From Quantum Gates to Quantum Learning: recent research and open problems in quantum circuits

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 www.icknowledge.com
1947 First point contact transistor by Bardeen
and Brattain
http//www.pbs.org/transistor/science/events/point
ctrans.html
3
Nano-systemHow 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

4
History

• 1970s and 1980s, introduction of quantum
  computers (Richard Feynmann, David Deutsch, and
  Paul Benioff)
• 1994, Peter Shors 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

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 SystemFrom R. Keyes,
IBM J. Res. Develop (1988) atoms to store a
bit dopant atoms/bipolar transistor
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EX Quantum Parallelism

• Quantum
• 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!!!

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Jiffy Quantum Theory
Info unit 1 bit. Physical system 2 states
1gt
0gt
0gt and 1gt

• 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 states 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 photons polarizations or
  an elementary particles spins.
• After encoding these distinguishable quantities
  into multiple-valued values, qutrit states are
  represented by basis states 0gt, 1gt and 2gt ,
  respectively.
• A qubit, used in binary QC uses only two basis
  states, 0gt and 1gt
• 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 ? are complex quantities.
• Pairs of qutrits are capable of representing nine
  distinct states,, , , , , , , , and , as well as
  all possible superpositions of these states.

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• Quantum Logic Circuits

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Quantum Logic
Single photon
Specchio
50
1
0
50
Optical sensor
13
strange behavior
0
1
0
1
14
Quantum Gate
0
1
0
1
1
0
1
0
NOT
15
Qubit
16
Qubit in a Ion Trap
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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
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Quantum Computation
A A1A2 A3A4
P A1A2 A3A42 A1A2 A3A42 2Re(A1A2A3A
4)
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 50-50 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 satisfying 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
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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

Re

0gt

(c)
Im
1gt
0gt

0gt
(d)
-
1gt
1gt
34
0gt
1gt
(b)
(a)
(c)
(d)
35
An arrangement like
is represented with a network like
36
(a)
cos?
-
sin?
cos?

sin?
(b)
37
0gt
0gt
00gt
0gt
00gt
01gt
1gt
01gt
1gt
1gt
0gt
10gt
10gt
11gt
1gt
11gt
(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.
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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
  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

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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

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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)

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Classical vs. Quantum Circuits
Classical adder
47
Classical vs. Quantum Circuits
Quantum adder
48
Reversible Circuits
49
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

i
i
Junk
Reversible Boolean Circuit
f(i)
Junk
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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.

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Reversible Circuits

• Reversible gate family Toffoli 1980

• Every Boolean function has a reversible
  implementation using Toffoli gates.
• There is no universal reversible gate with fewer
  than three inputs

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Quantum Gates
53
Quantum Gates

• One-Input gate NOT
• Input state c00? c11?
• Output state c10? c01?
• Pure states are mapped thus 0? ? 1? and 1? ?
  0?
• Gate operator (matrix) is
• As expected

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• One-Input gate NOT
• Input state c00? c11?
• Output state c10? c01?
• Pure states are mapped thus 0? ? 1? and 1? ?
  0?
• Gate operator (matrix) is
• As expected

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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/?22 1/2
• and 0? ? 1? with probability 1/
  ? 22 1/2
• Similarly, this gate randomizes input 1?
• But concatenation of two gates eliminates the
  randomness!

57
Quantum Gates

• One-Input gate Hadamard
• 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

?
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Quantum Gates

• Universal One-Input Gate Sets
• Requirement
• 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

59
Quantum Gates

• Two-Input Gate Controlled NOT (CNOT)

• CNOT maps x?0? ? x?x? and x?1? ? x?NOT
  x?
• x?0? ? x?x? looks like cloning, but its
  not. These mappings are valid only for the pure
  states 0? and 1?
• Serves as a non-demolition measurement gate

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(b)
(a)
(c)
(d)
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Quantum Gates

• 3-Input gate Controlled CNOT (C2NOT or Toffoli
  gate)

a?
a?
b?
b?
c?
ab ? c?
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(a)
(b)
000gt
000gt
001gt
001gt
010gt
010gt
011gt
011gt
100gt
100gt
(c)
101gt
101gt
110gt
110gt
111gt
111gt
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Quantum Gates

• General controlled gates that control some
  1-qubit unitary operation U are useful

etc.
U
U
U
C(U)
C2(U)
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 ? gt 0
  using a sequence of gates from G

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Quantum Gates

• Discrete Universal Gate Set
• Example 1 Four-member standard gate set

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

70
Quantum Circuits

• Example 1 Simulation

0? 1? x?
0? 1?
0? 1? x?
0? 1?
0? 1? Vx?
0? 1? x?
?

