Quantum Computing Made Simple

Measure superposition indirectly by analyzing outcomes from many identically prepared qubits; amplitudes may be negative or complex, and probabilities reveal their magnitudes.

Represent a qubit as a two-dimensional column vector with complex amplitudes alpha and beta, covering zero, one, and equal superposition, and apply linear algebra for vector addition and unit-length normalization.

Explore the main physical implementations of quantum computers, including photons, ion traps, and superconducting circuits, along with their pros, cons, and impact on qubit performance.

Explore real and complex numbers, the imaginary unit i with i squared equals minus one, and how complex numbers form the set C; amplitudes in quantum physics illustrate their use.

Represent complex numbers on the complex plane by real and imaginary parts, compute their modulus, and square it to obtain qubit state probabilities in a superposition that sums to one.

Visualize complex numbers as vectors in complex plane and express z as modulus times (cos φ + i sin φ), with angle φ or phase.

Represent complex numbers using modulus and argument in their exponential form with Euler's formula, using e, cosine, sine, and the example z = 1 + 4i.

Explore how complex numbers define a one-dimensional Hilbert space for a qubit, using H1 and H2, with two amplitudes in the complex plane.

Treat the qubit state as a complex-vector in Hilbert space, a model describing the qubit’s state, with dot products via complex conjugates and bra- and ket-vectors, including orthogonality.

Demonstrate how measurement devices define qubit states, collapsing superpositions to zero or one. Identify the standard basis, orthonormality, vector length, and the Hadamard basis.

learn the basics of linear algebra with matrices: add, subtract, and multiply matrices; understand size requirements, scalar and matrix multiplication, and how to apply the same rules to vectors.

Explore tensor product, which combines vectors and matrices into bigger results without size limits, and apply it to multi-qubit systems and quantum operators.

Identify pure qubit states: zero, one, and their superpositions described by bra and ket vectors with amplitudes alpha and beta. Contrast mixed states as statistical mixtures, not superpositions.

Describe density matrices as representations for pure and mixed quantum states, using bra-ket vectors and their tensor products, and note decoherence drives mixing.

Learn to compute the conjugate transpose, or dagger, by transposing a matrix and conjugating its entries, and see why Hermitian matrices equal their conjugate transpose in quantum computing.

Explore unitary matrices, where the conjugate transpose equals the inverse, so multiplying a matrix by its conjugate transpose yields the identity; some matrices are both hermitian and unitary.

Explains how the Hadamard gate, denoted by H, transforms state vectors into 50/50 superpositions and defines the Hadamard basis, while showing its hermitian and unitary properties.

Explore the pauli matrices, especially the x gate. See how the x gate flips qubit states like a quantum not, differs on superposition, and remains hermitian and unitary.

Explore how the Y gate introduces the complex phase i and flips basis states, mapping |0> to i|1> and |1> to −i|0>, with corresponding probabilities for superpositions.

The Z gate, the third Pauli matrix, changes the phase of the qubit without altering measurement probability. It flips the sign of the amplitude for state one, including in superposition.

Visualize a qubit state on the Bloch sphere, a three-dimensional unit sphere with x, y, z axes, illustrating pure and mixed states and amplitude phase reduction.

Describe the Bloch sphere for a qubit, where theta and phi define amplitudes and yield x, y, z via sin theta cos phi, sin theta sin phi, cos theta.

Explore how the CNOT (CX) gate, a two-qubit interaction, flips the target qubit when the control is one, and leaves it unchanged when the control is zero.

Explore quantum entanglement between two qubits, producing perfect correlations regardless of distance, and learn to prepare it with Hadamard and CNOT gates, or inverse correlations with an X gate.

Explore direct correlation and inverse correlation in Bell states built from entangled qubits. Learn to generate all four Bell states using Hadamard, CNOT, and X gates.

Explore how to swap two qubits using CNOT gates and the SWAP gate, including a three-gate sequence and the single matrix representation.

Explore the no-cloning theorem, proving that arbitrary unknown qubit states cannot be copied by any unitary operator, while showing swapping and teleportation preserve or move states.

Learn how to apply Python to quantum computing using the Qiskit library, which is common, simple, well documented, and widely used despite occasional syntax changes.

Explore Google Colab for researchers as a ready-to-use platform that runs programs, includes preinstalled data science libraries, and simplifies installing NumPy, SciPy, pandas, and scikit-learn for quantum computing work.

Learn to use python for quantum computing in jupyter notebook on colab, run code instantly, and mix code with text to document your experiments.

Install qiskit in colab, build a quantum circuit with quantum and classical registers, apply an x gate, measure, and run on a qasm_simulator with execute to obtain counts.

Create a two-qubit circuit in Qiskit, apply a hadamard gate to qubit zero and a cx gate to form a Bell state, then measure to reveal entanglement.

Learn quantum teleportation, a practical quantum computing protocol that transfers a qubit's state from Alice to Bob using entanglement, Bell states, and a three-qubit circuit.

Construct the quantum teleportation circuit using a cnot gate on an entangled pair and show how the state transfers to a distant qubit q2.

Describe how a CNOT operation and Hadamard gate transform qubits, express states with amplitudes, and factor out common terms using tensor products to reveal intermediate states.

Explain the four measurement outcomes from Alice's qubits in quantum teleportation and show how Bob applies X and Z gates to recover the psi state on his qubit.

Explore the IBM quantum computing service and circuit composer to teleport a qubit by entangling q1 and q2, using Hadamard and CNOT, then measuring and applying X or Z.