qram nn (handout)
TRANSCRIPT
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Quantum RAM Based Neural Networks
Quantum RAM-based Neural Networsk
Wilson Rosa de Oliveira
DEInfo-UFRPE
Seminrios do Grupo
Transporte Quntico em Sistemas MesoscpicosDF UFPE, 2009
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Quantum RAM Based Neural Networks
Abstract
A mathematical quantisation of a Random Access Memory (RAM)
is proposed starting from its matrix representation. This quantum
RAM (q-RAM) is employed as the neural unit of q-RAM-based
Neural Networks, q-RbNN, which can be seen as the quantisation
of the corresponding RAM-based ones. The models proposedhere are direct realisable in quantum circuits, have a natural
adaptation of the classical learning algorithms and physical
feasibility of quantum learning in contrast to what has been
proposed in the literature.
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Quantum RAM Based Neural Networks
Outline
1 Weighted Neural Networks2 RAM Based or Weightless Neural Networks
RAM classically defined
Weightless Neural Network3 Mathematical Quantisation and Quantum computation
Mathematical QuantisationQuantum Computation
A toolkit for a quantum programmer4 A new look at RAM5 Quantum WNN
Quantum Probabilistic Logic Node: q-PLN.Quantum Multi-valued PLN: q-MPLN.
Generalising the q-MPLN.6 Quantum Weighted NN7 Conclusion and Future Works
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Quantum RAM Based Neural Networks
Weighted Neural Networks
(It is not about) Weighted Neural Networks
McCulloch-Pitts Neuron
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Quantum RAM Based Neural Networks
RAM Based or Weightless Neural Networks
RAM classically defined
RAM Based or Weightless Neural Networks
RAM classically defined
A RAM with n inputs bits capable of string mbits can be seen asa finite map C : {0, ..., 2n} {0, ..., 2m}, where C[k] is the contentof the memory addressed by k:
s 11 . . . 1 C[2n 1]d 11 . . . 0 C[2n 2]
x1
...... y
... 00 . . . 1 C[1]xn 00 . . . 0 C[0]
Q t RAM B d N l N t k
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Quantum RAM Based Neural Networks
RAM Based or Weightless Neural Networks
Weightless Neural Network
Logical Neural Network(WNN)
Probabilistic Logic Neuron (PLN)
Its a RAM node that store a probability p
In PLN this probability can be 0, 1 or u = 0,5
y =
0, if C[I] = 01, if C[I] = 1random(0,1), if C[I] = u
Multi-Value Probabilistc Logic Neuron (MPLN)
It is a RAM node that store probability pThis probability can vary in range [0,1]
The output of a MPLN node will be 1 with probability p and 0
with probability 1 p
Q ant m RAM Based Ne ral Net orks
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Quantum RAM Based Neural Networks
Mathematical Quantisation and Quantum computation
Mathematical Quantisation
Mathematical Quantisation and Quantum
computationMathematical Quantisation
Nik Weaver in Mathematical Quantizationa: "The fundamental
idea of mathematical quantisation is sets are replaced with
Hilbert spaces".I would add: functions are replaced with linear maps.
categorically: Set is replaced with Hilb.
aN. Weaver. Mathematical Quantization. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton,
Florida, 2001
Quantum RAM Based Neural Networks
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Quantum RAM Based Neural Networks
Mathematical Quantisation and Quantum computation
Quantum Computation
Mathematical Quantisation and Quantum
computationQuantum Computation
classical bits {0, 1} as an orthonormal basis in a convenientHilbert space.
A general state of the system (a vector) can be written as:| = |0 + |1Multiple qubits are obtained via tensor products. By linearity
we just need to say how tensor behaves on the basis:
|i |j = |i |j = |ij , where i, j {0,1}
Operations on qubits are carried out only by unitary operators.Quantum algorithms on n bits are represented by unitary
operators U over the 2n-dimensional complex Hilbert space:
| U|.
Quantum RAM Based Neural Networks
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Quantum RAM Based Neural Networks
Mathematical Quantisation and Quantum computation
A toolkit for a quantum programmer
A toolkit for a quantum programmer includes:
Operators:
I =
1 0
0 1
I |0 = |0I |1 = |1 X =
0 1
1 0
X |0 = |1X |1 = |0
H = 1
2 1 1
1 1 H
|0
= 1/
2(
|0
+
|1
)
H |1 = 1/2(|0 |1)
Quantum Circuits:
......U ba
where
= X X
Quantum RAM Based Neural Networks
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Quantum RAM Based Neural Networks
A new look at RAM
A new look at RAM
RAM as Matrix
To view a RAM as circuit composed of matrices acting on cbits(bits represented as basis vectors) is instructive.
