Hadamard-basis states |+⟩ and |−⟩

Every single-qubit protocol that needs a second, mutually unbiased measurement basis reaches for |+⟩/|−⟩ first: they are equal-weight superpositions in the computational (Z) basis, yet perfectly distinguishable in the X basis, making them the standard tool for basis-mismatch protocols like BB84 and for phase-kickback tricks in oracle-based algorithms.

TakesNothingWhat joins here

Nothing goes in, and that is what this entry is: a pipeline starts here. There is no upstream to choose — what you pick is what comes after.

Nothing in the Atlas meets this end.

Returns2 qubitsWhat joins here

Returns a 2-qubit register a next stage can take. This is the end that joins.

12 entries meet this end. Named below.

Prepares a state. Nothing goes in, and what comes out is a register another stage can take. See all 13 →

Fits after this 12

The Hadamard-basis states are

$|+\rangle = \frac{|0\rangle+|1\rangle}{\sqrt2}, \qquad |-\rangle = \frac{|0\rangle-|1\rangle}{\sqrt2}.$

Eigenstates of $X$ . Direct matrix multiplication gives $X|+\rangle = |+\rangle$ and $X|-\rangle = -|-\rangle$ , so $\{|+\rangle,|-\rangle\}$ is the eigenbasis of the Pauli- $X$ operator, exactly as $\{|0\rangle,|1\rangle\}$ is the eigenbasis of $Z$ . Because $HXH=Z$ and $HZH=X$ ( $H$ conjugation swaps the two Pauli operators), the Hadamard gate is precisely the change-of-basis unitary between the $Z$ and $X$ eigenbases: $|+\rangle=H|0\rangle$ , $|-\rangle=H|1\rangle$ .

Phase kickback. Preparing an ancilla in $|-\rangle$ before a controlled- $U_f$ oracle is the standard trick that converts a classical bit-flip oracle $|x\rangle|y\rangle \mapsto |x\rangle|y\oplus f(x)\rangle$ into a phase oracle $|x\rangle|-\rangle \mapsto (-1)^{f(x)}|x\rangle|-\rangle$ , because $|-\rangle$ is an eigenstate of the flip with eigenvalue $-1$ . This is the mechanism used by Deutsch–Jozsa, Bernstein–Vazirani, Simon's algorithm, and Grover's diffusion oracle.

Stabilizer view. $|+\rangle$ is stabilized by $X$ (the unique $+1$ eigenstate) and $|-\rangle$ by $-X$ ; both are single-qubit stabilizer states, sitting alongside $|0\rangle,|1\rangle,|+i\rangle,|-i\rangle$ as the six single-qubit states reachable by Clifford operations from $|0\rangle$ — the vertices of the stabilizer octahedron on the Bloch sphere, in contrast to the off-axis magic states elsewhere in this catalog.

TakesNothingWhat joins here

Nothing goes in, and that is what this entry is: a pipeline starts here. There is no upstream to choose — what you pick is what comes after.

Nothing in the Atlas meets this end.

Returns2 qubitsWhat joins here

Returns a 2-qubit register a next stage can take. This is the end that joins.

12 entries meet this end. Named below.

Prepares a state. Nothing goes in, and what comes out is a register another stage can take. See all 13 →

Fits after this 12

Qubits: 1
Depth: 1 gate (H, or X+H)
Eigenvalue: X = ±1
|+⟩: (|0⟩+|1⟩)/√2, X-eigenvalue +1
|−⟩: (|0⟩−|1⟩)/√2, X-eigenvalue −1
Basis: Mutually unbiased with {|0⟩,|1⟩}

H on q[0] (|+⟩)
X on q[1] (|−⟩)
H on q[1] (|−⟩)

q[0] (|+⟩) Z-basis outcome50%

q[1] (|−⟩) Z-basis outcome50%

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.

FrameworkNative

plus_minus_states.py

import numpy as np
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector, Operator

def plus_state() -> QuantumCircuit:
    qc = QuantumCircuit(1)
    qc.h(0)
    return qc

def minus_state() -> QuantumCircuit:
    qc = QuantumCircuit(1)
    qc.x(0)
    qc.h(0)
    return qc

X = Operator.from_label("X").data
plus = Statevector.from_instruction(plus_state())
minus = Statevector.from_instruction(minus_state())

print("|+> =", plus.data)   # [0.7071, 0.7071]
print("|-> =", minus.data)  # [0.7071, -0.7071]
print("X|+> = +|+>:", np.allclose(X @ plus.data, plus.data))
print("X|-> = -|->:", np.allclose(X @ minus.data, -minus.data))

FINAL_CIRCUIT = plus_state()

Where the map uses this

This record is an instance of an object the map names, so these are the processes that consume or produce one. None of them is about this record in particular.

State you can prepare 5 of 34 processes

Quantum linear solve hands one back
Prepare an input state hands one back
Amplify a success branch hands one back
Estimate an observable takes one
Minimise the objective over the parameters hands one back