Introduction
What quantum computing changes, superposition, entanglement, interference, physical anchors, history, and the book’s roadmap.
Volume I · Foundations · A graduate problem book with full solutions
Quantum Computing
by Derivation
1Foundations
2Full derivations
3Circuits
4Verification
Can you derive the result—not merely recognize it?
The book
Volume I of Quantum Computing by Derivation is a problem-driven study and reference book for graduate students, instructors, researchers, and engineers who need more than an introductory survey.
It pairs mathematically explicit derivations with an entrance mock examination, worked problem banks, circuit practice, and structured verification. The aim is active fluency: knowing how a result is built, when it applies, and how to check it.
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Inside the book
Volume I contains thirteen chapters, beginning with an introduction and entrance mock examination before developing the mathematical and physical foundations of quantum computing.
Entrance Mock Exam
A rules-based diagnostic examination with problems and complete solutions.
Dirac Notation
Kets, bras, inner and outer products, adjoints, completeness, basis changes, operator reconstruction, and continuous bases.
Linear Algebra Foundations for Quantum Computing
Tensor products, Bell states, traces, commutators, vectors, norms, projectors, eigenvalues, and eigenvectors.
Single-Qubit Foundations for Spin-1/2
Spinors, superposition, normalization, measurement, eigenbases, time evolution, uncertainty, density matrices, and Bloch-sphere geometry.
Pauli Matrices and the Single-Qubit Operator Algebra
Pauli identities, projector representations, basis decomposition, commutation relations, exponentials, rotations, and gate conjugation.
Expectation Values and Measurement
Single- and two-qubit expectation values, variance, uncertainty, Born-rule examples, shot-based estimators, and Hamiltonian checks.
Quantum Gates and Circuit Operations
Single-, two-, and multi-qubit gates, unitarity, rotations, circuit identities, optical gates, interferometers, and controlled operations.
Quantum Information: Density Matrices, Channels, and Entropy
Purity, partial traces, entanglement measures, Kraus operators, Choi representations, state distances, entropy, correlations, and open systems.
Entanglement: Bell States, GHZ, and the EPR Paradox
Bell and GHZ preparation, measurement structure, reduced states, entropy, teleportation, swapping, separability, and CHSH checks.
Coupled Spins: The Heisenberg and Ising Models
Spectra, conservation, exchange, singlet states, coupled-spin dynamics, chain simulation, and transverse-field Ising problems.
Hamiltonian Dynamics, Time Evolution, and Product-Formula Simulation
Unitary evolution, Pauli exponentials, Larmor precession, Lie–Trotter and Suzuki formulas, accuracy targets, and slice counts.
Quantum Physics for Quantum Computing
The Stern–Gerlach experiment and the distinction between superposition and statistical mixture.
Pauli-String Expectation Estimation on a Quantum Computer
A practical appendix on measurement settings and expectation estimation for Pauli strings.
About the authors
Author
Shlomo Kashani
Shlomo Kashani is an interdisciplinary researcher examining deterrence, civilian protection, and political judgment under conditions of extreme uncertainty. His work sits at the intersection of strategic studies, artificial intelligence, and the ethics of emerging technology. His academic formation spans Defence and Strategic Studies at Missouri State University, advanced signal processing at Queen Mary University of London, and quantum physics and computing through Johns Hopkins University.
Author
Dr. Eli Bordo
Dr. Eli Bordo is a physicist by training and a technology leader by profession. He received his Ph.D. in quantum and ultrafast optics from the Technion, Israel Institute of Technology, studying light at the scales where classical intuition begins to give way to quantum mechanics. Today, he leads R&D efforts across quantum technologies, photonics, and other emerging fields, with a particular interest in turning elegant physics into working technology.
Author
Dr. Daniel Sattinger
Biography forthcoming.
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Three questions from the entrance mock exam
Try each problem before opening its worked solution. The samples move from a single-qubit warm-up to phase-sensitive projectors and bipartite entanglement.
Sample 01 · Expectation values
Basic expectation values and variance
Let
\[|\psi\rangle=\frac{|0\rangle+2i|1\rangle}{\sqrt5}.\]
This is a single Pauli operator applied to a two-component state, worked out one row at a time.
- Verify that \(|\psi\rangle\) is normalized.
- Compute \(\langle X\rangle\), \(\langle Y\rangle\), and \(\langle Z\rangle\) directly from the \(2\times2\) Pauli matrices.
- Compute \(\operatorname{Var}(Z)=\langle Z^2\rangle-\langle Z\rangle^2\), then check \(\langle X\rangle^2+\langle Y\rangle^2+\langle Z\rangle^2=1\).
Open worked solution+
1. Check the norm
\[\langle\psi|\psi\rangle=\frac{|1|^2+|2i|^2}{5}=\frac{1+4}{5}=1.\]
2. Apply the Pauli matrices
\[|\psi\rangle=\frac1{\sqrt5}\begin{pmatrix}1\\2i\end{pmatrix},\qquad \langle\psi|=\frac1{\sqrt5}\begin{pmatrix}1&-2i\end{pmatrix}.\]
X mean\(\langle X\rangle=0\)
Y mean\(\langle Y\rangle=\frac45\)
Z mean\(\langle Z\rangle=-\frac35\)
For example, \(Y|\psi\rangle=\tfrac1{\sqrt5}(2,i)^{\mathsf T}\), hence
\[\langle Y\rangle=\frac15\left(1\cdot2+(-2i)\cdot i\right)=\frac45.\]
3. Compute the variance and check
\[\operatorname{Var}(Z)=1-\left(-\frac35\right)^2=\boxed{\frac{16}{25}},\]
\[0^2+\left(\frac45\right)^2+\left(-\frac35\right)^2=1.\]
Sample 02 · Dirac notation
Overlaps, projectors, and a complex relative phase
Two normalized qubit states are
\[|\psi\rangle=\frac{2|0\rangle-i|1\rangle}{\sqrt5},\qquad |\phi\rangle=\frac{|0\rangle+e^{i\pi/4}|1\rangle}{\sqrt2}.\]
- Write both bras and compute the overlap \(\langle\phi|\psi\rangle\).
