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quant-ph9910066.txt |@word concept:3 homomorphism:2 consider:1 remarkable:2 induces:2 results:6 leads:1 scalar:4 commute:1 generalize:2 grateful:1 consists:1 follow:1 expressions:1 imbedding:10 gi:3 compact:2 whose:4 depend:3 referring:1 iii:6 concerned:1 equivalence:2 finally:2 under:5 phil:1 norm:10 p2:21 belonging:1 abelian:4 orthogonal:6 very:2 istituto:1 focus:2 electron:5 wishes:1 every:2 summarized:1 affect:1 χi:3 condition:1 dim:12 y1:2 turns:1 d2:3 deals:1 discrete:14 vector:5 small:1 corollary:1 shows:6 noted:1 zn:6 renormalized:3 measurable:1 approximation:3 go:4 imply:1 rigorous:2 further:2 dynamics:1 odd:1 canonical:9 spectral:3 established:1 what:3 appear:1 richard:1 fact:5 section:4 selected:1 brief:2 expressed:2 reasoning:1 varadhan:1 version:1 goes:4 obvious:2 construction:1 congruences:1 consisting:1 y2:3 relationships:1 led:2 respectively:4 proc:2 drew:1 ranges:1 following:8 explore:1 groups:1 others:5 november:1 appears:1 calculate:1 obtained:3 remark:8 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quant-ph9910069.txt |@word logarithms:1 consider:3 remarkable:1 global:1 focus:1 leads:2 eds:3 proposition:4 computation:9 follow:1 eζ:1 calculate:8 young:1 noncompact:1 manifolds:1 th:1 under:1 include:1 worth:1 very:2 corollary:4 wishes:1 nieto:1 hamiltonian:4 irreducible:5 list:1 leave:1 trx:1 eigenvalues:1 optics:1 degeneracy:1 2k3:1 cosh:24 fold:2 further:1 theories:4 eλa:7 canonical:3 aim:1 section:1 near:1 above:1 method:1 respectively:1 here:1 sinh:24 address:1 path:1 equations:1 arena:1 study:1 action:1 useful:1 projection:1 family:2 compare:1 unit:1 use:2 proceed:1 would:2 give:4 next:3 comparing:1 popescu:1 therefore:2 m2:2 type:3 more:1 becomes:1 operators:1 curvature:5 particular:1 known:1 given:7 balakrishnan:1 must:2 springer:1 oscillator:1 oct:1 fock:1 learn:1 following:1 making:2 example:5 f2:1 challenging:1 stated:1 predict:1 µa2:2 suggestions:1 states:6 degenerate:2 verlag:1 information:1 acts:1 utilize:1 write:1 far:1 classifying:1 parameter:2 simplicity:1 algebra:2 may:1 ζk:1 isospectral:3 applications:1 natural:2 outline:1 eµk:3 chern:3 explicit:2 su:2 so:7 representation:1 order:1 over:2 paper:3 including:1 japan:1 its:1 before:1 aκ:1 group:3 thank:1 lo:1 forms:2 mail:1 main:2 into:1 overcome:1 prime:1 non:2 good:2 uv0:1 rasetti:5 introduce:1 vect:1 they:2 now:9 easily:1 term:1 university:1 level:1 funahashi:1 each:1 crossing:1 side:1 holonomy:2 ek:7 ζa:1 idea:3 related:1 breakthrough:1 operator:4 toda:1 quantum:14 our:10 matrix:1 space:3 since:6 rev:1 denotes:1 formula:4 induced:2 math:2 sinh2:2 after:1 linear:1 diagonal:1 difficulty:2 iop:1 skagerstam:1 kazuyuki:1 geometry:1 tolerant:1 introduction:1 ltd:1 assign:1 first:2 separable:1 meet:1 number:1 proposing:1 open:1 wilczek:1 city:1 little:2 πe:4 ordering:1 their:5 calculations:2 siam:1 1m:2 gives:2 believe:1 target:3 stm:12 namely:1 annihilation:1 nakahara:1 future:3 result:2 zanardi:5 generalization:4 lie:2 k3:5 sciences:1 generalized:4 algorithms:1 degenerated:1 formulas:2 phase:2 class:2 seshadri:1 kf:6 cover:1 exp:1 physics:3 shor:2 notations:1 gauge:5 solutions:1 find:1 principal:1 based:1 just:1 parameters:2 explain:1 written:1 lemma:2 only:1 local:2 preliminaries:2 abelian:3 his:1 means:2 da:2 dv:6 aλ:9 du:2 dr:1 dp:4 disentangling:4 differential:1 factorization:1 qc:2 set:6 klauder:1 connections:2 see:8 computer:2 differentiable:1 close:1 dµ:9 theoretic:2 subject:3 phases:2 state:5 µk:3 progress:1 geometric:6 jp:1 however:5 topology:1 many:1 etc:2 connection:12 berry:4 holonomic:1 comment:2 cm:1 simple:1 harmonic:1 dimensional:2 anti:1 polynomial:1 moreover:1 cu:1 publishing:1 basis:2 been:1 arxiv:1 ph:8 abstract:1 define:2 obtained:3 quantities:1 coherent:8 c2:5 spin:1 those:1 case:1 tanh:7 truax:1 discrete:1 will:1 while:1 fault:1 squeezed:3 yokohama:3 unitary:5 simons:1 vm:1 author:2 make:1 parts:2 quant:8 widely:1 several:1 higher:1 dimc:1 calculation:4 singapore:3 hand:2 v0:4 v1:1 infinite:1 acknowledgement:1 model:4 stating:1 left:1 manifold:3 proposed:1 obtain:2 actions:1 λa:7 immediately:1 campbell:1 λj:1 λm:5 note:4 readers:1 character:1 other:1 easy:4 gave:1 misprints:1 decomposed:1 possible:3 know:2 vacuum:4 using:3 helpful:1 like:2 follows:1 adiabatic:7 become:1 terminology:3 mathematical:5 because:3 deal:1 stiefel:1 creation:1 some:3 back:2 sight:1 comments:1 does:2 scientific:3 power:1 bo:1 hilbert:1 tanh2:3 stage:1 world:3 recognized:1 compatible:1 map:4 important:2 preskill:1 insufficient:1 introduced:1 spiller:1 perelomov:3 h0:5 hermitian:1 right:1 references:1 elementary:1 hep:1 log:1 there:2 start:1 physicists:1 a55:1 suzuki:1 forn:1 form:12 shapere:1 construction:1 aµ:9 filed:1 pull:2 applying:1 whether:1 us:6 ωn:1 called:2 aa:1 ac:1 computing:1 defined:2 curvatures:1 universal:4 aj:1 covers:1 fujii:4 elog:2 dλ:17 grassmann:2 no:1 when:1 detail:1 field:2 interested:1 department:1 grm:9 briefly:1 intend:1 setting:1 bundle:4 hausdorff:1 generated:1 a2:1 bounded:1 baker:1 time:1 once:1
