A qubit state is a point on this sphere. Apply gates to rotate it.
The Bloch sphere is the standard geometric picture of a single quantum bit (qubit). Every pure state of a qubit corresponds to exactly one point on the surface of a unit sphere. This simulation lets you steer that point directly with sliders, or apply the fundamental quantum gates and watch the state vector rotate in real time.
|ψ⟩ = cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩
r = (sinθ cosφ, sinθ sinφ, cosθ)
P(|0⟩) = cos²(θ/2), P(|1⟩) = sin²(θ/2)
Although orthogonal quantum states are 90° apart in Hilbert space, on the Bloch sphere |0⟩ and |1⟩ sit at opposite poles — 180° apart. This factor-of-two "spinor" behaviour is the same mathematics that makes an electron return to itself only after a 720° rotation.
What is the Bloch sphere?The Bloch sphere is a geometric representation of the pure states of a single qubit. Every pure state corresponds to a unique point on the surface of a unit sphere, with the north and south poles representing the computational basis states |0⟩ and |1⟩.
How is a qubit state mapped onto the sphere?A pure qubit state is written |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩. The angles θ (polar) and φ (azimuthal) place the state vector at the point (sinθ cosφ, sinθ sinφ, cosθ) on the unit sphere.
What do the poles of the Bloch sphere mean?The north pole (+z) is the state |0⟩ and the south pole (−z) is the state |1⟩. Points on the equator are equal superpositions of |0⟩ and |1⟩, differing only by their relative phase φ.
The Pauli-X gate is a 180° rotation about the x-axis. It is the quantum NOT gate: it swaps |0⟩ and |1⟩, sending the north pole to the south pole and vice versa.
The Hadamard gate H is a 180° rotation about the axis halfway between x and z. It maps |0⟩ to (|0⟩+|1⟩)/√2 (the +x point on the equator), creating an equal superposition that is the starting point of many quantum algorithms.
The S gate (phase gate) rotates the state by 90° about the z-axis, and the T gate by 45°. They change the relative phase φ without altering the probabilities of measuring |0⟩ or |1⟩.
The probability of measuring |0⟩ is cos²(θ/2) and of measuring |1⟩ is sin²(θ/2). These depend only on the polar angle θ — the height of the state vector along the z-axis — not on the phase φ.
Because qubit states are equivalent up to a global phase, orthogonal states |0⟩ and |1⟩ sit at opposite poles, 180° apart on the sphere even though they are at 90° in the abstract Hilbert space. The factor of 1/2 reconciles the two pictures.
Yes. Pure states lie on the surface, while mixed states lie inside the ball. The fully mixed state sits at the centre. This simulation visualises pure states, so the vector always reaches the surface.
It turns abstract complex amplitudes into intuitive geometry: single-qubit gates become rotations, and algorithms become sequences of rotations. This makes it the standard mental model for reasoning about and debugging quantum circuits.
The Bloch sphere is a geometric representation of the pure quantum states of a single qubit. Every possible pure state of a qubit corresponds to a unique point on the surface of a unit sphere, with the north pole representing |0⟩, the south pole representing |1⟩, and all points on the equator representing equal superpositions of the two. This simulation lets you manipulate the state vector directly with sliders or by applying standard quantum gates (X, Y, Z, H, S, T), and watch the resulting rotation animate in real time on an interactive 3D sphere rendered with Three.js.
The Bloch sphere was formalised in the context of nuclear magnetic resonance by Felix Bloch in 1946, and has since become the standard mental model for single-qubit quantum computing. It underpins the design of quantum gates, error-correction protocols, and the analysis of qubit decoherence in real hardware such as superconducting transmons and trapped-ion qubits.
The Bloch sphere is a unit-sphere representation of all pure states of a single qubit. The north pole corresponds to the basis state |0⟩ and the south pole to |1⟩, while all other points on the surface encode quantum superpositions with different amplitudes and phases. It is the standard geometric tool for visualising and reasoning about single-qubit operations.
