Quantum Computing: An Applied Approach

Preface -- Acknowledgements -- Navigating this Book -- I. Foundations -- 1. Superposition, Entanglement and Reversibility -- 2. A Brief History of Quantum Computing -- 3. Qubits and Unitary Operators -- 4. Measurement and Quantum Circuits -- 5. Complexity Theory. II. Hardware and Applications -- 6. Building a Quantum Computer -- 7. Development Libraries for Quantum Computer Programming -- 8. Teleportation, Superdense Coding and Bell’s Inequality -- 9. The Canon: Code Walkthroughs -- 10. Quantum Computing Methods -- 11. Application and Quantum Supremacy -- III. Toolkit -- 12. Mathematical Tools for Quantum Computing I -- 13. Mathematical Tools for Quantum Computing II -- 14. Mathematical Tools for Quantum Computing III -- 15. Table of Quantum Operators and Core Circuits -- Works Cited.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  B. A. Bernstein.  A set of four independent postulates for Boolean algebras , 1916 .

[3]  M. Born.  Zur Quantenmechanik der Stoßvorgänge , 1926 .

[4]  Max Born,et al.  Das Adiabatenprinzip in der Quantenmechanik , 1927 .

[5]  H. S. Allen.  The Quantum Theory , 1928, Nature.

[6]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[7]  E. Schrödinger.  Discussion of Probability Relations between Separated Systems , 1935, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  Paul Adrien Maurice Dirac,et al.  A new notation for quantum mechanics , 1939, Mathematical Proceedings of the Cambridge Philosophical Society.

[9]  R. Tennant.  Algebra , 1941, Nature.

[10]  H. O. Foulkes.  Abstract Algebra , 1967, Nature.

[11]  Donald E. Knuth,et al.  Big Omicron and big Omega and big Theta , 1976, SIGA.

[12]  Volker Strassen,et al.  A Fast Monte-Carlo Test for Primality , 1977, SIAM J. Comput..

[13]  P. Benioff.  The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines , 1980 .

[14]  Tommaso Toffoli,et al.  Reversible Computing , 1980, ICALP.

[15]  R. Feynman.  Simulating physics with computers , 1999 .

[16]  Hong,et al.  Measurement of subpicosecond time intervals between two photons by interference. , 1987, Physical review letters.

[17]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[18]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[19]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[20]  Charles H. Bennett,et al.  Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. , 1992, Physical review letters.

[21]  Umesh V. Vazirani,et al.  Quantum complexity theory , 1993, STOC.

[22]  Aharonov,et al.  Quantum random walks. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[23]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[24]  S Lloyd,et al.  A Potentially Realizable Quantum Computer , 1993, Science.

[25]  Daniel R. Simon.  On the power of quantum computation , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[26]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[27]  J. Rotman.  A First Course in Abstract Algebra , 1995 .

[28]  S. Axler.  Linear Algebra Done Right , 1995, Undergraduate Texts in Mathematics.

[29]  Barenco,et al.  Elementary gates for quantum computation. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[30]  J. Cirac,et al.  Quantum Computations with Cold Trapped Ions. , 1995, Physical review letters.

[31]  S. Braunstein,et al.  Quantum computation , 1996 .

[32]  Edward Farhi,et al.  Analog analogue of a digital quantum computation , 1996 .

[33]  Weinfurter,et al.  Dense coding in experimental quantum communication. , 1996, Physical review letters.

[34]  Anthony J. G. Hey,et al.  Feynman Lectures on Computation , 1996 .

[35]  F. William Lawvere,et al.  Conceptual Mathematics: A First Introduction to Categories , 1997 .

[36]  Lov K. Grover.  Quantum Mechanics Helps in Searching for a Needle in a Haystack , 1997, quant-ph/9706033.

[37]  Gilles Brassard,et al.  Strengths and Weaknesses of Quantum Computing , 1997, SIAM J. Comput..

[38]  D. DiVincenzo.  TOPICS IN QUANTUM COMPUTERS , 1996, cond-mat/9612126.

[39]  A. Kitaev,et al.  Quantum codes on a lattice with boundary , 1998, quant-ph/9811052.

[40]  Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels , 1997, quant-ph/9710013.

[41]  D. Gottesman.  The Heisenberg Representation of Quantum Computers , 1998, quant-ph/9807006.

