Computability and Complexity of Unconventional Computing Devices

We discuss some claims that certain UCOMP devices can perform hypercomputation (compute Turing-uncomputable functions) or perform super-Turing computation (solve NP-complete problems in polynomial time). We discover that all these claims rely on the provision of one or more unphysical resources.

[1]  J. Onuchic,et al.  Funnels, pathways, and the energy landscape of protein folding: A synthesis , 1994, Proteins.

[2]  Victor Mitrana,et al.  Solving NP-Complete Problems With Networks of Evolutionary Processors , 2001, IWANN.

[3]  B. Lanyon,et al.  Towards quantum chemistry on a quantum computer. , 2009, Nature chemistry.

[4]  Victor Mitrana,et al.  Accepting Networks of Evolutionary Processors with Filtered Connections , 2009, J. Univers. Comput. Sci..

[5]  Matthias Troyer,et al.  A software methodology for compiling quantum programs , 2016, ArXiv.

[6]  Jacob biamonte,et al.  Quantum machine learning , 2016, Nature.

[7]  A. Harrow,et al.  Quantum Supremacy through the Quantum Approximate Optimization Algorithm , 2016, 1602.07674.

[8]  Susan Stepney,et al.  The natural science of computing , 2017, Commun. ACM.

[9]  Ashish Kapoor,et al.  Quantum algorithms for nearest-neighbor methods for supervised and unsupervised learning , 2014, Quantum Inf. Comput..

[10]  R. Jozsa,et al.  On the role of entanglement in quantum-computational speed-up , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[11]  Ron Unger,et al.  Finding the lowest free energy conformation of a protein is an NP-hard problem: Proof and implications , 1993 .

[12]  S. Lloyd.  Ultimate physical limits to computation , 1999, Nature.

[13]  D. Deutsch.  Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[14]  P D Kaplan,et al.  DNA solution of the maximal clique problem. , 1997, Science.

[15]  J. Edmonds.  Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.

[16]  Georg Seelig,et al.  Time-Complexity of Multilayered DNA Strand Displacement Circuits , 2009, DNA.

[17]  Robert W. Floyd,et al.  Nondeterministic Algorithms , 1967, JACM.

[18]  Ashley Montanaro,et al.  Quantum algorithms: an overview , 2015, npj Quantum Information.

[19]  Julian Francis Miller,et al.  Evolution in materio: looking beyond the silicon box , 2002, Proceedings 2002 NASA/DoD Conference on Evolvable Hardware.

[20]  Petrus H. Potgieter,et al.  Zeno machines and hypercomputation , 2004, Theor. Comput. Sci..

[21]  R. Blatt,et al.  Quantum simulations with trapped ions , 2011, Nature Physics.

[22]  Andrew S. Cassidy,et al.  A million spiking-neuron integrated circuit with a scalable communication network and interface , 2014, Science.

[23]  J. Cirac,et al.  Goals and opportunities in quantum simulation , 2012, Nature Physics.

[24]  Mikhail Smelyanskiy,et al.  High Performance Emulation of Quantum Circuits , 2016, SC16: International Conference for High Performance Computing, Networking, Storage and Analysis.

[25]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[26]  Ashish Kapoor,et al.  Quantum deep learning , 2014, Quantum Inf. Comput..

[27]  Scott Aaronson,et al.  The limits of quantum computers. , 2008 .

[28]  Rita Toth,et al.  Advances in unconventional computing , 2017 .

[29]  Eugene L. Lawler,et al.  The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization , 1985 .

[30]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[31]  Krysta Marie Svore,et al.  LIQUi|>: A Software Design Architecture and Domain-Specific Language for Quantum Computing , 2014, ArXiv.

[32]  B. Jack Copeland,et al.  The Church-Turing Thesis , 2007 .

[33]  Julian F. Miller,et al.  Computational Matter: Evolving Computational Functions in Nanoscale Materials , 2017 .

[34]  David Gosset,et al.  Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates. , 2016, Physical review letters.

[35]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[36]  Pierluigi Crescenzi,et al.  A compendium of NP optimization problems , 1994, WWW Spring 1994.

[37]  Zoran Konkoli,et al.  On Information Processing with Networks of Nano-Scale Switching Elements , 2014, Int. J. Unconv. Comput..

[38]  Keith Douglas,et al.  Learning to Hypercompute? An Analysis of Siegelmann Networks , 2013 .

[39]  John Watrous,et al.  Quantum Computational Complexity , 2008, Encyclopedia of Complexity and Systems Science.

[40]  T. Rado.  On non-computable functions , 1962 .

[41]  Ed Blakey.  Ray tracing - computing the incomputable? , 2012, DCM.

[42]  M. Hogarth.  Does general relativity allow an observer to view an eternity in a finite time? , 1992 .

[43]  Ed Blakey,et al.  Unconventional Computers and Unconventional Complexity Measures , 2017 .

[44]  Matthew B. Hastings,et al.  Hybrid quantum-classical approach to correlated materials , 2015, 1510.03859.

[45]  M. Gazzaniga,et al.  Understanding complexity in the human brain , 2011, Trends in Cognitive Sciences.

