Quantum complexity theory

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Quantum complexity theory is a part of computational complexity theory in theoretical computer science. It studies complexity classes defined using quantum computers and quantum information which are computational models based on quantum mechanics. It studies the hardness of problems in relation to these complexity classes, and the relationship between quantum complexity classes and classical (i.e., non-quantum) complexity classes.

A complexity class is a collection of problems which can be solved by some computational model under resource constraints. For instance, the complexity class P is defined to be the set of problems solvable by a Turing machine in polynomial time. Similarly, one may define a quantum complexity class using a quantum model of computation, such as a standard quantum computer or a quantum Turing machine. Thus, the complexity class BQP is defined to be the set of problems solvable by a quantum computer in polynomial time with bounded error.

Two important quantum complexity classes are BQP and QMA which are the bounded-error quantum analogues of P and NP. One of the main aims of quantum complexity theory is to find out where these classes lie with respect to classical complexity classes such as P, NP, PP, PSPACE and other complexity classes.

Quantum Query Complexity[edit]

It is one of the major field of Quantum Complexity Theory which deals with the number of queries to input in order to calculate the function. In query complexity model, input is given as oracle(Black Box). We get the information about the input by quering oracle. In the start of algorithm we start in some fixed quantum state and the state evolves as we query the oracle. We have to develop an algorithm which minimizes the number of queries required to calculate function. Clearly, Quantum Query Complexity is lower bound on overall time complexity of function and sometimes more important than time complexity.

Grover's algorithm for searching unstructured database is one of the best example depicting the power of Quantum Computing. Quantum Query Complexity of Grover Search is O(N1/2) which is quadratically better than best known classical query compleixty.