Physics
|
| Physics |
| Announcements: |
Syllabus:
| Date | Lecture | Reading | ||
|---|---|---|---|---|
| Week 1 | Tue 8/22 | 1. Intro |
Discussed past syllabus Fall'22, and overlaps with newer courses AEP 3100 (Spring, P. McMahon) and ECE 4950 (Fall, M. Wilde). Gave broad overview of past century of QM (1920s, 1960s, 1980s, 2000-now). 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.] and some recent relevant articles: 2105.02859 (QSVT), 2112.00778 (Q Adv), 1810.03787 (QCNN), 1910.00020 (RQC), 2207.14280 (MIC) began 1-qubit states and Bloch Sphere, deferred entanglement to next time. | |
| Thu 8/24 | 2. qubits and gates |
discussed direct product spaces, entangled states (M1.5), SVD criterion for bipartite entanglement (see these notes), Pauli σ matrices as X,Y,Z, some direct products, Born rule and generalized Born rule (M1.8-1.9), some examples, operators SWAP and CNOT (M1.2-1.4, more next time) (See also M.app.A for "Vector spaces: basic properties and Dirac notation"). | ||
| Week 2 | Tue 8/29 | 3. more gates | Continued M1.4: recalled cNOT, introduced the Hadamard H and Toffoli gate (= ccNOT), and more general controlled gates; mentioned relation to reversible computing; showed circuit for Bell pair and generalized GHZ states; relation between Pauli matrices and the quaternions (plus the Brougham/Broome bridge) 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. | |
| Thu 8/31 | 4. more SU(2) |
Additional discussion of cNOT, Hadamard, and CZ (M1.4) Continued SU(2)/Z2=SO(3) discussion of M.app.B, with numerous examples (and more discussion of fermions/bosons to be completed lec5). Started discussion of general computation process (M2.1, algorithms deferred to next time) | ||
| Week 3 | Tue 9/5 | 5. √X plus first speedup |
Some comments about SU(2) vs adjoint (SO(3)) actions of operators, including Hadamard as reflection about π / 8, using
half-wave plate, physical interpretation of SU(2) elements acting on spin-1/2 in 3d, and spin-statistics relation (for more, see pedagogic video) √X and quantum Toffoli gates (M2.6, pp 58-61) covered Deutsch's problem (1 bit to 1 bit functions, M2.2) | |
| Thu 9/7 | 6. platonic algorithms |
Began with discussion of ps1#5, √X in terms of the S operator, relations between Clifford group on 1 qubit and symmetry group of cube and permutation group on 4 elements; demo'd Ry(-π/2)Rx(-π/2) = Rn(-2π/3) with n=(1,1,1)/√3, more on H,T universality from standpoint of platonic solids (for practical implications, see also icosahedral qubit benchmarking) as lead in to discussion of universal gate set (A16.2, and see "The Solovay-Kitaev Algorithm" for technical review), and Clifford group on n qubits More on Deutsch's problem (M2.2, A17.3) and its Deutsch-Jozsa generalization (A17.4). 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 | ||
| Week 4 | Tue 9/12 | 7. BV→Simon |
Started with discussion of computational complexity in the context of linear speed-up for Bernstein-Vazirani problem; then Simon's problem f(x) = f(x⊕a) (M2.5, A18.2, and qiskit Simon's algorithm) Mentioned this video Straight From the Source (Peter Shor) describing how Simon's algorithm led to Shor's (See also comments from Daniel Simon) Simon's problem (M2.5 and appendix G) [now halfway through standard algorithms], and "birthday paradox" [see M app.G for "example of the purely mathematical postprocessing frequently necessary after the quantum algorithm returns its probabilistic results"] See also these covid-era written notes | |
| Thu 9/14 | 8. Simon→Shor |
Began with discussion of ps2#2d and Bloch Sphere (see also interactive demo), then stabilizer states for ps2#5 (see Lec28 of A.I and chpts 3-6 of A.II) Completed Simon's problem (M2.5 and appendix G) Began the discussion of period finding and Shor's algorithm (M3.1), and relation to factoring (M3.10) | ||
| Week 5 | Tue 9/19 | 9. RSA→QFT |
Discussed measurements in different bases, and 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) Outlined RSA encryption (M 3.2, 3.3; A 19.1) (including Euclidean algorithm, and the Die Hard 3 Water Jug problem with respect to the Extended Euclidean Algorithm) Returned to period finding problem and quantum Fourier transform (M 3.1, 3.4) | |
