DOCSLIB.ORG

Discriminating Non-Isomorphic Graphs with an Experimental Quantum Annealer

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Discriminating Non-Isomorphic Graphs with an Experimental Quantum Annealer Discriminating Non-Isomorphic Graphs with an Experimental Quantum Annealer Zoe Gonzalez Izquierdo,1, 2, ∗ Ruilin Zhou,3 Klas Markstr¨om,4 and Itay Hen1, 2 1Department of Physics and Astronomy, and Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA 2Information Sciences Institute, University of Southern California, Marina del Rey, California 90292, USA 3Department of Electrical and Computer Engineering, Northwestern University, Evanston, Illinois 60208, USA 4Department of Mathematics and Mathematical Statistics, Ume˚aUniversity, Ume˚a,Sweden We demonstrate experimentally the ability of a quantum annealer to distinguish between sets of non-isomorphic graphs that share the same classical Ising spectrum. Utilizing the pause-and-quench features recently introduced into D-Wave quantum annealing processors, which allow the user to probe the quantum Hamiltonian realized in the middle of an anneal, we show that obtaining thermal averages of diagonal observables of `classically indistinguishable' non-isomorphic graphs encoded into transverse-field Ising Hamiltonians enable their discrimination. We discuss the significance of our results in the context of the graph isomorphism problem. I. INTRODUCTION vides access to thermal properties of quantum, rather than classical, Hamiltonians. This new ability has al- Commercial quantum annealers [1{3] irrupted into the ready been tested with varying degrees of success [20{ quantum computing scene about a decade ago, bringing 22, 26]. along great promise|and just as much skepticism [4,5]. Another novel feature that has recently been intro- Whether these machines do or could in principle pro- duced into experimental quantum annealers is that of re- vide any advantages over their classical analogues|or verse annealing. Here, the initial Hamiltonian is set to be whether they could even be considered genuine quantum the Ising spin glass and the transverse field is ramped up processors [6{10]|is still a topic of much debate [11{14]. from zero to some intermediate value, and then brought While concrete evidence illustrating their usefulness back to zero again. This capability has been shown to as practical solvers of optimization problems is still help with the exploitation of regions of interest in the scarce [15, 16] the possibility that they can function as solution space of various optimization problems [27, 28]. efficient Boltzmann samplers [17{19] or as effective quan- In this work, we examine the possibility that mid- tum simulators [20{23] has been gaining traction. anneal probing through pause-and-quench may be uti- Alongside efforts to extend the range of potential ap- lized toward accomplishing an altogether different enter- plications for quantum annealers, improvements to these prise, namely, to solve instances of the Graph Isomor- devices are continuously being made. Notably, currently phism problem|the task of deciding whether two graphs available devices allow the user to exercise additional are isomorphic (i.e., identical up to vertex relabeling). control over various annealing parameters [24] and de- The possibility of using quantum annealers in this man- termine, to a certain extent, the schedule of the anneal, ner was first proposed almost a decade ago [29, 30] and which until recently could only be set to interpolate di- later experimentally attempted with somewhat limited rectly between a transverse-field at the beginning of the success on an older-generation experimental quantum de- anneal and a classical Ising spin glass at the end of it. vice that lacked the pause-and-quench features [31]. In particular, novel pause-and-quench capabilities enable Here, we study the power of quantum annealers some flexibility in determining the rates at which the clas- equipped with pause-and-quench capabilities to dis- sical Ising and transverse field Hamiltonians shift their re- tinguish between non-isomorphic graphs mapped onto spective strengths. This capability of pausing|the tem- transverse-field Ising Hamiltonians. We find that while arXiv:2005.01241v1 [quant-ph] 4 May 2020 porary halting of the anneal midway|deployed shortly measurements at the end of the anneal have only limited after the minimum gap, has already been demonstrated discrimination capabilities, mid-anneal measurements al- to increase the success rates of the quantum annealer low the extraction of sufficient information from the spec- when used as an optimizer [18, 25], by giving the system tra of strictly quantum Hamiltonians and are in turn time to thermalize after the minimum gap. powerful enough to distinguish all tested sets of non- Combining said pause with a sufficiently fast quench isomorphic graphs. to the end of the anneal, such that the evolution during