Physics
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| Physics |
| Announcements: | |
| – 3 Dec | ps7 posted, due Tue evening 10 Dec 2024 23:59 | – 16 Nov | ps6 problem 5 posted, due before or just after break (via Gradescope) |
Syllabus:
| Date | Lecture | Reading | |
|---|---|---|---|
| Week 1 | Tue 8/27 | 1. Intro |
Discussed past syllabus Fall'23, and overlaps with newer courses AEP 3100 (Spring, P. McMahon) and CS 4812 (Spring, N. Spooner). Gave broad overview of past century of QM (1905, 1920s, 1960s, 1980's-now, including offhand reference to Babbage's analytical engine). Mentioned a few possible topics [measures of entanglement (von Neumann entropy, Renyi entropy, purity, mutual information, negativity, strong subadditivity, other entropy inequalities), special quantum states (bosonic/fermionic Gaussian states, stabilizer states), evolution/manipulation of quantum states (unitary quantum circuits, projective measurements, weak measurements, quantum channels, Clifford group evolution), behavior of many-body systems (area- and volume-law entropy scaling, eigenstate thermalization hypothesis), quantum adiabatic algorithm, quantum error correction, matrix product states, quantum and shadow tomography.] began 1-qubit states |
| Thu 8/29 | 2. n-qubits and operators |
recalled parametrized 1-qubit state, photon polarizers, then discussed basics of linear algebra (M App.A) and direct product spaces, then multi-qubit states (M1.2), entangled and product states (M1.5) Pauli σ matrices as X,Y,Z (including relation between Pauli matrices and quaternions plus gratuitous mention of the Brougham/Broome bridge), Bloch sphere (see also interactive demo), expectation values of operators in states, and exp values of X,Y,Z operators in parametrized 1-qubit state (See also M.app.A for "Vector spaces: basic properties and Dirac notation") | |
| Week 2 | Tue 9/3 | 3. more gates | Some follow up on expectation values of operators and the Bloch sphere to set up problems 1-3 on ps1; introduced the phase operator S, then some higher qubit operators, starting with direct products of operators and then operators SWAP and CNOT (M1.2-1.4), continuing M1.4, introduced the Hadamard H and Toffoli gate (= ccNOT), and more general controlled gates; mentioned relation to reversible computing, and started drawing circuit diagrams (M1.7) |
| Thu 9/5 | 4. reversibility, entanglement |
Some comments on
reversible computing
and Landauer's principle [recent refs describing experimental verification of latter: Viewpoint (May 2018) describing pure quantum verification, and an older News and Views (Mar 2012) re classical tests. Plus a couple of popular refs: The Fundamental Physical Limits of Computation (Bennett and Landauer, SciAm Jul 1985), Demons, Engines and the Second Law (Bennett, SciAm Nov 1987)] Described half-wave plates, and Hadamard as reflection about π / 8 in optical quantum computing. More examples of direct product operators, and Sij=CijCjiCij identity (M 1.3), cNot as to create entanglement, circuit diagram (M 1.7) for Bell pair, finished with SVD criterion for bipartite entanglement (see these notes) | |
| Week 3 | Tue 9/10 | 5. Bell + Born |
Discussed some problems on ps1 Showed circuits for Bell pair and generalized GHZ states, and no causality violation; Mentioned cNOT,H,T as universal gate set (A16.2, and see "The Solovay-Kitaev Algorithm" for technical review), and Clifford group on n qubits Discussed difference between superposition and mixed states (M1.8, p.27), introduced density matrix ρ . Then more on Born rule and generalized Born rule (M1.8-1.9), plus some examples |
| Thu 9/12 | 6. √X, CZ, SU(2) |
√X in terms of the S operator, and
quantum Toffoli gates (M2.6, pp 58-61) Additional discussion of cNOT, Hadamard, and CZ (M1.4), and generalized control gates Started M.app.B on Pauli σ algebra and SO(3)=SU(2)/Z2. (See also M.app.A for "Vector spaces: basic properties and Dirac notation") Here are some relevant zoom era whiteboards written in real-time on a tablet. | |