Use the theta (polar) slider to move the state vector from the north pole (theta = 0, state |0⟩) to the south pole (theta = 180°, state |1⟩). The phi (azimuthal) slider rotates the vector around the z-axis, changing the quantum phase. Clicking the gate buttons (X, Y, Z, H, S, T) animates the rotation corresponding to that quantum gate. Use Pause to freeze the animation and Reset to return to the |0⟩ state. You can also drag the canvas to orbit the camera.
Every point on the equator (theta = 90°) is an equal superposition of |0⟩ and |1⟩, meaning a measurement has a 50% chance of returning each outcome. The different points on the equator differ only by their relative phase phi, which affects interference in multi-qubit algorithms but has no effect on single-qubit measurement probabilities. The +x point (phi = 0) is the |+⟩ state created by applying the Hadamard gate to |0⟩.
A pure qubit state is written |psi⟩ = cos(theta/2)|0⟩ + exp(i*phi)*sin(theta/2)|1⟩, where theta (0 to pi) is the polar angle and phi (0 to 2*pi) is the azimuthal phase. The corresponding Bloch vector is r = (sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta)), which is a unit vector pointing to the surface of the sphere. The factor of 1/2 in the angle argument arises because qubit states form a spin-1/2 system and must be rotated by 720° (not 360°) to return to themselves.
In superconducting qubit processors (such as those made by IBM and Google), single-qubit gates are implemented as microwave pulses that rotate the Bloch vector by a precise angle around a chosen axis. Engineers use the Bloch sphere to visualise pulse calibration and diagnose gate errors. In nuclear magnetic resonance (NMR) spectroscopy and MRI, the same picture describes spin-1/2 nuclei precessing in a magnetic field, with the Bloch equations governing relaxation back to equilibrium.
This is a common misconception. A qubit in superposition is in a definite quantum state — a specific point on the Bloch sphere — not simultaneously in two states. The superposition means that upon measurement, each outcome has a calculable probability, and the qubit's state is fully described by the two angles theta and phi. What superposition enables is quantum interference: the amplitudes can add or cancel when combined with other qubits, which is the resource exploited by quantum algorithms.
The representation is named after Swiss-American physicist Felix Bloch, who introduced the Bloch equations in his 1946 paper on nuclear induction, describing the dynamics of spin-1/2 particles in magnetic fields. The geometric sphere picture was implicit in that work and became standard in quantum optics and computing over subsequent decades. Bloch shared the 1952 Nobel Prize in Physics with Edward Purcell for their development of nuclear magnetic resonance techniques.
The Bloch sphere is closely related to the Poincare sphere, which represents the polarisation states of a photon using the same mathematical structure. It also connects to quantum entanglement simulations: while a single qubit lives on the Bloch sphere, two entangled qubits require a higher-dimensional space that cannot be visualised as simply. Related simulations include quantum circuit builders, spin precession in magnetic fields, and Rabi oscillation visualisers that show the qubit cycling between |0⟩ and |1⟩ under a resonant drive.
In quantum error correction, the Bloch sphere helps visualise decoherence: noise causes the state vector to shrink toward the centre of the ball (the fully mixed state), representing the loss of quantum information. T1 relaxation (energy decay) drives the vector toward the north pole, while T2 dephasing shrinks it toward the z-axis. Error-correcting codes such as the surface code aim to detect and reverse these drifts before they accumulate, and engineers use Bloch sphere tomography — applying many measurements in different bases — to reconstruct the current qubit state and benchmark gate fidelity.
Beyond the single-qubit Bloch sphere, researchers are developing higher-dimensional geometric representations for multi-qubit and qudit (d-level) systems, such as the Husimi Q-function and Wigner function on phase space. Geometric phases — most famously the Berry phase — arise when a qubit's Bloch vector is transported around a closed loop, and these are being exploited in holonomic quantum computing to build gates that are intrinsically robust to certain noise types. Understanding the geometry of quantum state space is also central to the field of quantum information geometry and the design of optimal quantum sensors.