qram nn (handout)

7/30/2019 qRAM NN (Handout)

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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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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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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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Weighted Neural Networks

(It is not about) Weighted Neural Networks

McCulloch-Pitts Neuron

UTSRHIPQx1w1

""EEE

EEEE

EEEE

EEE

UTSRHIPQx2 w2((PP

PPPPPPPP

PP

...yxwvrstu // y

UTSRHIPQxnwn

66nnnnnnnnnnnn


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RAM Based or Weightless Neural Networks




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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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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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



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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|.



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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



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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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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



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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.


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



|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

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