The paper is organized as follows. In Sec.II, we provide the quench is highly non-adiabatic, allows the user to ef- some background on the Graph Isomorphism problem, fectively measure the system at the time of the pause, and the means for solving instances of the problem using rendering the annealer a quantum simulator, which pro- quantum annealers. Sec.III relays the technical details of our experiments, including information about the specific parameters used. Results are presented and discussed in Sec.IV, and conclusions and final remarks laid out in ∗ [email protected] Sec.V. 2 II. DISCRIMINATING BETWEEN GRAPHS Calculating the full quantum Ising spectrum of a given USING A QUANTUM ANNEALER graph requires in general the diagonalization of a 2n 2n matrix, which quickly becomes intractable as n increases.× A graph G = (V; E), is a pair of sets, with V being the Quantum annealers, which implement the above quan- set of vertices and E the set of edges (unordered pairs of tum Ising Hamiltonian H(G; s), could alternatively be vertices), such that E [V ]2 [32]. We denote the number used to extract valuable information without resorting of vertices V by n.⊆ Two graphs G and G0 are said to to calculating its entire spectrum [29]. This can be done j j 0 if the annealing process is stopped midway at some in- be isomorphic, G ∼= G , when there exists a bijection between them that preserves vertex adjacency, that is, termediate s value, 0 < s < 1, and measurements are one can be converted to the other by a simple relabeling then taken in order to obtain an estimation of thermal of its vertices. averages of physical observables. Determining whether two graphs are isomorphic is Experimentally testing the conjecture that physi- known as the Graph Isomorphism (GI) problem [33]. cal quantum annealers can distinguish non-isomorphic The best general case algorithm for solving it is quasi- graphs poses however two main challenges. The first is polynomial in graph size [34]. One way of solving the that currently available processors only allow diagonal GI problem is by comparing graph invariants: maps that measurements to be performed, restricting to some ex- take a graph as their argument and assign equal values tent the amount of information accessible to the experi- to isomorphic graphs. A specific method to construct menter. Moreover, to ensure that the quantum annealer graph invariants results from encoding the structure of identifies isomorphic graphs as such, one must also choose the graph on an Ising Hamiltonian [29]. To do this, to to measure quantities that remain unchanged under ver- z z every edge (i; j) in the graph we assign a term σi σj in the tex relabeling. Hamiltonian, where σz represents an individual Pauli-z In this study, we measure the classical energy E i ≡ matrix on spin i. Optionally, we can add an external, Hp , the z-magnetization Mz , the spin-glass order pa- h i h i homogeneous longitudinal field to the spins. We call this rameter [29], the `problem Hamiltonian' of the graph: s n 1 X z z 2 X X Q2 = σi σj ; (4) H (G) = σzσz + σz: (1) N(N 1) h i p i j i − i6=j (i;j)2E(G) i=1 and the next-nearest neighbor interaction energy [31] The invariant of interest is the spectrum of Hp, and 0 two graphs G and G are said to be co-Ising if Hp(G) X 0 2 2 z z and Hp(G ) have the same spectra. The spectrum of Ω = [A (G)]i;jσi σj : (5) i;j Hp is however not a complete invariant, meaning that non-isomorphic, co-Ising graphs exist [35], and trying to The second challenge that current quantum anneal- use Hp as a graph discriminator to solve the GI problem would therefore fail in the general case. ing technology faces is that it only allows measurements Quantum annealers offer the exciting possibility of ex- to be taken at the end of the anneal, where only the tending the set of Hamiltonian graph invariants into the classical Hp component has non-zero strength. This set- quantum regime. This can most simply be done by ting is appropriate for the standard usage of a quantum adding a quantum term, such as a homogeneous trans- annealer as an optimizer, when the objective is finding verse field, to the classical Ising model onto which the the ground state of Hp, and the transverse field is just a graph is mapped. In this case, the Hamiltonian can be means to arriving at it. However, as was discussed above, written as for distinguishing between non-isomorphic graphs, sam- pling from a thermal state of a classical Hamiltonian is H(G; s) = A(s)Hd + B(s)Hp(G); (2) insufficient in general. To illustrate why this is so, con- sider Fig.1, which shows ∆ E = EG13 EG13p for one of where s is a rescaled, dimensionless time parameter, − s [0; 1]. A(s) and B(s) are functions of s such that the pairs of graphs we try to distinguish, denoted G13 2 and G13p, with n = 13 (small enough to be directly di- A(0) B(0) 0 and B(1) A(1) 0, and Hd is the standard transverse' field driver Hamiltonian' in quantum agonalized).
