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
1Foundations2Full derivations3Circuits4Verification
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 builds production AI and quantum systems — and studies the high-stakes decisions that ride on them. He ships large language models, retrieval, and agentic AI, and works hands-on across the quantum stack: quantum machine learning (QML), quantum algorithm design, and high-performance circuit simulation. His academic formation spans Defence and Strategic Studies at Missouri State University, an M.Sc in Quantum Physics & Computing at Johns Hopkins University, and an M.Sc in Digital Signal Processing at Queen Mary, University of London.
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
This section combines three detailed mock-exam questions with a full Hamiltonian-simulation example. Follow the Ising model from Pauli strings through Trotterization to an executable circuit, then try each question before opening its worked solution.
Extended mock-exam question · Hamiltonian simulation
From Pauli strings to an executable Trotter circuit
Consider the two-qubit transverse-field Ising Hamiltonian
\[H=-JZ_1Z_2-h(X_1+X_2).\]
Your goal is to turn the formal time-evolution operator \(U(t)=e^{-iHt}\) into gates that a quantum processor can execute.
- Write \(H\) explicitly as a sum of Pauli strings containing only \(I\), \(X\), and \(Z\).
- Compute the relevant commutator and explain why the interaction and field evolutions cannot simply be separated exactly.
- Derive one first-order Lie–Trotter step for \(\Delta t=t/r\), including its error scaling.
- Compile every factor into one- and two-qubit gates and draw the complete circuit for one Trotter slice.
- Explain how the same Pauli-string recipe extends to the \(XX\), \(YY\), and \(ZZ\) terms of a Heisenberg spin chain.
Open full worked solution+
Solution roadmap
A complete solution connects four representations: the physical model, its Pauli strings, a controlled approximation to \(e^{-iHt}\), and the gates that a processor can actually execute. The Ising derivation and Heisenberg extension below carry that chain all the way through.
Full example A · Transverse-field Ising model
Competing alignment and transverse rotation
12-spin latticeschematic view
\(hX\)
Physical model
\[H=-J\sum_{\langle i,j\rangle}Z_iZ_j-h\sum_iX_i.\]
The \(ZZ\) term rewards neighbouring spins that agree in the computational basis. The \(X\) term continuously rotates each spin away from that basis. Their competition produces nontrivial dynamics.
Key observationThe entire Hamiltonian is made from Pauli matrices and identities.
That local Pauli structure is exactly what lets us compile the evolution into one- and two-qubit gates.
01
Specialize to two qubits
Expose the Pauli words
\[H=-JZ_1Z_2-h(X_1+X_2)=-J\,ZZ-h\,XI-h\,IX.\]
Define \(H_{ZZ}=-JZZ\) and \(H_X=-h(XI+IX)\). Each word says exactly which local operator acts on each qubit.
02
Why approximation is needed
Check the commutator
\[[H_{ZZ},H_X]=2iJh(YZ+ZY)\neq0.\]
Because the layers do not commute, \(e^{-i(H_{ZZ}+H_X)t}\neq e^{-iH_Xt}e^{-iH_{ZZ}t}\) at finite time. An exact separation would discard commutator corrections.
03
Controlled product formula
Slice time into \(r\) pieces
\[\Delta t=\frac tr,\qquad U(t)\approx\left[e^{-iH_X\Delta t}e^{-iH_{ZZ}\Delta t}\right]^r.\]
The leading first-order global error scales as \(O(t^2/r)\), with its coefficient controlled by \(\lVert[H_{ZZ},H_X]\rVert\). More slices mean shorter steps and less ordering error.
04
Exact local compilation
Turn each exponential into gates
Field layer\(e^{+ih\Delta tX_i}=R_x^{(i)}(-2h\Delta t)\)
Interaction layer\(e^{+iJ\Delta tZZ}=\mathrm{CX}\,R_z(-2J\Delta t)\,\mathrm{CX}\)
The mappings are exact. Only the ordering of the noncommuting layers is approximate.
