Quantum Computing and Information
Introduction to the ideas and principles of quantum computing and information. Review of mathematical foundations. Discussion of quantum theory, architecture, and quantum gates. Basic algorithms in quantum computing. Theoretical computer science and computation. Applications of quantum computing to cryptography. Quantum information and error correction.
Study notes
Independent, AI-drafted study guide · Public sources linked below · Added November 5, 2026
Original guide aligned to the public Fall 2026 outline. The outline identifies course topics; explanations and exercises are independently written.
States, amplitudes, and measurement
A pure qubit state has the form alpha|0> + beta|1>, where complex amplitudes satisfy |alpha|^2 + |beta|^2 = 1. Measurement in the computational basis returns 0 or 1 with those squared magnitudes. Relative phase can affect later interference even when immediate measurement probabilities are unchanged. A global phase does not change observable predictions. A superposition is therefore not merely an ordinary coin whose hidden classical value has not yet been revealed.
References: IBM Quantum Learning — Basics of quantum information
Linear algebra and quantum operations
Closed-system pure-state evolution is represented by a unitary matrix, which preserves inner products and total probability. An observable is represented by a Hermitian operator; its eigenvalues are possible measurement values under the appropriate measurement model. Matrix multiplication order follows operation order and the chosen vector convention. A circuit is a structured description of these transformations. Tensor products combine systems, so two qubits require four computational-basis amplitudes rather than a pair of independent two-number descriptions.
References: IBM Quantum Learning — Basics of quantum information
Worked Bell-state circuit
Begin with |00>. Apply a Hadamard gate to the first qubit, giving (|00> + |10>)/sqrt(2), then a controlled-NOT with that qubit as control, giving (|00> + |11>)/sqrt(2). Measuring both in the computational basis yields 00 or 11, each with probability one half. Each local result is random while the pair is correlated. The joint state cannot be factored into one pure state per qubit, which distinguishes entanglement from a product state.
References: IBM Quantum Learning — Basics of quantum information
Entanglement, communication, and no-cloning
Entanglement creates correlations that are useful in communication protocols, but it does not allow a sender to choose a distant measurement outcome and signal instantly. Teleportation combines shared entanglement with classical communication; it transfers an unknown quantum state while consuming the original state in the protocol. A universal copying operation for arbitrary unknown states is impossible because quantum evolution is linear. Copying known classical basis states is compatible with this limitation and does not contradict it.
References: IBM Quantum Learning — Basics of quantum information
Interference and algorithmic speedups
A quantum algorithm arranges transformations so that undesired amplitudes cancel and useful outcomes become more likely. Merely evaluating many basis components in superposition does not make all answers readable. Grover-style search improves query complexity for unstructured search under its oracle model; factoring algorithms use additional algebraic structure. Distinguish a query bound, a gate count, and the engineering cost of implementing the oracle. A speedup claim must specify the classical comparison and resources being counted.
References: IBM Quantum Learning — Fundamentals of quantum algorithms
Quantum error correction and physical limitations
Quantum error correction encodes information across several physical systems so that certain errors can be detected and corrected without measuring the encoded logical information directly. Syndrome measurements reveal error information under the code's assumptions. Error correction does not clone an unknown state; it embeds it in a larger code space. Noise, imperfect gates, measurement errors, and resource overhead constrain implementation. Bell inequalities test limits of particular classical correlation models under stated experimental assumptions; they are not a license for faster-than-light communication.
References: IBM Quantum Learning — Basics of quantum information
Practice & worked answers
Original exercises for this guide.
1. Do |+> = (|0>+|1>)/sqrt(2) and |-> = (|0>-|1>)/sqrt(2) always behave identically?
Show worked answer
No. A computational-basis measurement gives equal probabilities for both, but applying Hadamard maps them to |0> and |1> respectively. Their relative phase changes interference, so one measurement basis cannot characterize all behavior.
2. Why can the Bell state not be written as independent single-qubit pure states?
Show worked answer
If (a|0>+b|1>)(c|0>+d|1>) had nonzero 00 and 11 coefficients, then a,c,b,d would all be nonzero. Its 01 and 10 coefficients would therefore also be nonzero, contradicting the Bell state's zero coefficients for those outcomes.
Public notes & references
Course sources establish the scope above. Reference materials support the explanations; linked lecture notes and books remain with their authors.
Prerequisites & course information
Precludes additional credit for MATH 4821 , MATH 5821 . Prerequisite(s): COMP 2804 . Lectures three hours a week
Sources
- Course catalogue ↗2026–2027
- Quantum Computing ↗2026