Load more

Recommended publications
  • Achievements of the IARPA-QEO and DARPA-QAFS Programs & The
    Achievements of the IARPA-QEO and DARPA-QAFS programs & The prospects for quantum enhancement with QA Elizabeth Crosson & DL, Nature Reviews Physics (2021) AQC 20201 June 22, 2021 Daniel Lidar USC A decade+ since the D‐Wave 1: where does QA stand? Numerous studies have attempted to demonstrate speedups using the D-Wave processors. Some studies have succeeded in A scaling speedup against path-integral Monte demonstrating constant-factor Carlo was claimed for quantum simulation of speedups in optimization against all geometrically frustrated magnets (scaling of classical algorithms they tried, e.g., relaxation time in escape from topological frustrated cluster-loop problems [1]. obstructions) [2]. Simulation is a natural and promising QA application. But QA was originally proposed as an optimization heuristic. No study so far has demonstrated an unequivocal scaling speedup in optimization. [1] S. Mandrà, H. Katzgraber, Q. Sci. & Tech. 3, 04LT01 (2018) [2] A. King et al., arXiv:1911.03446 What could be the reasons for the lack of scaling speedup? • Perhaps QA is inherently a poor algorithm? More on this later. • Control errors? Random errors in the final Hamiltonian (J-chaos) have a strong detrimental effect. • Such errors limit the size of QA devices that can be expected to accurately solve optimization instances without error correction/suppression [1,2]. • Decoherence? Single-qubit and ≪ anneal time. Often cited as a major problem of QA relative to the gate-model. • How important is this? Depends on weak vs strong coupling to the environment [3]. More on this later as well. [1] T. Albash, V. Martin‐Mayor, I. Hen, Q.
    [Show full text]
  • Why and When Is Pausing Beneficial in Quantum Annealing?
    Why and when pausing is beneficial in quantum annealing Huo Chen1, 2 and Daniel A. Lidar1, 2, 3, 4 1Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA 2Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA 3Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA 4Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA Recent empirical results using quantum annealing hardware have shown that mid anneal pausing has a surprisingly beneficial impact on the probability of finding the ground state for of a variety of problems. A theoretical explanation of this phenomenon has thus far been lacking. Here we provide an analysis of pausing using a master equation framework, and derive conditions for the strategy to result in a success probability enhancement. The conditions, which we identify through numerical simulations and then prove to be sufficient, require that relative to the pause duration the relaxation rate is large and decreasing right after crossing the minimum gap, small and decreasing at the end of the anneal, and is also cumulatively small over this interval, in the sense that the system does not thermally equilibrate. This establishes that the observed success probability enhancement can be attributed to incomplete quantum relaxation, i.e., is a form of beneficial non-equilibrium coupling to the environment. I. INTRODUCTION lems [23] and training deep generative machine learn- ing models [24]. Numerical studies [25, 26] of the p-spin Quantum annealing [1{4] stands out among the multi- model also agree with these empirical results.