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quant-ph9910070.txt |@word inverse:1 monterey:1 desired:1 code:1 sci:1 consider:13 springer:2 squared:1 varying:1 brownian:1 results:3 leads:2 outputs:1 entropy:1 amendola:1 consists:1 gn:4 expressions:1 research:2 viceversa:1 devoted:2 whose:3 c0:2 oscillating:3 paris:1 technique:2 contained:1 reverting:1 presents:3 vigier:3 finally:3 monotonically:2 promoting:1 those:2 th:1 stable:3 stochastic:29 nodes:8 necessary:2 case:17 already:3 straightforward:3 g0:2 very:6 observables:1 kω:1 rise:1 dell:2 wave:28 every:16 experimentally:1 manageable:1 logarithmically:1 semiaxes:2 condition:11 drag:1 entire:1 obvious:2 level:1 turns:1 solution:28 t0:23 t1:4 reminding:1 solved:5 marra:2 vector:1 phase:11 progress:1 signficant:1 relation:5 markov:2 added:1 inserting:1 work:1 sturm:6 consistent:1 guidelines:1 among:1 measurable:1 ornstein:2 second:1 epac:1 design:1 value:2 equal:1 notwithstanding:1 further:4 dynamics:11 correspond:1 canonical:1 even:3 established:1 what:4 plays:1 sub:6 rate:1 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quant-ph9910071.txt |@word aguilera:1 limited:3 consider:1 magnetic:3 results:23 leads:1 roy:1 grateful:1 c1:1 increase:1 spector:1 calculate:1 graph:1 technique:2 charge:1 present:1 under:2 anharmonic:2 worth:1 case:11 worse:1 straightforward:1 very:4 wave:3 witten:1 little:1 exact:14 condition:6 treating:1 dim:2 mc:2 vibrational:1 called:1 solution:2 correct:2 large:4 small:3 eigenvalues:11 noted:2 findings:1 estimates:1 direct:1 approximation:5 perspective:1 chen:1 ghatak:3 chem:4 linear:1 even:4 what:1 stands:1 giving:1 section:5 uniform:2 indian:1 referees:1 new:1 method:16 contrast:1 bi:12 degree:1 respectively:7 error:1 modify:1 here:4 atoms:2 behaviour:1 others:1 ley:2 quite:2 equations:2 kawazoe:1 obtained:9 roychoudhury:2 antisymmetric:9 institute:1 study:6 studied:2 commonly:1 suitable:3 makes:1 useful:4 explained:2 natural:1 highly:1 viz:5 apply:4 establish:1 unit:1 derivatives:2 use:1 asymptotic:4 working:1 found:1 france:1 few:1 ψi:5 therefore:1 taken:2 type:2 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quant-ph9910074.txt |@word represent:1 concept:1 partial:1 consider:1 scl:12 interference:1 demand:1 focus:1 illustrated:2 computation:1 edu:1 consists:1 follow:3 clarified:1 primer:1 localized:1 payoff:1 whose:1 dimensionality:1 const:4 calculate:1 smailagic:1 mutatis:1 deterministic:1 former:1 tn:4 finally:4 tj:1 those:3 ti:15 connecting:7 worth:1 case:10 amplitudes:4 string:2 without:1 very:2 than:2 emphasizes:1 wave:10 novel:2 orientations:1 polytechnic:1 difference:2 fractal:2 reading:1 condition:3 identity:1 entire:2 randomness:1 solves:1 level:2 positions:1 having:2 t2:4 solution:1 t0:2 t1:3 discrete:2 convenience:1 differential:1 progress:1 succession:1 rev:5 relation:1 lapse:3 configurations:1 possibility:1 preassigned:1 squared:1 neighborhood:1 approximation:1 aurilia:6 cost:2 perspective:1 further:2 dynamics:2 correspond:1 substitute:1 canonical:1 majors:2 what:2 plays:1 apparatus:1 section:5 selected:1 brief:1 wavelike:1 d3:6 version:1 above:13 new:13 net:1 appeal:1 ever:1 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quant-ph9910080.txt |@word entanglements:1 holds:1 wigner:5 ϕk:1 rev:4 λq:1 µp:1 russian:1 quadrature:1 offdiagonal:2 thanks:1 husimi:2 λk:1 lebedev:1 satisfies:1 before:1 dimensionality:1 projection:1 density:25 chosen:1 tr:1 passage:1 only:3 permanent:1 location:1 factor:1 pad:1 must:1 λj:1 oct:1 texas:2 odinger:1 real:1 mixture:1 return:1 greater:1 di:6 dependent:3 overall:1 expectation:2 introduce:2 federico:2 interior:1 ϕn:1 they:1 ϕj:2 oxford:1 using:1 matrices:3 dp:1 dq:1 term:3 necessary:1 like:1 particle:1 tra:1 alternatively:1 university:1 arxiv:1 sin:8 leave:1 vector:2 phase:8 sezione:1 sesquilinear:1 universit:2 optics:1 kano:2 set:1 ei:1 constructing:1 ed:1 sin2:6 principles:2 relative:2 elements:2 us:1 w1:1 convex:1 quantum:8 eiϕ:3 operation:2 nonnegative:1 bilinear:1 orthonormal:1 phases:2 fisiche:2 matrix:4 space:2 support:1 sum:2 since:1 interference:2 marmo:1 expressed:1 research:1 naturwissenschaften:1 representatives:1 state:8 istituto:1 get:2 proc:2 math:2 foundation:1 terms:5 linear:1 ρab:7 fisica:1 