No, it is not possible to build a universal quantum computer using only two qubits. In order to achieve universal quantum computation, a quantum computer needs a sufficient number of qubits and the ability to manipulate and control those qubits effectively.
A universal quantum computer is capable of simulating any other quantum system efficiently, as well as executing a wide range of quantum algorithms. To achieve this level of versatility, a quantum computer requires a sufficiently large number of qubits and the ability to perform operations on those qubits.
Quantum algorithms such as Shor's algorithm for factoring large numbers or Grover's algorithm for searching unsorted databases typically require a certain number of qubits to achieve their computational advantages. For example, Shor's algorithm requires a number of qubits that grows with the size of the number being factored.
While two-qubit systems, such as qubits represented by particles with two distinguishable states (e.g., spin-up and spin-down), can exhibit quantum behavior, they are limited in their computational power. Such systems can perform specific tasks and demonstrate quantum phenomena, but they lack the capacity for universal quantum computation.
Building a universal quantum computer capable of outperforming classical computers for a wide range of tasks typically requires a larger number of qubits and the ability to implement quantum gates and operations on those qubits. Current quantum computers being developed and researched aim to scale up the number of qubits to achieve the computational power necessary for universal quantum computation.