    [Show full text]
  • Arxiv:1908.04480V2 [Quant-Ph] 23 Oct 2020
    Quantum adiabatic machine learning with zooming Alexander Zlokapa,1 Alex Mott,2 Joshua Job,3 Jean-Roch Vlimant,1 Daniel Lidar,4 and Maria Spiropulu1 1Division of Physics, Mathematics & Astronomy, Alliance for Quantum Technologies, California Institute of Technology, Pasadena, CA 91125, USA 2DeepMind Technologies, London, UK 3Lockheed Martin Advanced Technology Center, Sunnyvale, CA 94089, USA 4Departments of Electrical and Computer Engineering, Chemistry, and Physics & Astronomy, and Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, CA 90089, USA Recent work has shown that quantum annealing for machine learning, referred to as QAML, can perform comparably to state-of-the-art machine learning methods with a specific application to Higgs boson classification. We propose QAML-Z, a novel algorithm that iteratively zooms in on a region of the energy surface by mapping the problem to a continuous space and sequentially applying quantum annealing to an augmented set of weak classifiers. Results on a programmable quantum annealer show that QAML-Z matches classical deep neural network performance at small training set sizes and reduces the performance margin between QAML and classical deep neural networks by almost 50% at large training set sizes, as measured by area under the ROC curve. The significant improvement of quantum annealing algorithms for machine learning and the use of a discrete quantum algorithm on a continuous optimization problem both opens a new class of problems that can be solved by quantum annealers and suggests the approach in performance of near-term quantum machine learning towards classical benchmarks. I. INTRODUCTION lem Hamiltonian, ensuring that the system remains in the ground state if the system is perturbed slowly enough, as given by the energy gap between the ground state and Machine learning has gained an increasingly impor- the first excited state [36{38].
    [Show full text]
  • Error Suppression and Error Correction in Adiabatic Quantum Computation: Techniques and Challenges
    PHYSICAL REVIEW X 3, 041013 (2013) Error Suppression and Error Correction in Adiabatic Quantum Computation: Techniques and Challenges Kevin C. Young* and Mohan Sarovar Scalable and Secure Systems Research (08961), Sandia National Laboratories, Livermore, California 94550, USA Robin Blume-Kohout Advanced Device Technologies (01425), Sandia National Laboratories, Albuquerque, New Mexico 87185, USA (Received 10 September 2012; revised manuscript received 23 July 2013; published 13 November 2013) Adiabatic quantum computation (AQC) has been lauded for its inherent robustness to control imperfections and relaxation effects. A considerable body of previous work, however, has shown AQC to be acutely sensitive to noise that causes excitations from the adiabatically evolving ground state. In this paper, we develop techniques to mitigate such noise, and then we point out and analyze some obstacles to further progress. First, we examine two known techniques that leverage quantum error-detecting codes to suppress noise and show that they are intimately related and may be analyzed within the same formalism. Next, we analyze the effectiveness of such error-suppression techniques in AQC, identify critical constraints on their performance, and conclude that large-scale, fault-tolerant AQC will require error correction, not merely suppression. Finally, we study the consequences of encoding AQC in quantum stabilizer codes and discover that generic AQC problem Hamiltonians rapidly convert physical errors into uncorrectable logical errors. We present several techniques to remedy this problem, but all of them require unphysical resources, suggesting that the adiabatic model of quantum computation may be fundamentally incompatible with stabilizer quantum error correction. DOI: 10.1103/PhysRevX.3.041013 Subject Areas: Quantum Physics, Quantum Information I.
