Download From Quantum Gates to Quantum Learning: recent research and

From Quantum Gates to
Quantum Learning:
recent research and open
problems in quantum circuits
Marek A. Perkowski,
Portland Quantum Logic Group,
Department of Electrical Engineering and Computer Science,
Korea Advanced Institute of Science and Technology, and
Department of Electrical and Computer Engineering,
Portland State University, USA.
The progress in classical computer
technology has been dramatic
Many researchers believe
an even greater revolution
is coming: quantum
1999 Pentium IIIB
1947
First
point
contact
transistor
computers
by Bardeen and Brattain
http://www.pbs.org/transistor/scie
nce/events/pointctrans.html
www.icknowledge.com
Nano-system
How small is a nanometer?
•
•
•
•
•
•
•
•
1 meter
10 mm
1 mm
10 nm
1nanometer
0.1 nm
1 picometer
1 femtometer
–
–
–
–
–
Size of red blood cell
= a millionth of a meter
Size of polio virus
= a billionth of a meter
Size of the hydrogen
atom
– = a trillionth of a meter
– = 10 -15 m, size of a
proton
Number of Atoms in a Useful System
From R. Keyes, IBM J. Res. Develop (1988)
# atoms to store a bit
# dopant atoms/bipolar transistor
History
• 1970s and 1980s, introduction of
quantum computers (Richard
Feynmann, David Deutsch, and
Paul Benioff)
• 1994, Peter Shor’s factoring
algorithm
• 1996, Lov Grover, searching
algorithm
• 1998, 1999, 2001 Isaac L.
Chuang, developed the world's
first 2-qubit, 3-qubit, 5-qubit and
7-qubit quantum computer
People
First
Ideas
(1982)
”
Turing
Machine
…(1936)”
A. Turing
R. Feynmann
“… Quantum
Circuits…(1985)”
D. Deutsch
P. Shorr
“…Factorization
…(1997)”
InfoJiffy
unit: 1 Quantum
bit. Physical system:
2 states
Theory
|0>
•
•
•
•
•
|1>
|0> and |1>
Quantum nature: a combination of both.
In preparing the initial state: only one of the 2 states
On measurement: only one state found.
Probability: the state’s component in the mix
Both preparation and measurement in contact with a macro
system
Qubit in a Ion Trap
Quantum
Logic
Circuits
How a single photon behaves in a
beam splitter?
Beam splitter
50%
1
Single photon
0
50%
Optical sensor
… strange behavior
0
1
0
1
Mach-Zehnder apparatus
Quantum Gate: square root of NOT
0
1
0
1
0
1
1
not
not
NOT
0
Simple theory of the beam-splitter
0
0
1
1
50%
50%
The simplest explanation is that the beamsplitter acts as a classical coin-flip,
randomly sending each photon one way or
the other.
Quantum Interference
0
0
1
1
100%
The simplest explanation must be wrong,
since it would predict a 50-50 distribution.
More experimental data collected to
improve the theory
0
1


0 sin  2 
2  
1 cos  
2
2

A new theory
The particle can exist in a linear combination
or superposition of the two paths
0
1
i
1
0 
1
2
2


0 sin  2 
2  
1 cos  
2
2

i
ei
0 
1
2
2
ei - 1
i(ei  1)
0 
1
2
2
Probability Amplitude and
Measurement
If the photon is measured when it is in the
state 0 0  1 1
then we get 0 with
2
2
probability  0 and |1 with probability 1
0

1

0
1
0
2
1
2
2
 0  1
2
1
Quantum Operations are linear
The operations are induced by the apparatus
linearly, that is, if 0  i 0  1 1
2
then
and
2
1
i
1 
0 
1
2
2
 i
 0 0  1 1   0 
0 
 2
i

  0
 1
2

1


1   1 
2 

1 

 0   0
2

1
i

0 
1
2
2 
1
i 
 1
1
2
2
Quantum Operations are unitary
Any linear operation that takes states
2
2
satisfying
0 0  1 1
 0  1  1
and maps them to states
'0 0  '1 1 satisfying
must be UNITARY
'
0

2
'
1

2
1
Linear Algebra notation for
quantum circuits
0
corresponds to
1
corresponds to
0 0  1 1 corresponds to
1
 
0
0
 
1
1
 0   0 
 0    1     
0
 1   1 
Linear Algebra notation for
quantum circuits
corresponds to

corresponds to






i
2
1
2
1 

2
i 

2
1 0 


i 
0 e 
Linear Algebra notation for
quantum circuits
0

corresponds to






i
2
1
2
1 

2
i 

2

1
0

 


i  
0 e  


i
2
1
2
1 

2
i 

2
1
 
0
What is unitary matrix?
u00 u01 
U

 u10 u11 
is unitary if and only if

u
u
u


00
01
t
00
UU  


u10 u11  u01*
*
u10   1 0
I
*   

u11  0 1 

*
Abstraction
The two position states of a photon in
a Mach-Zehnder apparatus is just one
example of a quantum bit or qubit
Except when addressing a particular
physical implementation, we will simply
talk about “basis” states 0 and 1
and unitary operations like
H
and

Qubits as binary Qudits
•
In multi-valued (MV) Quantum Computing (QC), the unit of memory
(information) is qudit.
•
For instance, ternary logic values of 0, 1, and 2 are represented by a
set of distinguishable different basis states of a qutrit.
•
These states can be a photon’s polarizations or an elementary
particle’s spins.
•
•
After encoding these distinguishable quantities into multiple-valued
values, qutrit states are represented by basis states |0>, |1> and |2> ,
respectively.
A qubit, used in binary QC uses only two basis states, |0> and |1>
•
Qubit and qutrit are then special cases of qudits
A
PA
B
QB
C
R  AB  C