One first-order Trotter slice
Compute parity, phase it, uncompute, then rotate
The first CNOT stores \(Z_1Z_2\) parity on the target. \(R_z\) attaches the interaction phase. The second CNOT removes the temporary parity, and the two \(R_x\) gates apply the transverse field.
q1q2+\(R_z(-2J\Delta t)\)+\(R_x(-2h\Delta t)\)\(R_x(-2h\Delta t)\)
Per slice2 CNOTPhase1 \(R_z\)Field2 \(R_x\)Global error\(O(t^2/r)\)
Full example B · Heisenberg spin chain
Exchange in all three spin directions
Five-spin open chain\(N=5\)
1XXYYZZ2XXYYZZ3XXYYZZ4XXYYZZ5
Open isotropic chain
\[H_{\mathrm H}=J\sum_{i=1}^{N-1}\left(X_iX_{i+1}+Y_iY_{i+1}+Z_iZ_{i+1}\right).\]
Unlike the transverse-field Ising model, each neighbouring pair exchanges spin information through \(X\), \(Y\), and \(Z\). For \(N\ge3\), terms on overlapping bonds need not commute, so product formulas again provide a systematic gate construction.
Layer X\(H_X=J\sum_iX_iX_{i+1}\)Hadamards map \(X\leftrightarrow Z\)
Layer Y\(H_Y=J\sum_iY_iY_{i+1}\)\(S^\dagger\) and \(H\) map \(Y\to Z\)
Layer Z\(H_Z=J\sum_iZ_iZ_{i+1}\)direct parity-phase gadget
First-order chain step
\[U(t)\approx\left[e^{-iH_X\Delta t}e^{-iH_Y\Delta t}e^{-iH_Z\Delta t}\right]^r,\qquad \Delta t=t/r.\]
One bond, one reusable recipe
- Rotate the chosen Pauli axis into \(Z\).
- Apply CNOT–\(R_z(2J\Delta t)\)–CNOT.
- Undo the basis rotation.
- Schedule even and odd bonds separately so gates never share a qubit.
The three worked questions
Try each problem before opening its worked solution. Every solution keeps the intermediate vectors, matrices, algebra, and independent checks visible.
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 \(\operatorname{Var}(Z)\), then check the pure-state Bloch-vector identity.
Open worked solution+
1. Normalize the state and form the bra
Complex conjugation changes \(2i\) into \(-2i\). Therefore
\[|\psi\rangle=\frac1{\sqrt5}\begin{pmatrix}1\\2i\end{pmatrix},\qquad \langle\psi|=\frac1{\sqrt5}\begin{pmatrix}1&-2i\end{pmatrix}.\]
Square each amplitude→Add the probabilities→Confirm unit norm
\[\langle\psi|\psi\rangle=\frac{|1|^2+|2i|^2}{5}=\boxed{1}.\]
2. Apply each Pauli matrix before taking the inner product
The safest route is to calculate \(P|\psi\rangle\) first and only then multiply by \(\langle\psi|\). This keeps the row-by-column bookkeeping visible.
X
\[X|\psi\rangle=\frac1{\sqrt5}\begin{pmatrix}2i\\1\end{pmatrix},\quad \langle X\rangle=0.\]
\(X\) swaps the amplitudes.
Y
\[Y|\psi\rangle=\frac1{\sqrt5}\begin{pmatrix}2\\i\end{pmatrix},\quad \langle Y\rangle=\frac45.\]
The factors of \(i\) cancel.
Z
\[Z|\psi\rangle=\frac1{\sqrt5}\begin{pmatrix}1\\-2i\end{pmatrix},\quad \langle Z\rangle=-\frac35.\]
\(Z\) flips the \(|1\rangle\) sign.
X mean\(0\)
Y mean\(4/5\)
Z mean\(-3/5\)
3. Compute the variance two ways
Since \(Z^2=I\), \(\langle Z^2\rangle=1\):
\[\operatorname{Var}(Z)=1-\left(-\frac35\right)^2=\boxed{\frac{16}{25}}.\]
Independently, \(p_0=1/5\), \(p_1=4/5\), \(\mu_Z=(+1)p_0+(-1)p_1=-3/5\), and \(\mathbb E[Z^2]=1\).
4. Verify the pure-state identity
\[0^2+\left(\frac45\right)^2+\left(-\frac35\right)^2=1.\]
Interpretation
The Bloch vector \((0,4/5,-3/5)\) has unit length, independently confirming a pure state and consistent arithmetic.
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 \(\langle\phi|\psi\rangle\).
- For \(\{P_\phi,I-P_\phi\}\), find the probability of “yes.”
