Quantum State Optimization and Computational Pathway Evaluation for Gate-Model Quantum Computers Laszlo Gyongyosi1,2,3
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Simulating Quantum Field Theory with a Quantum Computer
Simulating quantum field theory with a quantum computer John Preskill Lattice 2018 28 July 2018 This talk has two parts (1) Near-term prospects for quantum computing. (2) Opportunities in quantum simulation of quantum field theory. Exascale digital computers will advance our knowledge of QCD, but some challenges will remain, especially concerning real-time evolution and properties of nuclear matter and quark-gluon plasma at nonzero temperature and chemical potential. Digital computers may never be able to address these (and other) problems; quantum computers will solve them eventually, though I’m not sure when. The physics payoff may still be far away, but today’s research can hasten the arrival of a new era in which quantum simulation fuels progress in fundamental physics. Frontiers of Physics short distance long distance complexity Higgs boson Large scale structure “More is different” Neutrino masses Cosmic microwave Many-body entanglement background Supersymmetry Phases of quantum Dark matter matter Quantum gravity Dark energy Quantum computing String theory Gravitational waves Quantum spacetime particle collision molecular chemistry entangled electrons A quantum computer can simulate efficiently any physical process that occurs in Nature. (Maybe. We don’t actually know for sure.) superconductor black hole early universe Two fundamental ideas (1) Quantum complexity Why we think quantum computing is powerful. (2) Quantum error correction Why we think quantum computing is scalable. A complete description of a typical quantum state of just 300 qubits requires more bits than the number of atoms in the visible universe. Why we think quantum computing is powerful We know examples of problems that can be solved efficiently by a quantum computer, where we believe the problems are hard for classical computers. -
Pilot Quantum Error Correction for Global
Pilot Quantum Error Correction for Global- Scale Quantum Communications Laszlo Gyongyosi*1,2, Member, IEEE, Sandor Imre1, Member, IEEE 1Quantum Technologies Laboratory, Department of Telecommunications Budapest University of Technology and Economics 2 Magyar tudosok krt, H-1111, Budapest, Hungary 2Information Systems Research Group, Mathematics and Natural Sciences Hungarian Academy of Sciences H-1518, Budapest, Hungary *[email protected] Real global-scale quantum communications and quantum key distribution systems cannot be implemented by the current fiber and free-space links. These links have high attenuation, low polarization-preserving capability or extreme sensitivity to the environment. A potential solution to the problem is the space-earth quantum channels. These channels have no absorption since the signal states are propagated in empty space, however a small fraction of these channels is in the atmosphere, which causes slight depolarizing effect. Furthermore, the relative motion of the ground station and the satellite causes a rotation in the polarization of the quantum states. In the current approaches to compensate for these types of polarization errors, high computational costs and extra physical apparatuses are required. Here we introduce a novel approach which breaks with the traditional views of currently developed quantum-error correction schemes. The proposed solution can be applied to fix the polarization errors which are critical in space-earth quantum communication systems. The channel coding scheme provides capacity-achieving communication over slightly depolarizing space-earth channels. I. Introduction Quantum error-correction schemes use different techniques to correct the various possible errors which occur in a quantum channel. In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered [12-16], [19-22] however the efficient error- correction in quantum systems is still a challenge. -
Quantum State Complexity in Computationally Tractable Quantum Circuits
PRX QUANTUM 2, 010329 (2021) Quantum State Complexity in Computationally Tractable Quantum Circuits Jason Iaconis * Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA (Received 28 September 2020; revised 29 December 2020; accepted 26 January 2021; published 23 February 2021) Characterizing the quantum complexity of local random quantum circuits is a very deep problem with implications to the seemingly disparate fields of quantum information theory, quantum many-body physics, and high-energy physics. While our theoretical understanding of these systems has progressed in recent years, numerical approaches for studying these models remains severely limited. In this paper, we discuss a special class of numerically tractable quantum circuits, known as quantum automaton circuits, which may be particularly well suited for this task. These are circuits that preserve the computational basis, yet can produce highly entangled output wave functions. Using ideas from quantum complexity theory, especially those concerning unitary designs, we argue that automaton wave functions have high quantum state complexity. We look at a wide variety of metrics, including measurements of the output bit-string distribution and characterization of the generalized entanglement properties of the quantum state, and find that automaton wave functions closely approximate the behavior of fully Haar random states. In addition to this, we identify the generalized out-of-time ordered 2k-point correlation functions as a particularly use- ful probe of complexity in automaton circuits. Using these correlators, we are able to numerically study the growth of complexity well beyond the scrambling time for very large systems. As a result, we are able to present evidence of a linear growth of design complexity in local quantum circuits, consistent with conjectures from quantum information theory. -
