1. 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.

2. The computer as we know it?
1999 Pentium IIIB
1947 First point contact transistor
by Bardeen and Brattain

3. Nano-system How small is a nanometer?
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
= m, size of a proton

4. 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
Reddy

5. People First Ideas…(1982)” Turing Machine …(1936)” A. Turing
R. Feyemann
“… Quantum Circuits…(1985)”
“…Factorization …(1997)”
D. Deutsch
P. Shor

6. Number of Atoms in a Useful System From R. Keyes, IBM J. Res
Number of Atoms in a Useful System From R. Keyes, IBM J. Res. Develop (1988) # atoms to store a bit # dopant atoms/bipolar transistor

7. EX: 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!!!

8. Jiffy Quantum Theory Info unit: 1 bit. Physical system: 2 states
|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

9. 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

10. Qudits
Qudits exist in a linear superposition of states, and are characterized by a wave function .
As an example (), it is possible to have light polarizations other than purely horizontal or vertical, such as slant 45 corresponding to the linear superposition of .
In ternary logic, the notation for the superposition is , where , , and  are complex numbers.
These intermediate states cannot be distinguished, rather a measurement will yield that the qutrit is in one of the basis states, , , or .
The probability that a measurement of a qutrit yields state is , state is , and state is .
The sum of these probabilities is one.
The absolute values are required since, in general, ,  and γ are complex quantities.
Pairs of qutrits are capable of representing nine distinct states,, , , , , , , , and , as well as all possible superpositions of these states.

11. Quantum Logic Circuits

12. Quantum Logic
Single photon
Specchio
50% 1
50%
Optical sensor

13. … strange behavior 1 1

14. Quantum Gate 1 1 1 1
NOT

15. Qubit

16. Qubit in a Ion Trap

17. Deterministic Turing Machine
Initial State
Final State
Deterministic Turing Machine transits deterministically from initial to final state.

18. Probabilistic Turing Machine
Probabilistic output states P4
Probabilities of final output states P5 P1 P6 P2
P = P2P7 + P3P8 P7 P3 P8 P9

19. Quantum Computation A = A1A2 + A3A4 P = |A1A2 + A3A4|2
+2Re(A1A2A3A4) A1 A2 A3 A4

20. Decoherence

21. A beam-splitter
The simplest explanation is that the beam-splitter acts as a classical coin-flip, randomly sending each photon one way or the other.

22. Quantum Interference
The simplest explanation must be wrong, since it would predict a distribution.

23. More experimental data

24. A new theory
The particle can exist in a linear combination or superposition of the two paths

25. Probability Amplitude and Measurement
If the photon is measured when it is in the state then we get with probability

26. Quantum Operations
The operations are induced by the apparatus linearly, that is, if
and
then

27. Quantum Operations Any linear operation that takes states satisfying
and maps them to states
must be UNITARY

28. Linear Algebra
corresponds to
corresponds to
corresponds to

29. Linear Algebra
corresponds to
corresponds to

30. Linear Algebra
corresponds to

31. Linear Algebra
is unitary if and only if

32. 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 and
and unitary operations like
and

33. (a) (b) (c) (d) + |0> + - |1> |0> |0> + - |1> |1> ReIm
|1>
|0>
|0> +
(d) -
|1>
|1>

34. |0>
|1>
(b)
(a)
(c) - +
cos
sin
(d)

35. An arrangement like
is represented with a network like

36. (a) + - + -
cos -
sin
cos +
sin
(b)

37. |0> |0> (a) |0> |1> |1> |1> |0> |1> (b)+ - + -
|00>
|0>
|00>
|01>
|1>
|01>
|1>
|1> + - + -
|0>
|10>
|10>
|1>
|11>
|11>
(b)

38. More than one qubit If we concatenate two qubits
we have a 2-qubit system with 4 basis states
and we can also describe the state as
or by the vector

39. More than one qubit In general we can have arbitrary superpositions
where there is no factorization into the tensor product of two independent qubits.
These states are called entangled.