H
1
2
1
2
1 

2
-1 

2
0 0  1 1
Re
+
+
-
|0>
|1>
1
2
1
2
Im
1
2
1
2
|0>
|1>
+
1
-
1
|0>
2
2
|1>
Registertransfer
notation for
quantum
circuits
1 0 


i 
0 e 

cos
|0>
|1>
e i
-
sin
cos
+
sin
Registertransfer
notation for
quantum
circuits
From physical devices to abstracted
quantum circuits
An arrangement like
0

is represented with a network like
0
H

H
0
H
+
+

H
+
+
1
2
1
-
1
2
cos
-
sin
1
2
2
1
2
2
-
1
-
1
-
1
2
2
cos
sin
+
Register-transfer notation for
quantum circuits
Kronecker Product of Matrices
• Superposition property may be mathematically
described using the Kronecker product (tensor
product) operation 
• The Kronecker product of two matrices is defined
as follows:
a
c

b  x


d  z
 x
a

y   z


v   x
c
 z
y
v 
y
v 
x
b
z
x
d
z
y   ax
 
v    az

y    cx
 

v    cz
ay
av
cy
cv
bx
bz
dx
dz
by
bv
dy

dv
Register-transfer diagram
for two Hadamard gates in
parallel
|0>
|0>
H
H
(a)
|0>
|00>
+
+
1
2
1
+
+
|1>
|0>
|01>
|10>
1
-
1
2
+
1
2
1
1
-
1
|1>
|11>
1
-
1
2
2
|00>
2
|01>
2
|1>+
+
1
2
1
|10>
2
2
-
1
-
2
|1>
+
2
2
2
-
1
-
1
-
1
2
|11>
2
(b)
Quantum Parallelism
• Put all 7-bits into a superposition state
• superposition allows quantum computer
to make calculations on all 128 possible
numbers (27) in ONE iteration i.e.
finishes in 1 second.
• Tremendous possibilities… imagine
doing computations on even larger
sample spaces all at the same time!!!
Kronecker Products for more
than one qubit circuits
If we concatenate two qubits

0
0  1 1
 
0
0  1 1

we have a 2-qubit system with 4 basis states
0 0  00
0 1  01
1 0  10
1 1  11
and we can also describe the state as
00 00  01 01  10 10  11 11
or by the vector
  0 0 


  0    0    0 
           
 1 0  1  1
 
 1 1
More than one qubit: superposition
and entanglement
In general we can have arbitrary
superpositions
00 0 0  01 0 1  10 1 0  11 1 1
2
2
2
2
 00   01  10  11  1
where there is no factorization into the
tensor product of two independent qubits.
These states are called entangled.
Measuring multi-qubit systems
If we measure both bits of
00 0 0  01 0 1  10 1 0  11 1 1
we get x y
with probability  xy
2
Classical
Versus
Quantum
Classical vs. Quantum Circuits
• Goal: Fast, low-cost implementation of useful algorithms
using standard components (gates) and design techniques
• Classical Logic Circuits
–
–
–
–
–
Circuit behavior is governed implicitly by classical physics
Signal states are simple bit vectors, e.g. X = 01010111
Operations are defined by Boolean Algebra
No restrictions exist on copying or measuring signals
Small well-defined sets of universal gate types, e.g. {NAND},
{AND,OR,NOT}, {AND,NOT}, etc.
– Well developed CAD methodologies exist
– Circuits are easily implemented in fast, scalable and macroscopic
technologies such as CMOS
Quantum Circuits are different
• Quantum Measurement
– Measurement yields only one state X of the superposed
states
– Measurement also makes X the new state and so
interferes with computational processes
– X is determined with some probability, implying
uncertainty in the result
• States cannot be copied (“cloned”), implying that
signal fanout is not permitted
• Environmental interference can cause a
measurement-like state collapse (decoherence)
Decoherence
Classical versus Quantum
Circuits
• Quantum Logic Circuits
– Circuit behavior is governed explicitly by quantum mechanics
– Signal states are vectors interpreted as a superposition of binary
“qubit” vectors with complex-number
coefficients
2 n -1
 
c i
i0
i i0
i n -1 n-1
– Operations are defined by linear algebra over Hilbert Space and
can be represented by unitary matrices with complex elements
– Severe restrictions exist on copying and measuring signals
– Many universal gate sets exist but the best types are not obvious
– Circuits must use microscopic technologies that are slow, fragile,
and not yet scalable, e.g., NMR
More Quantum Circuit
Characteristics
• Unitary
Operations
– Gates and circuits must be reversible (informationlossless)
• Number of output signal lines = Number of input signal
lines
• The circuit function must be a bijection, implying that
output vectors are a permutation of the input vectors
– Classical logic behavior can be represented by
permutation matrices
– Non-classical logic behavior can be represented
including state sign (phase) and entanglement
Classical vs. Quantum Circuits
Classical adder
cn–1
a0
s0
b0
s1
a1
b1
s2
a2
b2
s3
a3
Sum
b3
cn
Carry
Classical vs. Quantum Circuits
Quantum adder
Feynman
gate
Reversible
Circuits
Reversible Circuits
• Reversibility was studied around 1980 motivated by
power minimization considerations
• Bennett, Toffoli et al. showed that any classical logic
circuit C can be made reversible with modest
overhead
Circuit
i
…


m outputs
“Junk” 
…
Reversible
Boolean
Circuit
…
…
“Junk”



… f(i)



n inputs
Generic
Boolean



i
…

f(i)
Reversible Circuits
• How to make a given f reversible
– Suppose f :i  f(i) has n inputs m outputs
– Introduce n extra outputs and m extra inputs
– Replace f by frev: i, j  i, f(i)  j where  is XOR
• Example 1: f(a,b) = AND(a,b)
a
b
c
Reversible
AND
gate
a
b
f = ab  c
a b c
a b f
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
1
0
• This is the well-known Toffoli gate, which realizes AND
when c = 0, and NAND when c = 1.
Reversible Circuits
• Reversible gate family [Toffoli 1980]
(Toffoli gate)
• Every Boolean function has a reversible
implementation using Toffoli gates.
• There is no universal reversible gate with fewer
than three inputs
Quantum
Gates
• One-Input
gate: NOT
– Input state: c0|0 + c1|1
– Output state: c1|0 + c0|1
– Pure states are mapped thus: |0  |1 and
|1  |0
NOT
– Gate operator (matrix) is
– As expected:

0 1
0 1  
1 0
1 01 0 0 1

0 1
1 0
1


0 
0
NOT
NOT
0


1 
1
One-Input gate: “Square root of
NOT”
– Some matrix elements are imaginary
– Gate operator (matrix): i / 1/ 2 1/ 1/ 2 

– We find:
1/ 1/ 2
i 1

1

 
i / 1/ 2 
2 1 i 
2
1 
i 1
1  1 
i  so |0  |0 with probability |i/2| = 1/2
2 1 i 0
2 1 and |0  |1 with probability |1/  2|2 = 1/2
Similarly, this gate randomizes input |1
– But concatenation of two gates eliminates the randomness!
1 
i 1
i 1  
0 i 
2 1 i 1 i  i 0
NOT
NOT
Quantum Gates
• One-Input gate: Hadamard
1
2

1 1 
1 -1
H
– Maps |0  1/  2 |0 + 1/  2 |1 and |1  1/  2 |0 – 1/  2 |1.
– Ignoring the normalization factor 1/  2, we can write
|x  (-1)x |x – |1– x
• One-Input gate: Phase shift