On the Complexity of Numerical Analysis
On the Complexity of Numerical Analysis Eric Allender Peter B¨urgisser Rutgers, the State University of NJ Paderborn University Department of Computer Science Department of Mathematics Piscataway, NJ 08854-8019, USA DE-33095 Paderborn, Germany [email protected] [email protected] Johan Kjeldgaard-Pedersen Peter Bro Miltersen PA Consulting Group University of Aarhus Decision Sciences Practice Department of Computer Science Tuborg Blvd. 5, DK 2900 Hellerup, Denmark IT-parken, DK 8200 Aarhus N, Denmark [email protected] [email protected] Abstract In Section 1.3 we discuss our main technical contribu- tions: proving upper and lower bounds on the complexity We study two quite different approaches to understand- of PosSLP. In Section 1.4 we present applications of our ing the complexity of fundamental problems in numerical main result with respect to the Euclidean Traveling Sales- analysis. We show that both hinge on the question of under- man Problem and the Sum-of-Square-Roots problem. standing the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program 1.1 Polynomial Time Over the Reals producing an integer N, decide whether N>0. We show that PosSLP lies in the counting hierarchy, and combining The Blum-Shub-Smale model of computation over the our results with work of Tiwari, we show that the Euclidean reals provides a very well-studied complexity-theoretic set- Traveling Salesman Problem lies in the counting hierarchy ting in which to study the computational problems of nu- – the previous best upper bound for this important problem merical analysis. -
Quantum Computation and Complexity Theory
Quantum computation and complexity theory Course given at the Institut fÈurInformationssysteme, Abteilung fÈurDatenbanken und Expertensysteme, University of Technology Vienna, Wintersemester 1994/95 K. Svozil Institut fÈur Theoretische Physik University of Technology Vienna Wiedner Hauptstraûe 8-10/136 A-1040 Vienna, Austria e-mail: [email protected] December 5, 1994 qct.tex Abstract The Hilbert space formalism of quantum mechanics is reviewed with emphasis on applicationsto quantum computing. Standardinterferomeric techniques are used to construct a physical device capable of universal quantum computation. Some consequences for recursion theory and complexity theory are discussed. hep-th/9412047 06 Dec 94 1 Contents 1 The Quantum of action 3 2 Quantum mechanics for the computer scientist 7 2.1 Hilbert space quantum mechanics ..................... 7 2.2 From single to multiple quanta Ð ªsecondº ®eld quantization ...... 15 2.3 Quantum interference ............................ 17 2.4 Hilbert lattices and quantum logic ..................... 22 2.5 Partial algebras ............................... 24 3 Quantum information theory 25 3.1 Information is physical ........................... 25 3.2 Copying and cloning of qbits ........................ 25 3.3 Context dependence of qbits ........................ 26 3.4 Classical versus quantum tautologies .................... 27 4 Elements of quantum computatability and complexity theory 28 4.1 Universal quantum computers ....................... 30 4.2 Universal quantum networks ........................ 31 4.3 Quantum recursion theory ......................... 35 4.4 Factoring .................................. 36 4.5 Travelling salesman ............................. 36 4.6 Will the strong Church-Turing thesis survive? ............... 37 Appendix 39 A Hilbert space 39 B Fundamental constants of physics and their relations 42 B.1 Fundamental constants of physics ..................... 42 B.2 Conversion tables .............................. 43 B.3 Electromagnetic radiation and other wave phenomena ......... -
Models of Quantum Complexity Growth
Models of quantum complexity growth Fernando G.S.L. Brand~ao,a;b;c;d Wissam Chemissany,b Nicholas Hunter-Jones,* e;b Richard Kueng,* b;c John Preskillb;c;d aAmazon Web Services, AWS Center for Quantum Computing, Pasadena, CA bInstitute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125 cDepartment of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125 dWalter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 ePerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5 *Corresponding authors: [email protected] and [email protected] Abstract: The concept of quantum complexity has far-reaching implications spanning theoretical computer science, quantum many-body physics, and high energy physics. The quantum complexity of a unitary transformation or quantum state is defined as the size of the shortest quantum computation that executes the unitary or prepares the state. It is reasonable to expect that the complexity of a quantum state governed by a chaotic many- body Hamiltonian grows linearly with time for a time that is exponential in the system size; however, because it is hard to rule out a short-cut that improves the efficiency of a computation, it is notoriously difficult to derive lower bounds on quantum complexity for particular unitaries or states without making additional assumptions. To go further, one may study more generic models of complexity growth. We provide a rigorous connection between complexity growth and unitary k-designs, ensembles which capture the randomness of the unitary group. This connection allows us to leverage existing results about design growth to draw conclusions about the growth of complexity. -