40. Measuring multi-qubit systems
If we measure both bits of
we get with probability

41. Classical Versus Quantum

42. 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 =
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

43. Classical vs. 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
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

44. Quantum Circuit Characteristics
Unitary Operations
Gates and circuits must be reversible (information-lossless)
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

45. Quantum Circuit Characteristics
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)

46. Classical vs. Quantum Circuits
cn–1 s0 s1 s2 s3 cn a0 b0 a1 b1 a3 b3 a2 b2
Sum
Carry
Classical adder

47. Classical vs. Quantum Circuits
Quantum adder

48. Reversible Circuits

49. Reversible Circuits          … … … … … i f(i) i f(i)
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 …   
n inputs
Generic
Boolean
Circuit
m outputs
f(i)       … i … i
“Junk”
Reversible Boolean
Circuit    … …   
f(i)
“Junk”

50. 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)
This is the well-known Toffoli gate, which realizes AND when c = 0, and NAND when c = 1.
a b c a b f
Reversible
AND
gate a b
f = ab  c c

51. 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

52. Quantum Gates

53. 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
Gate operator (matrix) is
As expected:
NOT
NOT

54. 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
Gate operator (matrix) is
As expected:
NOT
NOT

55. NOT =

56. Quantum Gates One-Input gate: “Square root of NOT”
Some matrix elements are imaginary
Gate operator (matrix):
We find:
so |0  |0 with probability |i/2|2 = 1/2
and |0  |1 with probability |1/  2|2 = 1/2
Similarly, this gate randomizes input |1
But concatenation of two gates eliminates the randomness!

57. Quantum Gates H  One-Input gate: Hadamard One-Input gate: Phase shift
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 H 

58. Quantum Gates U H Universal One-Input Gate Sets Requirement: |0
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 U
|0
Any state |y 2 H

59. Quantum Gates Two-Input Gate: Controlled NOT (CNOT) |x |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
Serves as a “non-demolition” measurement gate

60. (b) (a) (c) (d) |x |y |x  y |x |y |x  y CNOT |00> |01>
|10>
|11>
|x
|y
|x  y
(c)
(d)

61. Quantum Gates 3-Input gate: Controlled CNOT (C2NOT or Toffoli gate)
|b
|b
|c
|ab  c

62. (a) (b) (c) |a |b |c |ab  c |000> |000> |001> |001>
|010>
|010>
|011>
|011>
|100>
|100>
(c)
|101>
|101>
|110>
|110>
|111>
|111>

63. Quantum Gates
General controlled gates that control some 1-qubit unitary operation U are useful
etc. U U U U
C(U)
C2(U)

64. 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

65. Quantum Gates Discrete Universal Gate Sets
The error on implementing U by V is defined as
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

66. Quantum Gates Discrete Universal Gate Set
Example 1: Four-member “standard” gate set H S
/8
CNOT Hadamard Phase /8 (T) gate
Example 2: {CNOT, Hadamard, Phase, Toffoli}

67. Quantum Circuits

68. 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!

69. 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 V V† = U

70. Quantum Circuits = Example 1: Simulation V V† U ? |0 |1 |x |0 |1
V|x
|0
|1
|x V V† ? = U

71. Quantum Circuits = Example 1: Simulation (contd.)
|1
|x
U|x
|1
|x
|1
V|x
|1
|0
|1
|0
V|x
|1
|1
U|x V V† = U ?
Exercise: Simulate the two remaining cases

72. Quantum Circuits Example 1: Algebraic analysis x1 x2 x3V V† = U ? x1 x2 x3 U4 U2 U3 U1 U5 U0
Is U0(x1, x2, x3) = U5U4U3U2U1(x1, x2, x3)
= (x1, x2, x1x2  U (x3) ) ?

73. Quantum Circuits
Example 1 (contd);

74. Quantum Circuits
Example 1 (contd);

75. 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

76. Quantum Circuits Implementing a Half Adder Generic design: Uadd
Problem: Implement the classical functions sum = x1  x0 and carry = x1x0
Generic design:
|x1
|x1
|x0
Uadd
|x0
|y1
|y1  carry
|y0
|y0  sum

77. Quantum Circuits
Half Adder: Generic design (contd.)

78. Quantum Circuits Half Adder: Specific (reduced) design |x1 |x1 |x0
CNOT
C2NOT (Toffoli)
|x0
sum
|y
|y  carry

79. Walsh Transform for two binary-input many-valued variables
Classical logic
Quantum logic
Variable 1
Variable 1 + -
You need a butterfly H H
Butterfly is created automatically by tensor product corresponding to superposition
minterms

80. Computation 1
G(0 0) 1) 1
G(1
G(x) QC

81. Quantum Gate Arrays 1-bit full adder |c> |c> |x> |x>
|0>
|s>
|1>
|1>
|0>
|0>
|c’>
|0>
|0>
|1>
Let |c> = |1>, |x> = |0>, |y> = |1>
|s> = |0>, |c’> = |1>

