{"Affiliation":[{"label":"Affiliation","value":"Non UBC","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Contributor":[{"label":"Contributor","value":"University of British Columbia. Department of Physics and Astronomy","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/contributor","classmap":"dpla:SourceResource","property":"dcterms:contributor"},"iri":"http:\/\/purl.org\/dc\/terms\/contributor","explain":"A Dublin Core Terms Property; An entity responsible for making contributions to the resource.; Examples of a Contributor include a person, an organization, or a service."},{"label":"Contributor","value":"Workshop on Quantum Algorithms, Computational Models and Foundations of Quantum Mechanics","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/contributor","classmap":"dpla:SourceResource","property":"dcterms:contributor"},"iri":"http:\/\/purl.org\/dc\/terms\/contributor","explain":"A Dublin Core Terms Property; An entity responsible for making contributions to the resource.; Examples of a Contributor include a person, an organization, or a service."},{"label":"Contributor","value":"Pacific Institute for the Mathematical Sciences","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/contributor","classmap":"dpla:SourceResource","property":"dcterms:contributor"},"iri":"http:\/\/purl.org\/dc\/terms\/contributor","explain":"A Dublin Core Terms Property; An entity responsible for making contributions to the resource.; Examples of a Contributor include a person, an organization, or a service."},{"label":"Contributor","value":"Summer School on Quantum Information (10th : 2010 : Vancouver, B.C.)","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/contributor","classmap":"dpla:SourceResource","property":"dcterms:contributor"},"iri":"http:\/\/purl.org\/dc\/terms\/contributor","explain":"A Dublin Core Terms Property; An entity responsible for making contributions to the resource.; Examples of a Contributor include a person, an organization, or a service."}],"Creator":[{"label":"Creator","value":"Altshuler, Boris","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2010-11-22T19:08:42Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"2010-07-23","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Description":[{"label":"Description","value":"Understanding NP-complete problems is a central topic in computer science. This is why adiabatic quantum optimization has attracted so much attention, as it provided a new approach to tackle NP-complete problems using a quantum computer. The efficiency of this approach is limited by small spectral gaps between the ground and excited states of the quantum computer's Hamiltonian. We will discuss the statistics of the gaps using the borrowed from the theory of quantum disordered systems. It turns out that due to a phenomenon similar to Anderson localization exponentially small gaps appear close to the end of the adiabatic algorithm for large random instances of NP-complete problems. We will present the quantitative analysis of the small spectral gaps and discuss possible consequence of this phenomenon on the adiabatic optimization paradigm.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/30065?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"Adiabatic quantum optimization and Anderson localization Boris Altshuler Columbia University Hari Krovi, J\u00e9r\u00e9mie Roland NEC Laboratories America Outline 1. Introduction: Adiabatic optimization for NP-complete problems 2. Exact Cover 3 problem 3. Anderson Localization on the hypercube 4. Small gaps in the Adiabatic approach to EC3 5. Exponentially large number of solutions (Knysh and Smelyanskiy, 2010) Cook \u2013 Levin theorem (1971): SAT problem is NP complete Computational Complexity Classes Quantum algorithms for NP complete