Which type of mathematical problem is too complex for a classical computer to solve efficiently?

• Computing has progressed at an astonishing rate, with new advancements helping solve large and complex issues. • Moore’s Law – which has been the guiding framework for computing advancement – is approaching the physical and engineering limits of what is possible. • In its place, new varieties of computers are emerging, including quantum, high-performance, bio- and bio-inspired computing. • Next-generation computing can help solve previously unsolvable questions, but agencies should act now to take full advantage of the opportunities. Many experts believe that the biggest advances in computing capability are still ahead of us. For federal agencies — charged with tackling the world’s gnarliest and most pervasive problems — these fast-emerging capabilities in computational power present both an incredible opportunity as well as a significant potential threat. This trend, Computing the Impossible, explores both the positive and cautionary dimensions of the growing computing power coming into our grasp. Over the last five decades, the state of computing has progressed at a truly astonishing rate. Consider that in 1971, chip-maker Intel launched the Intel® 4004 processor, the first general-purpose programmable processor on the market. At that time, the 4004 was a technical marvel: the size of a small fingernail, it held 2,300 transistors — tiny electrical switches representing the 1s and 0s that are the basic binary language of computers. Each transistor on the 4004 was 10 mic...

By • What is quantum supremacy? Quantum supremacy is the experimental demonstration of a quantum computer's dominance and advantage over classical computers by performing calculations previously impossible at unmatched speeds. To confirm that quantum supremacy has been achieved, computer scientists must be able to show that a classical computer could never have solved the problem while also proving that the quantum computer can perform the calculation quickly. Computer scientists believe Quantum computing is consistently evolving. Quantum computers have not yet reached a point where they can show their supremacy over classical computers. This is mostly due to the huge amount of quantum qubits, required to perform meaningful operations on quantum computers. As the amount of necessary logic gates and number of qubits increases, so does the error rate. If the error rate gets too high, the quantum computer loses any advantage it had over the classical computer. To successfully perform useful calculations -- such as determining the chemical properties of a substance -- a few million qubits would be necessary. Currently, the largest quantum computer design is IBM's quantum computer, named Osprey, which Quantum computers vs. classical computers The primary difference between quantum and classical computers is in how they work. Classical computers process information as bits, with all computations performed in a Conversely, quantum computers use Qubits can theoretically outperform...

Is there a general statement about what kinds of problems can be solved more efficiently using quantum computers (quantum gate model only)? Do the problems for which an algorithm is known today have a common property? As far as i understand quantum computing helps with the hidden subgroup problem (Shor); Grover's algorithm helps speedup search problems. I have read that quantum algorithms can provide speed-up if you look for a 'global property' of a function (Grover/Deutsch). • Is there a more concise and correct statement about where quantum computing can help? • Is it possible to give an explanation why quantum physics can help there (preferably something deeper that 'interference can be exploited')? And why it possibly will not help for other problems (e.g. for NP-complete problems)? Are there relevant papers that discuss just that? I had asked this question before over on On computational helpfulness in general Without perhaps realising it, you are asking a version of one of the most difficult questions you can possibly ask about theoretical computer science. You can ask the same question about classical computers, only instead of asking whether adding 'quantumness' is helpful, you can ask: • Is there a concise statement about where randomised algorithms can help? It's possible to say something very vague here — if you think that solutions are plentiful (or that the number of solutions to some sub-problem are plentiful) but that it might be difficult to systematically ...

In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order (classical) systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human co...

• Computing has progressed at an astonishing rate, with new advancements helping solve large and complex issues. • Moore’s Law – which has been the guiding framework for computing advancement – is approaching the physical and engineering limits of what is possible. • In its place, new varieties of computers are emerging, including quantum, high-performance, bio- and bio-inspired computing. • Next-generation computing can help solve previously unsolvable questions, but agencies should act now to take full advantage of the opportunities. Many experts believe that the biggest advances in computing capability are still ahead of us. For federal agencies — charged with tackling the world’s gnarliest and most pervasive problems — these fast-emerging capabilities in computational power present both an incredible opportunity as well as a significant potential threat. This trend, Computing the Impossible, explores both the positive and cautionary dimensions of the growing computing power coming into our grasp. Over the last five decades, the state of computing has progressed at a truly astonishing rate. Consider that in 1971, chip-maker Intel launched the Intel® 4004 processor, the first general-purpose programmable processor on the market. At that time, the 4004 was a technical marvel: the size of a small fingernail, it held 2,300 transistors — tiny electrical switches representing the 1s and 0s that are the basic binary language of computers. Each transistor on the 4004 was 10 mic...

Is there a general statement about what kinds of problems can be solved more efficiently using quantum computers (quantum gate model only)? Do the problems for which an algorithm is known today have a common property? As far as i understand quantum computing helps with the hidden subgroup problem (Shor); Grover's algorithm helps speedup search problems. I have read that quantum algorithms can provide speed-up if you look for a 'global property' of a function (Grover/Deutsch). • Is there a more concise and correct statement about where quantum computing can help? • Is it possible to give an explanation why quantum physics can help there (preferably something deeper that 'interference can be exploited')? And why it possibly will not help for other problems (e.g. for NP-complete problems)? Are there relevant papers that discuss just that? I had asked this question before over on On computational helpfulness in general Without perhaps realising it, you are asking a version of one of the most difficult questions you can possibly ask about theoretical computer science. You can ask the same question about classical computers, only instead of asking whether adding 'quantumness' is helpful, you can ask: • Is there a concise statement about where randomised algorithms can help? It's possible to say something very vague here — if you think that solutions are plentiful (or that the number of solutions to some sub-problem are plentiful) but that it might be difficult to systematically ...

By • What is quantum supremacy? Quantum supremacy is the experimental demonstration of a quantum computer's dominance and advantage over classical computers by performing calculations previously impossible at unmatched speeds. To confirm that quantum supremacy has been achieved, computer scientists must be able to show that a classical computer could never have solved the problem while also proving that the quantum computer can perform the calculation quickly. Computer scientists believe Quantum computing is consistently evolving. Quantum computers have not yet reached a point where they can show their supremacy over classical computers. This is mostly due to the huge amount of quantum qubits, required to perform meaningful operations on quantum computers. As the amount of necessary logic gates and number of qubits increases, so does the error rate. If the error rate gets too high, the quantum computer loses any advantage it had over the classical computer. To successfully perform useful calculations -- such as determining the chemical properties of a substance -- a few million qubits would be necessary. Currently, the largest quantum computer design is IBM's quantum computer, named Osprey, which Quantum computers vs. classical computers The primary difference between quantum and classical computers is in how they work. Classical computers process information as bits, with all computations performed in a Conversely, quantum computers use Qubits can theoretically outperform...

In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order (classical) systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human co...