Quantum Computing Calculator

CNOT is a two-qubit gate -- it flips a target qubit's state conditioned on a separate control qubit's value, which requires two qubits with a defined control/target relationship to mean anything. This calculator's Quantum Gates mode only accepts a single input qubit, so there is no well-defined state for CNOT to act on here; it remains a real and important part of a universal gate set, just not something a single-qubit tool can apply.

Several standard single-qubit gates -- Pauli-Y, the S (phase) gate, and the T gate -- are defined with an imaginary unit i in their matrix, so applying them to an otherwise real-valued input state can rotate part of that state into the imaginary axis of the complex plane. This is a genuine feature of quantum mechanics, not a calculator quirk: the imaginary component is just as physically real as the real component and matters for how the qubit's state evolves under further gates.

The two algorithms solve fundamentally different kinds of problems. Grover's search speeds up an unstructured search from O(N) classical steps to O(√N) quantum steps -- a quadratic improvement that holds for any searchable problem. Shor's algorithm exploits number-theoretic structure specific to integer factoring to go from sub-exponential classical time to polynomial quantum time, a far larger, exponential-class speedup that doesn't generalize to arbitrary problems the way Grover's does.

Real quantum hardware qubits are noisy and decohere quickly, so a single physical qubit can't reliably hold information long enough to run a useful algorithm. Error correction spreads one logical qubit's information redundantly across many physical qubits so that errors on individual qubits can be detected and corrected without directly measuring (and destroying) the protected quantum state -- the more error-resistant the code, the more physical qubits that redundancy costs.