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

[47]  Scott Aaronson,et al.  Why Philosophers Should Care About Computational Complexity , 2011, Electron. Colloquium Comput. Complex..

[48]  Peter C. Cheeseman,et al.  Where the Really Hard Problems Are , 1991, IJCAI.

[49]  Hao Wang.  Proving theorems by pattern recognition — II , 1961 .

[50]  Fabio L. Traversa,et al.  Polynomial-time solution of prime factorization and NP-hard problems with digital memcomputing machines , 2015, ArXiv.

[51]  Alán Aspuru-Guzik,et al.  A variational eigenvalue solver on a photonic quantum processor , 2013, Nature Communications.

[52]  Edwin J. Beggs,et al.  An analogue-Digital Church-Turing Thesis , 2014, Int. J. Found. Comput. Sci..

[53]  Ryan Babbush,et al.  What is the Computational Value of Finite Range Tunneling , 2015, 1512.02206.

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

[55]  R J Lipton,et al.  DNA solution of hard computational problems. , 1995, Science.

[56]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[57]  Fabrizio Bonani,et al.  Memcomputing NP-complete problems in polynomial time using polynomial resources and collective states , 2014, Science Advances.

[58]  F. Nori,et al.  Quantum Simulation , 2013, Quantum Atom Optics.

[59]  Erik D. Demaine,et al.  Classic Nintendo games are (computationally) hard , 2015, Theor. Comput. Sci..

[60]  Jonathan M. Smith,et al.  Programming the quantum future , 2015, Commun. ACM.

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

[62]  I. Pak,et al.  Computational complexity and decidability of tileability , 2013 .

[63]  François W. Primeau,et al.  A million spiking-neuron integrated circuit with a scalable communication network and interface , 2014 .

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

[65]  Matthias Troyer,et al.  Recent developments in quantum annealing , 2015 .

[66]  L M Adleman,et al.  Molecular computation of solutions to combinatorial problems. , 1994, Science.

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

[68]  David H. Wolpert,et al.  Ubiquity symposium: Evolutionary computation and the processes of life: what the no free lunch theorems really mean: how to improve search algorithms , 2013, UBIQ.

[69]  Masoud Mohseni,et al.  Computational Role of Multiqubit Tunneling in a Quantum Annealer , 2015 .

[70]  David H. Wolpert,et al.  No free lunch theorems for optimization , 1997, IEEE Trans. Evol. Comput..

[71]  Andrew S. Cassidy,et al.  Convolutional networks for fast, energy-efficient neuromorphic computing , 2016, Proceedings of the National Academy of Sciences.

[72]  Ehud Shapiro,et al.  A mechanical Turing machine: blueprint for a biomolecular computer , 2012, Interface Focus.

[73]  F. Petruccione,et al.  An introduction to quantum machine learning , 2014, Contemporary Physics.

[74]  Peter J. Wangersky,et al.  Lotka-Volterra Population Models , 1978 .

[75]  Michael Sipser,et al.  Introduction to the Theory of Computation , 1996, SIGA.

[76]  B. Jack Copeland,et al.  Hypercomputation: philosophical issues , 2004, Theor. Comput. Sci..

[77]  Stan Wagon,et al.  The Banach-Tarski paradox , 1985 .

[78]  Alán Aspuru-Guzik,et al.  Computational Complexity in Electronic Structure , 2012, Physical chemistry chemical physics : PCCP.

[79]  G.E. Moore,et al.  Cramming More Components Onto Integrated Circuits , 1998, Proceedings of the IEEE.

[80]  Giovanni Viglietta.  Gaming Is a Hard Job, But Someone Has to Do It! , 2012, FUN.

[81]  Scott Aaronson,et al.  NP-complete Problems and Physical Reality , 2005, Electron. Colloquium Comput. Complex..

[82]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[83]  Thomas J. Naughton,et al.  Optical computing , 2009, Appl. Math. Comput..

[84]  Brendan Juba.  Computational Complexity and the Function-Structure-Environment Loop of the Brain , 2016 .

[85]  Julian Togelius,et al.  Measuring Intelligence through Games , 2011, ArXiv.

[86]  Gregory J. Chaitin.  How Much Information Can There Be in a Real Number? , 2012, Computation, Physics and Beyond.

[87]  H. Neven,et al.  Digitized adiabatic quantum computing with a superconducting circuit. , 2015, Nature.

[88]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[89]  Daniel Saunders,et al.  A Survey and Discussion of Memcomputing Machines , 2017, ArXiv.

[90]  Cristiano Chesi,et al.  Computational complexity in the brain. , 2014 .

[91]  R. Barends,et al.  Digital quantum simulation of fermionic models with a superconducting circuit , 2015, Nature Communications.

[92]  Daniel A. Lidar,et al.  Defining and detecting quantum speedup , 2014, Science.

[93]  Michael R. Douglas,et al.  Computational complexity of the landscape I , 2006, ArXiv.

[94]  B. Dickinson,et al.  The complexity of analog computation , 1986 .

[95]  Scott Aaronson,et al.  Quantum Computing since Democritus , 2013 .

