Following John Preskill’s 1998 Quantum Information and Computation notes. Introduction Landauer’s principle: It takes energy to erase information (since erasure always compresses phase space, such processes are irreversible) You can store a bit of information as one molecule in a box. If it’s on the left, it’s on, else it’s off. If you slowly compress the volume in half, you’re gaurenteed to be in the LHS. The change in entropy is k ln 2, which has some associated work that needs to be performed Logic gates used to perform computation are typically irreversible For instance, NAND is irreversible, since one bit of information is lost for each gate Charles Bennett observed that any computation can in principle be done reversibly You can construct a Toffoli gate: Input is (a,b,c) Output is $(a,b, c \oplus a ^ b)$ So a and b get mirrored, and the third bit gets flipped if the first two bits are both 1 (otherwise, it mirrors the input) You can in principle do any computation up to the end, print out a copy of the answer (logically reversible process), then step back the computation back to the beginning Maxwell’s Demon The aforementioned ideas allows a resolution of the Maxwell’s demon paradox The original formulation is as follows: You have a partitioned box (split into A and B parts) and some demon which observes the molecules in the box If a fast particle is moving from A to B and it will cross the partition, then the demon allows it through If a fast particle is moving from B to A across the partition, the demon blocks it Over time, you will get faster particles on the right and slower ones on the left with minimal work....

Referencing “Modern Compiler Implementation in ML” by Andrew W. Appel. Lexical Analysis This refers to breaking up the input source code into individual words/tokens A “token” is a sequence of characters that can be treated as a unit in the grammer of a programming language Some common tokens which are typical in programming languages: ID: think names of any kind (variable, function classes etc.) NUM: Think integers REAL: Think floating point RESERVED: Words which are reserved in the programming language....

Following the book “Understanding Digital Signal Processing” by Richard G. Lyons. Discrete Sequences and Systems Definitions and Notation Signal Processing: the science of analyzing time-varying physical processes Analog refers to waveforms that are continuous in time and can take on a continuous range o amplitude values Comes from analog computers, where physical electronic differentiators and integrators were used to model differential equations Discrete: the time variable is quantized This means that the signal at intermediate values is unknown....

Notes following this MIT course and is based on the xv6 toy operating system. Operating System Interfaces Processes and Memory Forking Exec Operating System Organization Abstracting Physical Resources CPU Modes Processes xv6 startup Security Page Tables Kernel Address space Xv6 Implementation Physical Memory Allocation Xv6 Physical Memory Implementation Process Address Space Sbrk (Xv6) Exec (Xv6) Real World Paging Traps and System Calls RISC-V traps User Space Traps Syscalls Kernel Space Traps Page Fault Exceptions Interrupts ad Device Drivers Console Input Console Output Timer Interrupts Locking Races Spinlocks Using Locks Deadlocks Spinlocks Scheduling Context Switching Scheduler Sleep and Wakeup Semaphores Lost Wake-Up Problem Pipes Process Locks Real World Scheduling File System Buffer Cache Layer Logging Layer Block Allocator Inode Layer Directory Layer Path Layer File Descriptor Layer Operating System Interfaces Operating systems serve a number of purposes Abstracts away lower level hardware, so that programs don’t have to care about what disk you are writing to Allows multiple programs to run “at the same time” Provide well defined interface for programs to interact with each other For the interface, we want a simple interface that is easy to implement, but allows high sophisticated operations to happen UNIX philosophy of composing many simple programs together helps keep this problem trackable Can divide operating system into kernel space and user space kernel space is the program which provides services to other programs....

Linear Regression You have a set of data $(x_{i},y_{i},\sigma_[i])$, where x is the independent variable, y is the dependent variable, and $\sigma$ is the uncertainty in the dependent variable You want to fit a line to the data ($\hat{y} = mx+b$) You also want the uncertainties of the parameters Why would you want to do this? To know the parameters of the model To extrapolate/interpolate values of the dataset that you don’t have How you do this depends on your viewpoint a frequentist approach would try to maximize the likelihood estimate (information theory) Suppose that you have a probability density for the data $y_{i}$ ranging from 1 to N Treat this density as a function of the parameters....