here:2 diagonal:3 nazionale:1 add:1 cos2:6 address:1 along:1 includes:1 hilbert:2 obtained:1 scienze:2 napoli:4 institute:1 vectors:3 introduces:1 purify:1 called:1 probabilities:1 impure:6 4th:1 eigenvalues:1 mixed:2 retain:1 austin:1 other:2 involves:1 tomogram:1 considerations:1 soc:2 formulated:1 convolution:1 probability:2 extra:1 variables:1 corresponds:2 rank:2 system:2 expect:1 multiplication:3 square:1 references:1 determine:1 operator:7 proportional:1 tombesi:1 selfadjoint:1 use:1 acknowledgements:1 phil:1 additional:2 usa:1 addition:9 dipartimento:2 there:4 three:1 construct:1 while:2 call:1 structure:1 way:1 basic:1 ph:1 was:1 partial:2 lett:2 function:4 constructed:1 corresponding:1 elsewhere:1 form:3 composite:1 nonzero:1 operators:11 cos:7 but:4 direct:1 coherent:2 construction:1 dirac:1 normalized:2 angles:2 criterion:1 schr:1 irrelevant:1 than:1 those:1 center:1 ρa:5 states:5 kind:2 ρb:5 trb:1 up:1 value:2 will:1 symbolically:1 determined:1 superposed:1 mehta:1 1d:1 ko1:1 distribution:1 root:1 russia:1 w2:1 mechanics:4 ab:5 vogel:1 jpn:1 defined:1 abstract:1 thus:2 annal:1 denote:1 ii:2 tomography:2 purity:1 need:3 i12:4 any:1 zaccaria:1 physical:1 information:1 functions:1 traditionally:1 italy:1 recovery:1 depending:1 tomograms:3 discussion:1 means:1 same:5 tells:1 note:3 how:1 unity:1 parts:1 quant:1 generalized:1 pure:12 department:1 ray:2 parameter:2 moscow:1 mostra:1 eiα:1 product:3 used:1 entanglement:3 superposition:2 trace:1 may:2 sudarshan:2 shown:1 nonorthogonal:1 parametrized:1 hence:1 involving:1 hospitality:1 variable:1 angle:1 risken:1 law:8 unless:1 normalization:1 man:2 implies:1 dynamical:2 nucleare:1 computations:1 whole:1 required:1 ko:1 correspond:1 together:1 purification:6 mancini:1 points:1 so:1 principle:1 subsystems:1 camb:1 physics:2 having:1
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quant-ph9910081.txt |@word limited:1 dynasty:5 entropy:3 asymp:1 consists:2 localized:4 looking:1 calculate:2 tween:1 swap:2 under:2 alternating:3 rise:1 connects:1 experimentally:3 convexity:2 induction:8 vector:1 force:1 29th:1 second:6 rigorous:3 qualitatively:2 estimated:2 machines:1 even:7 errors:2 pawel:1 above:20 resilient:2 new:4 ever:1 consisting:1 evolving:1 simultaneously:1 never:1 here:19 represented:1 path:15 logarithmic:3 contraction:1 michal:1 rigorously:1 study:6 stressed:1 suitable:1 projection:1 explained:4 julia:1 replace:2 unit:3 would:11 call:1 therefore:14 type:2 until:2 relax:1 composite:3 nauki:1 phone:1 berlin:1 hold:5 springer:2 ma:2 ticle:3 work:3 macroscopically:2 example:14 impose:1 give:4 bennett:3 want:6 david:3 end:2 provide:1 machine:1 how:6 answer:1 gate:7 ancestor:3 description:4 turing:2 after:14 gottesman:1 summarize:1 law:1 unentangled:3 elusive:1 third:1 complexity:1 maintain:2 algorithms:3 correlation:7 operate:1 order:11 operations:7 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quant-ph9910082.txt |@word consider:1 similarity:1 rev:10 both:3 results:3 leads:3 roy:1 scalar:7 generalize:1 bohm:2 consists:1 εn:5 particles:4 inverse:1 terms:13 concerned:1 program:1 marchand:3 th:2 under:5 stochastic:3 norm:2 prigogine:2 p2:10 conjecture:3 orthogonal:2 very:1 kω:1 translations:3 connects:1 wave:12 wishes:1 every:1 resonant:8 experimentally:1 difference:1 exact:4 condition:2 four:2 school:1 hamiltonian:5 notes:1 solution:2 occurs:1 deals:1 admissible:1 vector:8 manifestly:1 iassns:1 shows:1 equation:2 upper:1 mn:5 tel:1 foliating:1 dealing:1 estimates:1 direct:5 second:1 canonical:1 spectral:10 what:2 appear:1 clock:1 section:2 experiment:1 poles:1 subspaces:12 boundary:1 new:3 method:2 satisfy:2 outgoing:7 full:1 e731:1 lee:15 degree:1 respectively:3 fock:1 component:1 here:5 friedrichs:6 objects:1 galilean:5 represented:4 const:1 along:2 equations:2 appears:1 change:1 box:1 remark:2 incoming:6 parameters:1 shift:3 study:3 studied:2 chemical:1 resulting:2 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quant-ph9910085.txt |@word represent:1 monte:10 customary:5 wavevectors:1 rev:13 varying:3 amplifier:1 results:9 four:3 eds:1 quadrature:6 entropy:1 per:3 parameterization:2 consists:1 expressions:1 denotes:6 causes:1 particles:1 amplification:2 depend:3 technique:1 tr:6 finally:3 efficiencies:1 those:1 γl:5 analogous:2 worth:1 supported:2 orthogonal:4 very:3 frequencies:2 focus:1 figs:2 experimentally:2 difference:2 exact:1 northwestern:2 smithey:1 presented:2 evaluation:1 scalar:1 large:4 convenience:2 small:2 optics:1 iψ0:3 zl:5 zm:3 averaged:2 slower:1 bo:2 vasilyev:1 sin2:2 za:1 cosh:1 feasibility:1 zero:2 evaluated:4 greenberger:3 carlo:10 4x2:1 dψ1:1 dψ0:1 established:1 