If you work with a normal computer — a classical computer — you’re used to two foundational ideas: Some problems can be solved very efficiently with classical bits and classical gates, like four-function (+, -, ×, ÷) mathematics, while other problems can only be solved extremely inefficiently with classical bits and classical gates, with the traveling salesman problem being a legendary example of such a “hard” problem. If you bring in the additional possibilities associated with a quantum computer, however, you no longer have just classical bits and classical gates, but qubits and fundamentally quantum gates. One of the biggest questions in all of computer science is this: which questions and problems, and under which conditions, will the additional options provided by these qubits and quantum gates result in a major computational speedup, and by how much?
That speedup is known as either quantum supremacy or quantum advantage. Which useful problems will experience a quantum advantage, and by how much? That’s one of the main goals of the field of quantum complexity.
One of the central open problems in that field, the Aaronson-Ambainis conjecture, has just been solved: by an AI company. Here’s what it does — and doesn’t — mean. Many of the different computational complexity classes of problems are shown.
The innermost spaces require the smallest amounts of writeable memory in order to solve them, while the larger spaces require larger amounts of writeable memory in order to solve them. Examples, from innermore to outermore complexity classes, include problems that can be solved in logarithmic time, polynomial time, nondeterministic polynomial time, and exponential time, among others. What you see, above, is a map of what’s known as different complexity classes in computer science.
Any computational problem is one where you can identify a problem, write down an algorithm for solving that problem, computationally execute that step-by-step algorithm for the specific input associated with that problem, and get an answer. That answer can be a simple binary (yes/no; true/false) answer, which comes for problems that are decision-type problems. However, computational problems include ones that require a more complex quantitative answer: a function problem, a counting problem, or an optimization problem (such as the traveling salesman problem) are all examples of problems more complex than decision problems.
One of the most important complexity classes is known as P). P contains all of the problems that can be solved in polynomial time: generally regarded as “efficiently solvable” problems. If you have a deterministic Turing machine, you can solve a problem of class P using a polynomial amount of computational time.
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