meaning the sum of the squares of the absolute values of the amplitudes must be one. The entries of the state vector are these amplitudes. Which entry each of the amplitudes are correspond to the none-zero component of the component vector which they are the coefficients of in the linear combination description. As an equation this is described as \alpha \begin 1 \\ 0 \end + \beta \begin 0 \\ 1 \end = \begin \alpha \\ \beta \end or \alpha \left \vert 1 \right \rangle + \beta \left \vert 0 \right \rangle = \begin \alpha \\ \beta \end using Dirac notation. The state of the entire S(n) qubit system can be described by a single state vector. This state vector describing the entire system is the tensor product of the state vectors describing the individual qubits in the system. The result of the tensor products of the S(n) qubits is a single state vector which has 2^ dimensions and entries that are the amplitudes associated with each basis state or component vector. Therefore, 2^amplitudes must be accounted for with a 2^ dimensional complex vector which is the state vector for the S(n) qubit system. In order to obtain an upper bound for the number of gates required to simulate a quantum circuit we need a sufficient upper bound for the amount data used to specify the information about each of the 2^ amplitudes. To do this O(T(n)) bits of precision are sufficient for encoding each amplitude. So it takes O(2^T(n)) classical bits to account for the state vector of the S(n) qubit system. Next the application of the T(n) quantum gates on 2^ amplitudes must be accounted for. The quantum gates can be represented as 2^\times2^ sparse matrices. So to account for the application of each of the T(n) quantum gates, the state vector must be multiplied by a 2^\times2^ sparse matrix for each of the T(n) quantum gates. Every time the state vector is multiplied by a 2^\times2^ sparse matrix, O(2^) arithmetic operations must be performed. Therefore, there are O(2^T(n)^2) bit operations for every quantum gate applied to the state vector. So O(2^T(n)^2) classical gate are needed to simulate S(n) qubit circuit with just one quantum gate. Therefore, O(2^T(n)^3) classical gates are needed to simulate a quantum circuit of S(n) qubits with T(n) quantum gates. While there is no known way to efficiently simulate a quantum computer with a classical computer, it is possible to efficiently simulate a classical computer with a quantum computer. This is evident from the belief that \mathsf.


Quantum query complexity

One major advantage of using a quantum computational system instead of a classical one, is that a quantum computer may be able to give a polynomial time algorithm for some problem for which no classical polynomial time algorithm exists, but more importantly, a quantum computer may significantly decrease the calculation time for a problem that a classical computer can already solve efficiently. Essentially, a quantum computer may be able to both determine how long it will take to solve a problem, while a classical computer may be unable to do so, and can also greatly improve the computational efficiency associated with the solution to a particular problem. Quantum query complexity refers to how complex, or how many queries to the graph associated with the solution of a particular problem, are required to solve the problem. Before we delve further into query complexity, let us consider some background regarding graphing solutions to particular problems, and the queries associated with these solutions.


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 graphs, 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 G=(N,E), where N is the set of vertices, or nodes, and E is the set of edges.


Adjacency matrix model

When considering quantum computation of the solution to directed graph problems, there are two important query models to understand. First, there is the
adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...
model, where the graph of the solution is given by the adjacency matrix: M \in \a^ , with M_=1 , if and only if (v_,v_)\in E .


Adjacency array model

Next, there is the slightly more complicated adjacency array model built on the idea of
adjacency list In graph theory and computer science, an adjacency list is a collection of unordered lists used to represent a finite graph. Each unordered list within an adjacency list describes the set of neighbors of a particular vertex in the graph. This is ...
s, where every vertex, u '','' is associated with an array of neighboring vertices such that f_i: _i^+rightarrow , for the out-degrees of vertices d_i^+,...,d_n^+ , where n is the minimum value of the upper bound of this model, and f_i(j) returns the "j^ " vertex adjacent to i . Additionally, the adjacency array model satisfies the simple graph condition, \forall i\in j,j'\in j\neq j': f_i(j)\neq f_i(j') , meaning that there is only one edge between any pair of vertices, and the number of edges is minimized throughout the entire model (see
Spanning tree In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is ...
model for more background).


Quantum query complexity of certain types of graph problems

Both of the above models can be used to determine the query complexity of particular types of graphing problems, including the connectivity, strong connectivity (a directed graph version of the connectivity model),
minimum spanning tree A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. ...
, and single source shortest path models of graphs. An important caveat to consider is that the quantum complexity of a particular type of graphing problem can change based on the query model (namely either matrix or array) used to determine the solution. The following table showing the quantum query complexities of these types of graphing problems illustrates this point well. Notice the discrepancy between the quantum query complexities associated with a particular type of problem, depending on which query model was used to determine the complexity. For example, when the matrix model is used, the quantum complexity of the connectivity model in
Big O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund L ...
is \Theta(n^), but when the array model is used, the complexity is \Theta(n). Additionally, for brevity, we use the shorthand m in certain cases, where m=\Theta(n^2). The important implication here is that the efficiency of the algorithm used to solve a graphing problem is dependent on the type of query model used to model the graph.


Other types of quantum computational queries

In the query complexity model, the input can also be given as an oracle (black box). The algorithm gets information about the input only by querying the oracle. The algorithm starts in some fixed quantum state and the state evolves as it queries the oracle. Similar to the case of graphing problems, the quantum query complexity of a black-box problem is the smallest number of queries to the oracle that are required in order to calculate the function. This makes the quantum query complexity a lower bound on the overall time complexity of a function.


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 O, a quadratic improvement over the best possible classical query complexity O(N), which is a linear search. Grover's algorithm is asymptotically optimal; in fact, it uses at most a 1+o(1) fraction more queries than the best possible algorithm.


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 f:\^n\rightarrow\ is constant or balanced, those being the only two possibilities. The only way to evaluate the function f is to consult a black box or
oracle An oracle is a person or agency considered to provide wise and insightful counsel or prophetic predictions, most notably including precognition of the future, inspired by deities. As such, it is a form of divination. Description The word ...
. 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 2^n possible inputs, the query complexity of the most efficient classical deterministic algorithm is 2^+1. 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 1.


See also

* Quantum computing *
Quantum Turing machine A quantum Turing machine (QTM) or universal quantum computer is an abstract machine used to model the effects of a quantum computer. It provides a simple model that captures all of the power of quantum computation—that is, any quantum algori ...
* Polynomial hierarchy (PH)


Notes


References

* * * * Watrous J. (2009
Quantum Computational Complexity
In: Meyers R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY


External links


MIT lectures
by
Scott Aaronson Scott Joel Aaronson (born May 21, 1981) is an American theoretical computer scientist and David J. Bruton Jr. Centennial Professor of Computer Science at the University of Texas at Austin. His primary areas of research are quantum computing ...
{{emerging technologies, quantum=yes, other=yes Computational complexity theory Theoretical computer science