what:1 errors:8 section:1 selected:1 analogously:1 current:1 dunn:1 fock:1 reconstruction:4 new:2 increasing:1 method:24 satisfy:1 κm:4 dψl:1 behavior:1 respectively:7 here:5 diagonal:4 beck:1 inverse:1 cos2:1 sinh:1 eikx:2 κt:10 consideration:1 strong:3 z0:2 z1:2 z2:1 simulating:1 obtained:11 naval:1 suitably:1 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quant-ph9910090.txt |@word limited:1 consider:3 secondly:1 leads:1 computation:1 particles:9 boghosian:2 tv:1 alamos:1 tt:1 factoring:1 finally:1 under:2 must:1 worth:1 resources:2 p2:2 very:2 choice:1 every:1 matrices:1 hamiltonian:6 pˆ1:1 vector:1 fourier:2 small:4 2k1:1 direct:1 design:2 further:1 dynamics:3 richard:1 find:2 3n:1 above:5 new:1 increasing:1 method:8 body:3 edited:1 full:2 whose:1 respectively:3 realizations:1 here:1 diagonal:8 china:3 obtained:6 great:1 goldwasser:1 periodic:2 usually:1 suitable:1 useful:1 summation:2 unit:1 use:9 would:3 stated:1 visit:1 properties:3 connector:2 ψi:1 m1:2 therefore:4 m2:2 taken:1 more:4 formally:1 operators:6 odinger:7 ip2:1 schrdinger:1 virtue:1 known:1 foundations:1 fermionic:1 me:3 states:1 science:5 equation:9 oscillator:4 following:3 making:1 my:4 example:6 beijing:1 wang:2 give:2 process:1 high:2 sense:1 simple:1 tseng:1 needs:2 rather:1 get:1 orthogonality:1 how:4 symposium:1 fp:1 simplicity:1 product:4 designed:1 multiplication:1 efficiently:1 short:1 natural:2 register:3 so:6 fundamental:3 representation:4 order:4 δmn:1 acts:1 help:1 mainly:1 course:2 paper:3 including:1 kinetic:6 ieee:1 its:9 discretizing:1 rapidly:1 thank:1 chosen:2 coordinates:2 better:1 systems:6 derivative:2 easier:1 lx:1 them:1 practice:1 they:1 oxford:1 now:3 realize:2 easily:5 several:1 cory:1 university:3 l2:1 higher:2 2k:12 schemes:1 directly:2 series:1 energy:9 reduce:1 principles:3 operator:14 used:6 measurement:1 quantum:73 laboratory:2 eq:1 special:4 out:9 shown:1 network:20 space:7 furthermore:1 since:2 rev:3 research:1 increase:1 fellowship:1 pˆ2:1 performs:1 size:1 foundation:1 theory:4 written:3 little:1 free:7 eih0:1 milburn:1 interaction:2 just:3 impose:1 turn:1 exponentially:2 first:5 there:2 xn:2 features:1 symmetry:1 xm:38 number:1 iht:2 clear:1 done:3 approximately:1 another:1 lidar:1 zalka:1 specially:3 given:6 rewritten:1 πn:1 system:13 construct:1 realization:4 scheme:1 bits:1 store:1 academy:1 ih0:1 simulate:16 x2:1 x1:1 than:3 havel:1 wiesner:1 speaking:1 were:2 modern:1 need:4 seen:1 built:1 min:3 efficient:3 convenient:1 able:2 ideas:1 note:1 sciences:1 k1:2 potential:8 square:1 take:1 algorithms:2 multiple:1 reach:1 most:2 optimized:1 letter:3 successive:3 why:1 constructed:2 average:1 feynman:2 proceedings:1 exp:8 physics:2 fact:3 laws:1 shor:2 supported:1 registers:2 gauge:1 lloyd:3 networks:4 based:2 simulating:10 enough:4 should:1 jan:1 only:3 difficulties:1 factor:1 local:3 do:2 intractable:1 his:2 means:1 express:1 scalable:1 preparing:1 symmetrically:2 evolution:9 arxiv:1 activity:1 xm1:5 xm2:5 set:1 national:1 relative:2 see:1 computer:11 skill:1 said:1 exchange:1 state:7 c2a:1 various:2 simplest:1 drawn:1 boundary:2 conditions:2 discovery:1 barenco:2 terms:1 ability:1 extend:1 qubits:3 however:2 hefei:1 efficiency:2 key:1 both:1 last:1 many:1 taking:1 annual:1 qubit:3 whole:3 mechanics:4 point:1 reasons:1 ca:31 harmonic:4 dimensional:12 anti:1 polynomial:1 moreover:2 considers:1 complication:1 transformations:1 basis:1 addition:1 create:1 been:5 int:1 describes:1 basic:2 combine:1 ph:11 box:4 relations:1 precision:1 2in:1 external:2 graphics:1 present:1 case:2 eip:1 exponential:1 pacs:1 while:3 momentum:6 35th:1 discretize:1 unitary:2 ve:2 laflamme:1 vi:1 ir:11 thus:4 ih:1 reads:2 ia:10 technology:2 different:1 grant:1 perhaps:1 make:1 progressing:1 same:2 simultaneous:1 quant:11 finite:2 grows:1 difficult:1 extended:2 i1:1 i2:1 action:2 phy:1 uˆ:2 identity:1 task:3 schr:7 center:3 obviously:2 thought:2 simulated:5 coordinate:4 left:1 gates:1 proposed:5 among:1 generally:1 vˆe:1 obtain:1 using:3 facts:1 alternative:1 k2:3 seems:2 qho:1 iω:1 lattice:1 transformed:1 lett:1 easy:4 input:4 transformation:9 ought:1 identical:1 demands:1 rules:1 possible:1 know:1 world:1 necessary:1 like:3 particle:9 follows:1 steps:2 because:4 classical:5 people:1 some:7 examples:2 choosing:1 respect:2 ccast:1 scale:1 does:1 decomposition:1 umn:2 proc:1 bj:1 step:3 although:1 assemble:1 hilbert:4 bz:1 simpler:1 comparison:1 extension:2 simplify:2 discussed:1 o:3 simulation:1 practical:1 tremendous:1 wang1:1 soc:1 m0n:1 communication:1 within:2 appropriate:1 right:1 artur:1 elementary:2 deal:1 initial:2 u1:3 lond:1 u2:4 los:1 much:1 way:1 m0:1 form:3 simulator:1 but:5 hj:1 construction:1 highest:1 annihilate:1 society:1 hosting:1 ur:4 discretized:1 uj:2 ue:1 problem:2 similar:3 called:4 aa:2 taylor:1 constant:2 computing:8 defined:4 likewise:1 universal:4 describe:1 general:7 tˆ:1 single:1 pˆ:4 our:1 abrams:1 moves:1 physical:2 completeness:1 no:1 ekert:1 wavefunction:3 when:2 field:6 other:2 republic:1 eiht:3 department:1 theor:1 conclusion:1 picture:4 elements:2 problems:2 biham:1 prepared:2 meaning:1 important:3 matrix:2 implies:1 auxiliary:2 algorithm:3 cpu:3 mass:2 parallelism:1 time:16 n1:4 n2:4 original:1
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quant-ph9910091.txt |@word secondly:4 four:3 computation:2 consists:2 gh:1 inverse:2 calculate:1 attracted:1 finally:1 under:1 worth:1 very:3 unstructured:1 every:1 matrices:2 leads:1 large:2 deutsch:12 sign:1 second:3 design:5 implemented:1 aim:1 plays:2 find:1 above:1 new:4 method:3 full:1 respectively:3 realizations:6 china:3 others:1 represented:2 appears:1 simulating:6 obtained:5 search:2 antisymmetric:1 divisor:1 goldwasser:1 simultaneous:2 usually:3 implement:7 fourier:11 useful:2 beijing:1 compare:1 assemble:3 summation:3 total:3 select:1 alamitos:1 use:2 would:1 visit:1 negative:1 connector:1 few:1 therefore:5 means:1 los:1 more:2 operators:1 odinger:1 known:7 foundations:1 fermionic:1 me:5 science:4 equation:2 grover:6 values:3 following:6 making:2 arrive:1 my:1 example:2 beautiful:1 f1:4 f2:4 f3:4 f4:4 wang:2 give:3 calculation:1 high:1 acm:1 sense:1 needs:2 verify:2 divided:1 acts:1 how:2 vital:1 gate:3 explains:1 simplicity:1 product:5 act:3 mapping:2 multiplication:2 data:2 profited:1 natural:1 register:8 explicit:1 so:6 fundamental:3 order:3 vmn:1 help:1 including:7 ieee:1 its:8 thank:1 actually:2 t76:1 main:4 lx:1 them:4 good:1 practice:1 elementary:2 they:2 symposium:2 oxford:1 now:2 easily:3 term:1 university:3 each:5 found:2 2j:1 2k:21 weight:2 doing:2 series:3 operator:1 used:5 connect:3 measurement:4 quantum:91 laboratory:2 operation:1 eq:2 out:7 network:23 space:2 furthermore:2 since:5 rev:1 research:1 fellowship:1 acting:3 m2k:2 denoted:1 foundation:1 after:1 theory:1 diagonal:1 standard:3 complicated:2 e2:1 definition:2 r0:5 r1:2 r2:5 programming:1 times:2 exponentially:2 first:12 xj:2 initial:2 xn:3 fourthly:1 xm:6 number:2 intuitively:1 simplified:1 synchronously:1 given:3 rewritten:2 start:2 unknown:1 πn:2 relations:1 calculations:1 realization:7 siam:1 final:3 speaking:1 gives:1 academy:1 simulate:1 schr:1 than:2 center:1 showed:1 zero:2 modern:1 randomly:1 need:2 seen:2 seem:1 seek:1 min:3 efficient:1 convenient:2 able:2 note:2 sciences:1 finding:1 algorithms:7 play:1 scripta:1 most:1 letter:1 successive:3 feynman:1 proceedings:2 exp:11 physics:1 fact:1 shor:14 supported:1 registers:2 networks:5 proposed:4 parameters:2 written:1 jan:1 only:7 explicitly:1 factor:2 do:4 his:3 get:3 scalable:2 define:1 made:2 preparing:1 altogether:1 reveal:2 evolution:1 arxiv:1 factorization:5 2πi:1 common:1 national:1 δjm:1 hosting:1 computer:8 result:3 best:1 tensor:1 favor:1 state:10 c2a:1 simplest:2 drawn:1 barenco:3 terms:4 qubits:1 however:3 hefei:1 efficiency:1 jr:3 both:1 last:4 taking:1 annual:2 equal:2 qubit:13 whole:8 otherwise:1 simple:2 had:1 ca:34 corresponds:2 moreover:1 cq:5 described:4 transformations:4 basis:2 addition:1 create:1 three:1 been:2 strategy:1 int:1 basic:2 expected:3 ph:5 pn:3 pm:1 box:2 corresponding:1 formed:1 great:1 graphics:1 improved:1 present:1 input:7 ǫf:5 pacs:1 will:3 while:2 sometime:1 promising:1 35th:1 procedure:2 unitary:2 ir:10 thus:6 reads:1 ia:10 technology:2 different:1 grant:1 perhaps:1 make:4 progressing:1 read:1 quant:5 pauli:1 several:1 difficult:1 extended:1 i1:2 i2:3 action:1 levels:1 identity:2 task:2 kept:1 reflection:1 δ0m:1 well:2 roughly:1 obviously:3 thought:1 solve:2 greatest:1 usual:2 gates:4 just:2 obtain:7 press:1 speed:2 like:1 preprint:3 seems:2 except:1 add:1 transformed:1 lett:1 easy:7 role:3 transformation:20 rules:1 possible:4 period:1 five:1 know:2 judge:1 world:1 mod:1 necessary:1 helpful:1 follows:2 steps:8 δf:6 output:5 because:6 methods:1 classical:4 people:1 some:3 searching:1 repeat:1 28th:1 ccast:1 scale:2 measurements:3 umn:2 proc:1 equivalent:1 bj:2 step:3 although:1 hadamard:5 properties:2 about:1 o:2 modn:7 wang1:1 soc:1 m0n:1 communication:1 into:5 appropriate:1 determine:1 artur:1 balanced:3 iπn:1 prepare:3 there:2 u1:3 lond:1 u2:4 thirdly:3 physicists:1 lot:1 ed:1 was:1 m0:1 function:5 form:5 but:2 hj:4 construction:1 integrality:1 annihilate:1 consist:1 society:1 whether:1 up:2 ur:4 uj:2 problem:4 uf:1 similar:1 called:2 obtains:1 aa:2 constant:3 computing:17 defined:3 universal:6 measure:2 am:4 doesn:1 general:3 single:2 ar:1 our:1 journal:1 constructed:1 again:1 no:1 ekert:2 other:2 republic:1 department:1 theor:1 conclusion:1 picture:1 elements:2 kinds:1 superposition:3 calculating:1 prepared:2 important:3 matrix:5 auxiliary:3 algorithm:34 factors:1 together:2 original:1 time:2 far:1 starting:1 cpu:4 once:1