1 0 
0 e i 

Universal One-Input Quantum
Gate Sets
• Requirement:
|0
U
Any state |y
• Hadamard and phase-shift gates form a universal
gate set
• Example: The following circuit generates
|y = cos  |0 + ei sin  |1 up to a global factor
H
2
H

2

Quantum XOR gate
• Called also Feynman gate or Controlled Not gate.
• This gate allows inputs of |00> and |01> to appear
unchanged at the outputs, but interchanges the pairs
|10> and |11>.
• For example, consider the quantum XOR gate’s
operation for an input |10>.
1
0

0

0
0
1
0
0
0
0
0
1
0   0  0 





0   0  0 

1 1 0
   
0 0 1
|00>
|00>
|01>
|01>
|10>
|10>
|11>
|11>
Quantum XOR gate
|x
|y
CNOT
|x
|x  y
1
0
0

0
0
1
0
0
0 0
0 0
0 1

1 0
|x
|x
|y
|x  y
– CNOT maps |x|0  |x||x and |x|1  |x||NOT x
 |x|0  |x||x looks like cloning, but it’s not.
– These mappings are valid only for the pure states |0
and |1
1
0
0
0

0

0

0
0
0 0 0 0 0 0 0
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 0 1

0 0 0 0 0 1 0
|a
|a
|b
|b
|c
|ab  c
|000>
|000>
|001>
|010>
|010>
|011>
|011>
|100>
|100>
|101>
|101>
|110>
|110>
|111>
|111>
|001>
3-Input gate:
Controlled
CNOT
(C2NOT or
Toffoli gate)
Controlled Quantum Gates
• General controlled gates that control some
1-qubit unitary operation U are useful

u00 u01
u10 u11 
etc.
U
U
U
U
C(U)
C2(U)
Quantum Gates
Universal Gate Sets
• To implement any unitary operation on n
qubits exactly requires an infinite number of
gate types
• The (infinite) set of all 2-input gates is
universal
– Any n-qubit unitary operation can be
implemented using (n34n) gates [Reck
et al. 1994]
• CNOT and the (infinite) set of all 1-qubit
gates is universal
Quantum Gates
Discrete Universal Gate Sets
• The error on implementing U by V is defined as
E(U,V )  max (U - V ) 

• If U can be implemented by K gates, we can
simulate U with a total error less than  with a gate
overhead that is polynomial in log(K/)
• A discrete set of gate types G is universal, if we
can approximate any U to within any  > 0 using a
sequence of gates from G
Quantum Gates
Discrete Universal Gate Set
• Example 1: Four-member “standard” gate set
1
0
0

0
0
1
0
0
0 0
0 0
0 1

1 0
1 
1 1 
2 1 -1
H
CNOT
Hadamard

1 0
0 i 
S
Phase

1
0
0 
e i / 4 
/8
/8 (T) gate
• Example 2: {CNOT, Hadamard, Phase, Toffoli}
Quantum
Circuits
Quantum Circuits
• A quantum (combinational) circuit is a sequence
of quantum gates, linked by “wires”
• The circuit has fixed “width” corresponding to the
number of qubits being processed
• Logic design (classical and quantum) attempts
to find circuit structures for needed operations
that are
– Functionally correct
– Independent of physical technology
– Low-cost, e.g., use the minimum number of qubits or
gates
• Quantum logic design is not well developed!
Quantum Circuits
• Ad hoc designs known for many specific functions and gates
• Example 1 illustrating a theorem by [Barenco et al. 1995]:
Any C2(U) gate can be built from CNOTs, C(V), and C(V†)
gates, where V2 = U
=
U
V
V†
V
Simulation of Quantum Circuits
|0
|1
|x
U
|0
|1
|x
?
=
|0
|0
|0
|0
|0
|0
|1
|1
|1
|1
|1
|1
|x
V|x
V
|x
|x
V†
V
Simulation of Quantum Circuits
Simulation continued
|1
|1
|1
|1
|x
U|x
U
?
=
|1
|1
|1
|1
|1
|1
|1
|1
|0
|0
|1
|1
|x
V
V|x
V†
V|x
V
U|x
Algebraic Analysis of Quantum
Circuits
x1
?
=
x2
x3
U
V
U0
U1
V†
U2
U3
V
U4
U5
• Is U0(x1, x2, x3) = U5U4U3U2U1(x1, x2, x3)
= (x1, x2, x1x2  U (x3) ) ?
Quantum Circuits
Example 1 (contd);
U1  I1  C(V)
1
1 0 0




0 1 
0
0
0 0
1 0
0 v00
0 v10
1
0
0  0
0  0
 
v01  0
 
v11  0

0
0
0 0
0
1 0
0
0 v 00 v01
0 v10 v11
0 0
0
0 0
0
0 0
0
0 0
0
0
0
0
0
1
0
0
0
0 0
0 0
0 0
0 0
0 0
1 0
0 v00
0 v10
0 
0 
0 
0 
0 

0

v01

v11 
Quantum Circuits
Example 1 (contd);
U2  U4  CNOT(x1 , x2 )  I1
1
0
 
0

0
0
0
1
0
0
0
0
1
1
0
0
0
0
0 
1
0

 
 
1 0 1 0


0
0

0
0
0 0 0 0 0 0 0
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 0 0 1 0

0 0 0 0 0 0 1

0 0 0 1 0 0 0

0 0 0 0 1 0 0
Quantum Circuits
Example 1 (contd);
– U5 is the same as U1 but has x1and x2 permuted (tricky!)
– It remains to evaluate the product of five 8 x 8 matrices
U5U4U3U2U1 using the fact that VV† = I and VV = U
1
0
0
0

0

0

0
0
0 1
0 0
0 0
0 0

0  0

0 0

v01 0

v11 0
0 0 0 0 0 0 01
1 0 0 0 0 0 00
0 1 0 0 0 0 00
0 0 1 0 0 0 00

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 00
0 0 0
0
0
0
1 0 0
0
0
0
0 1 0
0
0
0
0 0 1
0
0
0
0 0 0 v00 v 01
0
0 0 0 v10
v11
0
0 0 0
0 0 0
0
0
0
0
v 00
v10
1
0
0
0
 
0

0

0
0
0
0 0 0
0
0
1
0 0 0
0
0
0
1 0 0
0
0
0
0 1 0
0
0
0
0 0 1
0
0
0
0 0 0
1
0
0
0
0 0 0
0 0 0
0 v00 v00  v10v10
0 v 01` v00  v11v10
0
0
0
0
0 0
0
1
0
0
0 0
0
0 v 00 v10
0 0
0
0 v 01
v11 0 0
0
0
0
0
1 0
0
0
0
0
0 1
0
0
0
0
0
0
0
0 0 v00
0 0 v 01



