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- Quantum Algorithmic Thinking · Quantum AlgorithmsThe complete input family for Deutsch’s problemOne coherent query on a superposition of inputsForms of access and what they do not includeDeutsch’s algorithm with one queryA target record prevents cancellationThe answer states before the final HadamardMajority voting amplifies a bounded advantageThe Deutsch–Jozsa circuit on n input qubitsResources for the Deutsch circuit as drawnThe design process applied to Deutsch’s problemTechniques of the course and where they are used
- Coherent Computation · Quantum AlgorithmsThe clean oracle contractDirect reversible translation leaves garbageAdding one modulo fourCompute, copy, uncomputeA phase oracle from a clean computationAn interference test separates clean from dirty oraclesWhat one gate costs depends on the gate setTrading time for space in reversible computationReflection about a prepared stateControlling a known circuit gate by gateA coherent output is not a table of values
- Phase Encoding · Quantum AlgorithmsPhase kickback from an eigenstateThe Hadamard testA phase becomes a biasA bit oracle from a controlled phase oracleThe signs of the two-qubit Hadamard transformThe Bernstein–Vazirani circuit for three input bitsThe zero outcome outside the promiseOne run of Simon’s algorithmSimon’s algorithm for n = 3 and s = 110A phase from a computed integerPhase kickback from modular additionPhase precision as a resource
- Fourier Methods · Quantum AlgorithmsThe eighth roots of unityA period-2 pattern becomes two frequenciesThe quantum Fourier transform on three qubitsExact and approximate Fourier transformsPhase estimation with three control qubitsPhase estimation for φ = 3/8 = 0.011 in binaryPhase estimation for a phase between grid pointsPhase estimation on a superposition of eigenstatesThe orbit of 1 under multiplication by 7 modulo 15Order finding by phase estimationRecovering r = 6 from the outcome 85/512Shor’s reduction from factoring to order findingWhere the cost of Shor’s algorithm goesHidden subgroup problems
- Amplitude Manipulation · Quantum AlgorithmsAmplification in the good–bad planeOne amplification iterationSuccess probability oscillates with the number of iterationsOptimal iteration countsExact amplification by lowering the angleOrdinary versus fixed-point amplificationThe Grover iterate as a rotation with complex eigenvectorsAmplitude estimationSampling versus amplitude estimationEncoding a mean as a success probabilityMinimum finding as amplification in a classical loopWhat amplification can and cannot do
- Quantum Walks · Quantum AlgorithmsA random walk on a four-cycleOne step of a coined walk on a cycleQuantum and classical walks after 60 stepsCoherent oscillation versus classical mixingEvolving through eigenvectorsEigenvalues of a cycle are cosinesOne step of Szegedy’s walkA spectral gap becomes a larger phase gapClassical and quantum walk searchElement distinctness on the Johnson graphWhat quantum walks achieve
- Operator Embedding · Quantum AlgorithmsA non-unitary matrix changes lengths unevenlyA block encoding with one ancillaPostselection for A = diag(1, 1/2)Linear combination of unitariesA block encoding of (I + X)/2An LCU block encoding of a two-qubit HamiltonianMultiplying encoded matricesHow block-encoding errors combineOblivious amplitude amplificationHow a matrix can be givenSpeedups under compatible access
- Polynomial Transformations · Quantum AlgorithmsResponses of repeated signal callsA quantum signal processing sequenceConstraints on achievable responsesFunctions of a Hermitian matrixThe qubitized walkSingular value transformationA polynomial filterA bounded approximation to the inverseWhy simulation degree is about αt
- Evolution Engineering · Quantum AlgorithmsThe Hamiltonian simulation problemFirst-order product formula for X + ZFirst-order Trotter error: bound and realityFirst- and second-order formulasExponential of a Pauli stringThree families of simulation methodsAn error budget for a simulationEnergies along an adiabatic pathSlow enough evolution follows the ground stateWhat adiabatic optimization achieves
- Variational Methods · Quantum AlgorithmsA one-parameter entangled ansatzThe energy landscape of the exampleShots needed to estimate an energyExact gradients from shifted circuitsTwo kinds of noise in variational methodsBarren plateaus: causes and responsesThe variational loopDepth-1 QAOA on a single edgeWhat kind of evidence supports a claim