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quant-ph9910092.txt |@word limited:4 concept:3 sci:1 foug:4 remarkable:1 caused:4 excellent:1 results:9 leads:2 consists:1 follow:1 systematic:1 research:1 n34:6 reasoning:1 academic:1 consider:1 compact:1 particles:1 griffin:1 repetition:2 calculate:2 concerned:1 tain:1 finally:4 efficiencies:1 helps:1 under:3 analogous:3 silicon:1 sent:1 applied:2 kπ:2 imation:1 straightforward:1 far:1 conditions:1 wave:5 reconstructions:1 should:3 dispersions:10 fall:1 experimentally:8 difference:17 condition:1 four:1 modulator:1 presented:4 ref:1 deals:1 discrete:3 detected:6 laser:6 indices:1 divides:1 referred:1 optics:7 upper:1 smaller:2 pragmatic:1 feeling:1 sin2:4 prediction:9 pulse:8 fold:1 zero:5 cost:6 value:18 further:1 estimated:14 port:2 quite:1 linear:2 even:2 established:1 what:1 appear:2 apparatus:1 incompatible:1 section:8 axes:1 lengthy:1 current:1 experiment:4 above:4 obvious:1 increasing:1 order:5 consisting:1 contrast:1 justification:1 simultaneously:2 pm1:3 pm2:1 dow:1 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quant-ph9910101.txt |@word represent:1 partial:3 consider:6 remarkable:1 rev:10 keep:1 varying:2 results:1 perelman:1 leads:2 meier:1 c1:13 cyclically:3 whose:1 anticipation:1 depend:3 producing:1 concerned:1 reverting:1 recurrence:4 finally:1 γn:15 under:3 circumstances:1 p2:2 chennai:2 coeffcients:1 frequencies:1 stroud:2 induce:1 wave:23 every:2 exact:1 condition:2 arrange:1 illustrated:1 vibrational:1 hamiltonian:15 turns:3 hr:4 discrete:3 science:1 small:1 zn:1 neighborhood:1 occurrence:2 periodicity:2 ghosh:1 pulse:1 cosh:1 second:1 datta:1 further:3 dynamics:5 choices:1 substitute:1 canonical:1 even:3 established:1 nonlinear:6 appear:2 fine:1 section:4 pandey:1 favour:1 expressed:1 uniform:1 indian:2 melo:1 above:2 fractional:22 coefficient:1 full:12 respectively:2 understanding:1 desired:1 situation:1 slater:1 along:1 yan:1 obtained:1 institute:3 shift:1 study:1 host:1 unitarily:1 suggests:1 periodic:3 action:2 suitable:3 cancellation:2 oschl:1 appropriately:1 spatially:1 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quant-ph9910102.txt |@word partial:1 consider:3 remarkable:1 rev:12 berman:2 γn:8 results:4 perelman:1 leads:1 roy:2 grateful:1 hyperboloid:3 research:2 contributions:1 whose:1 simon:1 fellowship:1 concerned:2 recurrence:7 quasi:1 squeezing:8 helps:1 under:5 stochastic:1 analogous:1 circumstances:1 mehta:1 chennai:1 very:2 frequencies:1 stroud:2 induce:1 every:3 mixing:1 experimentally:1 exact:3 hamiltonian:11 altering:1 t2:3 solution:1 t1:3 discrete:3 vector:1 science:1 small:2 optics:1 der:1 upper:1 zn:2 neighborhood:1 occurrence:2 periodicity:2 consistent:1 direct:1 cosh:1 zero:2 enclosed:2 further:2 dynamics:4 established:2 nonlinear:3 plays:1 stands:1 d2:1 sum:1 near:13 uniform:2 current:1 indian:1 melo:1 fractional:4 new:2 increasing:1 coefficient:1 full:1 christian:1 ibid:6 exchange:1 respectively:3 here:4 following:3 operating:1 sinh:3 excess:2 represented:1 slater:1 quite:1 extreme:1 solid:1 institute:1 involved:1 eberly:1 amount:1 periodic:5 walther:2 pick:1 changes:1 extra:1 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quant-ph9910103.txt |@word replacing:4 limited:1 positions:1 concept:1 consider:7 remarkable:2 rev:20 results:11 leads:4 scalar:1 grateful:1 stochastically:1 constants:1 follow:1 expressions:15 research:1 soc:2 ρss:15 whose:1 certainly:1 hold:2 depend:6 pointing:1 re:2 subjected:1 equivalence:2 tr:9 orthonormalized:1 passage:9 tn:12 finally:3 does:5 tk:78 th:14 ti:97 exit:10 tc:7 norm:2 detect:1 p2:1 chennai:1 straightforward:5 evolves:2 very:3 could:1 choice:2 englert:5 every:1 chooses:1 entries:2 experimentally:3 difference:2 exact:8 condition:7 entire:2 exits:5 level:2 evaluation:2 notes:1 t2:2 solution:5 t1:15 leave:2 discrete:6 convenience:2 demonstrated:1 small:9 relation:12 optics:4 noted:1 dis:1 regain:1 strongly:1 smaller:1 verifiable:2 publ:2 insignificant:1 a34:1 sin2:3 hillery:2 maintain:1 approximation:3 rate:13 evaluated:3 further:1 dynamics:10 choices:1 schenzle:1 goes:3 even:6 index:2 what:1 errors:1 lindblad:1 trapping:21 section:14 waiting:1 experiment:1 nex:77 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quant-ph9910104.txt |@word