Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
q m i
| |
S =1 if two ψ > with equal probability contribute to als it will be difficult to avoidthe quantum chaosregime
q i
|
one φ > and the maximal value S = n corresponds J > J . Therefore, it is also important to understand
m q c
|
to a mixture of all N states in one φ >. A mix- duringwhattimescalethequantumchaosbecomescom-
H m
|
ture of two states ψ > is already sufficient to mod- pletely developed. As for TBRIM this chaotic time scale
i
|
ify strongly the quantum register computations and it is τ isgivenbythedecayrateΓfromonequantumregister
χ
naturaltodeterminethecriticalcouplingJ bythecon- state to all others [24,25]:
cs
dition S (J ) = 1. The results for J dependence on
q cs cs
n and δ are shown in Figs. 4,5. They clearly show that τ χ 1/Γ, Γ J2n/δ, Γ J√n (8)
≈ ∼ ∼
J 0.13J 0.4δ/n ∆ .
cs c n
≈ ≈ ≫ wheretheexpressionsforΓaregivenforJ <δ andJ >δ
In the quantumchaosregime at J >J one eigenstate
c
respectively. For δ = ∆0 a similar estimate for τ was
is composed of ξ states ψ > mixed in the Breit-Wigner χ
i
width Γ J2/∆ J2n| /δ. As for the TBRIM the IPR given in [50].
c
∼ ∼ The detailed numerical studies of chaotic disintegra-
is exponentially large and is given by (5) [49]. For J >δ
tion of an initial state χ(t = 0)>= ψ >, correspond-
the interaction becomes too strong and Γ J√n [25]. i0
| |
comput∼ ing to the quantum register state i0, are done in [25].
The pictorialimage of the quantum er melting is
There for J > J the behavior of the projection proba-
shown in Fig. 6. For J > J the eigenstates become c
c bility F (t) = < ψ χ(t) > 2 is found and it is shown
very complex and the quantum computer hardware and ii0 i
| | |
that the probability to stay at the initial state F (t)
itsoperabilityaredestroyedbyresidualinter-qubitinter- i0i0
decays rapidly to zero with the time scale τ 1/Γ
action. χ ≈
(see Fig. 14 there). The growth of the quantum en-
tropy S(t)= F (t)log F (t) with time is shown
−Pi ii0 2 ii0
inFig. 7. Itclearlyshowsthatafterafinitetimecompa-
rable with τ exponentially many states are excited and
χ
the computer operability is quickly destroyed.
15
S(t) 1
10
0
0 1 2 3
5
FIG. 6. The quantum computer melting induced by the
0
coupling between qubits. Color represents the level of quan- 0 1 2 3 tδ 4
tumeigenstate entropySq,from bright red(Sq ≈11) toblue FIG.7. Time-dependenceof thequantumentropyS(t)for
(Sq =0). Horizontalaxisistheenergyofthecomputereigen- J/δ =0.4 > Jc/δ and n= 16 (diamonds), n=15 (squares),
states counted from the ground state to the maximal energy n=12(triangles),n=9(circles),n=6(*). Averageismade
(≈2n∆0). Vertical axis is thevalue of J/∆0, varyingfrom 0 over200initialstatesi0 randomlychoseninthecentralband.
to0.5. Heren=12,δ =∆0,Jc/∆0 =0.273,andonerandom Insert shows the same curves normalized to their maximal
realization of (6) is chosen. After[24]. value. After [25].
Above we discussed the emergence of quantum chaos
IV. TIME SCALES FOR QUANTUM CHAOS
induced by inter-qubit coupling in an isolated quantum
AND DECOHERENCE IN QUANTUM
computer. It is possible to assume that this process can
COMPUTING
model to a certain extend the effects of decoherence in-
duced by coupling to external world. Indeed, the two-
The results of above section definitely show that the
body nature of interaction is also valid for external cou-
optimal regime for quantum computing corresponds to
pling. In addition it is also important to discuss another
7
type of dangerfor quantum computing which is not nec- These results show that there are two important time
essary related to mixing and complex structure of eigen- scales τ and τ . The effect of external noise and deco-
χ φ
states. Indeed, the algorithms constructed for quantum herencecanbe alsocharacterizedby twodifferentscales.
computing, e.g. Shor’s orGrover’salgorithms,are based Indeed, from one side the noise gives some effective rate
on ideal qubits which have δ = 0. However, for δ > 0, Γ with which a multi-qubit state decaysto other states
T
and even if J = 0, there is a phase difference ∆φ be- thatdeterminesthetimescaleτ 1/Γ . Itisclearthat
χ T
∼
tweendifferentstatesinonebandwhichgrowswithtime Γ n since the noise acts on all qubits independently.
T
∝
as ∆φ ∆E t, where ∆E is the energy difference be- At the same time the noise gives some effective diffusion
tween s∼ tates in the band and its maximal value can be in energy which can be estimated as D = ω2Γ where
E T
estimated as ∆E δ√n. It is naturalto assume that as ωisatypicalenergychangeinducedbynoiseduringtime