- For the yes–no measurement \(\{P_\phi,I-P_\phi\}\), where \(P_\phi=|\phi\rangle\!\langle\phi|\), find the probability of “yes.”
- Write \(P_\phi\) as a \(2\times2\) matrix. Verify that it is Hermitian and idempotent, then confirm the probability using \(\langle\psi|P_\phi|\psi\rangle\).
Open worked solution+
1. Conjugate before taking the overlap
\[\langle\psi|=\frac1{\sqrt5}\begin{pmatrix}2&i\end{pmatrix},\qquad \langle\phi|=\frac1{\sqrt2}\begin{pmatrix}1&e^{-i\pi/4}\end{pmatrix}.\]
\[\langle\phi|\psi\rangle=\frac1{\sqrt{10}}\left(2-\frac{1+i}{\sqrt2}\right).\]
2. Apply the Born rule
\[p_\phi=|\langle\phi|\psi\rangle|^2=\boxed{\frac{5-2\sqrt2}{10}}\approx0.217.\]
3. Construct and test the projector
\[P_\phi=\frac12\begin{pmatrix}1&e^{-i\pi/4}\\e^{i\pi/4}&1\end{pmatrix}.\]
The off-diagonal entries are complex conjugates, so \(P_\phi^\dagger=P_\phi\). Normalization gives
\[P_\phi^2=|\phi\rangle\langle\phi|\phi\rangle\langle\phi|=P_\phi,\qquad \langle\psi|P_\phi|\psi\rangle=\frac{5-2\sqrt2}{10}.\]
Sample 09 · Entanglement
Reduced states and Schmidt coefficients
Consider the normalized two-qubit state
\[|\Psi\rangle=\frac{|00\rangle+|01\rangle+|11\rangle}{\sqrt3}.\]
- Find the complete computational-basis probability distribution and the probability that the two outcomes are equal.
- Compute \(\rho_A=\operatorname{Tr}_B(|\Psi\rangle\!\langle\Psi|)\).
- Use \(\rho_A\) to show that \(|\Psi\rangle\) is entangled.
- Find the two Schmidt coefficients.
Open worked solution+
1. Read the distribution
\[P(00)=P(01)=P(11)=\frac13,\qquad P(10)=0,\qquad P(\text{equal bits})=\frac23.\]
2. Trace out qubit B
With coefficient matrix \(C=\tfrac1{\sqrt3}\begin{pmatrix}1&1\\0&1\end{pmatrix}\),
\[\rho_A=CC^\dagger=\boxed{\frac13\begin{pmatrix}2&1\\1&1\end{pmatrix}}.\]
3. Test purity
\[\rho_A^2=\frac19\begin{pmatrix}5&3\\3&2\end{pmatrix},\qquad \operatorname{Tr}(\rho_A^2)=\frac79<1.\]
The reduced state is mixed, so the bipartite pure state \(|\Psi\rangle\) is entangled.
4. Find the Schmidt coefficients
\[\lambda_\pm=\frac{3\pm\sqrt5}{6},\qquad \boxed{s_\pm=\sqrt{\frac{3\pm\sqrt5}{6}}}.\]
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Derivation turns familiarity into command
The working sequence
01Framethe state, basis, and assumptions
→
02Derivethe result line by line
→
03Verifythe algebra and computation
→
04Generalizethe method to a new problem
The manuscript makes basis order, notation, intermediate steps, and acceptance criteria visible. Solutions are not compressed into answer-key form; they are written so a reader can reconstruct the reasoning and locate an error.
“The result matters. The route to the result is what makes it reusable.”
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Built for active problem-solving fluency
Passive familiarity
Recognize the vocabulary
Definitions and finished formulas can create a sense of recognition without testing whether the formalism can be retrieved and used.
Technical fluency
Reconstruct the argument
Worked derivations, problem banks, and computational checks make the reader choose a representation, carry the algebra, and test the result.
The progression runs from Dirac notation, linear algebra, and single-qubit systems to measurement, circuits, quantum information, entanglement, coupled spins, Hamiltonian dynamics, and quantum physics.
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From formalism to executable checks
A derivation should survive more than one form of scrutiny.
Analytical control
- Fix the basis order
- Track dimensions and phases
- State assumptions
- Check limiting cases
Does the implementation reproduce the derivation?
Computational control
- Construct the operator
- Run the reference case
- Compare observables
- Report tolerance and error
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One volume, three ways to use it
Learn
Study the derivation
Follow complete solutions with explicit notation, intermediate reasoning, and clearly stated assumptions.
Practise
Work the problem
Use chapter-aligned problem banks and the entrance mock examination for structured revision.
Verify
Test the result
Reproduce analytical claims with computational checks and portable implementation guidance.
Book details
Derive.
Then verify.
Kashani, Shlomo; Eli Bordo; and Daniel Sattinger.
Quantum Computing by Derivation. Volume I: Foundations.
A Graduate Problem Book with Full Solutions.
First edition, 2026.