82. Quantum Gate Arrays |x> |x> Uf |y>
It is possible to construct reversible quantum gates for any classical computable function f with m input and k output bits.
There exists a quantum gate array that implements the unitary transformation Uf : |x, y>  |x, y  f(x)>, where  indicates bitwise xor. Uf
|x>
|x>
|y>
|y  f(x)>

83. Quantum Gate Arrays The previous transformation Uf is reversible
Uf + = Uf Uf+ Uf = Uf Uf = I Uf Uf
|x>
|x>
|x>
|y  f(x)>
|y  f(x)  f(x)>
|y>
But |y  f(x)  f(x)> = |y>

84. Superposition of Quantum States
Consider the Tofolli Gate
|x>
|x>
|y>
|y>
|0>
|x>  |y>
Apply T, the Tofolli transform, to the superposition of all inputs.
T(H|0>  H|0>  |0> ) = ½(|000> + |010> + |100> + |111>)
Quantum parallelism – Applying the Tofolli transform to a superposition of all of the input states produces a superposition of all of the states in the “truth table”

85. Superposition of Quantum States
BUT
Only one of the superposed states can be extracted by measurement = 1
T(H|0>  H|0>  |0> ) = ½(|000> + |010> + |100> + |111>)
|000> + |010> + |100>
|111> 1
Measurement of the output projects the superposition onto the set of states consistent with the result

86. Quantum Parallelism
In order to take advantage of quantum parallelism one must:
1. Transform the state in such a way as to amplify the values of interest – so that they have a higher probability of being selected during measurement
Grover’s unstructured search algorithm
2. Find common properties of ALL the states of f(x)
Shor’s factoring algorithm

87. Where to learn more Web Page of Marek Perkowski
class see description of student projects
Portland Quantum Logic Group

88. 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:

89. Tensor Products
Similarly one can define tensor products for any size of matrices and in particularly for vectors representing superposed states.
As an example, consider two qutrits with and .
When the two qutrits are considered to represent a state, that state is the superposition of all possible combinations of the original qutrits, where:

90. 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.
In the above formula some coefficient can be equal to zero, so there exists a constraint bounding the possible states in which the system can exist (entanglement).
“Allowing d to be arbitrary enables a tradeoff between the number of qudits making up the quantum computer and the number of levels in each qudit”.
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.

91. Quantum Notation
An output of a gate is obtained by multiplying the unitary matrix of this gate by a vector of Hilbert space corresponding to this gate’s input state.
(Unitary matrix U is one such that U . U+ = I, where U+ is a Hermitian matrix of U. A Hermitian U+ is a conjugate transpose matrix of U).
A gate or a sub-circuit of a quantum circuit corresponds to a unitary matrix.
As shown below, 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 and creating efficient analysis, simulation, verification and synthesis algorithms for QC.
Generally, however, we believe that once the minimal amount of formalism is understood, logic researchers can quickly contribute to new designs, since 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

92. Quantum Circuits
In terms of logic operations, anything that changes a vector of qudit states to another qudit vector satisfying measurement probability properties can be considered as a quantum operator (unitary matrix).
These phenomena can be modeled using the analogy of a “quantum circuit” (called also quantum array).
In a quantum circuit, a wire does not carry ternary values but corresponds to a 3-tuple of complex values, , , and γ.
Quantum logic gates of the circuit map the complex values on their inputs to complex values on their outputs.
As mentioned, operation of all quantum gates and their assemblies is described by matrix operations.
Any quantum circuit is a composition of parallel and serial connections of blocks, from small to large.

93. Analysis of Quantum Circuits
Small blocks correspond to quantum gates that are easily directly realizable (like Pauli rotations) or are very simple and require just few basic quantum operations such as Feynman gates or Stroud/Muthukrishnan gates.
Serial connection of blocks corresponds to multiplication of their (unitary) matrices.
Parallel connection corresponds to Kronecker multiplication of their matrices.
So, theoretically, the analysis, simulation and verification are easy and can be based on matrix methods.
Practically they are tough because the dimensions of the matrices grow exponentially.
All these become much easier when one deals only with permutative matrices, which are equivalent to multi-output truth tables of completely specified functions. In such matrices there is exactly one “1” in every row and column.
An active research area is to represent operations on unitary matrices (in particular, the permutation matrices) by new efficient data structures and algorithms.