0

0
  U0
0

0

v 00v01  v10v11

v01v01  v11v11 
0
0 1
0 0
0 0
0 0

0  0

0 0

v10 0

v11 0
0 0 0 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 0 0 1
0 0 0 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
01
00
00
00

0 0

1 0

0 0

00
0
0
0
0 0
0
0 
1
0
0
0 0
0
0 v00
v01 0 0
0
0 v10
v11 0 0
0
0
0
0
1 0
0
0
0
0
0 1
0
0
0
0
0
0
0
0 0 v00
0 0 v10
0 
0 
0 
0 

0

v01

v11 
Quantum Circuits
• Implementing a Half Adder
– Problem: Implement the classical functions
sum = x1  x0 and carry = x1x0
• Generic design:
|x1
|x0
|y1
|y0
Uadd
|x1
|x0
|y1  carry
|y0  sum
Quantum Circuits
• Half Adder: Generic design (contd.)
1
0
0
0

0

0

0
0
U AD D 
0
0
0
0

0

0

0
0
0 0
1 0
0 1
0 0
0 0
0 0
0 0 0
0 0 0
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0 0
0 0
0 0
0 0
0 0 0
0 0 0
0 0 0 1 0
0 0 0 0 1
0 0
0 0
0
0
0
0
0

0

0

0
0
0
0
0
0

1

0

0
Quantum Circuits
• Half Adder: Specific (reduced) design
|x1
|x0
|y
|x1
C2NOT
(Toffoli)
CNOT
sum
|y  carry
Walsh Transform for two binary-input manyvalued variables
Classical logic
Quantum logic
Variable 1
•minterms
Variable 1
+
+
+
+
-
-
You need a
butterfly
•H
•H
Butterfly is created
automatically by
tensor product
corresponding to
superposition
Tremendous
potential for
truly innovative
research
Research Potential
• Our community should develop a systematic
methodology and CAD tools for synthesizing,
verifying, testing and simulating of quantum
computers.
• These methods and tools will be counterparts of what
exists now in binary CMOS.
– Re-use spectral approaches, DDs, XOR logic, etc.
• Development of these tools will require understanding
of real quantum circuit technology.
New Frontiers
Quantum Computer
Quantum Mechanics
Quantum Logic
Quantum Computers
Quantum Programming
Open Problems in Quantum
Circuits
• Synthesis of binary quantum
cascades with no garbage or small
garbage
– (Maslov, Dueck, Miller, Kerntopf,
Perkowski, Khlopotine,
Mishchenko, Curtis, Khan, Jha and
Agrawal, Hayes, Markov)
Fredkin
Toffoli
• Synthesis of multiple-valued
quantum cascades
– (Muthukrishnan and Stroud, Miller
et al, Khan, Perkowski, Curtis, Lee,
Denler)
– Universal gates, what are the
counterparts of Toffoli and Fredkin
gates?
Open Problems in Quantum Circuits
• What is the Fault Model for quantum circuits?
– Technology dependent?
• Formal Verification of quantum circuits
• Test Generation for quantum circuits
• Fault Localization of quantum circuits
• Synthesis of testable quantum circuits
• Synthesis of fault-tolerant, error correcting quantum circuits.
Open Problems in Quantum Circuits
• What are universal gates?
• How to calculate costs of elementary gates for each
quantum technology such as NMR or ion trap?
• What are the gates that can be truly realized in a
quantum technology?
• What are the synthesis, analysis and test methods
for sequential quantum circuits?
Open Problems in Quantum Circuits
• Invent new quantum algorithms.
• What are the principles to create quantum algorithms
• The nature of entanglement.
• Quantum computer architectures.
• Quantum formalisms. (Clifford algebras).
• Quantum Logic.
Example 1: First method to realize
MV quantum circuits
MV Tensor Products
• Analogous to binary quantum circuits.
• As an example, consider two qutrits.
• When the two qutrits are considered to represent a
state, that state is the superposition of all possible
combinations of the original qutrits, where:
y 12  y 1 y 2  1 2 00  1 2 01  1 2 02  1 2 10
 1 2 11  1 2 12   1 2 20   1 2 21   1 2 22
This approach to multi-valued quantum circuits requires measurements
with more than two basis states.
Also, new gates should be defined as well as the synthesis methods for
these gates.
Quantum MV Superposition
• The superposition property allows the qubit states to grow
much faster in dimension than classical bits, and the
qudits faster than qubits.
• In a classical system, n bits represent distinct states,
whereas n qubits correspond to a superposition of 2n
states and n qutrits correspond to a superposition of 3n
states.
• Because in contemporary quantum technologies every
qubit is costly, higher radices than 2 give an advantage of
improved processing and storage power at the same
realization cost.
• This is a strong argument for realization of multi-valued
logic in quantum circuits.
• In addition to standard advantages of mv logic, quantum
mv logic may be superior to binary one because of
different nature of entanglement.
Example 2: Second Method to
realize MV quantum circuits.
• The normalization ||2 + ||2 = 1
admits the parametrization  =
cos(/2) e j ,  = sin(/2) e j.
• | = e j (cos ( / 2) |0 + e j  sin
( / 2) |1 ).
• Since the global phase of | has
no observable effect, we may write
| = cos(/2) |0 + e j sin(/2) |1.
• The angles  and  define a point
on the surface of a unit sphere –
the Bloch sphere, see Fig. 1.
• The Bloch sphere provides an
excellent tool to visualize the state
vector of a qubit.
• This is a binary Bloch sphere, but a
multi-valued counterpart of it can be
also created.
Bloch Sphere
Second method to realize multi-valued logic using
binary quantum computing (cont).
• Figure shows the
location of 6 points, that
may correspond to
values of some multivalued algebras.
• For binary logic we use
|0 and |1.
• For quaternary logic we
use |0, |1, |0+|1, and
|0-|1.
• For 6-valued logic we
may use additionally |0
+ j |1 and |0 - j|1.
• A rotation or a
combination of rotations
leads from one value to
any other value.
Second Method to realize MV
quantum circuits (cont).
• Above we showed how multiple-valued logic can
be encoded in binary quantum computing.
• Quaternary logic requires two binary
measurements (readings).
• The first reading distinguishes states |0 and |1,
and the second reading uses additional rotation
gates to distinguish between states |0+|1, and
|0-|1.
• It can be shown that the logic with 2n values
requires n readings.
Example 3: Quantum Circuit
Simulation
• Simulation of quantum circuits plays absolutely
fundamental role in many areas of quantum physics and
engineering.
• Simulation is used to:
– verify correctness of the design,
– analyze its properties and
– find some interesting aspects that cannot be found by “hand and
pencil” methods.
– Fault simulation
– Evolutionary algorithms
• Researchers routinely use quantum simulators to help
them with inventions and verify their design guesses.
Fast simulation is extremely
important
• Matrix methods are slow.
• Acceleration is attempted to be achieved
by two fundamental methods:
– (1) acceleration of standard operations by
using special hardware emulators, parallel
computers or processor networks,
– (2) creating new advanced data structures
to represent quantum data more efficiently
using standard computers.
Example 4: Quantum Decision
Diagrams
• New data structures, such as QUIDDs [Viamontes, Markov,
Hayes] allow for implicit parallelism when executing
Kronecker multiplications on them.
• QUIDDs are based on ADDs and MTBDDs,
• so hopefully in future other decision diagrams may be used
to represent quantum circuits.
• It is also expected that basic software engines used
successfully in classical CAD (such as for instance SAT or
ATPG methods) may be used to deal with quantum circuits.
• Also, the fast simulators based on new types of decision
diagrams should be in future parallelized and possibly