[42]  M. Devoret,et al.  Quantum coherence with a single Cooper pair , 1998 .

[43]  E. Farhi,et al.  Quantum computation and decision trees , 1997, quant-ph/9706062.

[44]  Christoph Adami,et al.  Quantum Computation with Linear Optics , 1998, QCQC.

[45]  Y. Pashkin,et al.  Coherent control of macroscopic quantum states in a single-Cooper-pair box , 1999, Nature.

[46]  V. Roychowdhury,et al.  On Universal and Fault-Tolerant Quantum Computing , 1999, quant-ph/9906054.

[47]  細谷 暁夫,et al.  量子コンピュータの基礎 : lectures on quantum computation , 1999 .

[48]  D. DiVincenzo,et al.  The Physical Implementation of Quantum Computation , 2000, quant-ph/0002077.

[49]  Aharonov,et al.  Quantum Walks , 2012, 1207.7283.

[50]  R. Landauer,et al.  Irreversibility and heat generation in the computing process , 1961, IBM J. Res. Dev..

[51]  E. Knill,et al.  A scheme for efficient quantum computation with linear optics , 2001, Nature.

[52]  R Raussendorf,et al.  A one-way quantum computer. , 2001, Physical review letters.

[53]  Y. Pashkin,et al.  Rabi oscillations in a Josephson-junction charge two-level system. , 2001, Physical review letters.

[54]  Andris Ambainis,et al.  Quantum walks on graphs , 2000, STOC '01.

[55]  I. Chuang,et al.  Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance , 2001, Nature.

[56]  Edward Farhi,et al.  An Example of the Difference Between Quantum and Classical Random Walks , 2002, Quantum Inf. Process..

[57]  J. Rotman.  Advanced Modern Algebra , 2002 .

[58]  Julia Kempe,et al.  Quantum Random Walks Hit Exponentially Faster , 2002, ArXiv.

[59]  Hans J. Briegel,et al.  The one-way quantum computer--a non-network model of quantum computation , 2001, quant-ph/0108118.

[60]  J. Preskill,et al.  Topological quantum memory , 2001, quant-ph/0110143.

[61]  Hans-J. Briegel,et al.  Computational model underlying the one-way quantum computer , 2002, Quantum Inf. Comput..

[62]  Daniel A. Spielman,et al.  Exponential algorithmic speedup by a quantum walk , 2002, STOC '03.

[63]  L. Landau.  Fault-tolerant quantum computation by anyons , 2003 .

[64]  Yaoyun Shi.  Both Toffoli and controlled-NOT need little help to do universal quantum computing , 2003, Quantum Inf. Comput..

[65]  Julia Kempe,et al.  Quantum random walks: An introductory overview , 2003, quant-ph/0303081.

[66]  H. Briegel,et al.  Measurement-based quantum computation on cluster states , 2003, quant-ph/0301052.

[67]  Oxford ion-trap quantum computing project , 2003, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[68]  F. Verstraete,et al.  Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions , 2004, cond-mat/0407066.

[69]  Manindra Agrawal,et al.  PRIMES is in P , 2004 .

[70]  D. A. Edwards.  The mathematical foundations of quantum mechanics , 1979, Synthese.

[71]  Fuguo Deng,et al.  Quantum secure direct communication with high-dimension quantum superdense coding , 2005 .

[72]  Michael A. Nielsen,et al.  The Solovay-Kitaev algorithm , 2006, Quantum Inf. Comput..

[73]  N. Mermin.  Quantum Computer Science: An Introduction , 2007 .

[74]  G. Vidal.  Entanglement renormalization. , 2005, Physical review letters.

[75]  Scott Aaronson,et al.  The Limits of Quantum Computers , 2007, CSR.

[76]  G. Milburn,et al.  Linear optical quantum computing with photonic qubits , 2005, quant-ph/0512071.

[77]  Edward Farhi,et al.  A Quantum Algorithm for the Hamiltonian NAND Tree , 2008, Theory Comput..

[78]  Andrew D. Greentree,et al.  Diamond for Quantum Computing , 2008, Science.

[79]  Scott Aaronson,et al.  The Complexity Zoo , 2008 .

[80]  I. Chuang,et al.  Cavity sideband cooling of a single trapped ion. , 2009, Physical review letters.

[81]  Steve Mullett,et al.  Read the fine print. , 2009, RN.

