The number of qubits required to approximate a functional quantum computer depends on several factors, including the specific computational tasks you want to perform and the level of error correction and fault tolerance desired. It's important to note that creating a fully functional, error-corrected quantum computer is a significant technical challenge, and we are still in the early stages of its development.
To achieve practical computational capabilities, it is generally believed that quantum computers will need a large number of qubits, likely in the range of hundreds to thousands of qubits. This estimate takes into account the need for error correction and fault-tolerant operations to mitigate the impact of noise and errors inherent in quantum systems.
The exact number of qubits required can vary depending on the specific algorithms and applications. For some quantum algorithms, such as Shor's algorithm for factoring large numbers or Grover's algorithm for quantum search, there are known theoretical relationships between the number of qubits and the size of the problem being solved. These algorithms require a sufficient number of qubits to encode and manipulate the necessary information effectively.
However, it's worth mentioning that even with a relatively small number of qubits, so-called "noisy intermediate-scale quantum" (NISQ) computers can still be useful for certain applications. These devices have limited qubit counts and short coherence times, but they can still perform certain computations that are difficult for classical computers.
In summary, the number of qubits required to approximate a functional quantum computer capable of solving complex problems with error correction and fault tolerance is still an active area of research and development. While we don't have a precise answer yet, it is expected to be in the range of hundreds to thousands of qubits.