Lectures

Following is an outline of lectures given along with references and links to additional reading. Details of the textbooks NC, KLM and LR are available in reference page.

\( \def\PAL{\rm PAL} \def\Z{\mathbb{Z}} \def\NP{\mathsf{NP}} \)

Lec 1 (17 Jan, Wed) : Administrative information. Course overview, objectives and evaluation scheme. Classical computation via Boolean circuits. Basis of Boolean circuits. Universal gates. Classical reversible computation. Reversible gates - CNOT, CCNOT and DUP gates. Ancillia inputs and garbage outputs. Representation of reversible circuits via diagrams. Registers. Randomized reversible computation via COIN gate.

^{Reading : (Text NC) 1.3.1 Single qubit gates, 1.3.2 multi qubit gates, 1.3.4 Quantum circuits, 1.4.1 Classical computation on quantum computer}

Lec 2 (22 Jan, Mon) : Randomized reversible computation via COIN and COIN' gates. Viewing circuits as a sequence of instructions. Connections to quantum computing - all of reversible computation is possible. Qubits - definition, vector representation. Implementation of qubits. Multi-qubit systems - tensor products of quantum states, representation via linear algebra.

^{Reading : (Text NC) 1.2 Quantum bits, 1.3.6 Bell states, (Text KLM) Chapter 1.}

Lec 3 (24 Jan, Wed) : Braket notation, dual vectors, inner and outer products. Multi-qubit states that cannot be expressed via tensor products, entanglement, EPR pair. One qubit quantum gates and properties - linearity and length preserving. Showed why these imply that quantum gates are unitary and preserves inner product. Examples of one qubit quantum gates. $2$-qubit quantum gates via product of $1$-qubit gates. Tensor product of complex valued matrices. Visual representation of qubits via Bloch sphere. Revisiting the definition of a qubit. The road ahead - No cloning theorem, quantum teleportation, super dense coding.

^{Reading : (Text NC) 2.1.7 Tensor Products, 4.2 Single qubit operations, 1.3.1 Single qubit gates, (Text KLM) 2.2 Dual vectors, 2.6 Tensor Products}

Lec () (29 Jan, Mon) : Non-instructional day

Lec 4 (31 Jan, Wed) : Measurement in standard basis, Partial measurements and outcomes. No-cloning theorem - statement and proof. Quantum teleportation - sending a qubit using two classical bits and an EPR entanglement - protocol and correctness argument.

^{Reading : (Text NC) 1.3.5 Qubit copying circuit ?, 1.3.7 Quantum teleportation, (Text KLM) 5.2 Quantum Teleportation}

Lec 5 (05 Feb, Mon) : Measurement under orthonormal basis, Rotation unitaries, Superdense coding - sending two classical bits using an EPR entanglement - protocol and correctness.

^{Reading : (Text NC) 2.3 Application Superdense coding, (Text KLM) 5.1 Superdense coding}

Lec 6 (07 Feb, Wed) : Bomb testing - a non-trivial strategy with a success probability of 0.33. CHSH game - setup of the game, limitation of classical deterministic strategies.

^{Reading : Wiki page of Elitzur-Vaidman tester. (Notes of R. de Wolf) 17.2 CHSH Clauser-Horne-Shimony-Holt}

Lec 7 (12 Feb, Mon) : Limitation of randomized strategies. A quantum strategy for CHSH based on measuring in a suitable basis its analysis. Intuition behind the choice of basis used for measurement.

^{Reading : Class notes.}

Lec 8 (14 Feb, Wed) : Back to Bomb testing - a better strategy with a success probability of 0.99. Universal quantum gates - Solovay-Kitaev theorem (statement only). Introduction to quantum algorithms and the quantum query model. An illustrative example - correctly computing parity of two bits quantumly and the Deutsch problem. Why is the query model interesting ?

^{Reading : On exactly computing parity - here. Paper of Elitzur-Vaidman tester and class notes. (Text NC) 4.5 Universal Quantum gates and Appendix 3 - The Solovay-Kitaev theorem. (Text KLM) 6.3 The Deutsch Algorithm}

Lec 9 (19 Feb, Mon) : Computing parity of $N$ bits exactly in $N/2$ quantum queries (Farhi-Goldstone-Gutman-Sipser algorithm) - algorithm and correctness argument. The Deutsch-Jozsa problem - limitation of deterministic algorithms.

