What evidence do we have that suggests BQP might be more powerful than classical polynomial time, and what are some examples of problems believed to be in BQP but not in BPP?
One of the fundamental questions in quantum complexity theory is whether quantum computers can solve certain problems more efficiently than classical computers. The class of problems that can be efficiently solved by a quantum computer is known as BQP (Bounded-error Quantum Polynomial time), which is analogous to the class of problems that can be efficiently solved by a classical computer, known as BPP (Bounded-error Probabilistic Polynomial time).
There are several lines of evidence that suggest that BQP might be more powerful than BPP. One such line of evidence is based on the fact that quantum computers can efficiently solve certain problems that are believed to be intractable for classical computers. One example of such a problem is factoring large numbers. Shor's algorithm, a quantum algorithm, can factorize large numbers exponentially faster than the best-known classical algorithms, which rely on the factoring problem being difficult.
Another line of evidence comes from the study of quantum simulation. It has been shown that quantum computers can efficiently simulate quantum systems, which is believed to be a computationally hard problem for classical computers. This suggests that quantum computers have the potential to solve problems that are inherently quantum in nature more efficiently than classical computers.
Furthermore, there are problems in BQP that are believed to be outside the class BPP. One such problem is the simulation of quantum circuits. It is believed that simulating quantum circuits on a classical computer is exponentially hard, while it can be efficiently done on a quantum computer. This implies that there are problems that can be efficiently solved by a quantum computer but not by a classical computer.
Another example is the problem of approximating the Jones polynomial, which is a mathematical invariant of knots and links. It is believed that approximating the Jones polynomial is a complete problem for BQP, meaning that if a classical computer could efficiently approximate the Jones polynomial, it would imply that BQP is equal to BPP. However, there is currently no known efficient classical algorithm for approximating the Jones polynomial, suggesting that it might be in BQP but not in BPP.
There are several lines of evidence that suggest that BQP might be more powerful than classical polynomial time. These include the ability of quantum computers to efficiently solve problems that are believed to be intractable for classical computers, the efficient simulation of quantum systems, and the existence of problems in BQP that are believed to be outside the class BPP. These lines of evidence highlight the potential computational power of quantum computers and their ability to solve problems that are difficult for classical computers.
Other recent questions and answers regarding BQP:
- What are the open questions regarding the relationship between BQP and NP, and what would it mean for complexity theory if BQP is proven to be strictly larger than P?
- How can we increase the probability of obtaining the correct answer in BQP algorithms, and what error probability can be achieved?
- How do we define a language L to be in BQP and what are the requirements for a quantum circuit solving a problem in BQP?
- What is the complexity class BQP and how does it relate to classical complexity classes P and BPP?