From%20Quantum%20Gates%20to%20Quantum%20Learning:%20recent%20research%20and%20open%20problems%20in%20quantum%20circuits

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 progress in classical computer technology has
been dramatic
Many researchers believe an even greater
revolution is coming quantum computers
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
Number of Atoms in a Useful SystemFrom R. Keyes,
IBM J. Res. Develop (1988) atoms to store a
bit dopant atoms/bipolar transistor
5
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

6
People
First Ideas (1982)
Turing Machine (1936)
A. Turing
R. Feynmann
Quantum Circuits(1985)
D. Deutsch
P. Shorr
Factorization (1997)
7
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

8
Qubit in a Ion Trap
9

• Quantum Logic Circuits

10
How a single photon behaves in a beam splitter?
Beam splitter
50
1
Single photon
0
50
Optical sensor
11
strange behavior
0
1
0
1
Mach-Zehnder apparatus
12
Quantum Gate square root of NOT
0
1
0
1
1
0
1
0
NOT
13
Simple theory of the 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.
14
Quantum Interference
The simplest explanation must be wrong, since it
would predict a 50-50 distribution.
15
More experimental data collected to improve the
theory
16
A new theory
The particle can exist in a linear combination or
superposition of the two paths
17
Probability Amplitude and Measurement
If the photon is measured when it is in the
state then we get with probability
and 1? with probability
18
Quantum Operations are linear
The operations are induced by the apparatus
linearly, that is, if and then
19
Quantum Operations are unitary
Any linear operation that takes
states satisfying and maps them to
states satisfying must be UNITARY
20
Linear Algebra notation for quantum circuits
corresponds to
corresponds to
corresponds to
21
Linear Algebra notation for quantum circuits
corresponds to
corresponds to
22
Linear Algebra notation for quantum circuits
corresponds to
23
What is unitary matrix?
is unitary if and only if
24
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
25
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

26

Re

0gt

Im
1gt
Register-transfer notation for quantum circuits
0gt

0gt
-
1gt
1gt
27
0gt
1gt
Register-transfer notation for quantum circuits
28
From physical devices to abstracted quantum
circuits
An arrangement like
is represented with a network like
29
cos?
-
sin?
cos?

sin?
Register-transfer notation for quantum circuits
30
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

31
Register-transfer diagram for two Hadamard gates
in parallel
0gt
0gt
00gt
0gt
00gt
01gt
1gt
01gt
1gt
1gt
0gt
10gt
10gt
11gt
1gt
11gt
(b)
32
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!!!

33
Kronecker Products for more than one qubit
circuits
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
34
More than one qubit superposition and
entanglement
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.
35
Measuring multi-qubit systems
If we measure both bits of we get with
probability
36
Classical Versus Quantum
37
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

38
Quantum Circuits are different

• Quantum Measurement
• Measurement yields only one state X of the
  superposed states
• Measurement also makes X the new state and so
  interferes with computational processes
• X is determined with some probability, implying
  uncertainty in the result
• States cannot be copied (cloned), implying that
  signal fanout is not permitted
• Environmental interference can cause a
  measurement-like state collapse (decoherence)

39
Decoherence
40
Classical versus Quantum Circuits

• Quantum Logic Circuits
• Circuit behavior is governed explicitly by
  quantum mechanics
• Signal states are vectors interpreted as a
  superposition of binary qubit vectors with
  complex-number coefficients
• 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

41
More 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

42
Classical vs. Quantum Circuits
Classical adder
43
Classical vs. Quantum Circuits
Quantum adder
Feynman gate
44
Reversible Circuits
45
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
46
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.

47
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

48
Quantum Gates
49

• 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

50
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!

51
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

?
52
Universal One-Input Quantum 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

53
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.

54
Quantum XOR gate

• 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?

55
000gt

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

000gt
001gt
001gt
010gt
010gt
011gt
011gt
100gt
100gt
101gt
101gt
110gt
110gt
111gt
111gt
56
Controlled Quantum Gates

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

etc.
U
U
U
C(U)
C2(U)
U
57
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

58
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

59
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

60

Quantum Circuits
61
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!