Logistics Thermodynamics 0th Law 1st Law Response Functions 2nd Law Entropy 3rd Law Thermodynamic Potentials Enthalpy Helmhotz Free Energy Gibbs Free Energy Grand Potential Probability Definitions Important Probability Distributions Kinetic Theory of Gases Liouville’s Theorem Bogoliubov-Born-Green-Kirkwood-Yvon Hierarchy (BBGKY) Boltzmann Equation H-Theorem Classical Statistical Mechanics Microcanonical Ensemble Two State System Example 0th Law 1st Law 2nd Law Ideal Gas Mixing Entropy The Canonical Ensemble Grand Canonical Ensemble Conserved Ensemble Notes Interacting Particles Cumulant Expansion Mean Field Theory of Condensation Critical Points Quantum Statistical Mechanics Dilute Polyatomic Gases Quantized Vibrational Modes Quantized Rotational Modes Blackbody Radiation Formalism Identical Particles Grand Canonical Formulation Degenerate Fermi Gas $f_{m}^{\eta}(z)$ Denegerate Bose Gas Logistics Book(s) Landau and Lifshitz Kardar Pathria Thermodynamics 0th Law Thermodynamic equilibrium means that each object in thermal contact with each other have the same temperature Ideal gases, as you cool them down, all converge to a single point, which defines absolute zero....

The Basics Hibert Space Review In an infinitely dimensional space, we have some abstract vector $\Psi$. We can choose coordinate axes such that all values can be taken by the position x, so that we can describe $\Psi$ as as set of components $\Psi(x)$ Could have also chosen p instead for our coordinate axes $(v,w) \geq 0$, where (a,b) denotes an inner product This is linear in the right argument, and anti-linear in the first argument (ie....

IEEE Floating Point Standard Arrays Numerics Random Numbers LCM (Linear Congruent Method) Testing RNG Nonuniform Distributions Non-Analytic Nonuniform Distributions Interpolation Linear Interpolation Fitting Basis Coefficients Polynomial fits Fourier Basis Integration Trapezoid Rule Simpson’s Rule Rescaling Integrals Differentiation Scaling Linear Algebra LU SVD Eigensystems Uses Principal Component Analysis (PCA) Dimensional Reduction Root-Finding Minimization Quadratic Fit Golden Section Search Multi-Dimensional FFTs ODEs PDEs Initial Condition Problems Boundary Condition Problems MCMC (Markov Chain Monte Carlo) Markov Chain Metropolis Hastings Burn-in and Proposal Distribution IEEE Floating Point Standard $\pm 1+ 2^{e-127}\Sigma_{i=0}^{22} f_{i} 2^{-(i-23)}$ e is the exponent number $f_{i}$ represents the floating bits Some edge cases: e = 0, $f \neq 0$ are called subnormal numbers, where “1” above is replaced with 0 and $e-127$ is replaced by $e-126$ e=0 and f=0 is a signed zero e=255, f=0 is a signed $\inf$ e=255, $f\neq = 0$ is NaN Arrays TL;DR if you want to go fast, then use numpy arrays, since they are more contiguous in memory (and hence cache friendly) compared to python lists You also don’t have to worry about type checking since Python needs to check if operations There are also some built-in functions that numpy provides which are faster than Python built-ins Numerics TL;DR Keep numbers from being too small and too large and you’re good $E[\delta (f(a))^{2}] \approx E[[\frac{\partial f}{\partial a}]^{2} \delta a^{2}]$ Assuming similar errors for each operations, you find that the round-off error scales as $\sqrt{N}$ Making finer time steps yields a better approximation, but the round-off error degrades this....

Taken from Experimental Techniques in Modern High-Energy Physics: by Hanagaki, Tanaka, Tomoto, Yamazaki Why? Two basic categories of HEP experiments Find any new particle by converting collision energy to particles Done through finding resonances in the final state Investigate the nature of a particular interaction Done through precise measurements of other physical observables In theory, you work with a small number of states to keep problems tractable. Experimentally, you can get thousands of final states which you have to parse through The number of particles goes as $ln \sqrt{s}$ where s is the square of the center of mass energy of the collision This the angular density to go up, which create “jets” (ie....

Notes adapted from MIT Digital Electronics Lab Also from here Boolean Algebra Theorems Truth Tables Specific Functions 1-Bit Functions 2-bit Functions XOR Overview Verilog Basics Variables Parameters Operators and Loops Enums Syntax Oddities Combinational Logic Continuous Assignment Procedural Assignment Glitches! Gotchas Nesting Modules Combinational Versus Sequential Logic Latches Characteristic Equations Registers Verilog for Sequential Blocks Sequential Always Blocking and Non-Blocking Assignment External Signals (Metastability) FSMs FSM State Encoding Converting to Verilog Clocks PLLS The Treachery of Clocks Skew Jitter Proper usage Timing Timing Parameters Slack Video Black and White TV Color TV VGA HDMI Memory History of Memory Key parameters of memory FPGA SRAM Fabric Case Studies Off FPGA Memory EPROM Family DRAM Buffering DRAM FSM Modularization Pipelining Math Complexity Division CORDIC AXI AXI Signals Signed Number DSP48 CRCs Polynomial Arithmetic Boolean Algebra Theorems Let all variables be boolean in nature (ie....