Quantum Gate Calculator

A quantum gate is an operation that transforms a qubit's quantum state, and this calculator applies four of the most fundamental single-qubit gates to a chosen input state: Pauli-X (the quantum equivalent of a classical NOT gate, flipping |0⟩ and |1⟩), Pauli-Y and Pauli-Z (which apply different phase and bit-flip combinations), and the Hadamard gate (which creates an equal superposition, putting a definite |0⟩ or |1⟩ into a 50/50 mix of both). Each gate is represented internally as a 2×2 matrix, and the calculator multiplies that matrix by the input state vector to get the output amplitudes, then squares those amplitudes to get the measurement probabilities P(|0⟩) and P(|1⟩) -- this is the Born rule, the standard quantum mechanical link between amplitude and observable probability. One caveat specific to Pauli-Y: this calculator works entirely in real numbers, but the true Pauli-Y matrix uses imaginary entries (0, -i; i, 0). The probabilities come out correct either way, since |±i|² = |±1|², but the Output |0⟩ Amplitude and Output |1⟩ Amplitude figures shown for Pauli-Y are real magnitudes (∓1) rather than the true imaginary amplitudes (∓i) -- worth knowing if you're checking this against a textbook derivation.

All four gates implemented here are IDEAL, noiseless unitary operations, so applying any of them to an already-normalized input state always produces another normalized state -- P(|0⟩) + P(|1⟩) always sums to 1. The State Norm output confirms that normalization check; it always reads 1.000 because these are exact unitary matrices, not because this calculator is modeling (and passing) any real-world noise test. This calculator does not compute a genuine gate-fidelity metric, which would compare a real, physically imperfect gate's actual behavior against its ideal target -- a real quantum processor's measured gate fidelity is always somewhat less than 1 because of noise sources like decoherence, miscalibration, and thermal relaxation that this simplified model doesn't include.