[82]  Zheng Wang,et al.  Observation of unidirectional backscattering-immune topological electromagnetic states , 2009, Nature.

[83]  J. P. Home,et al.  Realization of a programmable two-qubit quantum processor , 2009, 0908.3031.

[84]  A. Harrow,et al.  Quantum algorithm for linear systems of equations. , 2008, Physical review letters.

[85]  Steve Simon.  Quantum computing with a twist , 2010 .

[86]  Thomas G. Walker,et al.  Quantum information with Rydberg atoms , 2009, 0909.4777.

[87]  Isaac L. Chuang,et al.  Quantum Computation and Quantum Information (10th Anniversary edition) , 2011 .

[88]  C. Kane,et al.  Topological Insulators , 2019, Electromagnetic Anisotropy and Bianisotropy.

[89]  Adiabatic theorem for finite dimensional quantum mechanical systems , 2009, 0909.0370.

[90]  Mohammad Hafezi,et al.  Robust optical delay lines with topological protection , 2011, 1102.3256.

[91]  Jacob Colbert.  Topological Quantum Computing , 2011 .

[92]  Scott Aaronson,et al.  The computational complexity of linear optics , 2010, STOC '11.

[93]  X. Qi,et al.  Topological insulators and superconductors , 2010, 1008.2026.

[94]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[95]  W. Marsden.  I and J , 2012 .

[96]  Zohar Ringel,et al.  Topological States and adiabatic pumping in quasicrystals. , 2011, Physical review letters.

[97]  M. Mariantoni,et al.  Surface codes: Towards practical large-scale quantum computation , 2012, 1208.0928.

[98]  Umesh Vazirani,et al.  Is Quantum Mechanics Falsifiable? A computational perspective on the foundations of Quantum Mechanics , 2012, 1206.3686.

[99]  John Preskill,et al.  Quantum computing and the entanglement frontier , 2012, 1203.5813.

[100]  P. Mataloni,et al.  Efficient experimental validation of photonic boson sampling against the uniform distribution , 2013, 1311.1622.

[101]  R. Hanson,et al.  Diamond NV centers for quantum computing and quantum networks , 2013 .

[102]  Neil B. Manson,et al.  The nitrogen-vacancy colour centre in diamond , 2013, 1302.3288.

[103]  M. Hafezi,et al.  Imaging topological edge states in silicon photonics , 2013, Nature Photonics.

[104]  O. Zilberberg,et al.  Observation of topological phase transitions in photonic quasicrystals. , 2012, Physical review letters.

[105]  M. Segev,et al.  Photonic Floquet topological insulators , 2012, Nature.

[106]  Yu-An Chen,et al.  Density matrix renormalization group , 2014 .

[107]  Gilles Brassard,et al.  Quantum cryptography: Public key distribution and coin tossing , 2014, Theor. Comput. Sci..

[108]  E. Farhi,et al.  A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem , 2014, 1412.6062.

[109]  Adam D. Bookatz.  QMA-complete problems , 2012, Quantum Inf. Comput..

[110]  T. Ohshima,et al.  A silicon carbide room-temperature single-photon source. , 2013, Nature materials.

[111]  Stuart A. Rice,et al.  Quantum information and computation for chemistry. , 2014 .

[112]  Xi Dong,et al.  Bulk locality and quantum error correction in AdS/CFT , 2014, 1411.7041.

[113]  A. F. Kockum.  Quantum optics with artificial atoms , 2014 .

[114]  J G Rarity,et al.  Experimental realization of a one-way quantum computer algorithm solving Simon's problem. , 2014, Physical review letters.

[115]  E. Farhi,et al.  A Quantum Approximate Optimization Algorithm , 2014, 1411.4028.

[116]  M. Hastings,et al.  Progress towards practical quantum variational algorithms , 2015, 1507.08969.

[117]  J. Meijer,et al.  Active charge state control of single NV centres in diamond by in-plane Al-Schottky junctions , 2015, Scientific Reports.

[118]  Yang Wang,et al.  Coherent Addressing of Individual Neutral Atoms in a 3D Optical Lattice. , 2015, Physical review letters.

[119]  Jay M. Gambetta,et al.  Building logical qubits in a superconducting quantum computing system , 2015, 1510.04375.