Marching Towards Quantum Supremacy
Princeton Center for Theoretical Science The Princeton Center for Theoretical Science is dedicated to exploring the frontiers of theory in the natural sciences. Its purpose is to promote interaction among theorists and seed new directions in research, especially in areas cutting across traditional disciplinary boundaries. The Center is home to a corps of Center Postdoctoral Fellows, chosen from nominations made by senior theoretical scientists around the world. A group of senior Faculty Fellows, chosen from science and engineering departments across the campus, are responsible for guiding the Center. Center activities include focused topical programs chosen from proposals by Princeton faculty across the natural sciences. The Center is located on the fourth floor of Jadwin Hall, in the heart of the campus “science neighborhood”. The Center hopes to become the focus for innovation and cross-fertilization in theoretical natural science at Princeton. Faculty Fellows Igor Klebanov, Director Ned Wingreen, Associate Director Marching Towards Quantum Andrei Bernevig Jeremy Goodman Duncan Haldane Supremacy Andrew Houck Mariangela Lisanti Thanos Panagiotopoulos Frans Pretorius November 13-15, 2019 Center Postdoctoral Fellows Ricard Alert-Zenon 2018-2021 PCTS Seminar Room Nathan Benjamin 2018-2021 Andrew Chael 2019-2022 Jadwin Hall, Fourth Floor, Room 407 Amos Chan 2019-2022 Fani Dosopoulou 2018-2021 Biao Lian 2017-2020 Program Organizers Vladimir Narovlansky 2019-2022 Sergey Frolov Sabrina Pasterski 2019-2022 Abhinav Prem 2018-2021 Michael Gullans -
Bohmian Mechanics Versus Madelung Quantum Hydrodynamics
Ann. Univ. Sofia, Fac. Phys. Special Edition (2012) 112-119 [arXiv 0904.0723] Bohmian mechanics versus Madelung quantum hydrodynamics Roumen Tsekov Department of Physical Chemistry, University of Sofia, 1164 Sofia, Bulgaria It is shown that the Bohmian mechanics and the Madelung quantum hy- drodynamics are different theories and the latter is a better ontological interpre- tation of quantum mechanics. A new stochastic interpretation of quantum me- chanics is proposed, which is the background of the Madelung quantum hydro- dynamics. Its relation to the complex mechanics is also explored. A new complex hydrodynamics is proposed, which eliminates completely the Bohm quantum po- tential. It describes the quantum evolution of the probability density by a con- vective diffusion with imaginary transport coefficients. The Copenhagen interpretation of quantum mechanics is guilty for the quantum mys- tery and many strange phenomena such as the Schrödinger cat, parallel quantum and classical worlds, wave-particle duality, decoherence, etc. Many scientists have tried, however, to put the quantum mechanics back on ontological foundations. For instance, Bohm [1] proposed an al- ternative interpretation of quantum mechanics, which is able to overcome some puzzles of the Copenhagen interpretation. He developed further the de Broglie pilot-wave theory and, for this reason, the Bohmian mechanics is also known as the de Broglie-Bohm theory. At the time of inception of quantum mechanics Madelung [2] has demonstrated that the Schrödinger equa- tion can be transformed in hydrodynamic form. This so-called Madelung quantum hydrodynam- ics is a less elaborated theory and usually considered as a precursor of the Bohmian mechanics. The scope of the present paper is to show that these two theories are different and the Made- lung hydrodynamics is a better interpretation of quantum mechanics than the Bohmian me- chanics. -
Bounds on Errors in Observables Computed from Molecular Dynamics Simulations
Bounds on Errors in Observables Computed from Molecular Dynamics Simulations Vikram Sundar [email protected] Advisor: Dr. David Gelbwaser and Prof. Alan´ Aspuru-Guzik (Chemistry) Shadow Advisor: Prof. Martin Nowak Submitted in partial fulfillment of the honors requirements for the degree of Bachelor of Arts in Mathematics March 19, 2018 Contents Abstract iii Acknowledgments iv 1 Introduction1 1.1 Molecular Dynamics................................1 1.1.1 Why Molecular Dynamics?........................1 1.1.2 What is Molecular Dynamics?......................2 1.1.2.1 Force Fields and Integrators..................2 1.1.2.2 Observables...........................3 1.2 Structure of this Thesis...............................4 1.2.1 Sources of Error...............................4 1.2.2 Key Results of this Thesis.........................5 2 The St¨ormer-Verlet Method and Energy Conservation7 2.1 Symplecticity....................................8 2.1.1 Hamiltonian and Lagrangian Mechanics................8 2.1.2 Symplectic Structures on Manifolds...................9 2.1.3 Hamiltonian Flows are Symplectic.................... 10 2.1.4 Symplecticity of the Stormer-Verlet¨ Method............... 