text stringlengths 0 8.13M |
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Moreover,inordertoimprovetheperformancewealso dx |
perform the temporal multiplexing. The temporal mul- =a(y x), (85) |
dt − |
tiplexing has been found to be useful to extract com- |
dy |
plex dynamics on the exponentially large hidden nodes =x(b z) y, (86) |
dt − − |
through the restricted number of the true nodes [38]. In |
dz |
temporal multiplexing, not only the true nodes just af- =xy cz, (87) |
dt − |
ter the time evolution U , also at each of the subdivided |
τ |
V time intervals during the unitary evolution U to con- |
τ with (a,b,c) = (10,28,8/3), (ii) the chaotic attractor of |
structV virtualnodes,asshowninFig.3(b). Aftereach |
Mackey-Glass equation, |
input by S , the signals from the hidden nodes (via the |
xk |
true nodes) are measured for each subdevided intervals d x(t τ) |
after the time evolution by U (v =1,2,...V), i.e., x(t)=β − γx(t) (88) |
vτ/V dt 1+x(t τ)n − |
− |
r(kτ +(v/V)τ) U S r(kτ). (82) |
≡ (v/V)τ xk with (β,γ,n) = (0.2,0.1,10) and τ = 17, (iii) R¨ossler |
attoractor, |
In total, now we have N V nodes, and the output is |
× |
defined as their linear combination: |
dx |
= y z, (89) |
N V dt − − |
(cid:88)(cid:88) |
y k = W jo ,u vtr¯ l(kτ +(v/V)τ). (83) dy |
=x+ay, (90) |
l=1v=1 dt |
dz |
By using the teacher data y¯ L, the linear readout =b+z(x c), (91) |
weights Wout can be determ{ ink e} dk by using the pseudo dt − |
j,v |
inverse. In Ref. [38], the performance of QRC has been |
with (0.2,0.2,5.7), and (iv) H´enon map, |
investigated extensively for both binary and continuous |
inputs. The result shows that even if the number of the x =1 1.4x +0.3x . (92) |
t+1 t t−1 |
qubits are small like 5-7 qubits the performance as pow- − |
erfulastheechostatenetworkofthe100-500nodeshave Regarding (i)-(iii), the time series is obtained by us- |
been reported both in short term memory and parity ing the fourth-order Runge-Kutta method with step size |
check capacities. Note that, although we do not go into 0.02, and only x(t) is employed as a target. For the time |
detail in this chapter, the technique called spatial multi- evolution of quantum reservoir, we employ a fully con- |
plexing [40], which exploits multiple quantum reservoirs nected transverse-field Ising model |
with common input sequence injected, is also introduced |
(cid:88) |
to harness quantum dynamics as a computational re- H = J X X +hZ , (93) |
ij i j i |
source. Recently, QRC has been further investigated in |
ij |
Refs. [41, 71, 74]. Specifically, in Ref. [71], the authors |
use quantum reserovir computing to detect many-body where the coupling strengths are randomly chosen such |
entanglement by estimating nonlinear functions of dein- that J is distributed randomly from [ 0.5,0.5] and |
ij |
− |
sity operators like entropy. h=1.0. Thetimeintervalandthenumberofthevirtual |
nodes are chosen to be τ = 4.0 and v = 10 so as to ob- |
tain the best performance. The first 104 steps are used |
D. Emulating chaotic attractors using quantum for training. After the linear readout weights are deter- |
dynamics mined, several 103 steps are predicted by autonomously |
evolving the quantum reservoir. The results are shown |
To see a performance of QRC, here we demonstrate in Fig. 4 for each of (a) Lorenz attractor, (b) the chaotic |
an emulation of chaotic attractors. Suppose x L is attractor of Mackey-Glass system, (c) R¨ossler attractor, |
a discretized time sequence being subject to a{ cok m} pk lex and (d) H´enon map. All these results show that training |
nonlinear equation, which might has a chaotic behavior. is done well and the prediction is successful for several |
In this task, the target, which the network is to output, hundredssteps. Moreover,theoutputfromthequantum |
is defined to be reservoir also successfully reconstruct the structures of |
these chaotic attractors as you can see from the delayed |
y¯ =x =f( x k ). (84) phase diagram. |
k k+1 { j }j=1 |
11 |
(a) Lorenz attractor |
1 |
0.9 0 .1 9 |
0.8 00 .. 78 |
000 ... 567 00000 ..... 23456 |
xk-15 |
0.4 0 . 01 |
000 ... 0123 9600t Qe Rac Cher 9800 10000 tim 10 e20 s0 tep 10400 10600 10800 11000 0 0.1 0.2 0.3 0.4 x 0. k5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0. x7 0. k8 0 -. 69 1 |
(b) Mackey-Glass time series |
1 0.9 |
0.9 0.8 |
0.8 0.7 |
0.7 0.6 |
0.6 |
00 .. 45 xk-15 00 .. 45 |
0.3 0.3 |
0.2 teacher 0.2 |
0.1 QRC 0.1 |
0 9600 9800 10000 10 e20 s0 10400 10600 10800 11000 0 0 0.1 0.2 0.3 0.4xk 0.5 0.6 0.7 0.8 0.9 |
tim tep |
(c) Rössler attractor |
1 |
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