Quantum Behaviors: synthesis and measurement Martin Lukac Normen Giesecke Sazzad Hossain and Marek...
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Transcript of Quantum Behaviors: synthesis and measurement Martin Lukac Normen Giesecke Sazzad Hossain and Marek...
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 computing basics
• 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)
Quantum computing basics
• 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
Various types of measurement
If nothing, If nothing, previous action previous action is continuedis continued
Various types of measurement
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
Simulations and results
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
...........
Simulations and results
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
Simulations and results
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
Simulations and results (contd.)
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
Simulations and results (contd.)
• 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
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
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