consider:6 ali:1 results:7 leads:6 roy:1 edu:1 systematic:1 whose:2 depend:4 calculate:3 rayleigh:5 tr:1 finally:1 present:1 analogous:1 worth:1 norm:1 pˆ:13 p2:11 belonging:1 abelian:1 very:1 2re:7 rise:1 geometrical:1 valued:2 l2µ:2 noncyclic:4 exact:3 condition:4 cmn:92 hamiltonian:48 zeit:1 positions:1 t2:7 solution:3 t1:10 discrete:1 vector:6 math:9 ml:1 small:1 eigenvalues:9 though:1 enforce:1 rohrlich:1 occurrence:2 consistent:4 kato:1 sign:7 approximation:3 second:8 fiber:1 abbreviates:1 pass:1 greenberger:3 dynamics:12 substitute:2 canonical:6 even:2 index:2 what:1 stands:2 section:5 brief:1 expressed:1 fock:1 en:45 above:3 referees:1 new:2 appeal:1 method:3 transportation:1 respectively:5 here:2 l20:6 enℓ:1 address:4 active:1 path:2 appropriate:1 equations:2 change:3 obtained:9 remark:1 shift:1 study:2 studied:3 resulting:2 bhandari:1 dra:3 changes:1 involves:1 orthonormality:1 requiring:3 replace:1 addressed:2 unitary:1 apply:2 total:1 relating:1 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quant-ph9910105.txt |@word consider:3 results:3 four:1 scalar:1 per:1 consists:1 expressions:4 nwo:1 contributions:1 layers:1 amplification:5 depend:2 re:3 tt:10 tr:6 extrapolate:1 supported:1 according:1 squeezing:7 th:3 materie:1 cumulants:7 characterised:1 presentation:1 photocount:7 advantage:1 returning:1 frequencies:2 springer:1 focus:1 every:1 figs:2 fom:1 difference:4 matrices:6 splitter:1 beating:3 presented:1 transmission:6 positions:2 large:2 vector:9 generalises:1 small:3 optics:2 der:1 application:1 abbreviated:1 averaged:1 disordered:4 direct:9 sign:2 approximation:3 threshold:2 rate:2 value:3 dsα:1 substitute:1 even:1 index:4 oll:2 abd:1 sub:3 section:1 expressed:1 current:2 3n:2 above:1 new:1 increasing:2 waveguide:12 method:1 satisfy:2 edited:1 outgoing:5 degree:3 here:4 specialise:1 squeezed:17 inverse:1 sinh:28 excess:1 path:3 strong:6 change:1 obtained:6 drops:1 incoming:3 averages:3 vectors:1 allows:1 simply:1 poisson:3 action:1 linearised:1 shaded:2 plotted:1 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quant-ph9910106.txt |@word represent:1 todays:1 consider:6 springer:1 maximum:2 stems:1 results:9 valid:1 four:14 consists:1 follow:2 expressions:3 research:2 contributions:1 gi:1 whose:1 certainly:1 depend:1 calculate:1 advantage:1 technique:1 opportunities:1 containing:1 transmittances:1 finally:4 according:5 present:2 under:4 connecting:2 include:4 sent:7 ingredient:1 overlaps:1 applied:2 supported:1 risk:1 confidential:2 orthogonal:5 very:2 ously:1 differently:1 splitting:2 summarized:1 matrices:1 condition:13 dim:2 obvious:2 level:1 transmission:13 turns:2 criterion:3 eve:36 replaced:1 large:5 solved:1 destroying:1 small:3 shows:3 optics:2 ups:1 work:2 occurrence:1 discriminates:1 dealing:1 pulse:1 cosh:3 equation:3 go:2 imply:3 accessing:2 evaluated:1 concludes:1 further:1 dynamics:1 odd:2 click:14 substitute:2 linear:5 even:8 aim:1 what:1 errors:5 sub:1 apparatus:1 displaying:2 sum:1 selected:1 jaeger:2 expressed:5 access:1 above:3 utkenhaus:3 shared:2 learned:1 increasing:2 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quant-ph9910107.txt |@word represent:1 logarithms:1 consider:6 global:3 focus:1 leads:1 broader:1 edu:1 go:1 relationships:1 depend:1 environment:1 compensate:1 helps:1 analogous:1 amplitudes:3 orthogonal:3 implicitly:1 continues:1 baton:1 visualized:2 turns:1 z12:1 solution:1 discrete:1 convenience:1 vector:5 zn:1 computation:1 grover:1 elegant:1 zero:5 cost:2 dordrecht:1 estimated:1 even:3 what:2 apparatus:2 axes:3 experiment:1 ongoing:1 method:1 implementable:1 evolving:1 degree:1 respectively:1 component:1 here:3 ranges:1 desired:4 directions:1 along:2 z1:1 z2:1 obtained:1 difficulty:1 vectors:6 allows:1 completely:2 defeat:1 advantages:1 suitable:1 makes:1 useful:3 outcomes:1 stern:1 unit:4 use:4 proceed:1 would:2 probabilities:3 taken:2 more:2 operators:2 controlling:1 normalized:1 known:1 foundations:1 zi1:1 challenge:1 work:1 remain:1 anywhere:1 values:1 concepts:1 following:2 example:2 performed:2 computers:1 process:1 aeiθ:1 states:9 numbers:1 information:4 end:1 get:1 pure:16 iθ2:1 gate:1 iθ1:1 simplicity:1 product:1 flip:1 transforming:1 after:1 plane:1 designed:1 dynamical:1 computations:1 register:3 so:3 fundamental:2 operations:3 reversal:1 over:1 installed:1 lsu:1 its:6 perfect:1 controllable:1 la:1 chosen:1 degrees:4 initializing:2 them:1 return:1 eiθ2:5 rotations:1 eiθ1:5 regarding:1 introduce:2 they:1 identically:1 nov:1 now:2 february:1 gets:1 equality:1 university:1 realistic:1 reasonable:1 each:5 2n:2 individually:1 schemes:1 ee:1 reduce:1 related:1 factorization:1 calibrate:1 