QC101 Quantum Computing & Intro to Quantum Machine Learning

Explore photon polarization as a property of light and its quantum behavior. See how polarized photons resemble quantum bits, exhibiting superposition and entanglement.

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.

Explore photon polarization and polarizing filters in light transmission. Show alignment yields full pass-through, anti-alignment blocks light, and angles between 0 and 90 degrees yield partial transmission.

When vertically polarized photons reach a 45-degree polarizer, 50% pass and emerge polarized at 45 degrees; unlike a red filter, the polarizer changes the photons' polarization.

Demonstrate how a 45-degree filter makes an initially opaque arrangement of photons partially transparent, with 50% passing the first stage and 25% reaching the screen, illustrating physics beyond classical explanations.

Explore quantum behavior of single photons and polarizing filters, where alignment at 0 degrees yields deterministic pass and 90 degrees yields deterministic block; intermediate angles produce probabilistic pass-through.

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.

Explains how a single photon with unknown polarization becomes either aligned or anti-aligned after passing through a calcite crystal, causing loss of the original polarization information.

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.

Explore how a single photon passing through calcite loses information about its prior polarization, transforming a real-number state into a single bit and complicating polarization inference.

Explore how measuring photon polarization yields only binary results—aligned or anti-aligned—and disturbs the state, reducing information to 1 bit and making the original polarization angle unknowable.

Explore how a calcite crystal acts as a quantum measurement that changes a photon's polarization, making subsequent identical measurements yield the same result across a three-crystal setup.

Examine an anti-aligned photon path through three aligned calcite crystals, where the first measurement fixes polarization and yields anti-aligned results in subsequent measurements.

Send a photon of unknown polarization to the first polarizing filter; if it passes, its polarization aligns with the second and third filters at the same angle.

Explore boolean algebra, the precise grammar for combining true or false statements using and, or, and not; learn how boolean expressions are manipulated.

Demonstrate boolean algebra with A and B using and, or, and not. Clarify the evaluation order of not before and before or, and show step-by-step simplification of expressions.

Explore truth tables to define boolean functions, map inputs to outputs for 2^n rows, and compare and, or, not, nand, and exclusive or.

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.

Examine the logic circuit diagram and its logical expression. Trace how the not operation is last, fed by C and the A and B result through an or gate.

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 not gate in logic circuits, a single-input, single-output device; the circle marks inversion, and the truth table shows 0 to 1 and 1 to 0.

Explore how multi-input gates are built from 2-input gates to form complex circuits, using examples of 3-input and gates and 3-input or gates that produce a final output.

Show how an and gate can be built from or and not gates, verify with a truth table, and use boolean algebra to conclude identical truth tables mean identical functions.

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.

Explore how XOR enables reversible assignment in quantum computing by initializing a zero register and XORing with B, replacing irreversible quantum assignment with repeatable quantum-XOR.

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.

Explore how xor-ing a bit sequence with the same input twice cancels the operation, showing that A xor B xor B equals A and the output depends only on A.

Quantum physics enables cryptography with completely secure messages, as quantum protocols resist known attacks while RSA can be broken by quantum computers, securing messages sent over insecure channels against eavesdropping.

Explore xor-based cryptography: convert messages to bits, use a shared secret bit sequence for sender and receiver, and apply xor to encrypt and transmit the message.

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.

Apply a bitwise xor of a random bit sequence with a message. Use a new random sequence for each encryption to avoid reusing the shared secret.

Explore the complete single-use shared secret protocol for encryption, where sender and receiver share a secret book of random bits and xor messages to achieve unbreakable quantum encryption.

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.

Explore predicting the likelihood of boolean statements with P(...) notation, and learn the probability of a boolean function being true or false, including and/or combinations.

Mutually exclusive events occur when one outcome rules out the others, and a complete set ensures exactly one true event with probabilities summing to one, as in quantum states.

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

Learn to express the probability of a boolean expression in terms of the probabilities of individual events, using both boolean algebra and the algebra of probabilities.

