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Quantum Lab

An interactive quantum mechanics learning platform and cryostat wiring co-design tool. From plain-language intuition to formal mathematics.

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Course outlineSingle-qubit gates and the Bloch sphere
Course Overview
States and Measurement
Qubits and state vectorsCore
Superposition and measurementCore
Wavefunction: the broader quantum idea
Operators, Evolution, and Uncertainty
Operators, eigenstates, and eigenvalues
The Schrödinger equation
The uncertainty principle
Gates, Phase, and Interference
Single-qubit gates and the Bloch sphereCore
Interference: why phase becomes visibleCore
Entanglement and Other Quantum Effects
EntanglementCore
Spin
Tunneling
Phase kickback and the road to algorithms
Course outline
Course Overview
States and Measurement
Qubits and state vectorsCore
Superposition and measurementCore
Wavefunction: the broader quantum idea
Operators, Evolution, and Uncertainty
Operators, eigenstates, and eigenvalues
The Schrödinger equation
The uncertainty principle
Gates, Phase, and Interference
Single-qubit gates and the Bloch sphereCore
Interference: why phase becomes visibleCore
Entanglement and Other Quantum Effects
EntanglementCore
Spin
Tunneling
Phase kickback and the road to algorithms
Home/Quantum Physics/Lessons/Single-qubit gates and the Bloch sphere
3

Gates, Phase, and Interference

The simulator makes this section concrete. A gate changes amplitudes. A phase change can look invisible until another gate turns it into a visible probability change. That conversion is interference, and it is the heart of quantum algorithms.

Single-qubit gates and the Bloch sphere

In one sentence: Single-qubit gates are reversible transformations, and the Bloch sphere gives you a geometric picture of how they move a qubit state.
Formula
∣ψ⟩↔(bx​,by​,bz​)
Simple intuition
The Bloch sphere turns an abstract state into a direction in space. Rotating that direction is often easier to picture than manipulating amplitudes directly.
Precise explanation
Pure one-qubit states can be represented as points on the surface of the Bloch sphere. Gates such as X, Z, and H correspond to rotations or reflections in this representation. When a qubit is entangled with others, its reduced state may move inside the sphere rather than on the surface.
Example or analogy
Example: |0⟩ sits at the north pole and |1⟩ at the south pole. Hadamard moves |0⟩ to an equatorial state, which matches what the simulator now shows in the Bloch sphere panel.
Common misconception
The Bloch sphere does not represent physical motion through ordinary space. It is a compact picture of the state of one qubit.
Why this matters
The new Bloch sphere view is one of the fastest ways to see the difference between a basis flip, a phase rotation, and a mixed state caused by entanglement.
Self-check
  • • What does it mean when the state vector sits on the equator of the Bloch sphere?
  • • Why can an entangled qubit appear inside the sphere instead of on its surface?
↗ Nielsen and Chuang, Quantum Computation and Quantum Information↗ MIT OCW 8.06: quantum computing notes▶ Watch the Bloch sphere
Operators, Evolution, and Uncertainty
The uncertainty principle
Gates, Phase, and Interference
Interference: why phase becomes visible