problems ? Circuit quantum computer - no good ideas Adiabatic quantum computer - promising 4 Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. \u201cQuantum computation by adiabatic evolution\u201d, 2000. arXiv:quant-ph\/0001106. Edward Farhi, Jeffrey Goldstone, Sam Gutmann, Joshua Lapan, Andrew Lundgren, and Daniel Preda. \u201cA quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem\u201d. Science, 292:472\u2013475, 2001. Adiabatic quantum optimization \u2022 Total time T (slower is better) \u2022 Gap \u2206(t) (larger gap is better) Slowly varying \u2192 system stays at the ground state ( )H\u0302 t Probability of excitation depends on: Probability to stay in the ground state 2 min T \u2206 \u221d Problem: Find minimum of a function f(x) 1.Choose initial Hamiltonian H0 with known ground state 2. Change Hamiltonian to HP \u201cmatching\u201d f(x) T large enough \u21d2 measuring reveals the minimum ( ) 00 \u02c6 0\u02c6 \u02c6 \u02c61 \u02c6p p H tt tH t H H T T H t T =\uf8eb \uf8f6= \u2212 + =\uf8ec \uf8f7 =\uf8ed \uf8f8 Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. \u201cQuantum computation by adiabatic evolution\u201d, 2000. arXiv:quant-ph\/0001106. Edward Farhi, Jeffrey Goldstone, Sam Gutmann, Joshua Lapan, Andrew Lundgren, and Daniel Preda. \u201cA quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem\u201d. Science, 292:472\u2013475, 2001. Adiabatic quantum optimization How powerful is it? \u2022 It is quantum! Unstructured search in time (cf Grover) \u2022 It is universal for quantum computation [vanDam-Mosca-Vazirani'01,Roland-Cerf'02] [Aharonov et al.'05] Good, but what about NP-complete problems? \u2022 Numerical simulations: promising scaling 8How powerful is it? \u2022 It is quantum! Unstructured search in time (cf Grover) \u2022 It is universal for quantum computation [vanDam-Mosca-Vazirani'01,Roland-Cerf'02] [Aharonov et al.'05] \u2022 But exponentially small gap \u2022 for specifically designed hard instances [Znidaric-Horvat'06,Farhi et al.'08] [vanDam-Vazirani'03,Reichardt'04] But maybe typical gaps are only polynomial? Good, but what about NP-complete problems? \u2022 Numerical simulations: promising scaling [Farhi et al.'00,Hogg'03,Banyuls et al.'04,Young et al.'08] \u2022 for bad choice of initial Hamiltonian 1 in 3 SAT = exact cover problem 1,2,...,i N= 1i\u03c3 = \u00b1bitslaterals Ising spins 1,2,..., 1 , ,c c c c M i j k N = \u2264 \u2264 { }, ,c c ci j kclausesN M Clause c is satisfied if one of the three spins is down and other two are up or or Otherwise the clause is not satisfied Task: to satisfy all M clauses Definition , 11 1, c c ci j k \u03c3 \u03c3 \u03c3= \u2212 = = 1 1, , 1 c c ci j k \u03c3 \u03c3 \u03c3= = =\u2212 1, , 11 c c ci j k \u03c3 \u03c3 \u03c3= = \u2212= 1,2,...,i N= 1i\u03c3 = \u00b1bitslaterals Ising spins 1,2,...,c M= { }, ,c c ci j kclausesN M Clause c is satisfied if one of the three spins is down and other two are up. Otherwise the clause is not satisfied Task: to satisfy all M clauses Important: We will considering ensemble of randomly selected clauses, not some families of artificially designed instances 1 in 3 SAT = exact cover 3 (EC3) problem 1,2,...,i N= 1i\u03c3 = \u00b1bitslaterals Ising spins 1,2,...,c M= { }, ,c c ci j kclausesN M Clause c is satisfied if one of the three spins is down and other two are up. Otherwise the clause is not satisfied Task: to satisfy all M clauses Size of the problem: , , MN M N \u03b1\u2192\u221e \u2192\u221e \u2192 \u03b1 No solutions Few solutionsMany solutions c\u03b1 s\u03b10 Clusters of the solutions 1 in 3 SAT = exact cover 3 (EC3) problem Easy Hard Very hard 1,2,...,i N= 1i\u03c3 = \u00b1bitslaterals Ising spins 1,2,...,c M= { }, ,c c ci j kclausesN M , , MN M N \u03b1\u2192\u221e \u2192\u221e \u2192 clustering threshold