94. Calculating output state of QC
Typically the symbols |0> and |1> are not present in the matrix formulation of the equations, only the probability amplitudes (i.e.  and ) are included; however, there are kept in Equation (1) for illustrative purposes.
(1)

95. Because the qubit probabilities must be preserved at the output of the quantum gate, all matrices representing them are unitary.
An important unitary matrix property is that of a full rank.
This property implies that quantum gate matrix rows and columns are orthonormal.
Therefore, past results from spectral methods for classic digital logic are directly applicable to quantum logic synthesis.
Furthermore, since quantum logic gates are represented using unitary orthonormal matrices, they represent logically reversible gates.
These observations mean that the single input/output quantum logic gates as represented in Equation (1) are rotation matrices characterized by some particular rotation angle , where, for example, a = cos, b = sin, c = -sin and d = cos.
With this viewpoint, it can be seen that there are, in fact, an infinite number of single input/output qubit gates.

96. Rotation Gates
However, three elementary gates can be used to generate any rotation [7].
These are the R, S, and T gates described in matrix notation by
(1a)

97. 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>.

98. XOR logic synthesis is useful for QC
In this example, the input is |10> = ((0)|0>+(1)|1>) ((1)|0>+(0)|1>), and the input vector is represented by the coefficients shown in parentheses.
It is a significant fact that the unitary gates described by Equations (1) and (2) can realize any quantum logic function (including standard binary).
There are several strong similarities of quantum logic to classic digital circuit design using AND/XOR logic.
Our research group has been heavily involved in AND/XOR logic circuit design as well as related algebraic and spectral methods for several years.
We found these experiences very useful in quantum circuit design.

99. Bloch Sphere
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.

100. Figure 1. Bloch Sphere with 6 values shown

101. The identity matrix and three Pauli matrices:
form a basis for the 2x2 density matrices.
So every density matrix can be written as p = ½ (I + ax X + ay Y + az Z).
We associate with every 1-qubit state p = ½ (I + ax X + ay Y + az Z) the vector (ax, ay, az). If p = | | for a state | = e j (cos ( / 2) |0 + e j  sin ( / 2) |1 ).
Then the corresponding vector is (ax, ay, az) = (sin  cos , sin  sin  , cos ).
It can be easily derived that the vectors (ax, ay, az) satisfy |ax|2 + |ay|2 + |az|2 = 1, which means that all pure states are located on the surface of the Bloch Sphere.
When there many identical quantum circuits working together they are described by density matrices and the (mixed) states may lay inside the sphere, not on the surface

102. One way to realize multi-valued logic using binary quantum computing.
Figure shows the location of 6 points, that may correspond to values of some multi-valued 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.

103. Important quantum gates
Because global phase does not count, the T gate can be also written as follows:
T (/8) =
H denotes the important Hadamard gate:

104. The Hadamard and the /8 gate can be used to approximate any given single-qubit unitary operation with arbitrary accuracy.
On the Bloch sphere, T and HTH are rotations by an angle /4 radians around the z- and x-axes, respectively.
The composition of these two operations gives a rotation by an angle , which is defined by cos/2 = cos2/8, around an axis n, which is defined by n = (cos /8, sin /8, cos/ 8).
Since  is irrational, any rotation around the -axis can be build, to arbitrary precision, from T and HTH.
Furthermore, since for  arbitrary H R n () H = R m () with m = (cos/8, - sin/8, cos/8 ) not collinear n, there are angles ,  ,  such that any given U can be written U = Rn() Rm () R n().
It can be also shown that any given 2-qubit gate can be composed from CNOT and a single qubit gate.

105. Similarly other universal sets of 1-qubit gates can be found and illustrated using Bloch Sphere.
This sphere is also useful to find operator identities (quantum generalizations of rules like “ Not (Not B) = B “ ) which play fundamental role in quantum circuit optimization.
Study of universality and power as well as quantum realization costs of these gates are still active research areas.
More study should be devoted to multi-valued Bloch Sphere, operators in it and their transformations and realization.

106. 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.
Another 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.

107. While several books and numerous papers have been published on binary quantum circuits [45,72,103] not much information on their multi-valued counterparts is available. In their pioneering paper,
Muthukrishnan and Stroud [68] developed in 2000 multi-valued logic for multi-level quantum computing systems and showed their realizability in linear ion trap devices.
However, no experimental data are known so far. In addition, this approach generates circuits that are too large and no procedure was proposed to minimize them.
In 2002, Brylinski and Brylinski [13] discussed the universality of n-qudit gates without giving any design algorithms.
Since 2001, PQLG group [2-5,51-55] proposed Galois Field approach to multi-valued quantum logic synthesis in several regular structures.
They used gates were ternary counterparts of classical binary Feynman and Toffoli gates. De Vos [23] proposed two ternary 1*1 gates and two ternary 2*2 gates, but again no synthesis method was proposed.
In 2002, Perkowski, Al-Rabadi, and Kerntopf [75] proposed a 2*2 Generalized Ternary Gate (GTG gate) based on the ternary conditional gate [68] and ternary shift gates [52-54] and showed the realization of ternary Toffoli gate using GTG gates. This work introduced for the first time the practical realizability of Galois Field circuits in realizable multi-valued quantum technology.

