Gates · Pauli operator
Pauli-Z gate
A phase-flip gate whose behavior is invisible to a basis measurement until interference is introduced.
Exact & formalUpdated
A small worked example, computed in your browser. It is not a result from this record's papers.
Z, the half-turn of phase
One qubit, put into |+⟩ = (|0⟩ + |1⟩)/√2 first, because on |0⟩ or |1⟩ alone this gate changes nothing a measurement or another gate can ever detect.
Step 1 of 2 · H
Probability of each outcome after this step
- 050%
- 150%
Puts the qubit into an equal superposition. Without this there is no second amplitude for the gate's phase to be measured against, and a phase on the whole state is not observable.
Probability spreads from 1 outcome to 2. All 2 are equally likely.
Multiplies the |1⟩ half of the state by a phase of π, and leaves |0⟩ alone. That is a sign flip: the largest phase there is, and the point past which turning further starts coming back. The two outcomes stay equally likely — the difference is carried entirely in the phase between them, which is what the panel beside the bars shows.
The outcome probabilities do not move at all. What this step changes is phase — 2 distinct relative phases across the state now — which no measurement can see until a later step turns it back into probability.
Readout Measuring gives 0 or 1, each half the time — exactly as it did before the gate. A single measurement of this qubit cannot detect this gate at all. Its effect becomes visible only when the phase is turned back into probability by a later interference step, which is what every algorithm that uses it does.
Open in Studio- Steps1
- Wires1
- Qubits1
- Depth1 gate
- Outcomesbasis-preserving
- Papers1
Pauli-Z is the smallest useful record for explaining why phase is operational even when probabilities look unchanged.
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 →
Pauli- is the phase-flip operator . On the computational basis, and , exactly the identity checked here by comparing the matrix action against both basis states directly. Because a measurement in the computational basis cannot distinguish from (both give outcome "1" with probability 1), is invisible on a bare basis state; its effect only shows up once the qubit is in a superposition, where the flipped sign changes how later amplitudes interfere — the point the entry's own explanation makes. is Hermitian and unitary with , so it is its own inverse, and it is a rotation of the Bloch sphere about the -axis. underlies the phase-kickback trick used in Deutsch–Jozsa-style oracles and phase-flip error models in stabilizer codes.
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
- basis-preserving
- Matrix
- [[1, 0], [0, -1]]
- Rotation
- π around Z
- Square
- I
Zon 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;
z q;Related entries
- Pauli-Y gate
Gates
- Hadamard gate
Gates
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.