accelerated in FPGA-based boards.
• Even before quantum computers will be available, their
emulations on standard computers and ASIC/FPGA may
prove useful to solve some practical problems.
Example 5: Testing and
diagnosis of quantum circuits
• Patel, Markov, and Hayes showed that reversible circuits
are much better testable than irreversible circuits.
– This is because every test covers half faults and every fault is
covered by half tests.
– The reversible circuits are then “transparent” to faults, making
them well observable and controllable.
• We showed that fault localization in reversible circuits is
easier.
• We presented preliminary results on testing binary
quantum circuits and on fault localization of quantum
circuits.
Testing Quantum Circuits (1)
• The good circuit is simulated.
• Next every possible quantum fault is inserted (our
fault model is inserting arbitrary matrix in place of
fault, this allows to simulate many different types of
faults) and the circuit with fault is simulated in
Hilbert space (no measurement).
• All possible measurement values are calculated
with their probabilities.
• The comparison of a measurement from the unitary
matrix of a correct circuit and a circuit with fault
determines which input combinations (tests) give
different measurements.
• In some cases the circuit is modified for multivalued realization in order to distinguish the values.
Testing Quantum Circuits (2)
• Observe that in contrast to standard testing
and reversible circuits testing, there are three
types of faults in quantum domain:
– (1) faults that can be detected deterministically,
– (2) faults that cannot be detected (like global
phase faults), and
– (3) faults that can be detected by repeated
application of tests, possibly with special
measuring gates.
• These faults can be detected only with
certain probability.
• Thus, quantum testing is probabilistic testing.
Research Challenges in Quantum Test
• Open problems include basically
everything:
– fault models,
– fault simulation,
– test generation,
– test minimization,
– fault coverage,
– fault localization using probabilistic adaptive
trees.
Quantum Computational Intelligence (QCI)
• The two most famous quantum
algorithms to date were created
by Peter Shor and Lov Grover.
• Shor’s algorithm is for factoring
integers:
– It produces an exponential
computational speedup over classical
algorithms
– It can break the RSA encryption
techniques.
• Grover’s algorithm searches an
unordered list of data, to find a
particular item.
– It has a provable quadratic speedup
over the best classical algorithm.
– It is like looking for name of a person
in yellow pages knowing only his
telephone number.
Research Challenges in Quantum
Algorithms for Computational Intelligence
• “How these algorithms can be used in the
field of Computational Intelligence?”.
• Quantum computing is in every particular
instance at least as powerful as standard
computing.
• It is therefore very reasonable to look for
quantum counterparts to all concepts created
in past in:
–
–
–
–
–
algorithm design,
Artificial Intelligence,
Machine Learning,
Computational Intelligence,
Soft Computing.
Future Applications in Structured Search
• Grover algorithm for searching an unstructured database started many
practical applications because of the generality of its main idea – phase
amplification.
• Grover himself extended his algorithm for the structured search problem,
one of the main tough research issues in AI, with a multitude of important
and practical applications, including in EDA.
• Many interesting papers about quantum search using problem structure
were written by Hogg and collaborators.
• Boyer developed bound for quantum searching algorithms.
• The class of NP-complete problems includes:
–
–
–
–
–
–
–
graph coloring,
satisfiability,
planning,
set covering,
combinatorial optimization,
tautology verification
and many other problems
that are useful for instance to solve the synthesis and optimization problems.
Generalizations of gates, circuits
and automata
• Because gates, the basic concept of quantum
computing, are a powerful generalization of gates in
standard computing, researchers are systematically
generalizing all the fundamental concepts of
computing to involve quantum concepts in one way
or another.
• And thus;
– a quantum circuit is a generalization of a combinational
Boolean circuit,
– Quantum Automata (various formalizations) generalize
Finite State Machines,
– Quantum Turing machine generalizes Turing Machines and
Probabilistic Turing Machines,
– and so on.
From CI to QCI
• The same tendency is seen in Computational Intelligence.
• Its concepts and algorithms are being generalized to those of
the Quantum Computational Intelligence (QCI).
• And thus;
– Quantum Neural Networks,
– Quantum Associative Memories,
– Quantum Bayesian Nets,
– Quantum Games,
– Quantum Markets,
– Quantum Agents,
– Quantum Formulas,
– Quantum Fuzzy Networks,
– Quantum Spectral Transforms and Networks,
– Quantum Evolutionary Algorithms,
– Quantum Braitenberg Vehicles,
– and many others
have been created and are actively investigated both theoretically, using
software simulators, hardware emulators and in real quantum circuits.
Importance of intelligent learning
理论计算机科学中的几个问题
应明生
清华大学计算机科学与技术系
智能技术与系统国家重点实验室
Research Challenges in NP
problems
• Because laws of quantum mechanics proved useful to
improve algorithmic performance of some NP problems,
there is a high probability that more problems will find
efficient solutions in quantum domain.
Quantum-Neural Algorithms:
• Quantum Associative Memories of Ventura and Martinez,
• Competitive Learning in Quantum System by Ventura and
Perus.
• While neural net processes real values, quantum NN
processes complex values.
• It includes therefore standard NN and binary computers as
special cases
• Thanks to superposition and entanglement can do much
more.
• Weights that are complex values will allow to express much
more and higher order information.
• Totally new algorithms can be invented for learning and using
such nets.
• QuAM is analogous to a linear associative memory but all
neurons are quantum mechanical gates.
Research Challenges in QCI
• There are dual influences of CI and quantum
computing.
– 1. The quantum ideas can be used to create
powerful quantum-inspired algorithms to solve
many types of problems in EDA, QDA and robotics.
– 2. The ideas and algorithms from many classical
computer science areas can be now used in
quantum domain or transformed and extended to
quantum domain.
• Very little operational software packages that
use these ideas.
Quantum Computational Intelligence
• Quantum Neural Nets
• Quantum Associative Memories
• Quantum Inspired Genetic Algorithms
• Learning by synthesis of quantum circuits
• Other models of learning based on quantum concepts.
• Quantum Braitenberg Vehicles.
In 2020 quantum computing will be
a reality
• As a community, we have a unique chance to work
on the forefront of the future dominating technology.
• Logic design community did not have this opportunity
in the past.
Quantum Circuit
Design and
Mathematics
QuantumQuantum
Information
Design
Automation
and logic
And
Technology
Quantum Computational Intelligence
Conclusions (1)
• Emerging new area of Quantum Design Automation