- Adaptive Measurement · Quantum AlgorithmsMeasurement with feedforwardDeferring a measurementA measurement-dependent Pauli correctionOne stage of iterative phase estimationIterative phase estimation for φ = 3/8 = 0.011 in binaryTwo scalings of phase estimationApplying a gate by one-bit teleportationMeasurement-based computation on a chainInjecting a T gate from a magic state
- Glossary · Quantum Algorithms
- Introduction · Quantum Algorithms
- Study guide · Quantum Algorithms
- Logic and Computability · Lecture slidesA proof is a network of justified stepsHilbert’s program: proving consistencyChecking, searching, and decidingEuclid’s algorithm: what stays fixed, what decreasesConfigurations contain what is needed to continueA Turing machine: tape, head, and finite controlFormal model equivalence and the Church–Turing thesisThe halting contradiction: run D on its own descriptionUndecidability and incompleteness concern different objectsDeterministic does not imply reversibleRetaining inputs preserves the distinctions AND discards
- Logic and Computability · Quantum ComputingA proof is a network of justified stepsHilbert’s program: proving consistencyChecking, searching, and decidingEuclid’s algorithm: what stays fixed, what decreasesConfigurations contain what is needed to continueA Turing machine: tape, head, and finite controlFormal model equivalence and the Church–Turing thesisThe halting contradiction: run D on its own descriptionUndecidability and incompleteness concern different objectsDeterministic does not imply reversibleRetaining inputs preserves the distinctions AND discards
- Boolean and Reversible Computation · Lecture slidesCircuit for x + y + xyXOR from four NAND gatesControlled reversible gatesCompute, copy a basis label, and uncompute
- Boolean and Reversible Computation · Quantum ComputingOne bit string, different interpretationsAND as a complete input–output specificationOR written as a polynomial over bitsCircuit for x + y + xyXOR from four NAND gatesThree different inputs mergeResetting an unknown bitLogical and physical questions are separatePreserve the input; update the targetA controlled operation has two branchesControlled reversible gatesCompute, copy a basis label, and uncompute
- Probability Amplitudes and Interference · Lecture slidesDeterministic transitions: NOT applied twiceNondeterministic transitions: keep or flip, applied twiceProbabilistic transitions: a fair coin step, applied twiceCancellation at a common endpointSigned Hadamard transition with weight cSquare root of NOT with weights a and bPath visualization of the cross-sum conditionRelative phase changes how paths combineA retained record keeps the endpoints distinctUndo the record before the final H
- Probability Amplitudes and Interference · Quantum ComputingA state of one bitDeterministic transitions: NOT applied twiceGeneric deterministic transitions across multiple states and stepsRESET merges two configurationsNondeterministic transitions: keep or flip, applied twiceGeneric nondeterministic transitions across multiple states and stepsProbabilistic transitions: a fair coin step, applied twiceGeneric probabilistic transitions across multiple states and stepsThe product rule along one pathSum the paths to a common endpointUpdate one component of the stateSigns multiply along a pathCancellation at a common endpointGeneric signed transitions across multiple states and stepsSigned Hadamard transition with weight cHadamard transitions and their ket notationQuantum transitions: the Hadamard step, applied twiceSquare root of NOT with weights a and bPath visualization of the cross-sum conditionRelative phase changes how paths combineA retained record keeps the endpoints distinctUndo the record before the final H
- Quantum States and Measurement · Lecture slidesA ket as a vector in the standard basisPaths from an initial state to a final stateSplitting a state along a directionInner product: amplitude along the final directionTransition amplitudes and matrix entriesGeometric action of a transition matrixAdjoint: reverse transitions and conjugate amplitudesComposition of U and its adjointA matrix and its adjoint acting on the same vectorUnitary evolution preserves state geometryMeasurement and its conditional stateMeasurement: project, square the length, then normalizeContinuous evolution interrupted by measurementTwo-qubit amplitudes on a complete bipartite graph
