Quantum Behaviors: synthesis and measurement Martin Lukac Normen Giesecke Sazzad Hossain and Marek Perkowski Department of Electrical Engineering Portland

Quantum Behaviors:Quantum Behaviors:synthesis and measurementsynthesis and measurement

Martin LukacNormen GieseckeSazzad Hossain

and Marek Perkowski

Department of Electrical EngineeringPortland State University1900 SW Fourth Avenue

Oregon, USAE-mail: {lukacm,mperkows}@ece.pdx.edu

Dong Hwa Kim

Dept. of Instrumentation and Control Engn.Hanbat National University,

16-1 San Duckmyong-Dong Yuseong-Gu,Daejon, Korea, 305-719.

E-mail: [email protected]

Overview

Motivations and Problem Definition

Quantum computing basics

Quantum Inductive Learning

Controlled [V/V*] gate synthesis

Measurement dependent synthesis

Simulations and results

Conclusion and future work

Motivations

Human-Human interaction is highly variable, individual, unique, non-repeating, etc.

Emotional Robot, Humanoid Robot

Quantum emotional state machine

Control logic for robotic quantum controllers in order to increase interactivity and quality of communication

Logic synthesis of such circuits is in the middle of this paper

Synthesis from Synthesis from examplesexamples

Quantum mappings – Quantum Braitenberg Vehicles – Arushi ISMVL 2007

Quantum Oracles such as Grover – Yale ISMVL 2007

Emotional State Machines – Lukac ISMVL 2007

Quantum Automata and Cellular Quantum Automata – Lukac ULSI 2007

Motion – Quay and Scott

Quantum Robots for Quantum Robots for TeenagersTeenagers

Quantum Emotional Quantum Emotional Facial GesturesFacial Gestures


• Units are qubits, quantum bits, represented by wave function, on real (observable bases) in the complex Vector Space H.

2cos

2sin

2sin

2cos

2sin

2cos2/

i

iXiIeR Xi

x

Unitary transformations on single and two qubits (rotations in the Complex Hilbert Space), example rotation around X axis :

Because quantum states are complex, they are measured (or observed) before they can be recorded in the real world. The measurement operation describe this fact: 1

5probability of observing output state |0>

45

probability of observing output state |1>

1|0||

1|5

20|

5

1|

||

||'|

*nn

n

MM

MMDifference of complete

measurement and expected measurement in a robot.

Because the coefficients of the states are complex positive and negative), interference occurs allowing to sum or subtract probabilities of observation of each state. Gates such as CV can be used to synthesize permutative functions with real state transition coefficients (boolean reversible functions)


• On of the particular properties of Quantum Computation is the superposition of states: allowing to synthesise quantum probabilistic logic functions

2

1

2

12

1

2

1

*

ii

i

iii

V

and entanglement (initially known as EPR)

11

11

2

1H

2

1

2

100

2

1

2

100

0010

0001

iii

ii

iVControlled

1||,||2

i

ii

i cic

11|00|2

1|

Meaning of entanglement in terms of gestures

Three Types of Quantum Inductive Learning

ab c 0 1

00

01

11

10

0

1

1

1

1

0

0

0

ab c 0 1

00

01

11

10

0

1

1

1

-

-

-

-

ab c 0 1

00

01

11

10

0

1

1

1

V1

V0

V0

V1

ab c 0 1

00

01

11

10

0

1

1

1

V0

V1

M0,1

M0,1

Classical Deterministic Learning

Probabilistic and Quantum Probabilistic Learning

Quantum Probabilistic and Measurement Dependent Learning

V0=V |0>=V * |1>

V1=V |1>=V * |0>

M0,1 - output reading dependent on wether the result is 0 or 1

Controlled [V/V*] gates

• V, V*, C-V, C-V*, are well know elements of quantum logic synthesis for pseudo-boolean (permutative) functions.