(QDA).
• Similarly as in design automation, there will appear
sub-areas of:
–
–
–
–
–
–
–
–
–
–
high level quantum synthesis,
logic level quantum synthesis,
quantum test,
quantum verification,
quantum simulation,
quantum software-hardware co-design,
quantum physical design,
automatic learning from examples,
data mining,
and so on.
Conclusions (2)
• At the moment, even a single paper has been not
published in many of these areas
• But surely they will appear in the forthcoming 10
years.
• We outlined some subjective choice of recent
papers as a potential base of future research in
QDA.
• Conventional logic synthesis, test and machine
learning methods, for both binary and multiplevalued logic, form a powerful base of new
approaches in quantum engineering.
Conclusions (3)
• Similarly the data structures like decision
diagrams or fundamental algorithms such as
satisfiability or reachability analysis continue
to have their role.
• Because of high numerical demands of
quantum logic there exist even higher
expectations on these methods.
• Growing mutual influence of QDA and QCI,
leading in long term to their unification.
My husband has not taken his
decision yet, he is not sure if he
should work on quantum computing
Additional slides for
questions
Multi-valued Quantum Circuit
Synthesis
• Let us first briefly summarize current results in binary
quantum circuit synthesis.
• This is the most advanced research area and there are
two gate models for synthesis (especially for permutative
circuits):
–
(1) The first gate model assumes that only gates with limited
number of inputs can be used (for instance universal Toffoli3 gate
that operates on three qubits; P=a, Q=B, R=abc).
– We will call it the limited qubit gate model.
– Observe that while in binary reversible logic all 2-bit gates are
linear and thus cannot be universal, in quantum logic there are
very many universal 2-qubit gates (theoretically infinite).
– They can be all used in the limited qubit gate model, but there are
no constructive methods yet to make use of this fact even for
binary case.
Multi-valued Quantum Circuit
Synthesis
• (2) The second gate model assumes that for any given
number of qubits N for which a function is realized, there
exist a Toffoli gate ToffoliN (or a similar universal gate in
which one data qubit is controlled by more than 2 control
qubits) that operates on N qubits.
• We will call it the unlimited qubit gate model.
• In the first model it was proved by Shende et al that every
N-qubit reversible function which is represented by an
even number of cycles, is realizable without constant wires
(ancilla bits) and every N-qubit function that is represented
by an odd number of cycles is realizable with N+1 wires
(one ancilla bit).
•
•
•
•
•
•
•
•
(Observe that every permutation matrix specifies the permutation of input/output
minterms, so it is a permutation and can be described as a set of cycles of
minterm numbers.
Ancilla bits are also called constant inputs, dummy variables or input garbages).
In general, synthesis using this model is more difficult, but the results are closer
to the minimum.
In the second model every function is realizable, regardless its cycles number.
But it is at the cost of expensive and not necessarily quantum realizable gates
(such gates may require many ancilla bits internally, so they tend to hide the
high cost of realizations obtained by the methods [27,28,65].)
Otherwise, there are methods to realize these complex gates with small ancilla,
but for large N the realization of each complex gate necessitates an exhaustive
number of limited-qubit realizable gates.
The model (2) should be thus combined with post-processing methods based on
local peephole optimization.
So far, not much comparisons between these various synthesis models,
especially for real quantum realizable gates, have been done.
Two ways to synthesize permutative circuits
• The permutative quantum circuit synthesis problems
are formulated in two ways:
– (a) A complete reversible function is specified (as a one-toone mapping, set of permutation cycles, or as a unitary
matrix)
– (b) A irreversible single or multi-output function is specified.
• Some subset of input signals should be returned unmodified as the
output signals.
• The final circuit, together with its constant inputs and garbage outputs
should be reversible.
• A special case of this model is a controlled gate where all inputs
except one have to be reconstructed on the output and there is no
ancilla bits.
• Usually however this model requires M ancilla bits, as many as the
original outputs of the specification function, one for every output.
• In some cases the number of ancilla bits can be smaller than M.
• The first method is more elegant and does not create garbage.
• It is restricted in that it assumes that a Boolean function has been
already converted to a reversible one (by appropriate adding of ancilla
bits).
• For some formulations (like evolutionary programming and search)
this method allows to be easily extended to non-permutative unitary
matrices.
– So far, however, only small circuits can be synthesized using this method,
even using very advanced algebraic and group-theoretic methods to
decompose a larger matrix to a composition of smaller matrices.
• Because of its formulation, the second way is more similar to
traditional logic synthesis.
• Methods developed previously for ESOPs, GFSOPs and similar forms
in the AND/XOR logic synthesis are used for larger circuits, rather than
methods specific to reversible design.
Research Challenges
• This “adapted” approach allows now to realize larger
functions than the approach from (a), but when applied to
multi-output functions usually leads to high garbage (one
ancilla bit for each output).
• In the long run, perhaps this kind of methods will be better
scalable since they use the structure of the function rather
than relying on heuristic search methods, especially that
there are no strong heuristics known so far.
• Finding structure in problems and finding good heuristics
are the interrelated problems for future research, which will
perhaps combine both ways (a) and (b).
• The problem of optimal conversion from irreversible to
reversible function has been not solved yet.
Four Synthesis Models
•
There exist the following synthesis models, both for
binary and multiple-valued logic:
1. limited qubit gate model and full reversible function (way
a). Usually zero or one ancilla bits are expected.
2. unlimited qubit gates and full reversible function (way a).
Usually zero or one ancilla bits are expected.
3. limited qubit gates and single output function (way b).
Usually at most M ancilla bits are expected.
4. unlimited qubit gates and irreversible input function (way
b). Usually at most M ancilla bits are expected.
• Comparing to binary quantum circuit synthesis, multiple-valued quantum
circuit synthesis is a relatively immature area of research.
• One can expect that it will repeat the history of development of algorithms in
binary reversible logic.
• In binary, model (1) has been developed in [84].
• As related to multiple-valued quantum circuits, the model (1) of reversible
quantum circuits synthesis above has been investigated by [20] and by a
Genetic Algorithm approach from [54].
