Andrew Childs University of Maryland the Origin of Quantum Speedup Quantum Computers Allow for Interference Between Computational Paths
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Durham E-Theses
Durham E-Theses Quantum search at low temperature in the single avoided crossing model PATEL, PARTH,ASHVINKUMAR How to cite: PATEL, PARTH,ASHVINKUMAR (2019) Quantum search at low temperature in the single avoided crossing model, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/13285/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk 2 Quantum search at low temperature in the single avoided crossing model Parth Ashvinkumar Patel A Thesis presented for the degree of Master of Science by Research Dr. Vivien Kendon Department of Physics University of Durham England August 2019 Dedicated to My parents for supporting me throughout my studies. Quantum search at low temperature in the single avoided crossing model Parth Ashvinkumar Patel Submitted for the degree of Master of Science by Research August 2019 Abstract We begin with an n-qubit quantum search algorithm and formulate it in terms of quantum walk and adiabatic quantum computation. -
Discrete-Query Quantum Algorithm for NAND Trees
THEORY OF COMPUTING, Volume 5 (2009), pp. 119–123 www.theoryofcomputing.org NOTE Discrete-Query Quantum Algorithm for NAND Trees Andrew M. Childs Richard Cleve Stephen P. Jordan David Yonge-Mallo Received: September 30, 2008; published: July 1, 2009. Abstract: This is a comment on the article “A Quantum Algorithm for the Hamiltonian NAND Tree” by Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, Theory of Comput- ing 4 (2008) 169–190√ . That paper gave a quantum algorithm for evaluating NAND trees with running time O( N) in the Hamiltonian query model. In this note, we point out that their algorithm can be converted into an algorithm using N1/2+o(1) queries in the conventional (discrete) quantum query model. ACM Classification: F.1.2, F.2.2 AMS Classification: 68Q10, 81P68 Key words and phrases: Quantum computation, quantum query complexity, formula evaluation, quan- tum walk, Hamiltonian simulation A NAND tree of depth n is a balanced binary tree whose internal vertices represent NAND gates. Placing bits x1,...,x2n at the leaves, the root of the NAND tree evaluates to the function fn(x1,...,x2n ), where 2n fn : {0,1} → {0,1} is defined recursively as follows. For n = 0, f0(x) = x, and for n > 0, fn(x1,...,x2n ) = ¬ fn−1(x1,...,x2n−1 ) ∧ fn−1(x2n−1+1,...,x2n ) . (1) The goal of the NAND tree problem is to evaluate fn(x1,...,x2n ), making as few queries to the bits 0.753 x1,...,x2n as possible. The optimal classical randomized algorithm for this problem makes Θ(N ) n queries, where N = 2 [9, 10, 11]. -
Hamiltonian Simulation with Nearly Optimal Dependence on Spectral Norm
Hamiltonian simulation with nearly optimal dependence on spectral norm Guang Hao Low∗ [email protected] July 12, 2018 Abstract We present a quantum algorithm for approximating the real time evolution e−iHt of an ar- bitrary d-sparse Hamiltonian to error , given black-box access to the positionsp and b-bit values 1+o(1) o(1) of its non-zero matrix entries. The complexity of our algorithm is O((t dkHk1 2) / ) queries and a factor O(b) more gates, which is shown to be optimal up to subpolynomial factors through a matching query lower bound. This provides a polynomial speedup in sparsity for the common case where the spectral norm kHk ≥ kHk1 2 is known, and generalizes previous approaches which achieve optimal scaling, but with respect to more restrictive parameters. By exploiting knowledge of the spectral norm, our algorithm solves the black-box unitary imple- 1=2+o(1) mentation problem { O(d ) queries suffice to approximate any d-sparsep unitary in the black-box setting, which matches the quantum search lower bound of Ω( d) queries and im- proves upon prior art [Berry and Childs, QIP 2010] of O~(d2=3) queries. Combined with known techniques,p we also solve systems of sparse linear equations with condition number κ using O((κ d)1+o(1)/o(1)) queries, which is a quadratic improvement in sparsity. Contents 1 Introduction . .2 2 Overview of algorithms . .5 3 Hamiltonian simulation by recursionp . .7 4 Sparse Hamiltonian simulation with t dkHk1!2 scaling . .9 5 Sparse Hamiltonian simulation lower bound . -
Report on Enabling the Quantum Leap
Enabling the quantum leap Quantum algorithms for chemistry and materials Report on a National Science Foundation workshop Alexandria, Virginia: January 21-24, 2019 Report on the NSF Workshop on Enabling Quantum Leap: Quantum algorithms for quantum chemistry and materials Bela Bauer1, Sergey Bravyi2, Mario Motta3,and Garnet Kin-Lic Chan4 1 Station Q, Microsoft Corporation, Santa Barbara, California 93106 USA 2 IBM T. J. Watson Research Center, Yorktown Heights, USA 3 IBM Almaden Research, San Jose, CA 95120, USA 4 California Institute of Technology, Pasadena, CA 91125, USA November 12, 2019 Acknowledgements This report is derived from an NSF workshop held between January 22-24 2019 in Alexandria VA. Funding for the workshop and for the production of the report was provided by the National Science Foundation via award no. CHE 1909531. The workshop participants and website are provided below. Workshop Participants Workshop Co-Chairs: • Garnet Kin-Lic Chan (California Institute of Technology) • Sergey Bravyi (IBM T. J. Watson Research Center) • Bela Bauer (Microsoft Station Q) Participant List: Chemistry and Molecular Science • Victor Batista (Yale) • Francesco Evangelista (Emory) •Tim Berkelbach (UChicago) • Joe Subotnik (UPenn) • Tucker Carrington (Queens U.) • Haobin Wang (CU Denver) • Garnet Chan (Caltech) •Dominika Zgid (UMich) • Gavin Crooks (Rigetti Computing) Materials Science, condensed-matter physics • Bela Bauer (Microsoft) • Barbara Jones (IBM Almaden) • Bryan Clark (UIUC) • Austin Minnich (Caltech) •Tom Deveraux (Stanford) •