| Week 4 | Tue 9/17 | 7. SU(2)/Z2 |
Continued SU(2)/Z2=SO(3) discussion of M.app.B, with numerous examples. Some comments about SU(2) vs adjoint (SO(3)) actions of operators, including Hadamard as reflection about π / 8 vs adjoint action as reflection by π about (x̂+ẑ)/√2, and demo'd Ry(-π/2)Rx(-π/2) = Rn(-2π/3) with n=(1,1,1)/√3. mentioned compact Lie groups (SO(n), SU(n), Sp(n), G2, F4, E6,7,8) and their algebras. physical interpretation of SU(2) elements acting on spin-1/2 in 3d, and spin-statistics relation (for more, see pedagogic video, and more discussion of fermions/bosons to be completed lec8) |
| Thu 9/19 | 8. Computation |
Some additional comments on spin-statistics,
action of Rn(θ) on Bloch sphere. Introduced general computation process (M2.1), then Deutsch's problem (M2.2, A17.3) and its Deutsch-Jozsa generalization (A17.4) (see also Qiskit D-J) | |
| Week 5 | Tue 9/24 | 9. BV→Simon |
no-cloning (A 7.2; M2.1[p.39-40]) then Bernstein-Vazirani problem (n bit to 1 bit function of the form f(x)=a·x, see M2.4, A18.1) See also these covid-era written notes and Qiskit: Phase Kickback, Deutsch-Jozsa Algorithm, Bernstein-Vazirani Algorithm. then started Simon's problem f(x) = f(x⊕a) (M2.5, A18.2, and qiskit Simon's algorithm) Mentioned "birthday paradox", and this video Straight From the Source (Peter Shor) describing how Simon's algorithm led to Shor's (See also comments from Daniel Simon) |
| Thu 9/26 | 10. Simon→Shor |
Simon's problem (M2.5) [see M app.G for "example of the purely mathematical postprocessing frequently necessary after the quantum algorithm returns its probabilistic results", now halfway through standard algorithms] See also these covid-era written notes why additional Qbits don't mess things up (M2.3, pp 46-50); quantum garbage collection (A17.2) Began the discussion of period finding and Shor's algorithm (M3.1) | |
| Week 6 | Tue 10/1 | 11. QFT→period |
Began with discussion of ps2#4, relations between Clifford group on 1 qubit and symmetry group of cube and permutation group on 4 elements; more on H,T universality from standpoint of platonic solids (for practical implications, see also icosahedral qubit benchmarking Returned to period finding problem and quantum Fourier transform (M 3.1, 3.4), continued to derivation of p(y) (M 3.7), and relation to diffraction grating (where d/λ ~ r/2n) See these lec11 written notes | Thu 10/3 | 12. period→RSA |
Discussed the CHSH game (A13.2, A14.1, though feel encouraged to read all of A13,A14) to set up ps3#4. Continued period finding and Shor's algorithm (M3.1), and relation to factoring (M3.10). Outlined RSA encryption (M 3.2, 3.3; A 19.1) (including Euclidean algorithm and Extended Euclidean Algorithm) included comments on continued fractions (see 355/116), see this notebook lec_RSA.ipynb for an overview of RSA and use of sympy for continued fractions, euclidean algorithm, gcd, etc. See these lec12 written notes |
| Week 7 | Tue 10/8 | 13. O(n2), and simplifications |
Comments on ps3, including 3-qubit stabilizer states and the "GHZ game" (M6.6). then the QFT implementation, showing circuit diagram with quadratic growth in number of gates (see also A 20.1.1). and additional simplifications: eliminate 2-qubit gates (M 3.6), unimportance of small phase errors (M 3.9), and qubit recycling (experimental implementation: arXiv:1111.4147; for more on "recycling qubits", see e.g., quant-ph/9903071, quant-ph/0001066, quant-ph/0601097, 1301.7007) mentioned modular exponent is O(n3) See these lec13 written notes |
| Thu 10/10 | 14. period → Grover |
