Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
∼
soon as the accumulated phase difference becomes com- 1/Γ . For example, noisy fluctuations of J in (6) give
T ij
parable to 1 the computational errors become too large transitions with an energy change ω δ. As a result,
for correct quantum simulations. Therefore, in addition ∆E D τ and ∆φ D 1/2τ 3/2∼ 1. This deter-
p E φ E φ
∼ ∼ ∼
tothechaotictimescaleτ ,thereexiststhephasecoher- mines two time scales for external decoherence:
χ
ence time scale τ which is determined by the condition
φ
∆φ(τ φ) ∆Eτ φ 1. This scale is finite even at J = 0 τ χ ≈1/Γ T, τ φ ≈1/(ω2Γ T)1/3 (9)
∼ ∼
and can be estimated as τ 1/(δ√n) since the band
φ
∼ where τ is related to the decay and relaxation, while
width is ∆E δ√n and the computer usually operates χ
∼ τ determines the phase coherence. Let us note that the
withallstatesinside theband. ForsmallJ thisestimate φ
is still valid and in this case τ < τ for J < δ/n1/4. phase decoherence was discussed for electrons in metals
φ χ
at low temperature [51] and was observed experimen-
On the contrary for δ = 0 and J > 0 the energy band
tally, see e.g. [52]. However, there Γ was independent
widthis∆E J√nandaccordingto(8)bothscalesare T
∼ ofnumberofelectrons,while forquantumcomputingΓ
comparable τ τ . While both time scales τ and τ T
φ χ χ φ
∼ is proportional to the number of qubits since the global
areimportantforthequantumcomputeroperabilityitis
coherence of multi-qubit states should be preserved.
clearthattheeffectsofquantumchaosonthescaleτ are
χ
much more dangerous since after this scale an exponen-
tially largenumber of quantum register states are mixed
as it is shown in Fig. 7. On the contrary on the scale τ
φ
onlythephasesarechangedbutnotthenumberofstates.
Therefore, it is natural to expect that phase spreading
can be more easily suppressed by error-correcting codes
than the onset of quantum chaos.
FIG.9. Sameas inFig. 8 butfor thecorrespondingquan-
tum system (11) with K = 5, k = 20,¯h = 1 realized on
the same computer. The vertical line marks the inversion of
∗
time made by replacement ψ →ψ . The quantum dynamics
remain perfectly reversible. After [53].
The time scales (9) play an important role for quan-
tum computing. Indeed, the gate operations should be
FIG. 8. Time-dependence of the energy E =< n2/2 > in fast enough comparing to these scales to allow to realize
the Chirikov standard map (10) in the chaotic regime with error-correctingcodes [7,8] and to avoida destruction of
K = 5, k = 20 for 1000 orbits homogeneously distributed
operabilityafterthetimescales(9). Here,itisimportant
at n = 0 for t = 0. The straight line shows the theoretical
tostressanimportantpropertyofquantumevolutionfor
diffusion E = k2t/4. All velocities are inverted after t=150
withcomputeraccuracy10−12thatdestroystimereversibility which there is no exponential growth of errors, contrary
to the classical dynamics. An illustration of this fact is
after 20 iterations. After[53].
basedonthedynamicsoftheChirikovstandardmap[53]:
8
n¯ =n+ksin(θ+Tn/2), θ¯=θ+T(n+n¯)/2 (10) many-body quantum systems with interaction and dis-
order are presented. The ˚Aberg criterion represents the
Here, (n,θ) is a pair of momentum and phase vari-
main condition for onset of quantum chaos and different
ables and bar denotes the new values of variables af-
checksperformedforvariousphysicalsystemsconfirmits
ter one period of perturbation. The classical dynam-
validity. The generalizationof these results allows to de-
ics depends only on the chaos parameter K = kT and
terminethe quantumchaosborderforquantumcomput-
for K > 0.9716.. the dynamics becomes globally diffu-
ing. In particular, it is shown that the critical coupling
sive in n. In this regime the classical trajectories are
betweenqubits,whichleadstoquantumchaosandquan-
exponentially unstable and have positive Kolomogorov
tumcomputerhardwaremelting,dropsonlylinearlywith
entropy. Due to thatthe computererrorsgrowexponen-
the number of qubits and is exponentially larger than
tially quickly in time that in practice destroys the time-
the energy levelspacing between eigenstates of quantum
reversibility of the map (10). This fact is illustrated on
computer. In this sense the ideal multi-qubit structure
Fig. 8 [54] where the iterations are done on a computer
is rather robust in respect to perturbations that opens
10−12.
with errorsof the order of Due to exponentialin-
broadpossibilities forrealizationofquantumcomputers.
stabilityofclassicalorbitsthetime-reversibilityfororbits
Of course, the optimal regime for quantum computer
with inverted momentum (n n) completely disap-
operability corresponds to the integrable regime below
→ −
pears after 20 iterations. The situation is absolutely dif-
the quantum chaos border. In this respect the quantum
ferentforthecorrespondingquantumdynamicsdescribed
chaos is a negative effect for quantum computing which
bytheunitaryevolutionoperatorforwavefunctionψ on
should be eliminated. Here it is possible to make an
one iteration [54]: