Quantum Discord — Quantum Evolution Builder

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The question

You know evolution is continuous. Now: can you control it precisely enough to land on a specific state?

Initial State

Hamiltonian H

        H determines the axis of rotation on the Bloch sphere. Engineers choose H by applying a physical field.
      

H matrix (Pauli)

σ_x = [[0, 1], [1, 0]]

Frequency ω1.0

        U = e−iHθ/2  →  higher ω scales rotation rate
      

Rotation Angle θ

0.000 rad · 0°

0 π/2 π 3π/2 2π

Jump to Named Gate

        Each gate is a named point along the continuous evolution. Click to jump.
      

Named gates

Rotation gates (set θ below)

State

Hamiltonian

Discover

Current state |ψ(θ)⟩

|ψ⟩ = 1|0⟩

Probabilities

|0⟩

100%

|1⟩

Relative Phase

φ

—

phase of β
relative to α

|α|² + |β|² =1.0000 ✓

Gate at this angle

Move to a named angle to reveal the equivalent gate.

continuous evolution Every angle θ produces a valid quantum state. Gates are the angles engineers stop at — nothing more.

What is the Hamiltonian?

physical meaning The Hamiltonian H encodes the energy of the system. For a qubit, H is determined by the physical field applied — a magnetic field, a laser, a microwave pulse. H tells the qubit which axis to rotate around.

Generator of evolution

The connection

U(θ) = e^−iHθ/2
H generates U. θ controls how long the field is on.

H for each axis

σ_x (X-axis rotation)

H = [[0, 1], [1, 0]] → Rx(θ)

σ_y (Y-axis rotation)

H = [[0, −i], [i, 0]] → Ry(θ)

σ_z (Z-axis rotation)

H = [[1, 0], [0, −1]] → Rz(θ)

Matrix exponential intuition

e^−iHθ/2 — what does it mean? For a Pauli Hamiltonian, the matrix exponential simplifies to:

e^−iσθ/2 = cos(θ/2)·I − i·sin(θ/2)·σ

This is exactly the rotation matrix. The exponential is not a coincidence — it is how rotation appears in quantum mechanics. H is the generator; e^−iHθ/2 is the rotation it generates.

Gates are controlled evolution

the engineering view An engineer applies a field (chooses H), controls the duration θ, and stops the evolution at a named point. That stopping point is what we call a gate.

Gates are not separate objects. They are controlled stops on continuous evolution.

Physical implementation

Superconducting qubit → microwave pulse → σ_x or σ_y
Trapped ion → laser beam → any axis
Spin qubit → magnetic field → σ_z (Larmor precession)
Same physics. Different hardware. Same mathematics.

Before moving the slider: if you apply H = σ_x and evolve for θ = π — what happens to |0⟩?

The state arrived at |1⟩ at exactly θ = π — but it traveled through every intermediate state on the way. No jumps. The gate X is just the name for where the evolution stopped. The continuous evolution produced the gate.

The probabilities stayed at 50% / 50% throughout. σ_z rotates around the Z axis — which only changes phase, not measurement probabilities. The S and T gates (at π/2 and π/4) are pure phase evolutions. Same continuous rotation, different physical effect.

At θ = π/2 with σ_x, you get √X — the square root of X. Apply it twice and you get X. It is a perfectly valid gate, used in real quantum computers. Every angle is a valid unitary. The named gates are just the ones engineers gave names to.

H = σ_y, θ = π → Ry(π) → the state lands at |1⟩, just like Rx(π). All three Pauli Hamiltonians, at θ = π, flip the population between |0⟩ and |1⟩ — through different paths on the Bloch sphere. The gate depends on which Hamiltonian was applied and for how long. That is the complete picture of quantum gate engineering.

Investigation complete.