    [Show full text]
  • High Energy Physics Quantum Computing
    High Energy Physics Quantum Computing Quantum Information Science in High Energy Physics at the Large Hadron Collider PI: O.K. Baker, Yale University Unraveling the quantum structure of QCD in parton shower Monte Carlo generators PI: Christian Bauer, Lawrence Berkeley National Laboratory Co-PIs: Wibe de Jong and Ben Nachman (LBNL) The HEP.QPR Project: Quantum Pattern Recognition for Charged Particle Tracking PI: Heather Gray, Lawrence Berkeley National Laboratory Co-PIs: Wahid Bhimji, Paolo Calafiura, Steve Farrell, Wim Lavrijsen, Lucy Linder, Illya Shapoval (LBNL) Neutrino-Nucleus Scattering on a Quantum Computer PI: Rajan Gupta, Los Alamos National Laboratory Co-PIs: Joseph Carlson (LANL); Alessandro Roggero (UW), Gabriel Purdue (FNAL) Particle Track Pattern Recognition via Content-Addressable Memory and Adiabatic Quantum Optimization PI: Lauren Ice, Johns Hopkins University Co-PIs: Gregory Quiroz (Johns Hopkins); Travis Humble (Oak Ridge National Laboratory) Towards practical quantum simulation for High Energy Physics PI: Peter Love, Tufts University Co-PIs: Gary Goldstein, Hugo Beauchemin (Tufts) High Energy Physics (HEP) ML and Optimization Go Quantum PI: Gabriel Perdue, Fermilab Co-PIs: Jim Kowalkowski, Stephen Mrenna, Brian Nord, Aris Tsaris (Fermilab); Travis Humble, Alex McCaskey (Oak Ridge National Lab) Quantum Machine Learning and Quantum Computation Frameworks for HEP (QMLQCF) PI: M. Spiropulu, California Institute of Technology Co-PIs: Panagiotis Spentzouris (Fermilab), Daniel Lidar (USC), Seth Lloyd (MIT) Quantum Algorithms for Collider Physics PI: Jesse Thaler, Massachusetts Institute of Technology Co-PI: Aram Harrow, Massachusetts Institute of Technology Quantum Machine Learning for Lattice QCD PI: Boram Yoon, Los Alamos National Laboratory Co-PIs: Nga T. T. Nguyen, Garrett Kenyon, Tanmoy Bhattacharya and Rajan Gupta (LANL) Quantum Information Science in High Energy Physics at the Large Hadron Collider O.K.
    [Show full text]
  • 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.
    [Show full text]
  • Daniel Lidar Interview - Special Topic of Quantum Computers Institutional Interviews Journal Interviews AUTHOR COMMENTARIES - from Special Topics Podcasts
    Home About Thomson Reuters Press Room Contact Us ● ScienceWatch Home ● Inside This Month... ● Interviews Featured Interviews Author Commentaries Special Topics : Quantum Computers : Daniel Lidar Interview - Special Topic of Quantum Computers Institutional Interviews Journal Interviews AUTHOR COMMENTARIES - From Special Topics Podcasts Quantum Computers - Published: March 2010 ● Analyses Interview Date: May 2009 Featured Analyses Daniel Lidar What's Hot In... From the Special Topic of Quantum Computers Special Topics According to our Special Topics analysis of quantum computers research over the past decade, the work of Dr. Daniel Lidar ranks at #4 by total ● Data & Rankings number of papers, based on 79 papers cited a total of 1,814 times between Sci-Bytes January 1, 1999 and December 31, 2009. Fast Breaking Papers In the Web of Science®, Dr. Lidar currently has 113 original articles, reviews, New Hot Papers and proceedings papers from 1998-2010, cited a total of 3,318 times. Six of Emerging Research Fronts these papers have been named as Highly Cited Papers in the field of Fast Moving Fronts Corporate Research Fronts Physics in Essential Science IndicatorsSM from Thomson Reuters. Research Front Maps Current Classics Top Topics Dr. Lidar is Associate Professor in the Departments of Electrical Engineering and Chemistry at the Rising Stars University of Southern California in Los Angeles. He is also the Director and co-founding member of the New Entrants USC Center for Quantum Information Science and Technology (CQIST). Country Profiles In this interview, he talks with ScienceWatch.com about his highly cited research on quantum ● About Science Watch computers. Methodology Archives What first drew your interest to the field of quantum computing? Contact Us As I was finishing my Ph.D.