^{Reading : The Farhi-Goldstone-Gutman-Sipser algorithm (Section IV). (Text KLM) 6.4 The Deutsch-Jozsa Algorithm.}

Lec 10 (21 Feb, Wed) : Deutsch-Jozsa problem - One query quantum algorithm and correctness, $c\log N$ query randomized algorithm with one-sided error probability of $2N^{-c}$. Simon's problem - $n$-query quantum algorithm and analysis.

^{Reading : (Text KLM) 6.4 The Deutsch-Jozsa Algorithm, 6.5 Simons algorithm}

Lec 11 (26 Feb, Mon) : Error analysis of Simon’s algorithm - estimating the probability for $n-1$ random vectors to be linearly independent over $\mathbb{F}_2^n$.

^{Reading : (Text KLM) 6.4 The Deutsch-Jozsa Algorithm, 6.5 Simon’s algorithm. Section 2.4.3 of notes by O’Donnell.}

Lec 12 (28 Feb, Wed) : Randomized algorithm for Simon’s problem succeeding with $O(\sqrt{N})$ time. Fourier basis - parities, orthogonality of the parity basis. Expressing Boolean functions in Fourier basis. Applying $n$-fold Hadamard on $U_f^{\pm}$ amounts to representing $f$ in the Fourier basis. Connection to Deutsch-Jozsa algorithm.

^{Reading : On Fourier representation - Notes by Wright. Optional- Randomized lower bound (based on Minmax principle, not covered in class. See here)}

Lec 13 (04 Mar, Mon) : Searching unstructured database - Grover's search. Algorithm. Grover iterate operator - intuition behind the definition via geometric and diffusion views.

^{Reading : Section 4 of Andrej’s notes and Notes by Wright.}

Lec 14 (06 Mar, Wed) : Properties of the Grover iterate operator. Using the properties to prove correctness of Grover's search algorithm. Grover's search with multiple targets. Simon's problem over $\Z_N$. Introduction to Fourier transforms over $\Z_N$. Characters of $\Z_N$ - existence, properties and orthonormality – statements and proof arguments.

^{Reading : Notes by Wright, see section 2.4 to 2.6 for analysis of Grover’s search. Section 2 of notes by O’Donnell for Fourier transforms over $\Z_N$.}

Lec 15 (11 Mar, Mon) : Fourier Transforms over $\Z_N$ - properties and quantum circuits. Period finding problem and analysis of Simon’s algorithm.

^{Reading : Section 2 of notes by O’Donnell for Fourier transforms over $\Z_N$. Notes by Wright.}

Lec 16 (13 Mar, Wed) : Completed analysis of Simon’s algorithm for period finding. Towards Shor’s factoring algorithm for finding a non-trivial factor of an $n$-bit number - overview. Efficient algorithms for basic arithmetic operations involving numbers.

^{Reading : Notes by Wright and O’Donnell}

Lec 17 (18 Mar, Mon) : GCD computation and Euclid’s algorithm, relative primes. Groups - definition and examples - $(\Z_M,+)$, $(\Z_M^*,\cdot)$. $\Z_M^*$ consists exactly of those integers that are relative prime to $M$. Necessary and sufficient condition for integers $a,b$ to be relatively prime.

^{Reading : Section 3.3, 3.4 and 4.1 in Shoup.}

Lec () (20 Mar, Wed) : Class cancelled due to EML. To be compensated on Mar 21, 28.

Lec 18 (21 Mar, Thu) : Subgroups, Cosets and Proof of Lagrange’s theorem. Order of an element. Ordering finding problem. Factoring algorithm (assuming efficient algorithm for order-finding) – description of algorithm and proof of its correctness.

^{Reading : Class notes and Libretext on Lagrange’s theorem. Notes by O’Donnell.}

Lec () (25 Mar, Mon) : Holiday due to Holi

Lec 19 (27 Mar, Wed) : Finding number of elements of even order in $\Z_M^*$. Chinese remainder theorem - bijection arguments. For a prime $p$, $\Z_p^*$ is cyclic. Relating order of an element in $\Z_M^*$ and order of the images under the Chinese remainder map when $M$ is a product of odd primes.