| Thu 9/21 | 10. O(n2) |
Showed discretized Bloch sphere videos,
Recalled derivation of p(y) (M 3.7), and relation to diffraction grating (where d/λ ~ r/2n) then Quantum Fourier Transform (see also A 20.1.1). then the QFT implementation, showing circuit diagram with quadratic growth in number of gates, mentioned 1905.09749 ("How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits") See these lec9-10 written notes | ||
| Week 6 | Tue 9/26 | 11. QFT simplifications |
Comments on ps3, including 3-qubit stabilizer states and the "GHZ game" (M6.6). final QFT implementation comments, 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) then covered how to factor N=15 (though using n=no rather than n=2no as in those notes), | |
| Thu 9/27 | 12. Grover |
Started with 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), including multiple marked states, and special case one of four items. (see lec12_notes) The optimality of Grover's algorithm is shown here, and implications here [Now finished the basic basic quantum algorithms.] Finished with comments on continued fractions, see this notebook lec17_RSA.ipynb (from last year) for an overview of RSA and use of sympy for continued fractions, euclidean algorithm, gcd, etc. | ||
| Week 7 | Tue 10/3 | 13. QEC I. 3-qubit |
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), 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 lec13 written notes | |
| Thu 10/5 | 14. QEC II. 5-qubit |
Illustrated how "measurement of Z operator" corresponds to conventional projective measurement in Z-basis, and how general interaction with environment corresponds to a state-valued unitary transformation Then commuting operator measurements and counted 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. See these lec14 written notes | ||
| Week 8 | Tue 10/10 | Fall Break | ||
| Thu 10/12 | 15. QEC III. 7-qubit, surface |
Started with (intentionally outdated) vendor roadmaps and difficulties realizing error correcting codes with current qubit fidelities Recalled 5-qubit code, then 7-qubit code (M 5.6-5.8, p.121-127), brief mention of multiqubit errors (M 5.6) and composite 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] Then started the surface code, see arXiv:1208.0928, sections II-VI, pp.3-10, and mentioned 1912.09410 See these lec15 written notes | ||
| Week 9 | Tue 10/17 | 16. surface, no hidden vars |
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; then more details on experimental realization of truncated 7-qubit surface code in 1912.09410 (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). Mentioned IBM's heavy-hex-lattice and associated heavy-hex-codes. closed with discussion of the GHZ state and implications for excluding hidden variables (M6.6). See these lec16 written notes | |
| Thu 10/19 | 17. teleportation, CHSH |
9-qubit code (A.27.2) Quantum teleportation (M 6.5; A 10.1; qiskit 3.11), Circuit diagram interpretation of teleportation (M 6.5), teleportation of entanglement (A 10.1, see fig 10.3) -- note this fun historical account, and experimental implementation (from 1997). Discussed the CHSH game (A13.2, A14.1, though feel encouraged to read all of A13,A14) See these lec17 written notes on teleportation | ||
| Week 10 | Tue 10/24 | 18. CHSH, qiskit |
Additional comments on CHSH and entanglement witnesses, why additional Qbits don't mess things up (M2.3, pp 46-50); quantum garbage collection (A17.2) Demoed the qiskit circuit composer (GHZ states, Xθ operators, etc), see this notebook for examples of extracting data from 7-qubit GHZ state run with 1024 shots. Also mentioned that "Identity check is QMA-complete" quant-ph/0305050 (i.e., extremely hard in the technical sense to show that a complicated-looking circuit might be trivial) | |