    [Show full text]
  • Books on Classical and Quantum Error-Correcting Codes
    Useful Articles and Books on Error-Correcting Codes Quantum Mechanics P. A. M. Dirac, Principles of Quantum Mechanics, Clarendon Press Oxford L. Landau and E. M. Lifshitz, Quantum Mechanics: the Non-relativistic Theory Albert Messiah, Quantum Mechanics, Dover Stephen Gasiorowicz, Quantum Mechanics, Wiley David J. Griffiths, Introduction to Quantum Mechanics, Cambridge University Press; undergraduate level J. J. Sakurai, Quantum Mechanics, (revised edition) Addison-Wesley John von Neumann, Mathematical Foundations of Quantum Mechanics", Prince- ton University Press, 1955. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics", Cambridge University Press, 1987 The Theory of Observation in Quantum Mechanics by Fritz London and Ed- mund Bauer, a set of lectures given at the Sorbonne in June 1939. Robert B. Griffiths, Consistent Quantum Theory, Cambridge University Press (2002) Yakir Aharonov and Daniel Rohrlich, Quantum Paradoxes Wiley-VCH (2005) 1 Classical Codes C. E. Shannon, A Mathematical Theory of Communication, The Bell System Technical Journal, Vol 27, 379-423, 623-656, July October 1948. Reissued De- cember 1957. L´eonBrillouin, La Science et la Th´eoriede l'Information, Editions´ Jacques Gabay, first published in the US in 1956 as \Science and Information Theory". Thomas M. Thompson, From Error-Correcting Codes Through Sphere Packings to Simple Groups", Mathematical Association of America, 1983. R. Hill, A First Course in Coding Theory", Clarendon Press, Oxford, 1986. W Wesley Peterson and E. J. Weldon, Error-Correcting Codes, MIT
    [Show full text]
  • Quantum Computing
    Fujitsu Laboratories Advanced Technology Symposium 2017 The Impact of Quantum Computing Daniel Lidar University of Southern California Quantum Computing - Origins Credit goes to Feynman: Quantum Physics: The theory invented to explain the smallest scales of our universe (Planck, Schrödinger, Einstein, Heisenberg - 1920s) Quantum Computing: Leverage quantum properties for computation – and solve problems deemed intractable by classical computing (Richard Feynman - 1980s) Quantum Computing - Origins Credit goes to Feynman: Quantum Physics: The theory invented to Yuri Manin explain the smallest scales of our universe (Planck, Schrödinger, Einstein, Heisenberg - 1920s) Russian meddling/Fake news? Quantum Computing: Leverage quantumRussian mathematician first proposed QCs in 1980 properties for computation – and solve problems Radio Moscow broadcast deemed intractable by classical computing (Richard Feynman - 1980s) Quantum Computing - Origins Credit goes to Feynman: Quantum Physics: The theory invented to explain the smallest scales of our universe (Planck, Schrödinger, Einstein, Heisenberg - 1920s) Quantum Computing - Origins Credit goes to Feynman: Quantum Physics: The theory invented to explain the smallest scales of our universe (Planck, Schrödinger, Einstein, Heisenberg - 1920s) This talk will: - Provide some background on quantum computing - Plant some seeds for the panel discussions and later talks - Speculate about where the field is going Quantum Computing - Origins Credit goes to Feynman: Quantum Physics: The theory invented to explain
    [Show full text]
  • Dynamical Decoupling and Quantum Error Correction Codes
    Dynamical Decoupling and Quantum Error Correction Codes Gerardo A. Paz-Silva and Daniel Lidar Center for Quantum Information Science & Technology University of Southern California GAPS and DAL paper in preparation 1 Dynamical Decoupling and Quantum Error Correction Codes (SXDD) Gerardo A. Paz-Silva and Daniel Lidar Center for Quantum Information Science & Technology University of Southern California GAPS and DAL paper in preparation 2 Motivation qMac 푯푺 3 Motivation 푯푩 qMac 푯푺 4 Motivation 푯푩 푯푺푩 푯푺푩 qMac 푯푺 푯푺푩 푯푺푩 5 Motivation 푯푩 푯푺푩 QEC + FT 푯푺푩 qMac 푯푺 푯푺푩 푯푺푩 6 Motivation 푯푩 Dynamical Decoupling 푯푺푩 QEC + FT 푯푺푩 qMac 푯푺 푯푺푩 푯푺푩 7 Motivation 푯푩 Dynamical Decoupling 푯′푺푩 푯푺푩 QEC + FT 푯푺푩 푯′푺푩 qMac 푯푺 푯푺푩 푯′푺푩 푯′푺푩 푯푺푩 8 −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 [[n,k,d]] QEC code −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 −푖(퐻∅,푒푓푓 푇+퐻푆퐵,푒푓푓 푂 푇 ) 푈퐷퐷 푇 = 푒 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 [[n,k,d]] QEC