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U J, the spacing between directly coupled states is
COMPUTER MODEL ∆ c∼ ∆0/n2, since each state is coupled to n(n 1)/2
∼ −
states in the energy band of the order of ∆0. There-
In [24] the standard generic quantum computer
fore, according to (3) the quantum chaos and ergodicity
(SGQC) model wasintroducedto describe a systemof n
emergeforJ >∆0/n2 ≫∆ nasitwasshownanalytically
qubits containing imperfections which generate a resid-
and numerically in [48].
ual inter-qubit coupling and fluctuations in the energy
5
A similar analysis can be done for the SGQC model C =3.16 and clearly show that J ∆ (see inserts in
q c n
≫
(6). Indeed, for δ ∆0 and J < δ the total spectrum Fig. 4). According to Fig. 4 the transition is sharp in
≪
iscomposedofnbandswithinter-bandspacing2∆0 and the limit of large n in contrast to the smooth crossover
bandwidth√nδ. Withinonebandonequantumregister in the TBRIM. This difference is due to local interac-
state is coupled to about n states in an energy window tion between particles in (6) contrary to the long range
of 2δ so that ∆ δ/n and the quantum chaos border is interaction in the TBRIM.
c
∼
given by [24] In the limit δ ∆0 and J ∆0 the coupling
≪ ≪
between different energy bands is negligibly small. In
J c =C qδ/n (7) this case to a good approximation the SGQC Hamilto-
nian (6) can be reduced to the renormalized Hamilto-
where C q is a numerical factor. All above arguments nian H = Σn+1Pˆ HPˆ where Pˆ is the projector on
remainvalid up to δ =∆0 when the bands become over- kthP sk o=1 thak quk cok
the band, t bits are upled only inside one
lapped. It is important that J ∆ and that J de-
c n c band. For an even n this band is centered exactly at
≫
creaseswithδ. Thelastpropertyisnaturalsinceatsmall
E = 0, while for odd n there are two bands centered at
δ the states in one band are quasi-degenerate and it is
E = ∆0, and we will use the one at E = ∆0. Such a
easier to mix levels. ± −
bandcorrespondsto the highestdensity ofstates,andin
a sense represents the quantum computer core. The de-
pendenceofcriticalcoupling,determinedviaη(J )=0.3,
c
1.0
η 0.0 on the number of qubits n is shown in Fig. 5 being in
log(J/∆ 0) goodagreementwiththetheoreticalquantumchaosbor-
-1.0
0.8 der (7). It is important to note that at δ = 0 the pa-
-2.0 rameter η = 0 and the eigenstates are chaotic [25] in
agreement with (7).
0.6
-3.0
-4.0 0
0.4 0.7 0.9 1.1
-1 log(n) log(J/δ )
0.2 -2
−1
-1 0
0.0
0.0 0.5 1.0 1.5 J/J 2.0
c
−2
FIG.4. Dependenceofη ontherescaled couplingstrength
J/Jc for the states in the middle of the energy band for
n=6(∗),9(o),12(triangles),15(squares); δ =∆0. The upper
insert shows log(Jc/∆0) (diamonds) and log(Jcs/∆0) (trian- −3
gles) versus log(n); the variation of the scaled multi-qubit
spacing ∆n/∆0 with log(n) is shown for comparison (+).
Dashed linegives thetheoretical formula (7) with Cq =3.16;
−4
the solid line is Jcs = 0.41∆0/n. The lower insert shows 0.8 0.9 1.0 1.1 1.2
log(Jcs/∆0) versus log(δ/∆0) for n = 6(∗),9 (o), 12 (trian- log(n)
gles); straight lines haveslope 1. After [24].
FIG. 5. Dependence of log(Jc/δ) (diamonds) and
log(Jcs/δ)(triangles)versuslog(n);thevariationofthescaled
The direct numerical simulations for the quantum multi-qubitspacing(log(∆n/δ))withlog(n)isshownforcom-
chaos border in quantum computer are done in [24] for parison (+). Dashed line gives the theoretical formula (7)
theSGQCmodel. Thechangeinthelevelspacingstatis- with Cq = 3.3; the solid line is Jcs = 0.41δ/n; the dotted
tics P(s) in the band center with the growth of residual curveis drawn to guidethe eyefor (+). After [25].
interaction J can be quantitatively characterized by the
parameter η. To suppress fluctuations P(s) is obtained The drastic change in the level spacing statistics is in
by averaging over 5 N 4 104 random realiza- factrelatedto aqualitative changeinthe quantumcom-
D
≤ ≤ ×
tions of Γ i,J ij so that the total spacing statistics was puted eigenstate structure. For J ≪ J c the eigenstates
104 < N S 1.6 ×105. Also P(s) is determined inside areverycloseto noninteractingmulti-qubitstates |ψ i >,
≤
oneofthesymmetryclassesof(6)withoddorevennum- while for J > J c each eigenstate φ m > becomes a mix-
|
ber of qubits up. The dependence of η on J is presented ture of exponentially many ψ i >. It is convenient to
|
inFig. 4. The variationofcriticalcouplingwith n andδ characterizethecomplexityofaneigenstate φ m >bythe
|
can be determined from the condition η(J c) = 0.3. The quantum eigenstate entropy S q = −PiW imlog 2W im,
data obtained are in a good agreement with (7) with where W im is the quantum probability to find the state
6
ψ > in the eigenstate φ > (W = < ψ φ > 2). the integrable region below chaos border J < J . How-
i m im i m c
| | | | |
In this way S =0 if φ > is represented by one ψ >, ever, it is possible that for certain experimental propos-