Gates · Single-qubit gate
Hadamard gate
The smallest useful reference for putting a qubit into an equal superposition.
Exact & formalUpdated
A small worked example, computed in your browser. It is not a result from this record's papers.
H turns a definite bit into an even superposition
One qubit starting at |0⟩, a state that reads 0 with certainty.
Step 1 of 1 · H
Probability of each outcome after this step
- 050%
- 150%
Splits the single certainty into two equal halves. Applying H a second time would put it back — H is its own inverse, so this is a change of basis rather than a loss of information.
Probability spreads from 1 outcome to 2. All 2 are equally likely.
Readout Measuring gives 0 or 1, each half the time. Almost every algorithm in the Atlas starts with a layer of these.
Open in Studio- Steps1
- Wires1
- Qubits1
- Depth1 gate
- Outcomes2
- Papers1
Hadamard is a good first record because its behavior is easy to inspect mathematically and in a measured distribution.
Takes1 qubitWhat joins here
Takes a 1-qubit register and states no assumption about what is on it. That is the whole condition: anything returning 1 qubits can feed it.
15 entries meet this end. Named below.
Returns1 qubitWhat joins here
Returns a 1-qubit register a next stage can take. This is the end that joins.
13 entries meet this end. Named below.
A unitary. The same register goes in and comes out, and this entry states no assumption about what is on it — which is what lets anything of the same width feed it. See all 29 →
Hadamard is the single-qubit unitary . It sends and , so a computational-basis measurement always returns or with probability regardless of which state it acted on — the difference is the relative sign in front of , a phase that only becomes visible once another gate lets the two branches interfere. is both Hermitian and unitary, so , and direct matrix multiplication gives , exactly the identity this record's matrix check confirms. Geometrically is a rotation of the Bloch sphere about the axis , exchanging the and eigenbases. It is the standard way to open superposition-based circuits such as Bell states, GHZ states, Grover diffusion, and the QFT.
Takes1 qubitWhat joins here
Takes a 1-qubit register and states no assumption about what is on it. That is the whole condition: anything returning 1 qubits can feed it.
15 entries meet this end. Named below.
Returns1 qubitWhat joins here
Returns a 1-qubit register a next stage can take. This is the end that joins.
13 entries meet this end. Named below.
A unitary. The same register goes in and comes out, and this entry states no assumption about what is on it — which is what lets anything of the same width feed it. See all 29 →
- Qubits
- 1
- Depth
- 1 gate
- Outcomes
- 2
- Input
- |0⟩ or |1⟩
- Output
- Equal-amplitude superposition
- Measurement
- P(0) = P(1) = 0.5
Hon q[0]
Quantum vs classical
Classical baseline
Use a classical state-vector or matrix simulation at the same width, precision, and measurement objective.
Quantum claim
The quantum record demonstrates a state or operator behavior; it does not make classical simulation or communication costs disappear.
How to compare
Compare fidelity, samples, gate depth, noise, memory, and the cost of preparing and reading the state.
Nobody has reviewed this record for gaps yet.
OPENQASM 3.0;
include "stdgates.inc";
qubit q;
h q;Related entries
- Bell state measurement
Basic circuits
- GHZ state preparation
Basic circuits
References
- Quantum Computation and Quantum Information: 10th Anniversary Edition
Michael A. Nielsen and Isaac L. Chuang · 2010
Standard graduate textbook covering the single- and two-qubit gate definitions, matrix identities, and circuit primitives this record states.