[96]  Artiom Alhazov,et al.  Five Nodes Are Sufficient for Hybrid Networks of Evolutionary Processors to Be Computationally Complete , 2014, UCNC.

[97]  Victor Mitrana,et al.  All NP-problems can be solved in polynomial time by accepting hybrid networks of evolutionary processors of constant size , 2007, Inf. Process. Lett..

[98]  M. Horodecki,et al.  Quantum entanglement , 2007, quant-ph/0702225.

[99]  Lulu Qian,et al.  Efficient Turing-Universal Computation with DNA Polymers , 2010, DNA.

[100]  Massimiliano Di Ventra,et al.  Polynomial-time solution of prime factorization and NP-hard problems with digital memcomputing machines , 2015, Chaos.

[101]  István Németi,et al.  0 Fe b 20 02 Non-Turing computations via Malament – Hogarth spacetimes , 2002 .

[102]  Edwin J. Beggs,et al.  An Analogue-Digital Model of Computation: Turing Machines with Physical Oracles , 2017 .

[103]  Oron Shagrir,et al.  Do Accelerating Turing Machines Compute the Uncomputable? , 2011, Minds and Machines.

[104]  Gregory J. Chaitin,et al.  A recent technical report , 1974, SIGA.

[105]  J. Olsen,et al.  The European Commission , 2020, The European Union.

[106]  Konstantine Arkoudas,et al.  Computation, hypercomputation, and physical science , 2008, J. Appl. Log..

[107]  Demis Hassabis,et al.  Mastering the game of Go with deep neural networks and tree search , 2016, Nature.

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

[109]  Robert L. Berger.  The undecidability of the domino problem , 1966 .

[110]  Selmer Bringsjord,et al.  P=np , 2004, ArXiv.

[111]  Juris Hartmanis,et al.  On the Weight of Computations , 1995, Bull. EATCS.

[112]  Hava T. Siegelmann,et al.  Evolving recurrent neural networks are super-Turing , 2011, The 2011 International Joint Conference on Neural Networks.

[113]  Fabio L. Traversa,et al.  Universal Memcomputing Machines , 2014, IEEE Transactions on Neural Networks and Learning Systems.

[114]  Celestine Preetham Lawrence,et al.  Evolution of a designless nanoparticle network into reconfigurable Boolean logic. , 2015, Nature nanotechnology.

[115]  Lei Qian,et al.  Thermodynamics of Computation , 2009, Encyclopedia of Complexity and Systems Science.

[116]  William J. Munro,et al.  Using Quantum Computers for Quantum Simulation , 2010, Entropy.

[117]  J. Whitfield,et al.  Quantum Simulation of Helium Hydride Cation in a Solid-State Spin Register. , 2014, ACS nano.

[118]  F. Verstraete,et al.  Computational complexity of interacting electrons and fundamental limitations of density functional theory , 2007, 0712.0483.

[119]  Susan Stepney,et al.  When does a physical system compute? , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[120]  S. Aaronson.  Computational complexity: Why quantum chemistry is hard , 2009 .

[121]  Peter Wittek,et al.  Quantum Machine Learning: What Quantum Computing Means to Data Mining , 2014 .

[122]  J. Doug Tygar,et al.  Computability and complexity of ray tracing , 1994, Discret. Comput. Geom..

[123]  Nikolay I. Zheludev,et al.  An optical fiber network oracle for NP-complete problems , 2014, Light: Science & Applications.

[124]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[125]  James V. Rauff.  The Pea and the Sun: A Mathematical Paradox , 2009 .

[126]  Victor Mitrana,et al.  Accepting Hybrid Networks of Evolutionary Processors , 2004, DNA.

[127]  José L. Balcázar,et al.  The Structure of Logarithmic Advice Complexity Classes , 1998, Theor. Comput. Sci..

[128]  Thomas J. Naughton,et al.  An optical model of computation , 2005, Theor. Comput. Sci..

[129]  T. Monz,et al.  An open-system quantum simulator with trapped ions , 2011, Nature.

[130]  G. Wendin.  Quantum information processing with superconducting circuits: a review , 2016, Reports on progress in physics. Physical Society.

[131]  H T Siegelmann,et al.  Dating and Context of Three Middle Stone Age Sites with Bone Points in the Upper Semliki Valley, Zaire , 2007 .

[132]  Lenore Blum,et al.  Computing over the Reals: Where Turing Meets Newton , 2004 .

[133]  Wolfgang Maass.  Energy-efficient neural network chips approach human recognition capabilities , 2016, Proceedings of the National Academy of Sciences.

[134]  Salil P. Vadhan,et al.  Computational Complexity , 2005, Encyclopedia of Cryptography and Security.

[135]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 1997, Texts in Computer Science.

[136]  Susan Stepney,et al.  Non-Classical Hypercomputation , 2009, Int. J. Unconv. Comput..

[137]  Yuriy Brun.  Solving satisfiability in the tile assembly model with a constant-size tileset , 2008, J. Algorithms.

[138]  Charles H. Bennett,et al.  The thermodynamics of computation—a review , 1982 .