The Hadamard gate H is the standard way to create a superposition from a definite computational-basis state. This simulator lets you pick a starting state, apply H, and watch exactly what happens to the Bloch vector and the measurement probabilities — including what happens when you apply it a second time.
• A choice of starting state, |0> or |1>. • A single H gate button that can be applied repeatedly, with a running application counter. • A live Bloch-sphere view of the vector before and after each application. • Live probability bars for P(0) and P(1). • Automatic recognition and labeling of |0>, |1>, |+> and |-> as they are reached. • The full 2x2 Hadamard matrix and its action on both basis states.
• Start at |0>, apply H once, and confirm the probability bars split to exactly 50/50 while the Bloch vector lands on the equator at |+>. • Apply H a second time and watch the vector return exactly to the north pole |0> with 100% probability — H is its own inverse. • Start at |1> instead and repeat the same experiment to see the mirror-image result at |->.
Because H turns a certain classical-looking input into a genuine superposition in a single step, it is the standard first move in circuits that need to explore multiple possibilities in parallel before later gates use interference to combine them into a useful answer. The Quantum Circuit and Bell-State-Preparation labs both use H as their opening gate for exactly this reason.
It takes a definite computational-basis state, |0> or |1>, and turns it into an equal superposition: H|0> gives |+> = (|0>+|1>)/sqrt(2), and H|1> gives |-> = (|0>-|1>)/sqrt(2). Geometrically, it rotates the Bloch vector from a pole onto the equator.
You get back exactly the state you started with. The Hadamard gate is its own inverse (H squared equals the identity), so applying it twice to |0> returns |0>, and twice to |1> returns |1> — you can confirm this directly with the application counter.
It is the standard way to create superposition from a definite starting state, letting a circuit explore multiple computational paths at once before later gates recombine them through interference. Nearly every quantum algorithm opens with a layer of Hadamard gates for exactly this reason.
No — unlike the Y gate or phase gates, every entry in the Hadamard matrix is a real number (plus or minus one over the square root of two), which is part of why it is usually the first gate introduced in quantum computing.