There are two main areas of interest in complexity theory that have been changed by quantum computing: quantum computational and communication complexity. They both deal with quantum materials, but they solve problems in different ways.
- Quantum Communication Complexity looks into how little quantum information (qubits) two or more people need to share in order to work together to solve a problem using quantum algorithms on a quantum computer.
- Quantum Computational Complexity looks into how hard it is to solve problems using quantum algorithms on a quantum computer.
In quantum computing, some techniques can be used in quantum communication, and in quantum communication, some methods can be used in quantum computing.
Relationship Between Quantum Computational and Communication complexity
Quantum computing and quantum communication are strongly linked:
Using quantum computing to make quantum communication better
Buhrman et al. (1998) came up with a way to turn quantum methods for computer problems into quantum communication protocols. The main ways to use this are:
- Finding Upper Bounds: If a quantum computer can solve a problem quickly, that means there is an upper bound on how hard quantum communication is.
- Finding Lower Bounds: If a problem needs a lot of quantum communication, it limits how hard it is for linked problems to use quantum computing.
For Example:
- If a quantum query algorithm figures out a function F(f) in t queries, then a quantum communication protocol can figure out the same function with t(2n + 4) qubits swapped.
- This change lets the results of how hard something is to compute have a direct effect on how hard it is to communicate.
Quantum communication as a way to make computations faster
It has been used to show that there are lower bounds on quantum computing through quantum communication complexity. One problem that limits how quickly a quantum computer can solve the same problem is one that needs a certain amount of quantum communication.
For example:
The disjoint Ness function (which checks if two sets are not the same) has a high quantum communication complexity, which means that related computing problems can’t be as hard.
Holevo’s Theorem and the Limits of Quantum Communication
Holevo’s Theorem is one of the most important results in quantum information theory. It says that sending n qubits can’t send more than n traditional bits of information. This could mean that quantum transmission doesn’t offer many big benefits. However, entanglement-assisted communication and quantum fingerprints are much better than traditional communication.
Future research continues to explore the limits of quantum algorithms and quantum networks
Quantum communication complexity and quantum processing complexity are two fields that are very similar and have an effect on each other. Quantum communication tries to cut down on the number of qubits that need to be sent between different parties. Quantum computing tries to speed up the process of fixing problems.
- The quantum processing complexity tells us what a quantum machine can do quickly.
- The complexity of quantum communication determines how well two parties can compute a function with little communication.
- The two fields overlap, with the results of quantum computation influencing quantum communication protocols and the results of quantum communication helping to set lower bounds for quantum computation.
For the creation of future quantum technologies, quantum networks, and global quantum computing, it is very important to understand how these two areas work together.
Quantum Computational complexity vs. Quantum Communication complexity
| Feature | Quantum Computational Complexity | Quantum Communication Complexity |
| Focus | Speed of solving problems on a quantum computer | Minimizing qubits exchanged between distant parties |
| Measure | Time (quantum gates), Space (qubits), Complexity classes (BQP, QMA, etc.) | Number of qubits exchanged (QCε(P), QC0(P), etc.) |
| Techniques | Quantum circuits, Quantum Turing machines | Quantum teleportation, Superdense coding, Entanglement |
| Use Cases | Algorithm efficiency, Speedup over classical computing | Reducing communication cost in distributed computing |
| Impact on Each Other | Provides lower bounds on communication complexity | Used to establish lower bounds in computational complexity |
Are quantum algorithms always more powerful than classical ones in communication complexity?
Some problems show exponential speedup, while others do not.
Can all quantum computational problems be turned into efficient quantum communication protocols?
Too much quantum communication might be needed to solve some problems in the real world.