satisfiability threshold c \u03b1 s\u03b1 0.546 < 0.644s\u03b1 < \u03b1 No solutions Few solutionsMany solutions c\u03b1 s\u03b10 Clusters of the solutions Easy Hard Very hard J. Raymond, A. Sportiello, & L. Zdeborova, Physical Review E 76, 011101 (2007). ( )2 , 1, 1 1, , 1 1 0 1, 1, 1 1 1 c c c c c c c c c c c c i j k i j k i j k i j k \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \uf8f1 \uf8fc= = = \uf8f4 \uf8f4 = = = \u21d4 + + \u2212 =\uf8f2 \uf8fd \uf8f4 \uf8f4= = \u2212 \u2212 = \u2212\uf8f3 \uf8fe ( )2 1 1 1 4 4 4 2 c c c M N i j k iji p i i j c i i j JM BH \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 = = \u2260 + + \u2212 = = \u2212 +\u2211 \u2211 \u2211 Otherwise ( )21 0c c ci j k\u03c3 \u03c3 \u03c3+ + \u2212 > Solutions and only solutions are zero energy ground states of the Hamiltonian { }i\u03c3 Bi \u2013 number of clauses, which involve spin i Jij \u2013 number of clauses, where both i and j participate ( ) { }( ) 2 1 1 0 1 1 4 4 4 2 \u02c6\u02c6 \u02c6 c c c M N i j k iji p i i j c i i j N x i i i JM BH H \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 = = \u2260 = + + \u2212 = = \u2212 + = \u2211 \u2211 \u2211 \u2211\uf072 Adiabatic Algorithm for EC3 Recipe: 1.Construct the Hamiltonian 2.Slowly change adiabatic parameter s from 0 to 1 { }( ) { }( ) ( ) { }( )0\u02c6 \u02c6\u02c6 \u02c6 \u02c6 1zs i p i iH sH s H\u03c3 \u03c3 \u03c3= + \u2212\uf072 \uf072 { }( ) { }( ) { }( )0 1\u02c6 \u02c6\u02c6 \u02c6 \u02c6 ;zi p i i s T tH H H s t\u03bb \u03c3 \u03c3 \u03bb \u03c3 \u03bb \u2212 \u2212 = + = = \uf072 \uf072 \u221e0 \u03bb ( ) { }( ) 2 0 1 1 1 \u02c6 \u02c6 \u02c6 1 \u02c6\u02c6\u02c6 \u02c6 \u02c6 \u02c6; 4 4 4 2 c c c z z zM N N i j k ijz z z xi p i i j i i c i i j i JM BH H \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 = = \u2260 = + + \u2212 = = \u2212 + =\u2211 \u2211 \u2211 \u2211\uf072 { }( ) { }( ) { }( )0 1\u02c6 \u02c6\u02c6 \u02c6 \u02c6 ;zi p i i sH H H s\u03bb \u03c3 \u03c3 \u03bb \u03c3 \u03bb \u2212 = + = \uf072 \uf072 Ising model (determined on a graph ) in a random parallel and a uniform perpendicular field \u03bb Adiabatic Algorithm for EC3 determines a site of N-dimensional hypercube ( ) { }( )2 0 1 1 1 \u02c6\u02c6 \u02c6\u02c6 \u02c6 \u02c6 \u02c6 \u02c6 \u02c6 \u02c61 ; c c c M N N z z z z z z x p i j k i i ij i j i i c i i j i H B J H\u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 = = \u2260 = = + + \u2212 = + =\u2211 \u2211 \u2211 \u2211\uf072 { }( ) { }( ) { }( )0 1\u02c6 \u02c6\u02c6 \u02c6 \u02c6 ;zi p i i sH H H s\u03bb \u03c3 \u03c3 \u03bb \u03c3 \u03bb \u2212 = + = \uf072 \uf072 Another way of thinking: { }i\u03c3 { }( )p iH \u03c3 onsite energy { }( )0 1 \u02c6\u02c6 \u02c6 \u02c6 \u02c6 \u02c6 N x x i i i H\u03bb \u03c3 \u03bb \u03c3 \u03c3 \u03c3 \u03c3+ \u2212 = = = +\u2211\uf072 hoping between nearest neighbors Adiabatic Algorithm for exact cover The simplest example: }3 qubits 1 clause \u21d2 3d cube, 3 solutions ( )1,1,1 ( )1,1, 1\u2212 ( )1,1,1\u2212 ( )1, 1,1\u2212 \u2212 ( )1, 1,1\u2212 ( )1, 1, 1\u2212 \u2212 \u2212 ( )1,1, 1\u2212 \u2212 ( )1, 1, 1\u2212 \u2212 1\u03b5 = 1\u03b5 = 1\u03b5 = 1\u03b5 = 4\u03b5 = determines a site of N-dimensional hypercube ( ) { }( )2 0 1 1 1 \u02c6\u02c6 \u02c6\u02c6 \u02c6 \u02c6 \u02c6 \u02c6 \u02c6 \u02c61 ; c c c M N N z z z z z z x p i j k i i ij i j i i c i i j i H B J H\u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 = = \u2260 = = + + \u2212 = + =\u2211 \u2211 \u2211 \u2211\uf072 { }( ) { }( ) { }( )0 1\u02c6 \u02c6\u02c6 \u02c6 \u02c6 ;zi p i i sH H H s\u03bb \u03c3 \u03c3 \u03bb \u03c3 \u03bb \u2212 = + = \uf072 \uf072 Another way of thinking: { }i\u03c3 { }( )p iH \u03c3 onsite energy { }( )0 1 \u02c6\u02c6 \u02c6 \u02c6 \u02c6 \u02c6 N x x i i i H\u03bb \u03c3 \u03bb \u03c3 \u03c3 \u03c3 \u03c3+ \u2212 = = = +\u2211\uf072 hoping between nearest neighbors Adiabatic Algorithm for exact cover Anderson model for Localization on N-dimensional cube Anderson Model \u2022 Lattice - tight binding model \u2022 Onsite energies \u03b5i - random \u2022 Hopping matrix elements Iijj i Iij Iij ={ I i and j are nearest neighbors0 otherwise-W < \u03b5i Ic Insulator All eigenstates are localized Localization length \u03b6loc Metal There appear states extended all over the whole system Anderson Transition extended localized 0 \u02c6 \u02c6 \u02c6H H V= + Conventional Anderson Model Basis: ,i i \u2211= i i iiH \u03b50\u02c6 \u2211 = = .., \u02c6 nnji jiIV Hamiltonian: \u2022one \u201cparticle\u201d, \u2022one level per site, \u2022onsite disorder \u2022nearest neighbor hoping labels sites ( ) { }( )2 0 1 1 1 \u02c6\u02c6 \u02c6\u02c6 \u02c6 \u02c6 \u02c6 \u02c6 \u02c6 \u02c61 ; c c c M N N z z z z z z x p i j k i i ij i j i i c i i j i H B J H\u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 \u03c3 = = \u2260 = = + + \u2212 = + =\u2211 \u2211 \u2211 \u2211\uf072 Adiabatic Quantum Algorithm for 1 in 3 SAT { }( ) { }( ) { }( )0 1\u02c6 \u02c6\u02c6 \u02c6 \u02c6 ;zi p i i sH H H s\u03bb \u03c3 \u03c3 \u03bb \u03c3 \u03bb \u2212 = + = \uf072 \uf072 Anderson Model on N-dimensional cube Usually: # of dimensions system linear size d const\u2192 L \u2192\u221e Here: # of dimensions system linear size d N= \u2192\u221e 1L = 6-dimensional cube 9-dimensional cube \u221e0 \u03bb extendedlocalized c\u03bb { }( ) { }( ) { }( )0\u02c6 \u02c6\u02c6 \u02c6 \u02c6 zi p i iH H H\u03bb \u03c3 \u03c3 \u03bb \u03c3= +\uf072 \uf072 Disorder W Zharekeschev & Kramer. Exact diagonalization of the Anderson model 1~ \u03bb\u2212 Avoided crossing gaps \u2022extended large \u2022localized small Significant amplitude Exponentially small amplitude Adiabatic transition: 1. Extended states \u2013 can be performed quickly 2. Localized states \u2013 need exponentially long time Otherwise \u2013 nonadiabatic Landau-Zienner transition \u03bb \u03bb< \uf025\u03bb \u03bb> \uf025 Localized states Exponentially long tunneling times Exponentially small anticrossing gaps localized extended 1) Anderson localization would imply As the size of the problem N increases 2) Anti-crossings between solutions and not-solutions \u2013 as long as exceeds we haveFor The algorithm fails (stuck in a local minimum) Our result: \u03bb en er gy \u03bb\u2217c\u03bb ( )1 logc N\u03bb = \u2126 \u03bb ( ) 1 8CN\u03bb \u2212\u2217 \u2248 ( ) 18cN C\u03bb \u2212 > c\u03bb \u03bb\u2217 < { }( ) { }( ) { }( )0\u02c6 \u02c6\u02c6 \u02c6 \u02c6 zi p i iH H H\u03bb \u03c3 \u03c3 \u03bb \u03c3= +\uf072 \uf072 2 1 E\u03b4 1 0 E \u03bb 3. Let us add one more clause, which is satisfied by but not by 1 0 When the gaps decrease even quicker than exponentially N \u2192\u221e 2. For \u03b1 is close to \u03b1s there typically are several solutions separated by distances . Consider two. 1. Hamiltonian is integrable: it commutes with all . Its states thus can be degenerated. These degeneracies should split at finite \u03bb since is non-integrable { }( )\u02c6 \u02c6 zp iH \u03c3 \u02c6 zi\u03c3 { }( )\u02c6\u02c6 iH\u03bb \u03c3\uf072 ( )O N 10 E \u03bb 0\u21d2 1\u21d2 { }( ) { }( ) { }( )0\u02c6 \u02c6\u02c6 \u02c6 \u02c6 zi p i iH H H\u03bb \u03c3 \u03c3 \u03bb \u03c3= +\uf072 \uf072 When the gaps decrease even quicker than exponentially N \u2192\u221e 3. Let us add one more clause, which is satisfied by but not by 1 0 1. Hamiltonian is integrable: it commutes with all . Its states thus can be degenerated. These degeneracies should be split by finite \u03bb in non-integrable 2. For \u03b1 is close to \u03b1s there typically are several solutions separated by distances . Consider two. { }( )\u02c6 \u02c6 zp iH \u03c3 \u02c6 zi\u03c3 { }( )\u02c6\u02c6 iH\u03bb \u03c3\uf072 2 1 ( )O N 2 1 1 0 E \u03bb 0\u21d2 1\u21d2 2 1 E\u03b4 1 0 E \u03bb Q1: Is the splitting big enough for to remain the ground state at largeE\u03b4 0\u03bb ? Q2: How big would be the anticrossing gap? ( ) ( )2k k k E N C \u03b1\u03b1 \u03bb \u03bb= \u2211 Q1: Is the splitting big enough for to remain the ground state at largeE\u03b4 0\u03bb ? Perturbation theory in \u03bb N M const N \u03b1 \u2192\u221e \u2192 = } ( )E N\u03b1 \u03bb \u221d Cluster expansion: ~N terms of order 1 1. is exactly the same for all states, i.e. for all solutions. In the leading order ( ) 1C \u03b1 ( )0 0E\u03b1 \u03bb = = 4E\u03b4 \u03bb\u221d 2. In each order of the perturbation theory a sum of terms with random signs. E\u03b4 ( )O N In the leading order in \u03bb 4E N\u03b4 \u03bb\u221d ( ) 2 4 64 6 4 6 , ... , ,... E N N \u03b4 \u03bb \u03bb \u03b4 \u03bb \u03b4 \u03b4 \u03b4 \uf8ee \uf8f9 = + +\uf8f0 \uf8fb \u221d 4\u03b4 6\u03b4 Numerical Simulations ( ) ( ) 1 81E CN\u03b4 \u03bb \u03bb \u2212\u2265 \u21d2 \u2265 In the leading order in \u03bb 4E N\u03b4 \u03bb\u221d Q1: Is the splitting big enough for to remain the ground state at finite E\u03b4 0 \u03bb ? Q1.1: How big is the interval in , where perturbation theory is valid\u03bb ? { }( ) { }( ) { }( )0 \u02c6\u02c6 \u02c6 \u02c6 \u02c6 z i p i i H H H \u03bb \u03c3 \u03c3 \u03bb \u03c3 = + \uf072 \uf072 Q1.1: How big is the interval in , where perturbation theory is valid\u03bb ? A1.1:1.Perturbation theory = locator expansion works as long as -Anderson localization !c\u03bb \u03bb< Cayley tree Anderson model W,I K \u2013 branching # # lnc WI K K = \u2248Q1.1: How big is the interval in , where perturbation theory is valid\u03bb ? 2.N-dimensional cube Cayley tree with brunching number K=N. no loopsalmost no loops 3. Abou-Chacra, Anderson, Thouless; PRL, 1973 1# ln lnc c WI O K K N \u03bb \uf8eb \uf8f6= \u21d2 = \uf8ec \uf8f7 \uf8ed \uf8f8 first extended state appears A1.1:1.Perturbation theory = locator expansion works as long as -Anderson localization !c\u03bb \u03bb< 1# ln lnc c WI O K K N \u03bb \uf8eb \uf8f6= \u21d2 = \uf8ec \uf8f7 \uf8ed \uf8f8 first extended state appears, i.e. it is a strong underestimation ( )1 81lnc O O NN\u03bb \u03bb \u2212 \u2217 \uf8eb \uf8f6= >> =\uf8ec \uf8f7 \uf8ed \uf8f8 Important: Perturbation theory is valid starting with \u03bb \u03bb\u2217>> A1.1:1.Perturbation theory = locator expansion works as long as -Anderson localization !c\u03bb \u03bb< Q2:How big is the anticrossing gap ? ( ) # # 1 8 ~ ~ exp # ln ~ # N NE E N N e N \u03b4 \u03bb \u03b4 \u03bb \u2212 \u2212 \uf8f1 \uf8fc \u21d2 \u2212 <<\uf8f2 \uf8fd \uf8f3 \uf8fe Adiabatic quantum computer badly fails at large enough N 4~ 10N N \u2217>c\u03bb \u03bb< Existing classical algorithms for solving 1 in 3 SAT problem work for ( ) 33 4 10N < \u00f7 \u00d7 The arguments are robust: { }( )p iH \u03c3 onsite energy { }( )0 \u02c6\u02c6 iH \u03c3\uf072 hoping between the neighboring sites of the hypercube N-dimensional hypercube }Correct for any Adiabatic optimization scheme for any problem This is not necessary ! For Anderson Localization it is sufficient that is local, i.e. contains only products of finite number of spins. Under this condition the arguments are valid for any adiabatic path in the Hamiltonian space { }( )0 \u02c6\u02c6 iH \u03c3\uf072 Sergey Knysh and Vadim Smelyanskiy \u201cOn the relevance of avoided crossings away from quantum critical point to the complexity of quantum adiabatic algorithm\u201d, arXiv:1005.3011v1[quant-ph] \u2022Away from \u03b1s the number of solutions is exponential: \u2022 For a typical solution \u2022 However the minimum over S0 solutions \u2022 # of states with energy 1 is \u2022 Energy difference between the true ground state and the E=1 