10

01** VVVV

V* V* X

V* V I V* V X

CNOT

2

1

2

100

2

1

2

100

0010

0001

iii

ii

iVControlled

01

10, ** VVXVVX

V*

*

V

V

• V, V*, C-V, C-V*, are well know elements of quantum logic synthesis for pseudo-boolean (permutative) functions.

2

1

2

100

2

1

2

100

0010

0001

iii

ii

iVControlled

01

10, ** VVXVVX

V*V M 0 or 1

Various types of measurement

2

1

2

100

2

1

2

100

0010

0001

iii

ii

iVControlled

V*V M0

0 or nothing

1 or nothingM1


If nothing, If nothing, previous action previous action is continuedis continued


2

1

2

100

2

1

2

100

0010

0001

iii

ii

iVControlled

V*V M

Operator built-in the measurement

V

V0 or V1

0 or 1

Measurement here would Measurement here would be non-deterministicbe non-deterministic

Measurement here is Measurement here is deterministicdeterministic

Symbolic Quantum synthesis

Assume function to be synthesized:

ab c 0 1

00

01

11

10

0

-

0

-

1

-

1

-

ab c 0 1

00

01

11

10

0

0

0

1

1

1

1

0

ab c 0 1

00

01

11

10

0

0

1

V1

V0

V0

V1

1In the case when all outputs are deterministic (using only CNOT, CV, CV*, V and V*, the parity of application of each V-type gate on the output must be of order 2n or 0.

When the output is specified by probabilities corresponding to V

0 or V

1 the parity of applying

the V-based gates is odd (2n-1, for n > 0).

Measurement dependent synthesis

• The measurement synthesis is interesting from the behavioral point of view: when a robotic controller generates commands all signals going to classical actuators must be completely deterministic.

• Because measurement is considered as action the robot must do to generate output, the function can be minimized with respect to M (measurement)

10,11,00,011011,1110,0001,0100 f

Assume completely defined reversible function:

With respect to the expected result after the measurement on the output qubits, the function can be written as:

)1()1(),1()1(),1()1(),1()1( 0111010 MMMMMMMMf o

Measurement dependent synthesis (contd.)

• Further introduction of entanglement into the output in the form of Bell bases states:

2

10|01|11|,

2

11|00|10|,

2

11|00|01|,

2

10|01|00|

Allows to rewrite the measurement based definition to a single qubit dependent form. Also note that M0 is the state of the system after being measured for 0, and m0 = 1 is the actual output (value 0) after this measurement :

Using, the fact that we have to measure only a single qubit to obtain a completely specified (not probabilistic) result. The output specification of the function requires in this case a real (not quantum) register holding the values of the measured qubit, allowing to determine whether the measurement operation yielded a correct result

110001

01

00

00

10

01

11

01

10

00

11

00 ,,,,,,,,, mmmmmmmmmmmmmmmmf


All methods have been simulated using Genetic Algorithm to test this approach.

In this case we tested specifically single qubit quantum functions. These are functions in which only one bit is truly quantum, other bits are permutative functions

The quantum symbolic synthesis is based on a circuit-type generator of the form:

abc

d g1

f1

g2

f2

gn

fn

...........


abc

d g1

f1

g2

f2

gn

fn

...........

Functions fi are “simple”:

a) Linear

b) Affine

c) Toffoli-like

They can be binary or multiple-valued

Functions gi are “square roots of unity”:

a) NOT

b) Square-root-of-NOT

c) Fourth-order-root-of-NOT

d) etc

They can be for realization of binary or multiple-valued logic

Exhaustive SearchA* search

Genetic AlgorithmIterative Deepening

e


Example 1:Example 1:

abcd 00 01

00

01

11

10

0

0

1

0

-

1

-

0 0 1

--

0 1

1 0

11 10

abc

000

001

011

010

110

111

101

100

-

VV*

VV

VV*

VV

VVVV*

VV

VV*

d dd

-

I

NOT

NOT

NOT

I

I

NOT *I

0

0

1

0

1

1

1

0

Symbolic synthesis Method

Classical Synthesis of reversible functions applied as a classical Machine Learning

Simulations and results (contd.)