- Quantum States and Measurement · Quantum ComputingA ket as a vector in the standard basisPaths from an initial state to a final stateSplitting a state along a directionInner product: amplitude along the final directionTransition amplitudes and matrix entriesGeometric action of a transition matrixAdjoint: reverse transitions and conjugate amplitudesComposition of U and its adjointA matrix and its adjoint acting on the same vectorGeometric action of AA dagger and A dagger AUnitary evolution preserves state geometryMeasurement and its conditional stateMeasurement: project, square the length, then normalizeContinuous evolution interrupted by measurementTwo-qubit amplitudes on a complete bipartite graph
- Quantum Model of Computation · Lecture slidesOne qubit: a gate followed by measurementRelative phase becomes a definite output
- Quantum Model of Computation · Quantum ComputingInput, unitary evolution, and readoutGates compose in time orderOne run gives one resultOne qubit: a gate followed by measurementRelative phase becomes a definite outputPhase becomes an output probabilityThe Pauli gates on one wireA matrix sum is not a gate sequence
- Multi-Qubit Computation · Quantum ComputingIndependent preparation on two wiresA superposition without entanglementA joint register and its readoutA complete two-qubit computationBell-state preparationA gate that prevents factorizationChoose the readout circuitCopying basis labelsThe proposed copying circuitLocal operations and remote readoutQuantum teleportationQuantum teleportationQuantum teleportationEvaluate f on every inputMeasure a parallel evaluationInterfere the input branches
- Quantum Queries and Phase Kickback · Quantum ComputingFour possible one-bit oraclesTwo classical value queriesThe reversible query gateEvaluate on a superpositionRead one input–value pairA workspace record changes interferenceFlip eigenstates on a wirePhase kickbackCompute, phase, uncomputeRead the relative signDeutsch algorithmPreparation and readout both matter
- Quantum Circuits and Universality · Quantum ComputingWire order and gate orderThe oracle inside a circuitQuantum teleportationRotations and phase gatesControlled Z from CNOTFour-item search circuitLocal gates preserve product statesLocal gates with entangling connectionsApproximate a gate sequenceCount gates, layers, and wiresRoute an interaction through a neighbourEncoding is not cloning
- Quantum Search and Amplitude Amplification · Quantum ComputingTest a candidate labelPrepare and mark four candidatesFour-item search circuitGroup outcomes of the search circuitReadout before amplificationOne pair of reflectionsRepeated Grover iterationsAdjust the repeated Grover circuitVerify a measured candidateAn arbitrary coherent preparationReflection about a prepared statePay for both preparation and its inverse
- Computational Complexity and Quantum Advantage · Quantum ComputingSpecify the input and success conditionA circuit family and its costCount queries separately from gatesChecking a certificateA bounded-error decision circuitSearch in its query modelThe circuit family covered by a query boundCompare hidden inputs in the same circuitQueries increase polynomial degreeCoherent operations before readoutA finite circuit still has a sampled outputA starting circuit for error correction
- Appendices · Quantum ComputingA stack remembers the unmatched prefixA derivation generates a stringUse conjugation when normalizingDifferent ensembles can yield the same density matrixQueries can increase polynomial degree
- Glossary · Quantum Computing
- Introduction · Quantum Computing
- Notation · Quantum Computing
- Revisions and corrections · Quantum Computing
- Study guide · Quantum Computing
- What Does a Secret Key Promise? · Quantum CryptographyBloch-circle picture of the states Eve must distinguishEve entangles the signal with an ancilla
- BB84: Preparing and Measuring · Quantum CryptographyOrder of messages in one BB84 blockBit counts for the six-signal blockBasis-resolved error rates against Eve's basis tiltHypergeometric tail against a hypothetical error count