state at finite \u03bb is ( )0 ~ exp 0sS N \u03b1 \u03b1\u03b7 \u03b7 \u2192 \u2212\uf8e7\uf8e7\uf8e7\uf8e7\u2192 ( )2 8~E E N\u03bb\u2212 ( )2 8 8 2min 0~ log ~E E N S N\u03bb \u03b7\u03bb\u2212 1 0~S NS ( )1 4 1 2min min ~ 1 logE E O N\u03bb \u03b7\u2212\u2212 \u2212 + ( ) 1 4 1 8log N\u03bb \u03b7\u2212\u2217> ( ) 1 4 1 8log N\u03bb \u03b7\u2212\u2217> Formally small in the limit, but \u2026 , 1N \u03b7\u2192\u221e << Moreover: 1.Under reasonable restrictions on the allowed sequences a typical sequence in the ensemble can have few solutions or even a unique one and be at the same time hard to solve. Zdeborova & Mezard, 2008, Krzakala & Zdeborova,2009 Zdeborova to be published However the exponentially small gap requires only that \u03bb is small ( ) 1 4 1 8log N\u03bb \u03b7\u2212\u2217> Formally small in the limit, but \u2026 , 1N \u03b7\u2192\u221e << Moreover: 2. Upper bound: given the evolution time T the state of the system would be a linear combination of low energy eigenstates separated by more than log T. Number of \u201cattempts\u201d is thus ~T rather than exp(\u03b7N). Provided that we return to more or less previous estimation. log 0T N \u2192 However the exponentially small gap requires only that \u03bb is small Conclusion Original idea of adiabatic quantum computation will not work Hopes \u2022Sampling different trajectories?Farhi, Goldstone, Gosset, Gutmann, Meyer, & Shor, arXiv:1004.5127 \u2022Maybe the delocalized ground state at finite \u03bb contains information that can speed up the classical algorithm? \u2022Large number of the solutions? \u2022Probability to find a solution. \u2022. . .","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. There is no restriction on the nature of this information, e.g., it could be plain text, hypertext, or an image; it could be a definition, information about the scope of a concept, editorial information, or any other type of information."}],"Genre":[{"label":"Genre","value":"Presentation","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","classmap":"dpla:SourceResource","property":"edm:hasType"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","explain":"A Europeana Data Model Property; This property relates a resource with the concepts it belongs to in a suitable type system such as MIME or any thesaurus that captures categories of objects in a given field. It does NOT capture aboutness"}],"IsShownAt":[{"label":"IsShownAt","value":"10.14288\/1.0103155","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","classmap":"edm:WebResource","property":"edm:isShownAt"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","explain":"A Europeana Data Model Property; An unambiguous URL reference to the digital object on the provider\u2019s website in its full information context."}],"Language":[{"label":"Language","value":"eng","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/language","classmap":"dpla:SourceResource","property":"dcterms:language"},"iri":"http:\/\/purl.org\/dc\/terms\/language","explain":"A Dublin Core Terms Property; A language of the resource.; Recommended best practice is to use a controlled vocabulary such as RFC 4646 [RFC4646]."}],"PeerReviewStatus":[{"label":"PeerReviewStatus","value":"Unreviewed","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#peerReviewStatus","classmap":"oc:PublicationDescription","property":"oc:peerReviewStatus"},"iri":"https:\/\/open.library.ubc.ca\/terms#peerReviewStatus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies whether or not the resource is peer-reviewed. Applies to published resources only."