# Introduction to quantum complexity classes: BQP, QMA, QIP

Sep 22, 2023

## Quantum complexity classes

Quantum complexity classes are a collection of computational problems that are solvable by a quantum computer. Quantum computers are devices that use the principles of quantum mechanics to perform computations. They are believed to be able to solve certain problems much faster than classical computers.

The most well-known quantum complexity class is BQP, which stands for Bounded-error Quantum Polynomial Time. A problem is in BQP if it can be solved by a quantum computer in polynomial time with an error probability of at most 1/3.

Another important quantum complexity class is QMA, which stands for Quantum Merlin Arthur. A problem is in QMA if there exists a quantum verifier that can verify a quantum proof of the solution to the problem. The verifier is a quantum computer that runs in polynomial time. The quantum proof is a quantum state that can be generated by a prover who knows the solution to the problem.

The quantum complexity class QIP stands for Quantum Interactive Polynomial time. A problem is in QIP if it can be solved by a quantum computer that interacts with a classical prover in polynomial time. The prover can send quantum states to the computer, and the computer can send classical messages back to the prover.

## Relationship between quantum complexity classes

The relationship between the quantum complexity classes is not fully understood. However, it is known that BQP is contained in QMA. This means that any problem that can be solved in BQP can also be solved in QMA. However, the converse is not known to be true. It is possible that there are problems that are in QMA but not in BQP.

The relationship between QMA and QIP is also not fully understood. However, it is known that QIP is contained in PSPACE, which is the class of all problems that can be solved by a classical computer in polynomial space. This means that any problem that can be solved in QIP can also be solved by a classical computer, but with more memory.

## Applications of quantum complexity classes

Quantum complexity classes have a number of potential applications. For example, they can be used to solve problems in cryptography, communication, and optimization.

In cryptography, quantum computers can be used to break certain classical cryptographic algorithms. For example, Shor’s algorithm can be used to factor large numbers in polynomial time. This means that the RSA and Diffie-Hellman algorithms, which are widely used in cryptography, would be vulnerable to attack by a quantum computer.

In communication, quantum computers can be used to create secure channels. For example, quantum key distribution (QKD) is a protocol that allows two parties to share a secret key over an insecure channel. QKD is secure against eavesdropping by a quantum computer.

In optimization, quantum computers can be used to solve certain optimization problems more efficiently than classical computers. For example, the quantum annealing algorithm can be used to find the ground state of a quantum system. This can be used to solve problems in scheduling, routing, and machine learning.

## Conclusion

Quantum complexity classes are a promising area of research. They have the potential to solve certain problems that are intractable for classical computers. As quantum computers become more powerful, quantum complexity classes will become increasingly important.