Quantum Computing
Quantum Computing
Quantum Computing
A quantum computer holds a state far larger than anything it can print, and a measurement returns only a handful of bits. This course is the study of what those bits can be made to say, and of the one mechanism — interference — that is the only way of making them say anything at all.
What the course is about
The screen beside this runs the smallest complete experiment in the course. One qubit is put into an even superposition, a phase is written between its two terms, and a second Hadamard brings the two amplitudes back into the same outcome so that they can add or cancel.
p(0) = cos2(φ ⁄ 2)
Measuring in the middle would learn nothing: both outcomes are one half there for every φ. Everything the readout knows arrives in the last gate. Watch the marker cross the fringe and the two bars swap heights in the same breath, because they are one quantity and not two.
The bars are one run of four hundred shots drawn from the probability under the marker, and the bit string in the corner is the first twelve of those shots rather than a second random stream. The white tick on each bar is the exact probability; the gap to it is what a finite run costs, and it closes as one over the square root of the shot count.
The seven chapters
A student arrives believing that a quantum computer tries every answer at once, and every later idea is harder to read while that sentence is still believed. Each chapter is either a way of writing a state down, a way of acting on it, or a way of arranging the one measurement that ends the computation — and every one of them answers to the same three replacements.
What the machine is for, how large the state really is, and the one interference experiment the rest of the course generalises.
Columns, inner products and what they conjugate; amplitude against phase; projectors, the tensor product, and where the exponential comes from.
The Born rule, what a reading leaves behind, the Pauli algebra, evolution as an exponential, and what a finite run of shots is worth.
The density operator and the two situations no state vector describes; channels, relaxation and dephasing; the partial trace; Bell states and CHSH.
One qubit drawn, and every operation on it as a motion of that drawing. The half angle, the gate set, and the two-qubit ordering that fails silently.
Depth against gate count, shots and error bars, what a compiler does — then teleportation and Grover search, worked end to end.
One mechanism, phase kickback, and the four algorithms built from it: through the Fourier transform and phase estimation to order finding and factoring.
Take it away
177 scenes, eleven laboratories and every practice question with its worked solution. One file, no network, works offline.
Open → PDFThe six chapters written out as a book, every derivation one step to a line, and Appendix A collecting every formula.
Download → PDFEvery question in the course with no answer and no solution, and room to work.
Download → PDFThe conventions, every formula the course establishes in the order it establishes them, and every symbol it defines.
Download →Two conventions decide every number here, and mixing them is silent rather than loud: a register is written |qn−1 … q1q0⟩, so entry x of the column is the amplitude of the basis state x read as a binary number; and a phase on the whole state may always be dropped while a phase between two terms may never be, because that is what interference is made of.