There are several classical algorithms for which no known efficient quantum version exists. Here are a few examples:
Grover's algorithm: Grover's algorithm is a quantum search algorithm that can speed up the search of an unsorted database quadratically compared to classical algorithms. However, there are certain search problems for which no known efficient quantum algorithm exists, such as searching an unordered list of elements with a unique property.
NP-complete problems: NP-complete problems are a class of computational problems for which no efficient classical algorithm is known. Examples include the traveling salesman problem, the knapsack problem, and the satisfiability problem. While there are quantum algorithms that provide some speedup for specific instances of these problems, there is no general-purpose quantum algorithm that solves all NP-complete problems efficiently.
Integer factorization: Factoring large integers into their prime factors is a problem that plays a crucial role in modern cryptography. Classical algorithms for integer factorization, such as the General Number Field Sieve (GNFS), have sub-exponential time complexity. Currently, there is no known efficient quantum algorithm for factoring large integers, which poses a challenge to the security of certain cryptographic systems like RSA.
Simulating quantum systems: Quantum systems can be notoriously difficult to simulate on classical computers. While there are quantum simulation algorithms like the quantum Monte Carlo method, they generally require exponential resources in terms of memory and computation time. So, efficiently simulating large-scale quantum systems remains a challenging problem for both classical and quantum computers.
It's worth noting that ongoing research is being conducted to explore the potential of quantum computing and to discover efficient quantum algorithms for various problems. However, at present, the examples mentioned above do not have known efficient quantum counterparts.