[120]  A. Lemaître,et al.  Macroscopic rotation of photon polarization induced by a single spin , 2014, Nature Communications.

[121]  O. Zilberberg,et al.  Topological pumping over a photonic Fibonacci quasicrystal , 2014, 1403.7124.

[122]  P. Coveney,et al.  Scalable Quantum Simulation of Molecular Energies , 2015, 1512.06860.

[123]  M. L. Wall,et al.  Quantum spin dynamics and entanglement generation with hundreds of trapped ions , 2015, Science.

[124]  P. Hayden,et al.  Holographic duality from random tensor networks , 2016, 1601.01694.

[125]  Alán Aspuru-Guzik,et al.  qHiPSTER: The Quantum High Performance Software Testing Environment , 2016, ArXiv.

[126]  Mohammad Hafezi,et al.  Measurement of topological invariants in a 2D photonic system , 2015, Nature Photonics.

[127]  E. Waks,et al.  A quantum phase switch between a single solid-state spin and a photon. , 2015, Nature nanotechnology.

[128]  C. T. Chan,et al.  Photonic crystals possessing multiple Weyl points and the experimental observation of robust surface states , 2016, Nature Communications.

[129]  William J. Zeng,et al.  A Practical Quantum Instruction Set Architecture , 2016, ArXiv.

[130]  H. Neven,et al.  Characterizing quantum supremacy in near-term devices , 2016, Nature Physics.

[131]  Alán Aspuru-Guzik,et al.  The theory of variational hybrid quantum-classical algorithms , 2015, 1509.04279.

[132]  Benjamin J Eggleton,et al.  Topological Optical Waveguiding in Silicon and the Transition between Topological and Trivial Defect States. , 2016, Physical review letters.

[133]  Guang-Can Guo,et al.  Simulating the exchange of Majorana zero modes with a photonic system , 2014, Nature Communications.

[134]  Aram W. Harrow,et al.  Quantum computational supremacy , 2017, Nature.

[135]  M. Troyer,et al.  Elucidating reaction mechanisms on quantum computers , 2016, Proceedings of the National Academy of Sciences.

[136]  Iordanis Kerenidis,et al.  Quantum Recommendation Systems , 2016, ITCS.

[137]  Masahito Hayashi,et al.  Secure uniform random number extraction via incoherent strategies , 2017, ArXiv.

[138]  Alán Aspuru-Guzik,et al.  Quantum Neuron: an elementary building block for machine learning on quantum computers , 2017, ArXiv.

[139]  Kevin P. Chen,et al.  Experimental observation of optical Weyl points and Fermi arc-like surface states , 2016, Nature Physics.

[140]  Masoud Mohseni,et al.  Commercialize quantum technologies in five years , 2017, Nature.

[141]  Blake R. Johnson,et al.  First quantum computers need smart software , 2017, Nature.

[142]  Stewart Allen,et al.  Reconfigurable and Programmable Ion Trap Quantum Computer , 2017, 2017 IEEE International Conference on Rebooting Computing (ICRC).

[143]  Jiannis K. Pachos,et al.  A Short Introduction to Topological Quantum Computation , 2017, 1705.04103.

[144]  S. Mittal,et al.  Topologically robust generation of correlated photon pairs , 2017 .

[145]  Peter Wittek,et al.  Quantum Enhanced Inference in Markov Logic Networks , 2016, Scientific Reports.

[146]  M. Segev,et al.  Photonic quantum walks with symmetry protected topological phases , 2017 .

[147]  Michael Broughton,et al.  A Universal Training Algorithm for Quantum Deep Learning , 2018, 1806.09729.

[148]  Liang Jiang,et al.  Deterministic teleportation of a quantum gate between two logical qubits , 2018, Nature.

[149]  Igor L. Markov,et al.  Quantum Supremacy Is Both Closer and Farther than It Appears , 2018, ArXiv.

[150]  Peter J. Love,et al.  A variational eigenvalue solver on a quantum processor , 2018 .

[151]  Yudong Cao,et al.  A framework for algorithm deployment on cloud-based quantum computers , 2018, 1810.10576.

[152]  Kevin P. Chen,et al.  Photonic topological boundary pumping as a probe of 4D quantum Hall physics , 2018, Nature.

[153]  Hartmut Neven,et al.  Classification with Quantum Neural Networks on Near Term Processors , 2018, 1802.06002.