}],"Provider":[{"label":"Provider","value":"Vancouver : University of British Columbia Library","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","classmap":"ore:Aggregation","property":"edm:provider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","explain":"A Europeana Data Model Property; The name or identifier of the organization who delivers data directly to an aggregation service (e.g. Europeana)"}],"Rights":[{"label":"Rights","value":"Attribution-NonCommercial-NoDerivatives 4.0 International","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/rights","classmap":"edm:WebResource","property":"dcterms:rights"},"iri":"http:\/\/purl.org\/dc\/terms\/rights","explain":"A Dublin Core Terms Property; Information about rights held in and over the resource.; Typically, rights information includes a statement about various property rights associated with the resource, including intellectual property rights."}],"RightsURI":[{"label":"RightsURI","value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#rightsURI","classmap":"oc:PublicationDescription","property":"oc:rightsURI"},"iri":"https:\/\/open.library.ubc.ca\/terms#rightsURI","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the Creative Commons license url."}],"ScholarlyLevel":[{"label":"ScholarlyLevel","value":"Faculty","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","classmap":"oc:PublicationDescription","property":"oc:scholarLevel"},"iri":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the scholarly level of the author(s)\/creator(s)."}],"Title":[{"label":"Title","value":"Adiabatic Quantum Optimization and Anderson Localization","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/title","classmap":"dpla:SourceResource","property":"dcterms:title"},"iri":"http:\/\/purl.org\/dc\/terms\/title","explain":"A Dublin Core Terms Property; The name given to the resource."}],"Type":[{"label":"Type","value":"Text","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/type","classmap":"dpla:SourceResource","property":"dcterms:type"},"iri":"http:\/\/purl.org\/dc\/terms\/type","explain":"A Dublin Core Terms Property; The nature or genre of the resource.; Recommended best practice is to use a controlled vocabulary such as the DCMI Type Vocabulary [DCMITYPE]. To describe the file format, physical medium, or dimensions of the resource, use the Format element."},{"label":"Type","value":"Moving Image","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/type","classmap":"dpla:SourceResource","property":"dcterms:type"},"iri":"http:\/\/purl.org\/dc\/terms\/type","explain":"A Dublin Core Terms Property; The nature or genre of the resource.; Recommended best practice is to use a controlled vocabulary such as the DCMI Type Vocabulary [DCMITYPE]. To describe the file format, physical medium, or dimensions of the resource, use the Format element."}],"URI":[{"label":"URI","value":"http:\/\/hdl.handle.net\/2429\/30065","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#identifierURI","classmap":"oc:PublicationDescription","property":"oc:identifierURI"},"iri":"https:\/\/open.library.ubc.ca\/terms#identifierURI","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the handle for item record."}],"SortDate":[{"label":"Sort Date","value":"2010-07-23 AD","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/date","classmap":"oc:InternalResource","property":"dcterms:date"},"iri":"http:\/\/purl.org\/dc\/terms\/date","explain":"A Dublin Core Elements Property; A point or period of time associated with an event in the lifecycle of the resource.; Date may be used to express temporal information at any level of granularity. Recommended best practice is to use an encoding scheme, such as the W3CDTF profile of ISO 8601 [W3CDTF].; A point or period of time associated with an event in the lifecycle of the resource.; Date may be used to express temporal information at any level of granularity. Recommended best practice is to use an encoding scheme, such as the W3CDTF profile of ISO 8601 [W3CDTF]."}]}