Explore mutually exclusive events and the rule that A implies not B and B implies not A, so P(A and B) equals zero.

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.

Explore mutually exclusive events and apply the addition rule to calculate P(A or B). Show that a complete set of mutually exclusive events sums to 1.

Apply the rule P(A or B)=P(A)+P(B)-P(A and B) for any relation between A and B, including mutual exclusivity and independence, and see how P(A and B) shapes P(A or B).

Explore an example of independent events by calculating P(A or B) using P(A) = 0.2 and P(B) = 0.3, showing P(A or B) = 0.44.

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.

Analyze two-bit systems and relate 00, 01, 10, 11 to single-bit probabilities, showing P(first bit = 1) = P(11) + P(10) and P(second bit = 1) = P(01) + P(11).

Visualize probability problems involving boolean expressions by using Venn diagrams to represent events and their P() probabilities within a sample space, drawing events A and B on the same diagram.

Explore how Venn diagrams represent composite events and identify P(A AND B) as the area of the red region, highlighting similarities to high school Venn diagrams.

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

Explore how not a is represented on a Venn diagram by shading the region outside a inside the rectangle that denotes the complete set of all possible events.

Visualize probability diagrams to compute P(A and not B), P(not A and B), and P(A or B) from P(A)=0.2, P(B)=0.3, and P(A and B)=0.05, yielding 0.15, 0.25, and 0.45.

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 conditional probability by analyzing events A and B and computing P(A and B)/P(A), illustrated with a rectangle diagram and scaling.

Illustrate conditional probability with formulas P(B/A)=P(A AND B)/P(A) and P(A/B)=P(A AND B)/P(B) through a grocery example: 100 buyers, 10 apples, 20 oranges, 5 both, giving 0.5.

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.

Map world events to random variables by assigning a number to each outcome in a complete, exclusive set. See how different mappings, like dice or car colors, alter the average.

Explore aggregate properties of random variables, focusing on the mean, which equals the average or expected value. See the dice example and the probability-weighted sum.

Examine loaded dice and their unfair probabilities, calculate the mean by multiplying each value by its probability, and compare to a fair dice to identify loading.

Define a random variable for an unfair coin with -1 for tails and +1 for heads, using p(-1)=0.48 and p(+1)=0.52 to compute the mean as 0.04, implying potential unfairness.

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 standard deviation as a measure of how far a random variable deviates from its mean. A fair dice example illustrates variance and how it becomes standard deviation.

Explore standard deviation through examples with fair and unfair coins and a loaded die, computing mean, variance, and standard deviation for different random variables.

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.

Explore how the product of two related random variables signals correlation, from constant outcomes to independence, and use the mean of XY to test correlation.

Explore complex numbers, the imaginary unit that acts as intermediate values to simplify quantum state transformations through linear operations, making math easier than using real numbers.

Define i as the square root of -1 and show that any fractional power can be written as a + bi, with a and b real, i.e., complex numbers.

Learn to add complex numbers by grouping real and imaginary parts, treating i as an unknown with i^2 = -1, as shown in examples like (1+2i)+(3+4i)=4+6i and (-6+7i)+(-1-2i)=-7+5i.

Learn to subtract complex numbers by removing parentheses and grouping real and imaginary parts, and compute results like 2+i, 7-10i, and 9-i.

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).

Show how to divide a complex number by a real number using examples such as (4+6i)/2 = 2+3i and (7+2i)/-3 = -7/3 - (2/3)i.

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.

Multiply complex numbers using standard algebra rules, treating i as an unknown and replacing i squared with -1. Identify the real and imaginary parts in examples like (-7+2i)(3-4i) and (6-2i)(-9+4i).

Master complex conjugates by flipping the sign of the imaginary part, turning a+bi into a-bi. Use this simple operation to divide complex numbers and see its role in quantum physics.

Master the squared magnitude of complex numbers using conjugates, where (a+bi)(a−bi)=a^2+b^2. See how the imaginary parts cancel and compute magnitudes for 2+3i and 7−6i.

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.