code −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 −푖(퐻∅,푒푓푓 푇+퐻푆퐵,푒푓푓 푂 푇 ) 푈퐷퐷 푇 = 푒 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 휂퐷퐷 < 휂0 [[n,k,d]] QEC code −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 −푖(퐻∅,푒푓푓 푇+퐻푆퐵,푒푓푓 푂 푇 ) 푈퐷퐷 푇 = 푒 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 휂퐷퐷 < 휂0 [[n,k,d]] QEC code DD DD DD DD DD Ng,Lidar,Preskill PRA 84, 012305(2011) • Enhanced fidelity of physical gates via appended DD sequences −푖(퐻 τ푚푖푛) 퐻 = 퐼⊗퐻퐵 + 퐻푆퐵 푈 τ푚푖푛 = 푒 휂0 = 퐻푆퐵 τ푚푖푛 푁+1 −푖(퐻∅,푒푓푓 푇+퐻푆퐵,푒푓푓 푂 푇 ) 푈퐷퐷 푇 = 푒 푁+1 휂퐷퐷(푁) = 퐻푆퐵,푒푓푓 푂 푇 휂퐷퐷 < 휂0 [[n,k,d]] QEC code DD DD DD DD DD Ng,Lidar,Preskill PRA 84, 012305(2011) • Enhanced fidelity of physical gates via appended DD sequences • Order of decoupling N cannot be arbitrarily large.
    [Show full text]
  • The Toric Code at Finite Temperature by Christian Daniel Freeman A
    The Toric Code at Finite Temperature by Christian Daniel Freeman A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Birgitta Whaley, Co-chair Professor Joel Moore, Co-chair Professor Irfan Siddiqi Professor Martin Head-Gordon Summer 2018 The Toric Code at Finite Temperature Copyright 2018 by Christian Daniel Freeman 1 Abstract The Toric Code at Finite Temperature by Christian Daniel Freeman Doctor of Philosophy in Physics University of California, Berkeley Professor Birgitta Whaley, Co-chair Professor Joel Moore, Co-chair Alexei Kitaev's toric code is a rich model, that has birthed and stimulated the develop- ment of topological quantum computing, error correction, and field theory. It was also the first example of a quantum error correcting code that could resiliently store quantum infor- mation in certain types of noisy environments without the need for active error correction. Unfortunately, the toric code loses much of its power as a noise-resilient quantum memory at any finite temperature. Many of the problems with the toric code at finite temperature are likewise shared among its cousin stabilizer codes. These problems can be traced to the proliferation of stringlike \defects" in these codes at finite temperature. The aim of this thesis is then twofold. First, I characterize both numerically and theoretically the failure modes of the toric code at finite temperature with fairly modest bath assumptions. I achieve this by numerically sampling the nonequilibrium dynamics of the toric code using a continuous time monte carlo algorithm.
    [Show full text]
  • On the Relation Between Quantum Computation and Classical Statistical Mechanics
    On the relation between quantum computation and classical statistical mechanics by Joseph Geraci A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright c 2008 by Joseph Geraci ! Abstract On the relation between quantum computation and classical statistical mechanics Joseph Geraci Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2008 We provide a quantum algorithm for the exact evaluation of the Potts partition function for a certain class of restricted instances of graphs that correspond to irreducible cyclic codes. We use the same approach to demonstrate that quantum computers can provide an exponential speed up over the best classical algorithms for the exact evaluation of the weight enumerator polynomial for a family of classical cyclic codes. In addition to this we also provide an efficient quantum approximation algorithm for a function (signed-Euler generating function) closely related to the Ising partition function and demonstrate that this problem is BQP-complete. We accomplish the above for the Potts partition function by using a series of links between Gauss sums, classical coding theory, graph theory and the partition function. We exploit the fact that there exists an efficient approximation algorithm for Gauss sums and the fact that this problem is equivalent in complexity to evaluating discrete log. A theorem of McEliece allows one to turn the Gauss sum approximation into an exact evaluation of the Potts partition function. Stripping the physics from this result leaves one with the result for the weight enumerator polynomial. The result for the approximation of the signed-Euler generating function was accomplished by fashioning a new mapping between quantum circuits and graphs.
    [Show full text]