108. 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 multi-valued computing.

109. Quantum Circuit Simulation
Simulation of quantum circuits plays absolutely fundamental role in many areas of quantum physics and engineering.
Similarly as in classical circuits design, 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.
It is amazing that the first quantum algorithms were invented without quantum simulators, but now the researchers routinely use quantum simulators to help them with inventions and verify their design guesses.
Quantum simulators are used to simulate a good circuit and a circuit with inserted faults, for test generation and fault localization.

110. The same is true for quantum fault simulation.
Moreover, because the search-based synthesis methods for quantum circuits such as exhaustive search, genetic algorithms, genetic programming, simulated annealing or heuristic search do not use deeper knowledge of circuit structure and properties, simulation is the only way (to be used as a part of the fitness function) to direct the search towards a circuit that satisfies the given requirements.
The results of the simulation are compared with the circuit specification many times in the loop of the search program.
The same is true for quantum fault simulation.
As we see, in all these applications the simulation of quantum circuits must be very fast and the computer memory should be large.
On the other hand, matrix operations on unitary matrices are slow, thus new methods and representations should be found to allow for very fast and low in memory usage simulation.
This is attempted to be achieved by two fundamental methods:
(1) acceleration of standard operations by using special hardware emulators, parallel computers or processor networks [71,73],
(2) creating new advanced data structures to represent quantum data more efficiently using standard computers.

111. 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.

112. 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.

113. 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).

114. (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.

115. 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-to-one 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.

116. 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.

117. What can we do?

118. Quantum Computers
Our community should 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.
Development of these tools will require understanding of real quantum circuit technology.

119. New Frontiers
Quantum Computer

120. Open Problems in Quantum Circuits
Synthesis of binary quantum cascades with no garbage or small garbage
(Maslov, Dueck, Miller, Perkowski, Khlopotine, Mishchenko, Curtis, Khan, Jha and Agrawal, Hayes, Markov)
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?
Fredkin
Toffoli

121. 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.

122. 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?

123. 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.

124. 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.

125. Four Synthesis Models
There exist the following synthesis models, both for binary and multiple-valued logic:
limited qubit gate model and full reversible function (way a). Usually zero or one ancilla bits are expected.
unlimited qubit gates and full reversible function (way a). Usually zero or one ancilla bits are expected.
limited qubit gates and single output function (way b). Usually at most M ancilla bits are expected.
unlimited qubit gates and irreversible input function (way b). Usually at most M ancilla bits are expected.

126. 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.

127. 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.

128. 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].

129. 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.

130. Figure 2. 3*3 Toffoli gate
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.

131. 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>.

132. 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>.

133. 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.

134. 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.

135. 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 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

136. Other problems in MV QC
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.
Development of methods based on unitary matrix decomposition, group theory, Lie groups and Clifford algebras,
Methods for incompletely specified functions, to be used in machine learning and data mining,
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),
Efficient methods for local optimization of quantum circuits on many levels of description,
High-level quantum hardware description languages that will play in QDA a role similar to VHDL and Verilog in EDA,
Development of formalisms and synthesis methods for sequential circuits.

137. 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.

138. 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 multi-valued realization in order to distinguish the values.

139. 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.

140. 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.

141. 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.

142. 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.

143. 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 from section 2 of this paper.

144. 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 NP-complete 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].

145. 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.

146. 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.

147. 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
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

148. 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.

149. 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.

150. 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.

151. 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.

152. 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.

153. 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.

154. Open Problems in Reversible Non-quantum Circuits
Reversible adiabatic gates and circuits.
Are they different than in quantum? Why?
Are garbage and ancilla bits important?
Realization of adiabatic reversible circuits and synthesis for them.
Applications in practical architectures for low power.
Applications in practical architectures for standard computing (FFT, butterflies, DSP circuits).
Testing and Test localization.
Realization in DNA and other nano-technologies (fluidic)
Optical realizations.

155. 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 Information and
Quantum Computational Intelligence
Quantum Circuit
Design
And Technology
Mathematics
and logic
Quantum Design
Automation

156. 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.

157. 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 multiple-valued logic, form a powerful base of new approaches in quantum engineering.

158. 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.

159. My husband has not taken his decision yet, he is not sure if he should work on quantum computing