Yes, there is a theoretical limit to the number of qubits a quantum computer can have. The limit is determined by several factors, including the physical resources available for constructing and operating the quantum computer, as well as the amount of noise and errors introduced during quantum operations.
One major challenge in building large-scale quantum computers is maintaining the coherence of the qubits. Qubits are extremely delicate and can easily lose their quantum state due to interactions with the surrounding environment, a phenomenon known as decoherence. The larger the number of qubits, the more difficult it becomes to control and protect them from decoherence.
Another limiting factor is the requirement for quantum error correction. Quantum error correction is a technique used to protect the fragile quantum states from errors and noise that naturally occur during quantum computation. It involves encoding logical qubits into a larger number of physical qubits and performing error correction operations. As the number of qubits increases, so does the overhead in terms of physical qubits and computational resources needed for error correction.
Currently, quantum computers with a few hundred qubits are being developed by various research organizations and companies. However, building quantum computers with thousands or millions of qubits is still a significant engineering and scientific challenge.
It's worth noting that there are also alternative approaches to quantum computing, such as topological quantum computing, where the qubits are encoded in non-local properties of physical systems. These approaches may have different limitations and possibilities for scalability compared to the more commonly pursued gate-based quantum computing paradigm.