A = I 0
0 X
where
A |00 = |0 I|0A |10 = |1 X |0
RAM as Circuit (read only mode)
|a ba|b ba|0 U0 U1 U2 U3 ba o
where Ui = I or X
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Quantum RAM Based Neural Networks
A new look at RAM
A new look at RAM
RAM as Circuit (read/write mode)
A two-bit RAM with the more general and capable of "storing" cbits
matrix A:
|a pi|b pi
|0
A A A A pi o
|s |s |s |s
Quantum RAM Based Neural Networks
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Q
Quantum WNN
Quantum Probabilistic Logic Node: q-PLN.
Quantum Probabilistic Logic Node: q-PLN.
The values stored in a PLN Node 0, 1 and u are, respectively, represented
by the qubits |0 , |1 and H|0 = |u. The probabilistic output generator ofthe PLN are represented as measurement of the corresponding qubit.
The random properties of the PLN are guaranteed to be implemented by
measurement of H |0 = |u from the quantum mechanics principles.Learning is to change the matrix. How to achieve this?Universal deterministic Programmable Quantum Gate Arrays are not
possible in generala but the q-PLN requires only three possible types of
matrix.
A =
I 0 0 0
0 X 0 0
0 0 H 0
0 0 0 U
where
A |000 = |00 I|0A |010 = |01 X |0A |100 = |10 H|0
aM. A. Nielsen and I. L. Chuang. Programmable quantum gate arrays. PRL, 79(2):321324, Jul 1997.
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Quantum WNN
Quantum Multi-valued PLN: q-MPLN.
Quantum Multi-valued PLN: q-MPLN.
In the classical each MPLN node stores the probability of the node having
the output 1. We do the same for the q-MPLN. In order to accomplish that
we simply define the matrices Up:
Up = p1 p p
p p1 p With this notation it is routine to check that U0 |0 = |0, U1 |0 = |1 andU1
2|0 = |u are the corresponding values in a q-PLN node.
A general q-MPLN node has p varying in the (complex) interval [0, 1] whichis useful for the quantisation of the pRAM networks.
A =
0BBBBB@
Up1 0 0 00 Up2 0 0 0...
. . .. . .
. . ....
0 0 0 Upn1 0
0 0 0 0 Upn
1CCCCCA
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Quantum WNN
Quantum Multi-valued PLN: q-MPLN.
Quantum Multi-valued PLN: q-MPLN.
|a pi m1|a
pi
m2
|0
A A A A
pi os11 s21 s31 s41...
......
...
s
1
k
s2
k
s3
k
s4
k
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Quantum WNN
Generalising the q-MPLN.
Generalising the q-MPLN.
| #"! #"! || #"! #"! |
|
A A A A
U|s11 s21
s31 s41
......
......
s1k s
2k s
3k s
4k
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Quantum Weighted NN
Quantum Weighted NN
Altaisky: Quantum neural network
M. V. Altaisky. Quantum neural network. Technical Report, Joint Institute for Nuclear
Research, Russia, 2001.
|y = Fn
j=
1
wjxj,
Kouda: Qubit NN
Noriaki Kouda at.al.: Qubit neural network and its learning effciency. Neural Comput &
Applic (2005) 14:114121, 2005.
f( )1
f( )k
f()1
x1
xk
f(y)arg(u)u z
.g()2
y
.
.
.
.
.
.
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Conclusion and Future Works
Conclusion and Future Works
The q-RAM proposed above are directly realisable in quantum
circuits (quantum hardware) and can be used for the
quantisation of all the (classical) RAM-based neural neural
networks and their learning algorithms.
Quantisation of the WISARD, GSN, pRAM and GRAMT arebeing investigated.
Encouraging results on the natural adaptation of the learning
algorithms and physical feasibility of quantum learning for the
q-MPLN models are reported ina.
Another line of research being pursued is the relationshipbetween quantum neural networks and the quantum
automata.
aW. R. de Oliveira et.al. Quantum logical neural networks. SBRN 2008. 10th Brazilian Symposium on Neural
Networks, 2008, pages 147152, Oct. 2008