measurement:1 quantum:17 our:1 operation:2 out:4 shor:1 matrix:1 space:1 furthermore:3 since:4 excepting:2 may:1 calcite:1 correct:1 theory:2 written:1 standard:1 clarify:1 reason:2 formation:1 could:1 transition:2 βe:1 subhash:1 initial:5 probability:4 number:1 done:2 impossible:2 zi:1 given:1 introduction:1 unknown:7 system:5 realistically:1 their:1 assumption:1 observations:1 siam:1 final:1 1n:4 exactly:1 authors:1 constituent:1 than:1 imply:2 preparation:2 matter:1 determined:1 outcome:2 steer:2 extracting:1 dynamically:1 locations:1 manner:2 seen:1 any:4 viewed:1 computationally:1 latter:1 note:1 take:1 algorithms:3 needle:1 allow:1 kak:3 object:1 most:1 phase:13 considered:1 average:1 implications:2 points:1 principle:1 ative:1 normally:1 fact:2 lumping:1 emerging:1 drastically:1 random:1 rotating:1 relation:1 gerlach:1 rouge:1 ensembles:1 spite:1 implementation:5 distributed:1 state:42 should:2 only:2 circle:1 local:1 do:3 means:3 between:1 cannot:4 preparing:2 words:2 new:1 arranged:1 prime:1 evolution:2 resolve:1 arxiv:1 yield:1 paper:1 common:1 view:2 ignored:2 set:2 achieved:1 maximum:1 relative:5 depends:1 computer:2 theoretic:1 subject:1 phases:9 tensor:1 ways:1 review:1 currently:2 definite:1 behind:1 unable:1 αe:1 correctly:1 simplest:1 available:1 qubits:11 however:1 undetermined:1 cos:2 many:1 uncertainty:10 equal:1 qubit:19 point:1 dimensional:1 polynomial:1 respect:2 speak:1 transformations:3 basis:6 due:1 been:3 change:1 basic:1 expected:1 ph:1 louisiana:1 correction:1 direction:1 indeterminacy:2 photon:1 θd:1 ensuring:1 angles:2 worked:1 spin:1 diffuse:1 applied:2 frozen:1 engineering:1 value:1 will:17 superposed:1 error:2 unitary:7 averaging:1 thus:1 elicit:1 different:1 author:1 make:2 indistinguishable:1 complex:3 quant:1 several:1 used:1 initialization:1 conceived:1 hand:1 purpose:1 entanglement:2 angle:3 recent:1 academic:1 without:1 components:1 gates:1 permissible:1 proposed:1 obtain:2 mistaken:1 questions:1 rotation:9 polarization:1 kluwer:1 conclusions:1 mathematically:1 transformed:1 easy:1 haystack:1 transformation:4 real:1 lacking:1 department:1 entangled:2 correcting:1 rotate:2 specific:2 yields:1 arbitrary:2 become:1 essential:1 because:4 methods:1 sequence:1 some:3 searching:1 examples:2 ignore:2 lead:1 eiθ:1 measurements:1 does:2 provides:1 assuming:1 limitations:1 noise:1 equivalent:1 permitted:1 simpler:1 about:2 discussed:1 remains:1 discard:1 own:1 letters:1 presence:1 into:4 obtaining:4 bound:1 uncontrolled:1 appropriate:3 references:1 operator:5 rely:1 prepare:1 assumed:6 there:3 question:1 way:4 was:2 assumes:1 function:3 form:2 but:6 repeated:1 deployed:1 applying:1 up:1 us:3 terized:1 crystal:1 problem:5 stages:1 clear:1 mechanics:2 associated:2 computing:5 defined:2 likewise:1 abstract:1 doesn:1 general:1 peres:1 examine:1 lumped:2 physical:3 codes:1 no:3 whereas:2 when:2 envisaged:1 application:1 other:5 electrical:1 test:2 realizability:1 superposition:8 prepared:3 important:1 individual:3 assume:3 physics:1 together:2 impossibility:1 weights:1 time:1 observables:2 starting:4 original:1
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quant-ph9910108.txt |@word interpretation:6 consider:1 suggests:2 obtain:3 shape:1 japan:1 gr:1 compact:1 particles:1 kanagawa:1 spin:1 relativity:1 only:5 easy:1 mail:1 analogous:3 do:1 kent:1 combination:1 possible:1 geometrical:2 electron:1 unification:1 totsuka:1 combinations:2 using:1 nov:1 mystery:2 worlds:4 like:4 particle:3 hamiltonian:1 arxiv:1 obius:2 sugiyama:1 solved:1 qc:1 common:1 integrals:1 each:3 set:2 exists:2 some:2 prediction:3 sθ1:1 second:1 decided:1 radius:2 measure:1 quantum:8 event:9 linear:1 rw:5 space:11 since:3 boundaries:1 state:11 provides:1 between:1 new:2 boundary:1 measuring:1 theory:14 jp:1 path:1 november:1 properties:1 interpret:2 many:7 transition:1 quantization:1 etc:1 think:1 probability:7 point:4 number:3 because:1 references:1 approximately:1 another:2 strip:2 elementary:24 unit:6 use:1 observer:1 introduction:1 there:7 system:1 construct:2 assumption:6 call:5 too:2 m2:4 ph:1 was:1 cancel:1 define:2 function:6 quantities:5 curvature:2 construction:1 line:1 true:1 count:1 kouji:1 rθ:3 different:4 minimum:6 suppose:9 problem:1 example:1 loop:1 yokohama:1 sw:1 abstract:1 describe:1 thus:6 states:4 loops:1 assumptions:2 exist:1 something:1 need:1 physical:2 functions:1 no:3 provide:1 make:1 ne:1 same:2 quant:1 finding:1 out:1 events:6 conclusion:1 effect:2 sx1:1 connected:2 st1:1 phase:6 wave:10 probabilistic:3 structure:2 torus:6 lines:1 points:1 exp:1 time:2 position:9 quantity:6
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