Brief reprise of RSA encryption (M 3.2, 3.3; A 19.1), mentioned of RSA_numbers, specifically 768 | 232 (Feb 2010), 795 | 240 (Dec 2019), 829 | 250 (Feb 2020). Unfortunately quantumcryptopocalypse.com (based on 1710.10377) is no longer up, but see older screengrabs in written notes. See also Post-quantum_cryptography for non-RSA paths forward. Discussed efficient circuit for modular exponent Uf (M 3.8, see also written notes), and Euclidean algorithm (M App.J). Some comments on P vs NP (A 22.0, 24.3), then Grover's Algorithm (M chp4, A 22.1.1-22.1.4, qiskit grover, see also pedagogical reviews: Grover's and Lavor et al.'s). [Have now introduced all of the basic basic quantum algorithms.] See these lec14 written notes | |
| Week 8 | Tue 10/15 | Fall Break | |
| Thu 10/17 | 15. 15, GHz, many marked |
For ps4, covered how to factor N=15
(though using n=no rather than n=2no as in those notes);
mentioned Fermat primes. Following up on ps3, redid the "GHZ game" using states rather than stabilizers, plus discussion of implications for excluding hidden variables (M6.6). Brief follow-up on Grover's algorithm, generalization to many marked states. See these lec15 written notes | |
| Week 9 | Tue 10/22 | 16. Grover → QKD | Special case of Grover (M 4.5), and
some comments on N1/2 optimality of Grover (A 23.1), and the
N1/3 quantum collision algorithm (A 24.1) (The optimality of Grover's algorithm is shown here, and implications here) Then started Quantum Key Distribution (A 8.2; M 6.2, and as well QKD wikipedia entry for overview) See these lec16 written notes |
| Thu 10/24 | 17. QKD, teleportation |
Completed discussion of quantum key distribution, see QKD wikipedia entry for an overview
of experimental implementations, including over 144km in the Canary Islands (2007), 307km optical fiber (2015), 1203km via satellite (2017), and a list of existing distribution networks).
Quantum teleportation (M 6.5; A 10.1; qiskit 3.11),
-- note this fun historical account, and experimental implementation (from 1997). See these lec16 written notes, lec17 written notes | |
| Week 10 | Tue 10/29 | 18. teleportation, QEC I |
Completed quantum teleportation, including circuit diagram interpretation of teleportation (M 6.5), teleportation of entanglement (A 10.1, see fig 10.3). Discussed decoherence due to environment (M 5.1), Schroedinger's cat, referred to fig. 2 and table 1 of 1905.13641 for semi-recent data on superconducting flux qubit coherence times and gate fidelities. Started quantum error correction (A 27.1-27.2), qiskit 5.1, mentioned again 1905.09749 in the context of error correction 3-qubit bit-flip code (M 5.1-5.2) See these lec17 written notes, lec18 written notes |
| Thu 10/31 | 19. QEC II | Recapped 3-qubit bit-flip code (M 5.1-5.2), then Stabilizer interpretation of 3-qubit bitflip correction code (M 5.2), Introduced measurement of operators (M 5.4), then general form of single qubit errors (M 5.3, A 27.2) See these lec19 written notes | |
| Week 11 | Tue 11/5 | 20. QEC III. 5-qubit, 7-qubit |
Mentioned how "measurement of Z operator" corresponds to conventional projective measurement in Z-basis, recalled count of 2(3n+1) < 2n for n >= 5 (M5.4); then the 5-qubit error correction code (M 5.5) see 1907.04507 (2019) for recent superconducting qubit implementation of 5-qubit error-correcting code (and see NMR implementation from 2001) note that error correcting operator algebras can also be manipulated using python or matlab, e.g., this sample code. See Gottesman (2009) for a (slightly, but not extremely, technical) review of stabilizer formalism and fault tolerance. Then started the 7-qubit code (M 5.6-5.8, p.121-127). See these lec20 written notes |
| Thu 11/7 | 21. QEC IV. 7-qubit, surface | Continued 7-qubit code (M 5.6-5.8, p.121-127), including mention of multiqubit errors (M 5.6) and composite H and cNOT operators (M 5.7) in 7-qubit code. Then started the surface code, see arXiv:1208.0928, sections II-VI, pp.3-10, and mentioned 1912.09410 (experimental realization of truncated 7-qubit surface code with 4 data qubits and 3 measure qubits, extending the coherence time of a single logical qubit), and more recently 2207.06431 (25 data qubits and 24 measure qubits), and most recently 2408.13687 (49 data qubits and 28 measure qubits). Finished with (intentionally outdated) vendor roadmaps and difficulties realizing error correcting codes with current qubit fidelities See these lec21 written notes | |