- Construct \(P_\phi\), test it, and confirm the probability by matrix methods.
Open worked solution+
1. Form the bras carefully
Taking a bra conjugates every coefficient: \(-i\mapsto+i\) and \(e^{i\pi/4}\mapsto e^{-i\pi/4}\).
\[\langle\psi|=\frac1{\sqrt5}\begin{pmatrix}2&i\end{pmatrix},\qquad \langle\phi|=\frac1{\sqrt2}\begin{pmatrix}1&e^{-i\pi/4}\end{pmatrix}.\]
\[\begin{aligned}\langle\phi|\psi\rangle&=\frac1{\sqrt{10}}\left(2-i e^{-i\pi/4}\right)\\&=\frac1{\sqrt{10}}\left[\left(2-\frac1{\sqrt2}\right)-\frac{i}{\sqrt2}\right].\end{aligned}\]
2. Apply the Born rule without skipping the modulus
\[\begin{aligned}p_\phi&=\frac1{10}\left[\left(2-\frac1{\sqrt2}\right)^2+\left(\frac1{\sqrt2}\right)^2\right]\\&=\boxed{\frac{5-2\sqrt2}{10}}\approx0.217157.\end{aligned}\]
Meaning of the answer
The probability is the squared length of the component of \(|\psi\rangle\) along \(|\phi\rangle\), not the complex overlap itself.
3. Construct and test the projector
\[P_\phi=\frac12\begin{pmatrix}1&e^{-i\pi/4}\\e^{i\pi/4}&1\end{pmatrix}.\]
Hermitian\(P_\phi^\dagger=P_\phi\)
Off-diagonal entries are conjugates.
Unit trace\(\operatorname{Tr}P_\phi=1\)
One eigenvalue is one.
Rank one\(\det P_\phi=0\)
The other eigenvalue is zero.
4. Verify idempotence and the probability
\[P_\phi^2=|\phi\rangle\underbrace{\langle\phi|\phi\rangle}_{1}\langle\phi|=P_\phi,\qquad \langle\psi|P_\phi|\psi\rangle=\boxed{\frac{5-2\sqrt2}{10}}.\]
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 basis distribution.
- Compute \(\rho_A\).
- Use \(\rho_A\) to show entanglement.
- Find the Schmidt coefficients.
Open worked solution+
1. Read every amplitude before squaring
\[|\Psi\rangle=\frac1{\sqrt3}\begin{pmatrix}1\\1\\0\\1\end{pmatrix},\qquad P(00)=P(01)=P(11)=\frac13,\quad P(10)=0.\]
The equal outcomes are \(00\) and \(11\), so \(P(\text{equal bits})=2/3\). The probabilities sum to one.
2. Organize the coefficients before tracing out B
\[|\Psi\rangle=\frac1{\sqrt3}\left[|0\rangle_A(|0\rangle_B+|1\rangle_B)+|1\rangle_A|1\rangle_B\right].\]
\[C=\frac1{\sqrt3}\begin{pmatrix}1&1\\0&1\end{pmatrix},\qquad \rho_A=CC^\dagger=\boxed{\frac13\begin{pmatrix}2&1\\1&1\end{pmatrix}}.\]
3. Validate the reduced state
Hermitian\(\rho_A^\dagger=\rho_A\)
It is real and symmetric.
Normalized\(\operatorname{Tr}\rho_A=1\)
Diagonal entries sum to one.
Positive\(\det\rho_A=1/9>0\)
Both eigenvalues are positive.
4. Use purity to diagnose entanglement
\[\rho_A^2=\frac19\begin{pmatrix}5&3\\3&2\end{pmatrix},\qquad \operatorname{Tr}(\rho_A^2)=\frac79<1.\]
The global state is pure but the reduced state is mixed. Therefore \(|\Psi\rangle\) is entangled.
5. Derive the Schmidt coefficients
\[\det(\rho_A-\lambda I)=\lambda^2-\lambda+\frac19=0,\qquad \lambda_\pm=\frac{3\pm\sqrt5}{6}.\]
\[\boxed{s_\pm=\sqrt{\frac{3\pm\sqrt5}{6}}},\qquad s_+^2+s_-^2=1.\]
Entanglement in one number
The reduced-state entropy is approximately \(0.550\) bits: nonzero, but below the one ebit of a Bell state.
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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.