- Entanglement and Security Proofs · Quantum CryptographyActual and virtual descriptions of one BB84 roundZero crossing of the rate against reconciliation overheadThe chain from physical overlap to key lengthIdeal BB84 rate as error increases
- From Raw Bits to a Usable Key · Quantum Cryptography
- Sources, Detectors, and Quantum Links · Quantum CryptographyPhoton-number content of two weak coherent pulsesWhere the detected clicks come fromObserved error rate against signal strengthWhich components each security model trusts
- Beyond Key Distribution · Quantum CryptographyPauli encryption decorrelates a message from its referenceA coherent attack on a commitment schemeTwo purifications of one hiding stateThe audit chain and the step that must be cited
- Quantum Cryptography · Quantum Cryptography
- Protecting an Unknown Quantum State · Quantum Error Correction
- Correctable Errors · Quantum Error Correction
- Stabilizer Codes · Quantum Error Correction
- CSS Codes and Classical Coding Theory · Quantum Error Correction
- Logical Operators and Code Distance · Quantum Error Correction
- Syndrome Extraction and Decoding · Quantum Error Correction
- Surface Codes · Quantum Error Correction
- Fault-Tolerant Operations · Quantum Error Correction
- Universal Logical Computation · Quantum Error Correction
- Thresholds and Resource Estimates · Quantum Error Correction
- Quantum Error Correction · Quantum Error Correction
- From Clicks to Evidence · Quantum ExperimentsAnimated two-outcome quantum experiment: preparation, transformation, and measurementBinomial distribution, standard-error scaling, and detector efficiency biasDetector-swap calibration protocol and explicit assumption ledgerA detector, one trial at a time
- Energy Comes in Steps · Quantum ExperimentsPhotoemission, line spectra, and inelastic collisions2.2 Linear stopping potentials and level differences2.3 Energy quantization versus field quantizationThe photoelectric energy balance
- Matter Waves and the Double Slit · Quantum ExperimentsLocalized arrivals and two-aperture geometry3.2 Superposing amplitudes and de Broglie wavelength3.3 Disentangling path records from phase driftBuild an interference pattern
- Stern–Gerlach and Sequential Measurements · Quantum ExperimentsMagnetic gradients and sequential spin analyzers4.2 Half-angle spin projections and cascaded filtering4.3 Projective state update and postselection accountingRotate the analyzer
- Polarization and Three Filters · Quantum ExperimentsCrossed polarizers and intermediate state projection5.2 Malus’s law and noncommuting projections5.3 Separating wave projection from photon statisticsThree polarizers
- Single Photons and Coincidences · Quantum ExperimentsHanbury Brown–Twiss beam-splitter correlations6.2 Second-order correlation and the antibunching dip6.3 Dead time, accidental subtraction, and source claimsCorrelations between clicks
- Mach–Zehnder: Adding Alternatives · Quantum ExperimentsTwo-arm recombination in a Mach–Zehnder interferometer7.2 Beam-splitter matrices and fringe visibility7.3 Phase scans, drift controls, and single-photon fringesRecombine the alternatives
- Einstein’s Recoiling Slit · Quantum ExperimentsCoupling path alternatives to a mechanical pointer8.2 Pointer overlap and the visibility–distinguishability tradeoff8.3 Entanglement with the apparatus versus classical noiseA record becomes distinguishable
- Wheeler’s Delayed Choice · Quantum ExperimentsSelecting path readout or recombination after entry9.2 Incompatible observables on a single two-path state9.3 Spacelike switching and experimental timing auditsChoose the final measurement
- Quantum Erasure and Conditional Fringes · Quantum ExperimentsEntangled path markers and coincidence sorting10.2 Basis rotation of the marker and antiphase fringes10.3 No-signalling and coincidence-window disciplineSort the same signal records
- The Elitzur–Vaidman Bomb Tester · Quantum ExperimentsBlocking one arm of a dark-port interferometer11.2 Three-outcome branching and certification efficiency11.3 Imperfect extinction, false positives, and Zeno extensionsKeep every outcome in view
- Tunnelling Through a Barrier · Quantum ExperimentsSub-barrier scattering and scanning tunnelling probes12.2 Evanescent decay constant and exact rectangular transmission12.3 Separating elastic tunnelling from thermal activationTransmission through a barrier
