quantum state

A quantum state is the complete description of a quantum system at a moment in time. Think of it as the system's full "address": once you know the state, you know everything physics allows you to know about that system. For a quantum computer, the state tells you exactly what each qubit is doing and, crucially, how the qubits relate to one another. We write it as a vector, often called |psi>, living in an abstract mathematical space (a Hilbert space). A single qubit's state is just a blend of the two basic outcomes, |0> and |1>, with two numbers (amplitudes) saying how much of each.

The catch is how fast this grows. One qubit needs two amplitudes. Two qubits need four. Three need eight. In general, n qubits need 2^n complex amplitudes to write down completely, because the state can describe every combination of the qubits at once and how those combinations interfere. This is where quantum computing's huge state space comes from, and also why simulating a quantum computer on an ordinary computer gets impossibly expensive past a few dozen qubits: even storing the list of numbers would fill more memory than exists.

It is tempting to read that big number as "the computer holds all 2^n answers and checks them in parallel," but that is the wrong picture. You never get to read out all those amplitudes. When you measure, the state collapses and you get just one outcome, with a probability set by the size of its amplitude. The real skill in quantum computing is arranging the amplitudes so that, through interference, the answer you want ends up large and the wrong answers cancel out. The large state space is the raw material; useful interference is what turns it into a result.