• Model (2), investigated for binary case in [27,28,63,65,66], has been not yet
investigated for multiple-valued logic (although [78] explains how it can be
done).
• Model (3) is researched in paper [55] and some other preliminary results
appear also in [78].
• Model (4) has been investigated in [4,50-55,59,60].
• It is important to distinguish among these four models, to avoid unrealistic
claims of superiority of one method over another, since obtaining solutions in
some of these models is much easier than in the other ones.
Research Challenges
• Objective comparison of the methods on many
large examples and using standardized
benchmarks should be a topic of further
research.
• Much work is left to be done in defining new
universal multi-valued quantum gates and the
(partially regular) structures to be build from
them.
• Approaches that use known universal gates
have the benefit of prior research (such as logic
synthesis using Galois Field operations), but can
be very costly and inefficient.
•
•
•
•
Below we give a complete characteristics of papers in multi-valued quantum logic
synthesis. Khan and Perkowski adapted the GFSOP (Galois Field Sum of Products)
method to permutative (ternary) quantum circuits [52,53].
The algorithm is based on finding a ternary decision diagram, and flattening it to
quantum cascade-realizable GFSOP.
In another work [54] these authors use Genetic Algorithm to synthesize multi-output,
no-garbage cascades of arbitrary ternary quantum gates.
The approach presented by Miller et al [65] is an extension of their greedy algorithm
for binary circuits [27,28,63]. A non-published extension to their work presents also a
method to encode ternary logic using standard binary qubits [66]. Observe that while
binary quantum logic uses 1800 rotation, and the quaternary logic from [49] uses 900
rotations, they use 1200 rotations for one vertical plane of Bloch Sphere in ternary
logic. While both ternary and quaternary model use two measurements to distinguish
encoded signals, the quaternary method is more efficient. A paper [49] based on SAT
and reachability analysis uses quaternary quantum logic to synthesize exact
minimum binary circuits from Feynman, Inverter, Controlled-V and Controlled-V+
gates. (V is called a “square-root-of-NOT” since its repeated application negates the
input signal, V V = NOT). A simple adaptation of this method allows to realize also
quaternary quantum circuits with arbitrary input and output signals [78].
Research Challenges
• Recent works suggest that many uniform general
methods can be created to realize various multiple-valued
logics that will use generalized rotations with respect to 3
orthogonal basis axes, rotations by angles 2/k, where
k>1 is a natural number.
• In general, rotations with respect to any axis n can be
used, but using some of Z, X, and Y simplifies gates.
• Every existing algorithm for binary quantum circuit design
can be extended to its various multiple-valued quantum
counterparts, but these generalizations are not trivial and
algorithms that use these gates are numerically very
challenging.
• These problems form then a good base for new research
by people who understand search-based EDA algorithms
and multiple-valued logic.
Figure 2. 3*3 Toffoli gate
A
PA
B
QB
C
R  AB  C
• Figure 2 presents a standard binary reversible
Toffoli gate.
• Its ternary counterpart has Galois Field 2
operations of multiplication and addition replaced
with Galois Field(3) operations.
• Observe that the internal structure of this gate is complex when
using quantum realizable gates (Figure 3). The Controlled-V gate
works like this: when the control (top) signal is |0>, the data input is
forwarded to output with no change. When the control signal is |1>
the operation of the lower box (so-called V) is executed. In our case
this is a square-root-of-NOT operation. Thus if two Controlled-V
gates in series are controlled by the same signal A, if A=1 then their
qubit data line is a negation. Two such gates in series serve then as
a controlled-NOT or Feynman gate. Also, the operation of V and V+
annihilate ( V V+ = I ) . The reader can simulate “by hand” the circuit
from Figure 3a to see that it truly realizes the Toffoli3 gate. Let us
observe that the circuit from Figure 3a can be redrawn to one from
Figure 3b. This circuit emphasizes that both CNOT, CV and C V+
are Controlled-Gates that leave data signal unchanged when the
control is |0> and apply its internal transformation (the symbol of this
transformation is in the input to multiplexer) when the control is |1>.
•
•
•
•
•
•
•
•
•
Observe that the internal structure of this gate is complex when using
quantum realizable gates (Figure 3).
The Controlled-V gate works like this: when the control (top) signal is |0>,
the data input is forwarded to output with no change.
When the control signal is |1> the operation of the lower box (so-called V) is
executed.
In our case this is a square-root-of-NOT operation.
Thus if two Controlled-V gates in series are controlled by the same signal A,
if A=1 then their qubit data line is a negation.
Two such gates in series serve then as a controlled-NOT or Feynman gate.
Also, the operation of V and V+ annihilate ( V V+ = I ) .
The reader can simulate “by hand” the circuit from Figure 3a to see that it
truly realizes the Toffoli3 gate.
Let us observe that the circuit from Figure 3a can be redrawn to one from
Figure 3b.
This circuit emphasizes that both CNOT, CV and C V+ are Controlled-Gates
that leave data signal unchanged when the control is |0> and apply its
internal transformation (the symbol of this transformation is in the input to
multiplexer) when the control is |1>.
• Observe that any single-qubit operation can be written in the
box, and also that any single qubit operation can be inserted
to the control and data lines.
• The control can be from top (as in the Figure 3b) or from the
bottom.
• The composition of this kind of multiplexed operations
allows to create arbitrary permutative gate of reversible logic
[55,59].
• Also, an arbitrary two-qubit quantum gate (described by a
unitary matrix) can be constructed from such gates.
• These methods can be used to hierarchically synthesize
larger circuits [55,59] and can be generalized to ternary (or
in general multi-valued) logic (see Figure 4) for the
realization of universal ternary permutative controlled gate.
• Universal quantum gate is created when operations are
single-qubit ternary rotations.
Figure 3. Smolin/DiVincenzo
realization of Toffoli gate as a
prototype of a regular controlled
quantum structure: (a) standard
notation, (b) notation used in this
paper to emphasize the similarity
•Observe that any single-qubit
operation can be written in the
box, and also that any single
qubit operation can be inserted
to the control and data lines.
•The control can be from top
(as in the Figure 3b) or from
the bottom.
•The composition of this kind of
multiplexed operations allows
to create arbitrary permutative
gate of reversible logic [55,59].
•Also, an arbitrary two-qubit
quantum gate (described by a
unitary matrix) can be
constructed from such gates.
A
P
B
R
C
V
V+
V
(a)
S
P
A
R
B
+1
+1
C
S
V
V
V+
(b)
Figure 4: Conceptual ternary
multiplexer
op = Logical Operations:
+0, +1, +2 represent Galois