^{Reading : Notes by Conrad on Chinese remainder theorem and cyclic groups and Notes by Vazirani}

Lec 20 (28 Mar, Thu) : Proof of $\Z_p^*$ is cyclic for a prime $p$. Using this to show that for an odd prime $p$ and for any $e \ge 1$, $\Z_{p^e}^*$ is cyclic. Estimating the number of even order elements in $\Z_M^*$. Overview of the argument to show that randomly picked $a \in \Z_M^*$ of even order satsify $a^{ord(a)/2} \not \equiv \pm 1 \mod M$ with high probability.

^{Reading : Notes by Conrad and Shin on cyclic groups and Notes by Vazirani.}

Lec 21 (01 Apr, Mon) : Completed the proof that a randomly picked $a \in \Z_M^*$ of even order satisfy $a^{ord(a)/2} \not \equiv \pm 1 \mod M$ with high probability - definition of a bad set $S$ and associated subgroup $T$ containing $S$; showed that $T$ is a strict subgroup of $\Z_M^*$. Introduction to Order finding problem. Overview of Shor’s quantum algorithm for order finding and its connection to period finding.

^{Reading : Proof based on argument due to Bach (page 294). Also notes by O’Donnell.}

Lec 22 (03 Apr, Wed) : Recap of Simon’s period finding algorithm and modifications for the order finding problem. Estimation of sampling probability when the order need not divide $N$. Definition of a good gamma. Argument for showing that a good gamma can be sampled with almost uniform probability. Estimating the period from a good gamma - the continued fraction algorithm and a worked out example. Completed the classical and quantum parts of Shor’s Factoring algorithm.

^{Reading : Notes by O’Donnell and Section 6.2 of Preskill.}

Lec 23 (08 Apr, Mon) : Using Grover search algorithm to speedup recursive decomposition based exact algorithms for $\NP$. Doing Grover search when the number of solutions are not known before (Boyer-Brassard-Høyer-Tapp algorithm).

^{Reading : Paper of Fürer, local copy and paper of Boyer-Brassard-Høyer-Tapp}

Lec () (10 Apr, Wed) : Holiday due to Eid-al-Fitr. Compensatory class on 11 April.

Lec 24 (11 Apr, Thu) : Completed the Boyer-Brassard-Høyer-Tapp algorithm. Search vs Decision problems (Search and Decision version of Grover) and Promise vs Total problems (Simon’s problem and Element Distinctness). Exponential speedup is not possible for total problems (Statement only). Lower bounds - overview of the polynomial method. Query lower bound for exact quantum algorithms for parity. Properties of real multilinear polynomials agreeing with Boolean functions.

^{Reading : Paper of Boyer-Brassard-Høyer-Tapp and notes by O’Donnell}

Lec 25 (15 Apr, Mon) : From quantum algorithm making $t$ queries to a real multilinear polynomial of degree at most $2t$. Query lower bound for quantum algorithm computing parity that can err. Key idea - symmetrizing a multilinear polynomial and proof overview.

^{Reading : Notes by Childs and O’Donnell}

Lec 26 (17 Apr, Wed) : Symmetrization. A lower bound for computing $OR$ function via Markov Brother’s inequality. Grover search is optimal. Element distinctness problem - showed classical deterministic bound of $\Theta(N\log N)$. Showed a quantum query lower bound of $\Omega(\sqrt{N})$ by reduction to computing $OR$.

^{Reading : For query lower bound on $OR$ function, see O’Donnell. For a lower bound for element distinctness, see Section 3 of Buhrman et al.. An nice article on the inequality on derivatives of a univariate polynomial (due to Markov) used in the lower bound argument.}

Lec 27 (22 Apr, Mon) : Quantum algorithms and lower bounds for computing OR of ANDs. Amplitude amplification. Application - an $O(N^{3/4}\log N)$ query quantum algorithm for element distinctness.

^{Reading : Notes on amplitude amplification by Dieter. Also Section 3 of Buhrman et al..}

Lec 28 (24 Apr, Wed) : Correctness of the quantum algorithm for element distinctness using amplitude amplification. Adversary method. Overview of the basic adversary method - statement and proof overview. Proving lower bounds using the adversay method for a few of the problems seen before.

^{Reading : Notes by O’Donnell. For more on total variation distance and its connection to distinguishing probability distributions, see here.}

Lec 29 (29 Apr, Mon) : Completed the proof of basic adversary method. What we did not see - connection to quantum physics and quantum information theory and related directions. Course summary and a glimpse beyond.

^{Reading : Notes by O’Donnell and class notes.}