QC101 Quantum Computing & Intro to Quantum Machine Learning

Explore how photon polarization encodes qubits, using vertical and horizontal polarization and polarizing filters to illustrate 0 and 1 states and probabilistic transmission at angles.

Observe how a calcite crystal splits incoming photons into two streams: one aligned with the crystal and one anti-aligned at 90 degrees. Contrast this with polarizing filters, where only the aligned path emerges, and when input is neither aligned nor anti-aligned, exit paths follow probabilistic rules with probabilities between 0 and 1.

Explore how loss of information arises in quantum measurements by demonstrating the angle of polarization using a polarizing filter and a calcite crystal, showing aligned versus anti-aligned paths.

Map boolean true to 1 and on; map false to 0 and off. Use gates to produce on or off outputs, including and, or, not operations.

Demonstrate the two-input and gate with inputs A and B, whose output is on only when both inputs are on, and show its connection to the next or gate.

See how the or gate outputs on when inputs are on, and off when inputs are off, with an and-gate feeding one input and C the other.

Explore the nand gate as a universal gate, deriving and implementing and, or, and not gates from nand through truth tables.

Explore how the nand gate operates, its truth table, and how nand alone can implement not, and, or gates as a universal gate.

Explore the exclusive-or gate, its xor truth table and circuit symbol, and why in quantum architectures we apply b onto a atomically, sacrificing a's original data.

Demonstrate element-wise XOR of bit sequences, showing how each corresponding pair yields an output bit according to the XOR truth table, with step-by-step examples.

Encrypt and decrypt with a shared secret, a bit sequence known to the sender and the receiver. XORing the message with this secret, then XORing again, restores the original message.

Learn how probability theory models uncertain future events using likelihoods and probability distributions. See how quantum physics relies on probabilistic outcomes and data-driven estimates to predict often unpredictable results.

Identify independent events as unrelated outcomes whose occurrences do not influence each other; two dice throws and two coin tosses are classic examples.

Explore independent events and how A and B do not influence one another. Calculate P(A and B) = P(A) × P(B); with P(A)=0.3 and P(B)=0.4, P(A and B)=0.12.

Compute P(A) from P(A and B) and P(A and not B) using boolean algebra and the algebra of probabilities for composite events, illustrated by Alice's red and blue car example.

Explore probability with a Venn diagram by identifying A, B, and their union P(A or B) as the red region.

Explore translating probability expressions into diagram areas with Venn diagrams, compute P(A AND B), and relate AND, OR, NOT to intersection, union, and complement.

Explore how statistics analyzes probabilities of boolean and real-valued random functions, from true or false events to weather variables like maximum temperature and wind speed.

Explore random variables as uncertain outcomes mapped to numbers, using dice and coin tosses. Analyze aggregate behavior and compute the average value, revealing a zero mean over many trials.

Show how the mean alone fails to capture a random variable's behavior by comparing coin and die mappings, highlighting spread and magnitude of outcomes.

Explore how combining random variables forms a new variable and how its mean equals the sum of individual means, illustrated by coin toss and dice examples.

Examine correlation by analyzing independent, dependent, and mutually exclusive events, showing when P(A and B) equals or differs from P(A) times P(B) and related outcomes.

Multiply a real number by a complex number and apply scalar multiplication to a+bi as ka + kbi, with i treated like an unknown and examples like 3(2+3i) and (-4)(2-3i).

Learn to multiply complex numbers by treating i as an unknown, use i^2 = -1, and express results in standard form a + bi, as shown with (3+4i)(5+6i) = -9+38i.

Learn to divide complex numbers by multiplying numerator and denominator by the complex conjugate, turning the denominator into a real number (its squared magnitude), then simplify the complex numerator.

Practice solving complex division problems using the complex conjugate method, yielding real denominators and simplified results like -11/85 - (58/85)i and -2 + 5i.

Use polar form to find roots of complex numbers and compute the square root of i as 1/root(2) + i/root(2) via Euler's formula.

Understand matrices as tables of numbers and learn that a matrix’s dimension is the number of rows by columns. Remember: rows first, followed by columns, written as rows by columns.

Learn to add matrices by ensuring same dimensions, add corresponding elements, and work through real and complex number examples in 2x2 matrices.

Explore matrix subtraction by performing per-element operations on equal-sized matrices, illustrated with a 2x2 example and verifications, including subtraction with complex numbers.

Work through a complex numbers example to compute a matrix element as the product of a yellow row and orange column, yielding 9+i, 16+2i, 14, and 10.

Explore multiplying non-square matrices, such as a 3 by 2 example, and follow along with the animation to see how the matrix product is computed.

Explore the outer product by multiplying a column matrix with a row matrix to form a table of all possible products, illustrated with an animation.

Explore the concept of inverses in real numbers and matrices, including the notation x^-1 and A^-1, and how A times A^-1 equals the identity matrix.

Learn how the transpose transforms a matrix by interchanging rows and columns, represented by T, and visualized as mirroring about the diagonal.

Explore the transpose operation through additional examples in quantum physics, and practice by mirroring and interchanging rows and columns.

Explore the adjoint as the transpose of the complex conjugate, denoted by dagger, and its role in quantum physics, with guided practice through examples.

Explore how unitary matrices model reversible and irreversible quantum state transformations, and why the adjoint equals the inverse, since their product yields the identity.

Explore Hermitian matrices, the self-adjoint operators for irreversible quantum state transformations, compare them with unitary matrices, and verify the adjoint equals itself (A† = A).

Explain why hermitian and unitary matrices matter in quantum computing, highlighting initialization as an irreversible hermitian process, a reversible unitary core, and measurement as an irreversible hermitian final step.

Explore how matrix transformations rotate points in 2d, using anti-clockwise rotations of (1,0) by 30 and 90 degrees to illustrate matrix multiplications as linear transformations on a coordinate space.

Explore how a transformation matrix yields eigenvectors and eigenvalues that define invariant directions, like the lines y=x and y=-x, using unit vectors, and connect to quantum state transformations.

Explore how sequential polarizing filters affect photons, showing that polarization acts as a quantum measurement that alters light and can yield unexpected transmission through intermediate angles.

Encode data as polarized photons with vertical or horizontal polarization in the BB84 protocol, and understand how eavesdropping compromises security and the shared secret.

Explore how quantum mechanics uses complex numbers to model subtractive probabilities, illustrated by the double-slit interference pattern where probabilities add or cancel, and where square roots of probabilities become complex.

Explore how a quantum state is represented as a column matrix, or vector, with two elements for a qubit, whose complex numbers' square magnitudes yield spin-down and spin-up probabilities.

Analyze how a quantum bit's state is represented as a two-element column vector and how measurement transforms this state, yielding deterministic outcomes after the first probabilistic result.

Explore how measurements alter a quantum system using a black box analogy: drawing an apple or an orange collapses the state, changing future probabilities.

Explore Dirac bra-ket notation to express qubit vectors algebraically, with |0> and |1> representing off and on states, and the equal superposition 1/√2(|0>+|1>) illustrating a superposed qubit.

Debunks the myth that quantum states are random by showing well-defined electron spin and apparatus states, with randomness arising only during measurements in misaligned directions, as with photon polarization.

Explore how a not gate acts as a reversible transformation, swapping qubit states via matrix multiplication. Note the pattern: irreversible initialization, reversible gates, and irreversible measurement.