[154]  G. Yocky,et al.  Decoherence , 2018, Principles of Quantum Computation and Information.

[155]  Alán Aspuru-Guzik,et al.  qTorch: The quantum tensor contraction handler , 2017, PloS one.

[156]  Robert König,et al.  Quantum advantage with shallow circuits , 2017, Science.

[157]  Ewin Tang,et al.  A quantum-inspired classical algorithm for recommendation systems , 2018, Electron. Colloquium Comput. Complex..

[158]  T. Bradley.  What is Applied Category Theory , 2018, 1809.05923.

[159]  Stephen Brierley,et al.  A Generalised Variational Quantum Eigensolver , 2018 .

[160]  Edo Waks,et al.  A single-photon switch and transistor enabled by a solid-state quantum memory , 2018, Science.

[161]  Simone Severini,et al.  Quantum linear systems algorithms: a primer , 2018, ArXiv.

[162]  A. Peruzzo,et al.  Quantum interference of topological states of light , 2018, Science Advances.

[163]  Feng Li,et al.  Weyl points and Fermi arcs in a chiral phononic crystal , 2017, Nature Physics.

[164]  M. Dutt,et al.  Single-photon heralded two-qubit unitary gates for pairs of nitrogen-vacancy centers in diamond , 2018, Physical Review A.

[165]  M. Lukin,et al.  Quantum optimization of maximum independent set using Rydberg atom arrays , 2018, Science.

[166]  Ryan Babbush,et al.  Barren plateaus in quantum neural network training landscapes , 2018, Nature Communications.

[167]  Ramis Movassagh,et al.  Efficient unitary paths and quantum computational supremacy: A proof of average-case hardness of Random Circuit Sampling , 2018, 1810.04681.

[168]  Peter Wittek,et al.  Open source software in quantum computing , 2018, PloS one.

[169]  G. Vidal,et al.  Tensor networks as conformal transformations , 2018, 1805.12524.

[170]  John Preskill,et al.  Quantum Computing in the NISQ era and beyond , 2018, Quantum.

[171]  Kristan Temme,et al.  Supervised learning with quantum-enhanced feature spaces , 2018, Nature.

[172]  Fei Yan,et al.  A quantum engineer's guide to superconducting qubits , 2019, Applied Physics Reviews.

[173]  Ryan LaRose,et al.  Overview and Comparison of Gate Level Quantum Software Platforms , 2018, Quantum.

[174]  Mark Saffman,et al.  Quantum computing with neutral atoms , 2017, National science review.

[175]  M. Soljačić,et al.  Topological photonics , 2014, Nature Photonics.

[176]  Marco Tomamichel,et al.  Quantum Advantage with Noisy Shallow Circuits in 3D , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[177]  Maria Schuld,et al.  Quantum Machine Learning in Feature Hilbert Spaces. , 2018, Physical review letters.

[178]  Aishwarya Kumar,et al.  Stern–Gerlach detection of neutral-atom qubits in a state-dependent optical lattice , 2018, Nature Physics.

[179]  V. Negnevitsky,et al.  Encoding a qubit in a trapped-ion mechanical oscillator , 2018, Nature.

[180]  S. Brierley,et al.  Variational Quantum Computation of Excited States , 2018, Quantum.

[181]  John Chiaverini,et al.  Trapped-ion quantum computing: Progress and challenges , 2019, Applied Physics Reviews.

[182]  D. Englund,et al.  Optical coherence of diamond nitrogen-vacancy centers formed by ion implantation and annealing , 2018, Physical Review B.

[183]  Ryan Babbush,et al.  Decoding quantum errors with subspace expansions , 2019, Nature Communications.

[184]  Ievgeniia Oshurko.  Quantum Machine Learning , 2020, Quantum Computing.

[185]  Alan Aspuru-Guzik,et al.  Variational Quantum Generators: Generative Adversarial Quantum Machine Learning for Continuous Distributions , 2019, Advanced Quantum Technologies.

[186]  P. Alam.  ‘K’ , 2021, Composites Engineering.

[187]  P. Alam.  ‘T’ , 2021, Composites Engineering: An A–Z Guide.

[188]  Craig Gidney,et al.  How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits , 2019, Quantum.

[189]  P. Alam.  ‘S’ , 2021, Composites Engineering: An A–Z Guide.