| Thu 10/26 | 19. QKD,ρ | Some additional comments on Clifford simulation (see A.I chpt 28 and A.II chpts 3-6, in particular see A.II.6.7 for exponential complexity in number of T gates). Also mentioned Mon 30 Oct physics colloquium (4pm Schwartz auditorium) "Title: Measurement Induced Criticality in Monitored Quantum Systems". Then Quantum Key Distribution (A 8.2; M 6.2, and the QKD wikipedia entry has 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). Started discussion of density matrix ρ. See these lec19 written notes on QKD | ||
| Week 11 | Tue 10/31 | 20. ρ, entropy |
Continued exposition of density matrix, expectation values of operators, and mixed states on Bloch sphere (A chapt 6) See these lec20 notes on Shannon info | |
| Thu 11/2 | 21. entropy, partial trace |
More on
Von Neumann entropy,
relation to entanglement, and maximally entangled states; Then Subsystems and partial trace | ||
| Week 12 | Tue 11/7 | 22. subsystems |
Subsystems and entanglement entropies; mutual information (see last page of lec20 notes, and see also this "Mini-Introduction To Information Theory" arXiv:1805.11965), more on monogamy of entangement, and started entanglement measure arXiv:quant-ph/9907047 | |
| Thu 11/9 | 23. tangles |
Continued entanglement measure
arXiv:quant-ph/9907047, with
examples of tangles and exercises in partial traces (see lec23_addendum.pdf) started "classical shadows" arXiv:2002.08953 | ||
| Week 13 | Tue 11/14 | 24. shadows |
Discussion of ps6 probs1,2, then Bayes' Rule for problem 4, recommended Physicists Observe 'Unobservable' Quantum Phase Transition", 2023 (and mentioned preceding quantum brain, 2016) and continued "classical shadows"arXiv:2002.08953 | |
| Thu 11/16 | 25. shadows 2 |
Covered the bulk of arXiv:2002.08953,
emphasizing that the choice of Clifford measurements permits efficient (non-exponential) storage of shadows, but does not limit the accuracy of the expectation values,
and that the logarithmic sample complexity follows from use of the (non-quantum) median of means statistical methodology. Started examples: quantum fidelity estimation, entanglement verification, expectation values of local observables | ||
| Week 14 | Tue 11/21 | 26. shadows 3 | Reiterated key points from last time, and finished examples
from arXiv:2002.08953, with comments on the space of 3-qubit states, and how to construct entanglement witnesses for them, drawing from quant-ph/0103025 (Acin et el, 2001). See also these visualizations of n=3 GHZ and W-state density matrices estimated as averages of shadows. For more on shadows from one of the coauthors, see this video (Preskill, 2021, roughly the first 20 minutes. The rest also worthwhile: at the 20:55 point covers later work 2103.07510 [same authors, 2021], on methodology adapted to specific observables to "derandomize" the meausurements, further reducing the number of measurements needed for given accuracy. Software available here (Huang, 2021) | |
| Thu 11/23 | Thanksgiving | break | ||
| Week 15 | Tue 11/28 | 27. QCNN |
Brief overview of classical CNN,
then QCNNs 1810.03787 (Cong/Choi/Lukin, 2018), including brief description of QPR (quantum phase recognition)
and more on learning 9-qubit QEC optimized for specific noise models Finished with relation between swapping subsystems and the Rényi entropy | |
| Thu 11/30 | 28. BQP et al. |
Continued discussion of computational complexity (A 23), see also P, NP, et al. and
BQP et al.; then
P vs. NP, and
3-SAT (as well as Preskill notes chpt 6, pp. 5-10, 22-24, 26-28, chapts 6-10 of "Quantum Computing since Democritus", Complexity Zoo, and abbreviated Petting Zoo). Mentioned 2101.12319 on universality of Quantum Hamiltonians, and "Quantum Algorithm Zoo": compendium of quantum algorithms. |
Course Topics:
Academic integrity policy:
You are expected to abide by the
Cornell University Code of Academic Integrity.
(In particular, the work you submit for course assignments must be your own.
You may discuss homework assignments with other students at a high
level, by for example discussing general methods or strategies to
solve a problem, but you must cite the other student in your submission.
Any work you submit must be your own understanding of the solution,
the details of which you personally and individually worked out, and
written in your own words.)