| Week 12 | Tue 11/12 | 22. QEC V |
Reprised discussion ofcomposite cNOT operators (M 5.7) in 7-qubit code. Mentioned the 29:28--29:43 section in this video (J. Martinis talk at Google LA, Oct '13), with claim that "if you can be at a .1% error here, and make a few thousand qubits, you can hold a qubit state, this fragile quantum state, for the lifetime of the universe." [note that entire video is worth watching]. More detail on the 1912.09410 (2019 experimental realization of truncated 7-qubit surface code), More on surface code: operators, more qubits -- see arXiv:1208.0928, sections VIII, XIII, XIV therein, especially pp 29-30, XIV C,D for cNOT; Mentioned IBM's heavy-hex-lattice and associated heavy-hex-codes. Then returned to the recent realization of distance 7 code reported in 2408.13687, and finished with a discussion of leakage in quantum processors (see this google quantum blog entry). See also these lec22 written notes |
| Thu 11/14 | 23. Density matrix | Introduced density matrix ρ, relation to expectation values of operators, mixed states, criterion for pure state, and geometry of mixed states on Bloch sphere (A chapt 6). Introduced reduced density matrices and partial trace (Nielsen/Chuang 2.4.3). | |
| Week 13 | Tue 11/19 | 24. Entropy |
Clarified breakdown of surface code due to line of errors half-way across surface, requiring more physical qubits per logical qubit to suppress (see the discussion around Fig.5 on p.12 of 1208.0928). Then more discussion of partial trace (as covered in Nielsen/Chuang 2.4.3), and more examples of reduced density matrices. Then as prelude to entanglement entropy introduced Von Neumann entropy S = -trρlogρ, and gave quick overview of its relation to Shannon information -- see these notes (see also this "Mini-Introduction To Information Theory" arXiv:1805.11965). |
| Thu 11/21 | 25. info gain, max entanglement |
Discussed these videos
comparing a few different 1-qubit strategies using measurements along any axis (i.e., not just X,Y,Z; and with different strategies for choosing them) to illustrate use of information gain. Then more on Von Neumann entropy, subsystems, more examples of partial traces, relation to entanglement entropies, and maximally entangled states. | |
| Week 14 | Tue 11/26 | 26. How to build a QSC |
Gave an overview of arXiv:2411.10406 (Nov 2024: "How to Build a Quantum Supercomputer: Scaling Challenges and Opportunities"), including: necessary fidelity for QEC to work scalably, problematic fat tail of lower qubit fidelities, different challenges at different numbers of qubits (10k-100k, 100k-1M, >1M), required fidelity for optical interconnects between dilution refridgerators (necessary for >100k), and resources for quantum chemistry calculations (examples of p-benzyne and FeMoco). For the latter, discussed "Trotterization" (and Baker-Campbell-Hausdorff), and introduced quantum phase estimation (see also these notes and lec26_PhaseEstimation.ipynb) |
| Thu 11/28 | Thanksgiving | break | |
| Week 15 | Tue 12/3 | 27. tangles |
Reprised Hamiltonian simulation and related resources from end of previous lecture, including VQE (= Variational Quantum Eigensolver, see also IBM tutorial). then more on monogamy of entangement, and entanglement measure arXiv:quant-ph/9907047, with examples of tangles and exercises in partial traces (see lec27_addendum.pdf) |
| Thu 12/5 | 28. entanglement as work |
Completed details of entanglement measure arXiv:quant-ph/9907047, see lec27_addendum.pdf for full notes of calculations. Then discussed how erasing a system can exchange entanglement for a net gain of work, focusing on Sec IIA of arXiv:1009.1630 (see figs 1,2,3 on pp.3-5). See "Quantum Algorithm Zoo" for a compendium of quantum algorithms. |
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