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quenceofthetwo-bodynatureofinteraction. Inasimilar
with exponentially large Hilbert space. The paper is or-
way for few particles n 3 the border U ∆2. This
c
ganizedasfollows. InthenextSectiontheanalyticaland ∼ ∼
conclusion was confirmed in [39].
numericalresultsarepresentedforemergenceofquantum
chaosinmany-bodysystems. InSectionIIIthestandard
1 1
generic quantum computer (SGQC) model is described
andthe results of SectionII aregeneralizedforquantum
chaos border for quantum computing. In Section IV the
time evolutionintheregimeofquantumchaosisstudied 2
and different time scales imposed by chaos and decoher- 2 2
ence on quantum computing are discussed. The paper
ends by the concluding remarks in the last Section.
3 3
II. ˚ABERG CRITERION FOR EMERGENCE OF
QUANTUM CHAOS IN MANY-BODY SYSTEMS
Following[38]letusconsiderfirstacaseofthreeparti- FIG.1. Diagramfortheeffective3-particlematrixelement
cles located on m one-particle orbitals with energy level U3 in (1), after [38].
spacing ∆ V/m. For simplicity we assume the parti-
∼
clestobedistinguishablethathoweverisnotofprinciple Let us now consider a more general case of TBRIM
importance for m 1. The level spacing between 2,3- [29–31] with n Fermi particles distributed over m en-
≫
3
′
ergy orbitals ǫ m′, m = 1,2,...,m. These energies are exponentially larger than the level spacing ∆ between
n
randomly and homogeneously distributed in the inter- multi-particle states.
val [0,m∆] with spacing ∆. The total number of many-
6.0
body states is N = m!/(n!(m n)!) and they are cou-
−
pled by random two-body matrix elements which value -1.6
)
is uniformly distributed in the interval [ U,U]. Due to ∆
the two-body nature of interaction the number − of multi- / U c -2.0
5.0 (goL
particle states coupled by interaction, or the number of )
2
U
direct n)(m −n transitions, −1)/4[31]. is K All = these 1+n(m tran− sitions n)+n(n occu− rinside− 1)(m a / ξ (goL -2.4 1.4 Log( 1 ρ. 8 ∆ )
two-body energyinterval B =(2m 4)∆aroundthe en- c
− 4.0
ergyofinitialmulti-particlestate. Forlargemandnthe
number oftransitions K is muchsmallerthan the size of
the matrix N but is muchlargerthanthe number ofdif-
ferenttwo-bodymatrixelementsN2 m2/2. TheFermi
≈ 3.0
energy of the systems is ǫ n∆ and the level spacing 2.0 3.0 4.0 5.0 6.0
F
in the middle of the total en≈ ergy band is exponentially Log( ρ ρ )
c n
small ∆ (m n)n∆/N.
n
≈ − FIG. 3. Dependence of the rescaled IPR ξ/U2 on ρcρn:
layer model data for n = 3, ∆ = 1 and 40 ≤ m ≤ 130 (o);
10−1 n= 4 and 30 ≤ m≤ 60 (diamonds). The straight line gives
theory (5). Insert shows Uc/∆ vs. ρc∆ in log-log scale for
thesameparameters;thestraightlineisthefitUc =0.62/ρc.
After[43].
10−2
n=2 The most direct way to detect the emergence of quan-
B/Uc n=3 tum chaos is by the change in the probability distribu-
n=4 tion P(s) of nearest level spacings s, where s is mea-
n=5
sured in units of average spacing. Indeed, for inte-
10−3 n=6
grable systems the levels are noncorrelated and char-
n=7
n=8 acterized by the Poisson distribution P P(s) = exp( s),
−
while in the quantum chaos regime the statistics is close
to the Wigner surmise P (s) = (πs/2)exp( πs2/4)
W
100 101 102 103 104 −
K [41]. To identify a transition from one limiting case
to another it is convenient to introduce the parameter
s0(P(s) s0(P
η = P (s))ds/ (s) P (s))ds, where
R0 W R0 P W
FIG. 2. Dependence of the rescaled critical interaction − −
strength Uc/B, above which P(s) becomes close to the s P0 = (s).0. I4 n72 th9. i. s. wis ayth ηe vi an rt ie er ss fe rc ot mion p (Poi (n st =of PP P (( ss )) ta ond
1 ) ) 0
Wigner-Dyson statistics, on the number of directly coupled W P
(P(s) = P (s) ) and the critical value of U can be de-
states K for 4 ≤ m ≤ 80 and 1/40 ≤ n/m ≤ 1/2. The line W c
termined by the condition η(U )=η =0.3 [40]. In fact
shows thetheory (3) with C =0.58, after [40]. c c
the choice of η affects only the numerical constant C in
c
At the middle of spectrum the density of directly cou- (3). The chosen η c = 0.3 is close to the value η c 0.2
≈
pled states is ρ = K/B. According to the usual pertur- for the critical statistics at the Anderson transition on
c
bation theory, these states, if they would be alone, are a 3-dimensional disordered lattice [42]. The results of
mixed if the coupling U is largerthan their levelspacing extensive numerical studies, performed in [40], are pre-
∆ = 1/ρ . This is the ˚Aberg criterion [34,35], which sented in Fig.2 and confirm the ˚Aberg criterion (3) in a