- Rabi Oscillations and Coherent Control · Quantum ExperimentsResonant driving of a two-level transition13.2 Rotating-frame Hamiltonian, pulse area, and detuning13.3 Amplitude scaling, leakage, and control calibrationRotate a state with a pulse
- Ramsey, Decoherence, and Echo · Quantum ExperimentsSeparated oscillatory fields and Hahn spin echo14.2 Transverse decay envelopes and relaxation bounds14.3 Noise spectroscopy and reduced-state interpretationA clock made from a superposition
- Reconstructing a Quantum State · Quantum ExperimentsMeasuring fresh copies across Pauli bases15.2 Bloch vector inversion, positivity, and purity15.3 Detector calibration and ensemble ambiguityMeasure along three axes
- Two-Photon Interference and Entanglement · Quantum ExperimentsHong–Ou–Mandel coincidence suppression at a beam splitter16.2 Bosonic mode algebra and the overlap dip16.3 Baseline normalization and entanglement witnessesOverlap two photons
- EPR, Bohm, and Bell Tests · Quantum ExperimentsSinglet anticorrelations and switched analyzer settings17.2 Local hidden-variable factorization and the CHSH bound17.3 Loopholes, no-signalling, and assumption packagesFour correlations, one bound
- GHZ and Contextuality · Quantum ExperimentsThree-party parity products and compatible operator grids18.2 Sign parity contradictions in GHZ and Peres–Mermin tables18.3 Mermin inequalities and operational context independenceFollow the product constraints
- Teleportation and Entanglement Swapping · Quantum ExperimentsJoint Bell measurement and classical feed-forward19.2 Four Pauli-frame branches and entanglement swapping19.3 Classical measure-and-prepare benchmarks and heraldingFour messages, one recovered state
- Schrödinger’s Cat and the Measurement Problem · Quantum ExperimentsAmplifying a microscopic superposition into pointer records20.2 Reduced density matrices and environmental decoherence20.3 Decoherence versus the definite-outcome problemRecords and coherence
- Wigner’s Friend · Quantum ExperimentsAn isolated laboratory inside a superobserver’s apparatus21.2 Uncomputing a CNOT premeasurement and joint interference21.3 Environmental record leakage and observer assumptionsInside the laboratory; outside the laboratory
- Extended Friends and Consistency · Quantum ExperimentsTwo entangled laboratories and superobserver basis choices22.2 Three-term amplitude cancellation and the one-twelfth probability22.3 Dissecting no-go assumption packagesAdd signed amplitudes
- Quantum Sensing and Atomic Clocks · Quantum ExperimentsRamsey frequency interrogation and clock feedback loops23.2 Shot-noise error propagation and optimal interrogation time23.3 Systematic shifts, dead time, and stability metricsFind a useful interrogation time
- Capstone: Audit an Interference Claim · Quantum ExperimentsProvenance and structure of a phase-scan dataset24.2 Fourier-moment visibility estimation and plug-in variance24.3 Writing a reproducible experimental audit reportAudit a synthetic phase scan
- Quantum Experiments · Quantum Experiments
- What a Physical Theory Must Say · Quantum Foundations
- States as Predictions · Quantum Foundations
- Effects and Measurements · Quantum FoundationsFour state spaces with two perfectly distinguishable states
- Composite Systems · Quantum Foundations
- Classical Theory and Its Limits · Quantum Foundations
- Continuous Reversibility · Quantum Foundations
- The Qubit · Quantum Foundations
- Larger Systems · Quantum Foundations
- Symmetry and Dynamics · Quantum Foundations
- Observables and Uncertainty · Quantum Foundations
- Measurement and State Update · Quantum Foundations
- Contextuality and the Born Rule · Quantum Foundations
- Entanglement and Nonlocality · Quantum Foundations
- Indistinguishable Systems · Quantum Foundations
- Purification and Open Systems · Quantum Foundations
- Infinite Dimensions · Quantum Foundations
- What Was Assumed · Quantum Foundations
- Quantum Foundations · Quantum Foundations
- Physical Quantum Computation · Quantum Hardware
- Physical Qubits · Quantum Hardware
- Coherent Control · Quantum Hardware
- Qubit Interactions · Quantum Hardware
- Initialization and Readout · Quantum Hardware
- Noise and Decoherence · Quantum Hardware
- Calibration and Characterization · Quantum Hardware
- Superconducting Circuits · Quantum Hardware
- Trapped Ions · Quantum Hardware
- Neutral Atoms · Quantum Hardware
- Semiconductor Spins · Quantum Hardware
