All files are in PDF format
John Bollinger (NIST)
Lecture 1: Trapped ion quantum computing
Reviews for basic tools of ion trap quantum computing:
 D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, J. Res. Nat. Inst. Stand. Tech. 103, 259 (1998)
 M. Sasura and V. Buzek, J. Mod. Opt. 49, 1593 (2002)
 D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003)
 H. Häffner, C. F. Roos, and R. Blatt, Physics Reports 469, 155 (2008)
Example experiments:
 “High Fidelity Universal Gate Set for 9Be+ Ion Qubits”, J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, Phys. Rev. Lett. 117, 060505 (2016).
 “14qubit entanglement: creation and coherence”, T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, R. Blatt, Phys. Rev. Lett. 106, 130506 (2011).
 “Demonstration of a Small Programmable Quantum Computer with Atomic Qubits,” S. Debnath, N. M. Linke, C. Figgatt, K. A. Landsman, K. Wright, and C. Monroe, Nature 536, 63 (2016).
Lecture 2: Trapped ion quantum simulation
Review
 R. Blatt and C.F. Roos, Nature Physics 8, 277 (2012).
Example experiment, linear rf trap:
 “Observation of a Discrete Time Crystal,” J. Zhang, P.W. Hess, A. Kyprianidis, P. Becker, Lee, J. Smith, G. Pagano, I.D. Potirniche, A.C. Potter, A. Vishwanath, N.Y. Yao, C. Monroe, Nature 543, 217 (2017).
Example experiment, Penning trap:
 “Quantum spin dynamics and entanglement generation with hundreds of trapped ions,” J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, A. M. Rey, M. FossFeig, J. J. Bollinger , Science 352, 1297 (2016).
 “Measuring outoftimeorder correlations and multiple quantum spectra in a trapped ion quantum magnet,” M. Gärttner, J. G. Bohnet, A. SafaviNaini, M. L. Wall, J. J. Bollinger, A. M. Rey, Nat. Phys., 13, 781 (2016).
Lecture 3: Trapped ion quantum sensing
Al^{+} quantum logic clock:
 “Frequency ratio of Al+ and Hg+ singleion optical clocks; Metrology at the 17th decimal place,” T. Rosenband, D. B. Hume, A. Brusch, L. Lorini, P. O. Schmidt, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, N. R. Newbury, W. C. Swann, W. H. Oskay, W. M. Itano, D. J. Wineland, and J. C. Bergquist, Science 319, 1808  1812 (2008).
 C. Chou, D.B. Hume, M.J. Thorpe, D.J. Wineland, and T. Rosenband, “Quantum coherence between two atoms beyond Q=1015,” Phys. Rev. Lett. 106, 160801 (2011).
Weak force sensing:
 “Amplitude sensing below the zeropoint fluctuations with a twodimensional trappedion mechanical oscillator”, K.A. Gilmore, J.G. Bohnet, B.C. Sawyer, J.W. Britton, J.J. Bollinger,Phys. Rev. Lett., 118, 263602 (2017).
Tony Cubitt (UCL)
Recommended reading
The AroraBarak book gives an excellent, modern treatment of the theory of computation and complexity, going far beyond what’s covered in this short course. The proof of Kitaev’s theorem in the course closely follows the original from the KitaevSchenVyalyi book. The other references in this list are review papers on Hamiltonian complexity, which may also be of interest.
 Arora and Barak, “Complexity Theory: A Modern Approach, Cambridge University Press
 Kitaev, A., Shen, A., and Vyalyi M. “Classical and Quantum Computation”, American Mathematical Society
 Aharonov, D. and Naveh, T. “Quantum NP  a Survey”
 Gharibian, S., Huang, Y. and Landau, Z. “Hamiltonian Complexity”
The following is a selective and incomplete list of links to the arXiv versions of papers that proved key results in Hamiltonian Complexity and Computability theory postKitaev.
QMAcompleteness with stronger locality conditions, and related results
Proves QMAcompleteness of the klocal Hamiltonian problem for k=3k=3.
Proves QMAcompleteness of the klocal Hamiltonian problem for k=2k=2. Introduces the perturbation gadget technique.
Proves QMAcompleteness of the klocal Hamiltonian problem for nearestneighbour interactions (k=2k=2) between qubits on a 2D square lattice. (Interactions are not translationallyinvariant). Developes stronger perturbation gadget techniques.
 D. Aharonov, D. Gottesman, S. Irani and Julia Kempe, “The power of quantum systems on a line” (2007)
Proves QMAcompleteness of the klocal Hamiltonian problem for nearestneighbour interactions (k=2k=2) bewteen quddits on a line, for d=13d=13. (Interactions are not translationallyinvariant.) Later improved to d=8d=8 by Nagaj et al.
Proves QMAEXPcompleteness of the local Hamiltonian problem for translationallyinvariant, nearestneighbour interactions (k=2k=2) between quddits on a line, with a fixed Hamiltonian and d≈106d≈106. (The only remaining parameter in the problem is the length of the chain!)
Proves the GottesmanIrani result for d=42d=42.
QMAcompleteness with restricted types of interaction, and related results
Proves QMAcompleteness of the 2local Hamiltonian problem for qubit Hamiltonians containing only XZ, X and Z interactions (amongst other similar results).
Proves a complete complexity classification for the klocal Hamiltonian problem with 2qubit interactions, according to the type of interactions allowed. (A quantum analogue of Schaeffer’s dichotomy theorem for boolean constraint satisfaction problems.)
Tightens the CubittMontanaro classification by proving one of the four classes in the classification is equal to stoqMA (a highly nontrivial improvement!), amongst other results.
Undecidability in physics
Shows that questions about the dynamics of a particle bouncing around a 3D box with linear and parabolic reflectors is undecidable.
Contains all the technical details and proofs for the above result, and more.
 TC, D. PerezGarcia and M. Wolf, “Undecidability of the Spectral Gap (full version)” and (short version)” (2015)
Proves undecidability of the spectral gap problem for translationallyinvariant, nearestneighbour interactions between quddits on a 2D square lattice in the thermodynamic limit, with d≈10100d≈10100 (or maybe a bit smaller). Makes key use of ideas from the GottesmanIrani result, amongst (many) other ingredients.
Daniel Gottesman (Perimeter)
arXiv:0904.2557 [quantph] covers a lot of the materials in the lectures
Emanuel Knill (NIST)
Lectures 1 & 2: States in context
 H. Barnum et al., A SubsystemIndependent Generalization of Entanglement, Phys. Rev. Lett. 92 107902 (2004)
 M. Tichy et al., Essential Entanglement for Atomic and Molecular Physics, J. Physics B 44 192001 (2011)
 E. Witten, Notes on some entanglement properties of QFT, arXiv:1803.04993
Lecture 3: Statistics in context
 T. Dorigo, Extraordinary claims: the 0.000029% solution, EPJ Web of Conferences 95, 02003 (2015)
 Examples from the literature