U
71
Quantum Circuits
Example 1 Simulation (contd.)
1? 1? x?
1? 1? Vx?
1? 0?
1? 0? Vx?
1? 1?
1? 1? Ux?
?

• Exercise Simulate the two remaining cases

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Quantum Circuits
Example 1 Algebraic analysis

• 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
• Problem Implement the classical functions sum
  x1 ? x0 and carry x1x0
• Generic design

x1?
x1?
x0?
x0?
Uadd
y1?
y1? ? carry
y0?
y0? ? sum
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Quantum Circuits

• Half Adder Generic design (contd.)

78
Quantum Circuits

• Half Adder Specific (reduced) design

x1?
x1?
CNOT
C2NOT (Toffoli)
x0?
sum
y?
y? ? carry
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Walsh Transform for two binary-input many-valued
variables
Classical logic
Quantum logic
Variable 1
Variable 1
Butterfly is created automatically by tensor
product corresponding to superposition

• minterms

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Computation
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
G(0 0 0)
G(0 0 1)
G(0 1 0)
G(0 1 1)
G(1 0 0)
G(1 0 1)
G(1 1 0)
G(1 1 1)
G(x) QC
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Quantum Gate Arrays
1-bit full adder
cgt
cgt
xgt
xgt
ygt
ygt
0gt
sgt
1gt
1gt
0gt
0gt
cgt
0gt
0gt
1gt
Let cgt 1gt, xgt 0gt, ygt 1gt
sgt 0gt, cgt 1gt
82
Quantum Gate Arrays
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, ygt ? x, y ?
f(x)gt, where ? indicates bitwise xor.
xgt
xgt
ygt
y ? f(x)gt
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Quantum Gate Arrays
The previous transformation Uf is reversible
Uf Uf Uf Uf Uf Uf I
xgt
xgt
xgt
y ? f(x)gt
y ? f(x) ? f(x)gt
ygt
But y ? f(x) ? f(x)gt ygt
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Superposition of Quantum States
Consider the Tofolli Gate
xgt
xgt
ygt
ygt
0gt
xgt ? ygt
Apply T, the Tofolli transform, to the
superposition of all inputs.
T(H0gt ? H0gt ? 0gt ) ½(000gt 010gt 100gt
111gt)
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
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Superposition of Quantum States
BUT
Only one of the superposed states can be
extracted by measurement
0 1
T(H0gt ? H0gt ? 0gt ) ½(000gt 010gt 100gt
111gt)
000gt 010gt 100gt
0
111gt
1
Measurement of the output projects the
superposition onto the set of states consistent
with the result
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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
Grovers unstructured search algorithm
2. Find common properties of ALL the states of
f(x)
Shors factoring algorithm
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Where to learn more

• Web Page of Marek Perkowski
• class 572 - see description of student projects
• Portland Quantum Logic Group

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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

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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

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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 gates 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 ?.
• 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 0gt and 1gt 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 00gt and 01gt to
  appear unchanged at the outputs, but interchanges
  the pairs 10gt and 11gt.
• For example, consider the quantum XOR gates
  operation for an input 10gt.

98
XOR logic synthesis is useful for QC

• In this example, the input is 10gt
  ((0)0gt(1)1gt) ?((1)0gt(0)1gt), 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 ax2 ay2 az2 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? - j1?.
• 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 11 gates and two
  ternary 22 gates, but again no synthesis method
  was proposed.
• In 2002, Perkowski, Al-Rabadi, and Kerntopf 75
  proposed a 22 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

• 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 Pa, QB, Rab?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 N1
  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 kgt1 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. 33 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 0gt, the
  data input is forwarded to output with no change.
  When the control signal is 1gt 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 A1 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 0gt and apply its internal
  transformation (the symbol of this transformation
  is in the input to multiplexer) when the control
  is 1gt.

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 0gt, the data input is
  forwarded to output with no change.
• When the control signal is 1gt 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 A1 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 0gt and apply its
  internal transformation (the symbol of this
  transformation is in the input to multiplexer)
  when the control is 1gt.

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-gt1, 1-gt0, and 2-gt2

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.
• Shors algorithm is for factoring integers
• It produces an exponential computational speedup
  over classical algorithms
• It can break the RSA encryption techniques.
• Grovers 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