Subject overview

Quantum

Simulate a single qubit with quantum gates, explore complex amplitudes and measurement probabilities, and follow state rotations on an interactive 3D Bloch sphere.

Qubit

Variables

αcomplex amplitude of |0⟩: 1dimensionless
βcomplex amplitude of |1⟩: 1dimensionless
θpolar angle: radradian
φazimuthal angle: radradian

Key Formulas

Pure state

|\psi\rangle=\alpha|0\rangle+\beta|1\rangle,\quad |\alpha|^2+|\beta|^2=1

Bloch representation (up to global phase)

|\psi\rangle\sim\cos\frac\theta2|0\rangle+e^{i\phi}\sin\frac\theta2|1\rangle

Bloch vector

\mathbf r=(2\operatorname{Re}(\alpha^*\beta),\;2\operatorname{Im}(\alpha^*\beta),\;|\alpha|^2-|\beta|^2)

Gate action

\begin{pmatrix}\alpha'\\\beta'\end{pmatrix}=G\begin{pmatrix}\alpha\\\beta\end{pmatrix}

Z-basis probabilities

P(0)=|\alpha|^2,\quad P(1)=|\beta|^2

X = \begin{pmatrix}0&1\\1&0\end{pmatrix}

Y = \begin{pmatrix}0&-i\\i&0\end{pmatrix}

Z = \begin{pmatrix}1&0\\0&-1\end{pmatrix}

√X

\sqrt{X} = \frac12\begin{pmatrix}1+i&1-i\\1-i&1+i\end{pmatrix}

√Z

\sqrt{Z} = S = \begin{pmatrix}1&0\\0&i\end{pmatrix}

H = \frac{1}{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}

Rₓ

R_x(\gamma)=\begin{pmatrix}\cos\frac\gamma2&-i\sin\frac\gamma2\\-i\sin\frac\gamma2&\cos\frac\gamma2\end{pmatrix}

Rᵧ

R_y(\gamma)=\begin{pmatrix}\cos\frac\gamma2&-\sin\frac\gamma2\\\sin\frac\gamma2&\cos\frac\gamma2\end{pmatrix}

R_z(\gamma)=\begin{pmatrix}e^{-i\gamma/2}&0\\0&e^{i\gamma/2}\end{pmatrix}

U(\theta,\phi,\lambda)=\begin{pmatrix}\cos\frac\theta2&-e^{i\lambda}\sin\frac\theta2\\e^{i\phi}\sin\frac\theta2&e^{i(\phi+\lambda)}\cos\frac\theta2\end{pmatrix}

Requires complex numbers, vectors and matrix multiplication. This simulator describes one pure, isolated qubit with ideal unitary gates. It excludes noise, mixed states, entanglement and measurement collapse. Probabilities refer to measurement in the computational (Z) basis.

X, Y and Z rotate the Bloch vector by π about their respective axes; √X and √Z rotate it by π/2. H rotates by π about (x + z)/√2. Global phase is retained in the amplitudes but is invisible on the sphere and does not affect measurement probabilities.

For U, θ is in [0, π] and φ, λ are in [0, 2π). Gate animations illustrate continuous unitary paths, not physical pulse timing. The azimuth φ is undefined at the poles.

IBM Quantum Learning — Bits, gates, and circuits

Calculate

Pauli gates

Square-root gates

Hadamard

Rotation gates

γ rotation angledefault π/2

General gate

θ polar rotationdefault π/2

φ output phasedefault 0

λ input phasedefault 0

Loading interactive 3D view…

Current state

\displaystyle |\psi\rangle \allowbreak =\nobreak \alpha|0\rangle + \beta|1\rangle \allowbreak \approx\nobreak \begin{pmatrix}1.00\\0.00\end{pmatrix}

Z-basis probabilities

\displaystyle P(0)\allowbreak =\nobreak |\alpha|^2\allowbreak \approx\nobreak 1.00,\quad P(1)\allowbreak =\nobreak |\beta|^2\allowbreak \approx\nobreak 0.00

Bloch coordinates

\displaystyle (x,y,z)\allowbreak \approx\nobreak (0.00,0.00,1.00)

\displaystyle \theta\allowbreak \approx\nobreak 0.00\,\mathrm{rad},\quad \phi \allowbreak =\nobreak \text{undefined at the poles}