The number of qubits required to solve the Schrödinger equation exactly for a given system depends on several factors, including the complexity of the system and the desired level of accuracy.
In general, the number of qubits needed to represent a quantum system grows exponentially with the size of the system. If a system has N quantum states, then representing it accurately typically requires a quantum computer with at least 2^N qubits. This exponential scaling is known as the "curse of dimensionality."
For larger systems, solving the Schrödinger equation exactly becomes computationally intractable even for quantum computers due to the exponentially increasing resource requirements. The reason is that the number of quantum states grows exponentially with the number of particles or degrees of freedom in the system.
To overcome this challenge, researchers often rely on approximate methods, such as variational algorithms or quantum simulation techniques, to tackle the Schrödinger equation on quantum computers. These approaches aim to find approximate solutions with a smaller number of qubits or by exploiting certain symmetries or properties of the system.
In summary, the number of qubits needed to solve the Schrödinger equation exactly for a specific system grows exponentially with the size of the system, making it impractical for large systems. Approximate methods are typically employed to address the challenges posed by the computational complexity of exact solutions.