• Circuit for the function from previous slide, realizing a symmetric function on the output (D) qubit:

V V V*V

a

b

c

d

dcbaSf ),,(3,2

Observe:● All controls are linear only

● All targets are square roots and their adjoints only

Observe that this is a generalization of the well-known realization of Toffoli invented by Barenco et al

We can create this type of functions for any number of variables

They are inexpensive in quantum but complex in Reed-Muller


Example 2:Example 2: ab c 0 1

00

01

11

10

-

0

-

1

0

-

1

-

ab c 0 1

00

01

11

10

V

VV

V

VV

-

V

-

V

ab c 0 1

00

01

11

10

V0

NOTNOT

-

V1

-

V1

V0

ab c 0 1

00

01

11

10

V0

01

0

V1

1

V1

V0

Synthesized function matches all required cares

V V

a

b

c

Multi-valued (quaternary) Synthesis of quantum functions

applied as a new Quantum Machine Learning


• Another solution (completely deterministic) is just slightly more complicated:

,2

111|000|111|,

2

101|010|110|

,2

101|010|001|,

2

111|000|000|

ab c 0 1

00

01

11

10

-

0

-

1

0

-

1

-

V V

a

b

c

0 This function can also be easily

synthesized using entanglement and measurement. Simply generate an entanglement circuit creating these bases states:

And define measurement criteria satisfying for each care in the K-map the desired output value:

1111|

,1110|

,1001|

,1000|

00

11

11

00

mM

mM

mM

mM

Measurement dependent Measurement dependent synthesissynthesis

Example 2 Example 2 (contd.):(contd.): Multi-valued (quaternary)

Synthesis of quantum functions applied as a new Quantum Machine Learning

Learning error

Conclusion

Quantum Circuit Measurement

Environment

Measurement dependent Learning

Symbolic quantum learning

Boolean Inputs

Probabilities

Symbolic method assumes known hidden states, predicts probabilistically the output events

We proposed two complementary mechanisms for learning: Symbolic and Measurement Dependent.

Measurement Dependent method assumes known output events and their probabilities – there are several unitary matrices for the same input-output probabilistic behavior (H or V)

Conclusion (contd.)

The quantum symbolic method is ideal for single output reversible functions, heuristics and AI search methods can be easily applied

The measurement dependent method requires the external register of size 2n (or of size of the desired input-output set of data), however the synthesis part is trivial.

Any entanglement circuit can be automatically build for any reversible function using only gates H, CNOT and X.

Future work: extensions to multi-qubit quantum functions, d-level functions

implementation and verificationimplementation and verification of these mechanisms in the Cynthea robotic framework

Future workGrover

Loop

Had

amard

s

Co

nstan

ts

Measu

remen

ts Grover SearchGrover Search

Oracle orQuantum

Circuit

Inp

uts-

senso

rs

Measu

remen

ts

Quantum Braitenberg Quantum Braitenberg VehicleVehicle

Ou

tpu

ts - actu

ators

Inp

uts-

senso

rs

Measu

remen

ts

New New ConceptConcept of of

Real-time Real-time Quantum Quantum

SearchSearch

Ou

tpu

ts - actu

ators

Grover Loop

Co

ntro

lled

Had

amard

s

Co

nstan

ts

ControlControl LEARNINGLEARNING

Additional Slides

Toffoli Gate as an example of Toffoli Gate as an example of composition of affine control composition of affine control

gates and rotation target gates and rotation target gatesgates

Controlled gatesControlled gates

Synthesis Synthesis of Majorityof Majority

Documents

Quantum Behaviors: synthesis and measurement Martin Lukac Normen Giesecke Sazzad Hossain and Marek Perkowski Department of Electrical Engineering Portland