Notes on statistical concepts by Manny (2015). Available by request from Manny.
Cindy Regal (JILA)

A.M. Kaufman, M.C. Tichy, F. Mintert, A.M. Rey, C.A. Regal, “The HongOuMandel effect with atoms” Advances In Atomic, Molecular, and Optical Physics 67, 377 (2018).

Y. Wang, A. Kumar, T. Y. Wu, David S. Weiss, “Singlequbit gates based on targeted phase shifts in a 3D neutral atom array”, Science 352, 1562 (2016).

M Saffman, “Quantum computing with atomic qubits and Rydberg interactions: progress and challenges” J. Phys. B: At. Mol. Opt. Phys. 49, 202001 (2016).

R. Islam, R. Ma., P. M. Preiss, E. M. Tai, A. Lukin, M. Rispoli, M. Greiner “Measuring entanglement entropy in a quantum manybody system”, Nature 528, 758 (2015).
Renato Renner (ETH)
Brian Swingle (University of Maryland)
For background on AdS/CFT, I recommend John McGreevy’s notes and Mark Van Raamsdonk’s notes. See also this review of the RT formula.
Barbara Terhal (Delft University of Technology)
 B.M. Terhal, Quantum Error Correction for Quantum Memories, https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.87.307 and https://arxiv.org/abs/1302.3428
 E. Campbell, B.M. Terhal, C.Vuillot, Roads towards faulttolerant universal quantum computation, https://www.ncbi.nlm.nih.gov/pubmed/28905902 and https://arxiv.org/abs/1612.07330
 A. Fowler, M. Mariantoni, J.M. Martinis, A.Cleland, Surface codes: Towards practical largescale quantum computation, https://journals.aps.org/pra/abstract/10.1103/PhysRevA.86.032324 and https://arxiv.org/abs/1208.0928
 V. Albert et al., Performance and Structure of Bosonic Codes, https://journals.aps.org/pra/abstract/10.1103/PhysRevA.97.032346 and https://arxiv.org/abs/1708.05010
 QEC Boulder Exercise Answers
Frank Verstraete (Vienna)
 symmetry breaking:
 “Symmetry breaking and the geometry of reduced density matrices”, http://iopscience.iop.org/article/10.1088/13672630/18/11/113033/pdf
 general MPS framework:
 “Matrix product states, projected entangled pair state and variational renormalization group methods for quantum spin systems”, https://arxiv.org/pdf/0907.2796
 MPS manifold
 “Timedependent variational principle for quantum lattices”, https://arxiv.org/pdf/1103.0936
 “Tangent space methods for matrix product states”, https://pdfs.semanticscholar.org/6aaa/f66b776d4edecaf8500b6ce59e17c364ce71.pdf
 “Geometry of matrix product states: Metric, parallel transport, and curvature”, https://arxiv.org/pdf/1210.7710
 Fermionic MPS:
 Fermionic matrix product states and onedimensional topological phases, https://arxiv.org/pdf/1610.07849
 symmetries and the fundamental theorem of MPS
 “String order and symmetries in quantum spin lattices”, https://arxiv.org/pdf/0802.0447
 “Classifying quantum phases using matrix product states and projected entangled pair states”, https://arxiv.org/pdf/1010.3732
 topological order and PEPS
 “PEPS as ground states: degeneracy and topology”, https://arxiv.org/pdf/1001.3807
 “Anyons and matrix product operator algebras”, https://arxiv.org/pdf/1511.08090