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c c
was also independently proposed in [40]. This relation largeparameterrangewithC =0.58[43]. Ofcourse,the
is also in agreement with the arguments given above for data[40]areobtainedforamuchlargerparameterrange
few particles. As a result, the onset of quantum chaos is than in [34,35]. Nevertheless, the above C value is in
expected for U >U : a good agreement with a numerical factor found in [35]
c
(see Eq. 22 there), which gives C 0.7, if to take into
B 2C account that C is defined via the sa≈ me average values of
U =C∆ =C (3)
c c K ≈ ρ2n2 square matrix element. A similar value is also found in
more advanced studies for n = 3,4 in the layer model
where C is a numerical constant to be found and ρ2
≈ approximation with the states selected in the energy in-
N2/B m/4∆isthetwo-particledensityatm n 1.
≈ ≫ ≫ terval∆(seeinsertinFig. 3)[44]. Alsotheabovestudies
It is important to stress that the critical coupling U is
c
4
[34,35,40,44] show that contrary to the sharp Anderson spacingsbetweenthetwostatesofonequbit. TheHamil-
transition [42] a smooth crossover from one statistics to tonian of this model reads:
another takes place at U U (see, however, [45] and
c
references therein). ≈ H =XΓ iσ iz +XJ ijσ ixσ jx, (6)
The ˚Aberg criterion can be applied not only to ex- i i<j
cited states but also to low energy excitations near the
where the σ are the Pauli matrices for the qubit i and
Fermi level. Suppose we have Fermi gas with a tem- i
thesecondsumrunsovernearest-neighborqubitpairson
perature T ǫ . Then, according to the Fermi-Dirac
≪ F a two-dimensional lattice with periodic boundary condi-
distribution, the number of effectively interacting parti-
tionsapplied. Theenergyspacingbetweenthetwostates
cles isδn Tn/ǫ withthe densityofthese two-particle
ts ota tate ls eρ x2 ci∼ t∼ atiT on/∆ eF 2 nea rn gyd iρ sc δ∼ ρ2δn T2 δn∼ ∆( TT 2/ /∆ ∆) .3. AT sh ae o df isa triq bu ub ti et dis inre tp hr eese inn tt ee rd vab ly [∆Γ i 0r −an δd /o 2m ,∆ly 0a +nd δu /2n ]i .for Tm hly e
E
∼ ∼ detuning parameter δ gives the width of the distribu-
result the interaction induced/dynamical thermalization
tion near the average value ∆0 and may vary from 0
and the quantum chaos set in [40] only for
to ∆0. Fluctuations in the values of Γ appear gener-
i
δE >δE ∆(∆/U)2/3; T >T ∆(∆/U)1/3 (4) ally as a resultof imperfections, e.g. local magnetic field
ch ch
≈ ≈ and density fluctuations in the experimental proposals
Theserelationsfollowalsofromthe˚Abergpapers[34,35] [11,12]. The couplings J represent the residual static
ij
evenif they were not written directly there. The numer- interaction between qubits which is always present for
ical constant in (4) corresponds to η = 0.3 [40]. Below reasons explained in the introduction. They can origi-
c
the border (4) the eigenstates are not ergodic and the nate from spin-exciton exchange [11,12], Coulomb inter-
interaction is too weak to thermalize the fermions even action [9], dipole-dipole interaction [17], etc... To catch
if the multi-particle level spacing is exponentially small thegeneralfeaturesofthedifferentproposals,J arecho-
ij
(∆ exp( 2.5(δE/∆)1/2 [47]). After [34,35,40] the sen randomly and uniformly distributed in the interval
n
∝ −
dependence (4) was also obtained in [46]. [ J,J]. This SGQC model describes the quantum com-
−
In the quantum chaos regime U > U the local den- puter hardware, while the gate operation in time should
c
sity of states is described by the Breit-Wigner distribu- include additionaltime-dependentterms inthe Hamilto-
tion with the energy width Γ given by the Fermi golden nian (6) and will be studied separately. At J = 0 the
rule Γ = 2πU2ρ /3 [44]. The value of Γ determines the noninteracting eigenstates of the SGQC model can be
c
spreadingwidthofeigenstatesmixedbyinteraction. The presented as ψ >= α1,...,α > where α =0,1 marks
i n k
| |
number of noninteracting eigenstates contributing to a the polarization of each individual qubit. These are the
given eigenstate can be measured through the inverse idealeigenstatesofaquantumcomputer calledquantum
participation ratio (IPR) ξ = 1/ a 4, where a are register states. For J = 0, these states are no longer
Pi| i | i 6
probability amplitudes in the noninteracting eigenbasis. eigenstates of the Hamiltonian, and the new eigenstates
The mixing of all levels in the interval Γ gives [44] arenowlinearcombinationsofdifferentquantumregister
states. The termmulti-qubitstatesisusedto denotethe
ξ Γρ n 2U2ρ cρ n. (5) eigenstatesofthe SGQC modelwithinteractionbutalso
≈ ≈
forthecaseJ =0. Itisimportanttostressthatthequan-
where the numerical factor is taken in analogy with the
tumcomputeroperatesinthemiddleofenergyspectrum
known result for band random matrices. This analytical
wherethedensityofstatesisexponentiallylargeanditis
relation is in a good agreement with the numerical data
naturalthatthequantumchaosinitiallysetsinthisbulk
shown in Fig. 3. It is important to note that at U =U
c part of the spectrum. In this respect low energy exci-
exponentiallymanystatesaremixedbyinteraction. The
tations are not important contrary to fermionic systems
widthΓ hasalsoanotherimportantphysicalmeaning: it
discussed above.
determines the chaotic time scale τ 1/Γ after which
χ ≈ It is interesting to note that when one site in (6) is
aninitialnoninteractingeigenstatedisintegratesoverex-
coupled with all other sites and δ = 2∆0 then the sys-
ponentially many (ξ) eigenstates of interacting system
tem becomes equivalent to the quantum version of the
(here and below ¯h=1).
classical Sherrington-Kirpatrick spin glass model stud-
ied in [48]. For such a quantum spin glass shard in the
middle of the spectrum the coupling matrix element is
III. STANDARD GENERIC QUANTUM