- Defect Centers · Quantum Hardware
- Photonic Computation · Quantum Hardware
- Bosonic Encodings · Quantum Hardware
- Topological Approaches · Quantum Hardware
- Control Infrastructure · Quantum Hardware
- Processor Architecture · Quantum Hardware
- Fault-Tolerant Hardware · Quantum Hardware
- Hardware Evaluation · Quantum Hardware
- Quantum Hardware · Quantum Hardware
- Education
- Distinguishability and Testing · Quantum Information TheoryComputational-basis measurement versus optimal Helstrom measurement for |0⟩ and |+⟩
- Quantum Channels and Dilations · Quantum Information Theory
- Entropy and Data Processing · Quantum Information Theory
- Entanglement as a Resource · Quantum Information Theory
- Quantum Compression · Quantum Information Theory
- Classical Communication over Quantum Channels · Quantum Information Theory
- Quantum Communication and Decoupling · Quantum Information Theory
- Recovery and Quantum Error Correction · Quantum Information Theory
- Quantum Resource Theories · Quantum Information Theory
- Research Practicum · Quantum Information Theory
- Quantum Information Theory · Quantum Information Theory
- Information and Distinguishability · Quantum InformationHow 100 model trials divide between cards and detector reportsHow 100 model trials divide between cards and detector reports
- Classical Information · Quantum Information
- Quantum States · Quantum Information
- Composite Systems · Quantum Information
- Mixed States · Quantum Information
- Quantum Operations · Quantum Information
- Quantum Measurement · Quantum Information
- State Discrimination · Quantum Information
- Information Constraints · Quantum Information
- Entanglement Protocols · Quantum Information
- Quantum Entropy · Quantum Information
- Quantum Compression · Quantum Information
- Entanglement Transformations · Quantum Information
- Quantum Error Correction · Quantum Information
- Quantum Communication · Quantum Information
- Quantum Cryptography · Quantum Information
- Quantum Information · Quantum Information
- What Needs Interpreting? · Interpretations of Quantum Mechanics
- Decoherence and the Classical Record · Interpretations of Quantum Mechanics
- Bell, Contextuality, and the Limits of Classical Accounts · Interpretations of Quantum Mechanics
- Operational Accounts and Consistent Histories · Interpretations of Quantum Mechanics
- Everett and Relative States · Interpretations of Quantum Mechanics
- Bohmian Mechanics · Interpretations of Quantum Mechanics
- Objective Collapse and Experimental Alternatives · Interpretations of Quantum Mechanics
- Relational Quantum Mechanics and QBism · Interpretations of Quantum Mechanics
- Quantum-State Ontology and the PBR Argument · Interpretations of Quantum Mechanics
- Quantum Observers and Comparing Interpretations · Interpretations of Quantum Mechanics
- Interpretations of Quantum Mechanics · Interpretations of Quantum Mechanics
- Quantum Programming Languages · Quantum Programming
- The Qiskit Environment · Quantum Programming
- Circuit Construction · Quantum Programming
- States and Operators · Quantum Programming
- Reusable Quantum Programs · Quantum Programming
- Sampling and Estimation · Quantum Programming
- Oracles and Arithmetic · Quantum Programming
- Algorithm Implementation · Quantum Programming
- Testing and Verification · Quantum Programming
- Compilation and Transpilation · Quantum Programming
- Quantum Hardware Execution · Quantum Programming
- Dynamic Circuits · Quantum Programming
- Noise and Reliability · Quantum Programming
- Hybrid Quantum Programs · Quantum Programming
- Advanced Algorithm Construction · Quantum Programming
- Reproducible Quantum Projects · Quantum Programming
- Quantum Programming · Quantum Programming
- What a Quantum Model Is · Quantum Machine LearningThe pipeline of a quantum machine learning modelFour gates an advantage claim must pass
- Encoding Data into Quantum States · Quantum Machine LearningKernel against feature difference for two encodings
- Quantum Kernels and Kernel Methods · Quantum Machine LearningShots needed for a fixed relative precisionKernel spread against qubit count
- Variational Quantum Models · Quantum Machine LearningEight Pauli terms grouped into three measurement settingsTwo-parameter slice of a variational loss landscape
- Quantum Machine Learning · Quantum Machine Learning