12 2.2 Backward Error Analysis.............................. 12 2.2.1 Symplectic Methods and Backward Error Analysis.......... 14 2.2.2 Bounds on Energy Conservation..................... 14 3 Integrable Systems and Bounds on Sampling Error 16 3.1 Integrable Systems and the Arnold-Liouville Theorem............. 17 3.1.1 Poisson Brackets and First Integrals................... 17 3.1.2 Liouville’s Theorem............................ 18 3.1.3 Action-Angle Coordinates......................... 20 3.1.4 Integrability and MD Force Fields.................... 22 3.2 Sampling Error................................... 23 3.2.1 Equivalence of Spatial and Time Averages............... 23 3.2.2 Bounding Sampling Error......................... 24 3.2.3 Reducing Sampling Error with Filter Functions........... -
Booklet of Abstracts
Booklet of abstracts Thomas Vidick California Institute of Technology Tsirelson's problem and MIP*=RE Boris Tsirelson in 1993 implicitly posed "Tsirelson's Problem", a question about the possible equivalence between two different ways of modeling locality, and hence entanglement, in quantum mechanics. Tsirelson's Problem gained prominence through work of Fritz, Navascues et al., and Ozawa a decade ago that establishes its equivalence to the famous "Connes' Embedding Problem" in the theory of von Neumann algebras. Recently we gave a negative answer to Tsirelson's Problem and Connes' Embedding Problem by proving a seemingly stronger result in quantum complexity theory. This result is summarized in the equation MIP* = RE between two complexity classes. In the talk I will present and motivate Tsirelson's problem, and outline its connection to Connes' Embedding Problem. I will then explain the connection to quantum complexity theory and show how ideas developed in the past two decades in the study of classical and quantum interactive proof systems led to the characterization (which I will explain) MIP* = RE and the negative resolution of Tsirelson's Problem. Based on joint work with Ji, Natarajan, Wright and Yuen available at arXiv:2001.04383. Joonho Lee, Dominic Berry, Craig Gidney, William Huggins, Jarrod McClean, Nathan Wiebe and Ryan Babbush Columbia University | Macquarie University | Google | Google Research | Google | University of Washington | Google Efficient quantum computation of chemistry through tensor hypercontraction We show how to achieve the highest efficiency yet for simulations with arbitrary basis sets by using a representation of the Coulomb operator known as tensor hypercontraction (THC). We use THC to express the Coulomb operator in a non-orthogonal basis, which we are able to block encode by separately rotating each term with angles that are obtained via QROM. -
Lecture 6: Quantum Error Correction and Quantum Capacity
Lecture 6: Quantum error correction and quantum capacity Mark M. Wilde∗ The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally \quantum" theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum com- munication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information. The no-cloning theorem provides the intuition behind quantum error correction. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information|otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H(B) − H(E)|a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain. -
Interactions of Computational Complexity Theory and Mathematics
Interactions of Computational Complexity Theory and Mathematics Avi Wigderson October 22, 2017 Abstract [This paper is a (self contained) chapter in a new book on computational complexity theory, called Mathematics and Computation, whose draft is available at https://www.math.ias.edu/avi/book]. We survey some concrete interaction areas between computational complexity theory and different fields of mathematics. We hope to demonstrate here that hardly any area of modern mathematics is untouched by the computational connection (which in some cases is completely natural and in others may seem quite surprising). In my view, the breadth, depth, beauty and novelty of these connections is inspiring, and speaks to a great potential of future interactions (which indeed, are quickly expanding). We aim for variety. We give short, simple descriptions (without proofs or much technical detail) of ideas, motivations, results and connections; this will hopefully entice the reader to dig deeper. Each vignette focuses only on a single topic within a large mathematical filed. We cover the following: • Number Theory: Primality testing • Combinatorial Geometry: Point-line incidences • Operator Theory: The Kadison-Singer problem • Metric Geometry: Distortion of embeddings • Group Theory: Generation and random generation • Statistical Physics: Monte-Carlo Markov chains • Analysis and Probability: Noise stability • Lattice Theory: Short vectors • Invariant Theory: Actions on matrix tuples 1 1 introduction The Theory of Computation (ToC) lays out the mathematical foundations of computer science. I am often asked if ToC is a branch of Mathematics, or of Computer Science. The answer is easy: it is clearly both (and in fact, much more). Ever since Turing's 1936 definition of the Turing machine, we have had a formal mathematical model of computation that enables the rigorous mathematical study of computational tasks, algorithms to solve them, and the resources these require.