Explore Euler's formula e power i x equals cosine x plus i sine x and express complex numbers with squared magnitude 1 as cos x plus i sin x.

Generalize a+bi to polar form as r e^{i x}, with r as magnitude and e^{i x} (cos x + i sin x) having unit magnitude, useful in quantum physics.

Convert complex numbers between polar and standard form (a+bi) using e power i x = cos x + i sin x, with 10 e power i pi/4.

Compute powers of complex numbers, including fractional powers, by converting to polar form. Use e power i x and 2 pi periodicity to reveal cube roots of 1.

Explore the complex cube roots of unity by solving x^3 = 1, and learn to compute fractional powers by expressing complex numbers in polar form and adding 2 pi multiples.

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.

Represent complex numbers as points on 2d coordinate plane, with real part x and imaginary part y, and convert between a+bi and polar form r e^(i theta) (r = sqrt(a^2+b^2)).

Explore linear algebra and matrix operations as essential tools for quantum systems, using real and complex numbers in matrices to perform addition, subtraction, and multiplication.

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.

Learn how to multiply matrices by scalars, applying real or complex numbers to every element. See 2x2 examples with scalars such as 2 and -4, including complex scalars.

Learn how to multiply matrices by taking rows from the first matrix and columns from the second, apply the row-by-column rule, and see how matrix multiplication models quantum state transformations.

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.

Work through a sequence of multiplication problems in quantum computing, pausing between each exercise to solve and verify the results.

Follow an animation to multiply a 3 by 3 matrix by another 3 by 3 matrix, and pause the video to compute each element yourself.

Pause the video to work through exercises and verify your computation against the provided answers, with another exercise to practice in quantum computing and quantum machine learning.

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.

Apply the matrix multiplication rule: the first matrix's columns must equal the second matrix's rows; visualize their dimensions to confirm the equal middle numbers.

Determine matrix multiplication feasibility by comparing row and column lengths; with a 2x2 matrix and a 3x2 matrix, the dimensions do not align, so multiplication is not possible.

Learn that matrix multiplication is not commutative: the order matters, as AB is possible while BA is not, and even when both exist, AB and BA may differ.

Explore the associative and distributive properties of matrix multiplication, showing that A(B+C)=AB+AC and (A+B)(C+D)=AC+AD+BC+BD, while AB ≠ BA.

Apply the matrix multiplication rule by matching inner dimensions; multiply m by n with n by p to yield an m by p result, as shown by the red box.

Learn how to multiply odd-shaped matrices in quantum physics by aligning inner dimensions, illustrated with a 3x2 and 2x1 example; practice the steps and verify the result.

Follow along an animated demonstration of odd shaped multiplications, then pause to verify you can perform the multiplications on your own.

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.

Work through another exercise, pause the video to verify your answer, and ensure you understand the steps before proceeding.

Explore the inner product by multiplying a row vector with a column vector to yield a scalar, contrasting it with the outer product and noting ad+be+cf.

Practice matrix multiplication by multiplying a square matrix with a column matrix, a common operation in quantum computing. Verify your results as you work through the exercises.

Explore the identity matrix I, its role as the matrix equivalent of 1, and how AI and IA leave matrices unchanged in 2x2, 3x3, and 4x4 cases.

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.

Demonstrate that the transpose of the product AB equals B transpose times A transpose, by computing AB and its transpose and comparing with B transpose times A transpose.

Learn how to compute the complex conjugate of matrices by taking the conjugate of each element, reversing the sign of imaginary parts while real numbers remain unchanged.

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).

Discover how a matrix can be both hermitian and unitary by confirming it equals its inverse and its adjoint, and that A squared equals the identity.

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 vectors as column matrices and learn how linear transformations map coordinates using a square matrix with coefficients A, B, C, D, and introduce eigenvectors.

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 swaps coordinates to produce reflections about y=x and y=-x, and learn how eigenvectors identify these special directions on the unit circle.

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 eigen vectors and eigen values of a 3x3 matrix, noting non-normalized directions and how matrix multiplication reveals corresponding eigen values, including real and complex cases in quantum computing.