Addition of constants 0, 1,
and 2, respectively
01, 02, 12 represent logical
replacement i.e. a 01
operation will replace 0->1,
1->0, and 2->2
•Also, an arbitrary two-qubit quantum gate
(described by a unitary matrix) can be
constructed from such gates.
•These methods can be used to
hierarchically synthesize larger circuits
[55,59] and can be generalized to ternary
(or in general multi-valued) logic (see
Figure 4) for the realization of universal
ternary permutative controlled gate.
•Universal quantum gate is created when
operations are single-qubit ternary rotations
Other problems in MV QC
(1)New models of gates, such as above, that will be close to
realization and at the same time would allow creation of
efficient synthesis algorithms, also for large circuits.
(2)Development of methods based on unitary matrix
decomposition, group theory, Lie groups and Clifford
algebras,
(3)Methods for incompletely specified functions, to be used in
machine learning and data mining,
(4)Geometrical and topological visualization methods to help
intuition of designers to design multi-qubit circuits (for
instance generalizations of Bloch sphere, QUIDDs and
Karnaugh Maps),
(5)Efficient methods for local optimization of quantum circuits
on many levels of description,
(6)High-level quantum hardware description languages that
will play in QDA a role similar to VHDL and Verilog in EDA,
(7)Development of formalisms and synthesis methods for
sequential circuits.
•
•
•
•
•
•
•
•
•
•
The name NP means non-deterministically polynomial, because there are no deterministic
algorithms to solve NP problems in polynomial time (w.r.t the size of the problem).
Any problem in the NP-complete class can be transformed into any other problem in this NPcomplete class using polynomial number of steps.
The quantum search algorithms can be used to solve the ”constraint satisfaction problems” into
which all other NP-complete algorithms can be reduced [17].
In a constraint satisfaction problems (SAT is the simplest example, graph coloring is another one)
we deal with multi-valued variables and constraint rules on value relations between values of
subsets of variables (relations like, “two adjacent nodes in a graph should have different colors”).
In other words, one has to find assignment of values b to all a variables so that all constraints are
satisfied.
All such problems can be reduced theoretically to SAT, but this is not necessarily the best way to
solve them.
On a classical computer O(ba) assignments must be searched before finding a valid solution, if
any.
Using heuristics, or domain-dependent knowledge of a particular problem’s structure, the search
can be dramatically speeded up to O(bka), where k1 and is problem dependent.
Grover’s quantum search algorithm for structured problems further reduces the number of states
searched to O(bak/2), which means a polynomial speedup over classical algorithms.
This may be enough to solve many currently intractable problem instances [58].
Quantum Gate Arrays
1-bit full adder
|c>
|c>
|x>
|x>
|y>
|y>
|0>
|s>
|1>
|1>
|0>
|c’>
|0>
|0>
Let |c> = |1>,
|0>
|x> = |0>,
|1>
|y> = |1>
|s> = |0>,
|c’> = |1>
Computation
0 0 0
G(0 0 0)
0 0 1
G(0 0 1)
0 1 0
0 1 1
G(x)
G(0 1 0)
G(0 1 1)
1 0 0
G(1 0 0)
1 0 1
G(1 0 1)
1 1 0
1 1 1
QC
G(1 1 0)
G(1 1 1)
Research Challenges on MV
quantum
• There are very few papers on:
–
–
–
–
realization of multiple-valued quantum circuits,
design of practical MV quantum circuits,
algorithms using MV quantum circuits,
Quantum Computational Learning based on MV logic
• No known work on:
– testing,
– simulation and
– algorithms for multiple-valued quantum circuit exist and
• Develop respective theories and QDA tools.
• Develop Binary-encoded model of MV quantum
computing.
• Develop truly multi-valued quantum model of multivalued computing.
Research Challenges in QCI
• Because previous work on computational learning and particularly
constructive induction designs arbitrary structures of arbitrary gates, it is
applicable also to these structures and new algorithms can be created
that generalize Ashenhurst Curtis decomposition.
• QuAMs are worse than classical algorithms on generalization, and our
algorithms are very good in generalization.
• Therefore we believe that by extending model of QuAMs, a more
general quantum structures will be found that will have good properties
of QuAMs such as storing exponential number of patterns but will be
also good in generalizing. It is well known that there exist animals with
very few neurons, such as nematode worms.
• Still they can exhibit much more complex behaviors that a robot
controlled by few neurons.
• The neuron used in NN theory is thus a big simplification of real neuron,
and it is possible that quantum computing is used in brains of animals.
• In any case, the fact that actual neurons are more powerful than their
current models is a powerful argument to investigate generalized models
of neurons - especially quantum neurons.
Research Challenges in QCI
•
•
•
•
•
•
•
•
•
•
Applicability of quantum paradigms in order to improve a Genetic Algorithm for
solving the traveling salesman problem.
The results of simulating quantum Genetic Crossover operators suggest that
indeed quantum computation can speed up the search for solutions to the
traveling salesman problem.
Several successful experiments of various variants of Quantum-inspired GA
have been described for several applications [40].
In [30] quantum algorithms for searching trees are discussed, there are
examples of trees for which the classical algorithm requires time exponential in
n, but for which the quantum algorithm succeeds in polynomial time.
Spectral Associative Memories (SAMs) are classical networks inspired by
quantum mechanics and proposed by Spencer.
They are quantized frequency domain formulations of conventional Contents
Addressable Memories (CAMs).
Non-local connectivity is made virtually by spectral convolution.
In classical CAMs attractors scale quadratically or polynomially.
In contrast, SAMs scale linearly with memory dimension. One model of the
neuron [61,62] is based on quantum holography [19].
Phase is not only the essential parameter of physical significance, as in the
postulated model of quantum neural information processing, but the essential
means by which holograms i.e. the 3 dimensional representations of objects
may be encoded, decoded or transmitted.
Quantum Notation
• As shown, the resultant unitary matrix of an arbitrary quantum
circuit is created by matrix multiplications or Kronecker
multiplications of matrices of its composing sub-circuits.
• Various quantum notations contribute to the difficulty in
understanding the concepts of quantum computing.
• Generally, however, we believe that once the minimal amount
of formalism is understood, logic researchers can quickly
contribute to new designs.
• Much can be learned from the history of Electronic Design
Automation as well as from MV logic theory and design.
• The lessons learned there should be used to design efficient
QDA tools for MV quantum computing.
• Here we include the absolute minimum amount of formalism
sufficient to start independent software development by people
who have sufficient background in EDA tools and algorithms
such as search or evolutionary programming