Explore how bb84 enables secure communication through photon polarization and light's quantum properties. Learn how photons, polarization angles, and vertical or horizontal orientations on the xy plane underpin quantum cryptography.

Explain how a polarizing filter selects photon polarization: 50% of 45-degree photons pass and align with the filter, while measurement changes the photon's polarization, illustrating quantum randomness.

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.

Explore the no-cloning theorem by showing why measuring properties alters quantum states, why unknown systems cannot be cloned, and how identical copies exist only when system parameters are known.

Explore quantum cryptography concepts, including measurement disturbance, no-cloning, and aligned measurements, then learn XOR as a reversible boolean operation where the output is 1 when inputs differ.

Use a random, single-use shared secret to encrypt a message by XORing it, then recover it by XORing again, while noting reuse breaks security and quantum methods establish the secret.

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

Alice randomizes bit encoding between vertical/horizontal and 45-degree polarizations to prevent eavesdropping, and Bob's guesses decode only the correctly aligned bits while wrong guesses produce random, corrupted bits.

Demonstrate exchanging polarization angle encoding over a classical channel in BB84, where Alice and Bob identify correctly decoded bits, discard wrong ones, and distill a shared secret using XOR.

Reveal encodings after Bob has received all photons to prevent Eve from eavesdropping, detect corruption with a checksum over the classical channel, and establish a shared secret.

Alice and Bob share a secret key over insecure channels via a quantum-based protocol, enabling unbreakable one-use shared-secret encryption, based on quantum photon behavior. RSA’s quantum vulnerability underscores physics-based security.

Learn how calculus and mathematical models describe classical systems governed by Newton's laws, then develop an analogous quantum physics model by examining clues from physical experiments.

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 the qubit's superposition requires matrix algebra to define a quantum system, using a collection of numbers to model states.

Represent quantum states as column matrices of complex numbers and show how transformation matrices change them; non-superposition yields a single nonzero entry with squared magnitudes summing to 1.

Explore how electron spin, measured as up or down, serves as a quantum bit or qubit, connecting magnetic properties to the quantum version of a classical bit.

Explain the spin qubit basis using up and down, showing orthogonality, measurement outcomes, and probability rules, and represent states as a two element column matrix.

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.

Explore how a qubit state vector yields probabilistic measurements in superposition and how measurement changes the state, with deterministic outcomes when probabilities are 0 or 1.

Align the apparatus along the z-axis to measure an electron's spin and report up or down, showing how measurement collapses a superposition and yields the same result in subsequent measurements.

Explore how measuring an unknown quantum state's spin yields probabilistic outcomes and collapses the state, making subsequent measurements deterministic, then examine hypothetical tilts and probabilities without triggering a real measurement.

Rotate the apparatus to horizontal to measure spin, changing the electron’s state and yielding up or down with equal probability; after rotating back to the z-axis, measurements have equal probability.

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.

Measurement changes the state of the system regardless of the apparatus, and the math model explains how the system is altered during measurement; no apparatus can measure without changing it.

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 for inner products by using the bra <A| and ket |E>; the bra is the adjoint of A, the ket is E, giving the inner product.

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.

Analyze spin measurement results as the apparatus orientation changes, deriving the vector [1/root(2) 1/root(2)] and showing the probability of measuring up is 1/2, matching experiment.

Examine how measuring electron spin with a given apparatus updates its state vector, yielding up or down, and use Dirac bracket notation to relate orthogonality to probabilities.

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.

Explain why measurement is irreversible: it changes the quantum state and destroys information about the prior state, as illustrated by the apple-and-orange box analogy.

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.

Analyze two-qubit systems by listing the four basis states 00, 01, 10, 11 and their ket vectors |00>, |01>, |10>, |11>, then examine measurement probabilities in superposition via square magnitudes.

Demonstrate entanglement in the two-qubit state (|00> + |11>)/√2, giving 50% for 00 and 11, 0% for 01/10, and a composite state that cannot be written as a tensor product.