62
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

63
Simulation of Quantum Circuits
0? 1? x?
0? 1?
0? 1? x?
0? 1?
0? 1? Vx?
0? 1? x?
?

U
64
Simulation of Quantum Circuits
Simulation continued
1? 1? x?
1? 1? Vx?
1? 0?
1? 0? Vx?
1? 1?
1? 1? Ux?
?
65
Algebraic Analysis of Quantum Circuits

• Is U0(x1, x2, x3) U5U4U3U2U1(x1, x2, x3)
• (x1, x2, x1x2 ? U (x3) ) ?

66
Quantum Circuits

• Example 1 (contd)

67
Quantum Circuits

• Example 1 (contd)

68
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

69
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
70
Quantum Circuits

• Half Adder Generic design (contd.)

71
Quantum Circuits

• Half Adder Specific (reduced) design

x1?
x1?
CNOT
C2NOT (Toffoli)
x0?
sum
y?
y? ? carry
72
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

73

Tremendous potential for truly innovative research
74
Research Potential

• Our community should develop a systematic
  methodology and CAD tools for synthesizing,
  verifying, testing and simulating of quantum
  computers.
• These methods and tools will be counterparts of
  what exists now in binary CMOS.
• Re-use spectral approaches, DDs, XOR logic, etc.
• Development of these tools will require
  understanding of real quantum circuit technology.

75
New Frontiers

• Quantum Computer

76
Open Problems in Quantum Circuits

• Synthesis of binary quantum cascades with no
  garbage or small garbage
• (Maslov, Dueck, Miller, Kerntopf, Perkowski,
  Khlopotine, Mishchenko, Curtis, Khan, Jha and
  Agrawal, Hayes, Markov)
• 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
77
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.

78
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?

79
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.

80
Example 1 First method to realize MV quantum
circuits
MV Tensor Products

• Analogous to binary quantum circuits.
• As an example, consider two qutrits.
• When the two qutrits are considered to represent
  a state, that state is the superposition of all
  possible combinations of the original qutrits,
  where

This approach to multi-valued quantum circuits
requires measurements with more than two basis
states. Also, new gates should be defined as
well as the synthesis methods for these gates.
81
Quantum MV Superposition

• The superposition property allows the qubit
  states to grow much faster in dimension than
  classical bits, and the qudits faster than
  qubits.
• In a classical system, n bits represent distinct
  states, whereas n qubits correspond to a
  superposition of 2n states and n qutrits
  correspond to a superposition of 3n states.
• Because in contemporary quantum technologies
  every qubit is costly, higher radices than 2 give
  an advantage of improved processing and storage
  power at the same realization cost.
• This is a strong argument for realization of
  multi-valued logic in quantum circuits.
• In addition to standard advantages of mv logic,
  quantum mv logic may be superior to binary one
  because of different nature of entanglement.

82
Bloch Sphere
Example 2 Second Method to realize MV quantum
circuits.

• The normalization ?2 ?2 1 admits the
  parametrization ? cos(?/2) e j? , ? sin(?/2)
  e j?.
• ?? e j? (cos (? / 2) 0? e j ? sin (? / 2)
  1? ).
• Since the global phase of ?? has no observable
  effect, we may write ?? cos(?/2) 0? e j?
  sin(?/2) 1?.
• The angles ? and ? define a point on the surface
  of a unit sphere the Bloch sphere, see Fig. 1.
• The Bloch sphere provides an excellent tool to
  visualize the state vector of a qubit.
• This is a binary Bloch sphere, but a multi-valued
  counterpart of it can be also created.

83
Second method to realize multi-valued logic using
binary quantum computing (cont).

• Figure shows the location of 6 points, that may
  correspond to values of some 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.

84
Second Method to realize MV quantum circuits
(cont).

• Above we showed how multiple-valued logic can be
  encoded in binary quantum computing.
• Quaternary logic requires two binary measurements
  (readings).
• The first reading distinguishes states 0? and
  1?, and the second reading uses additional
  rotation gates to distinguish between states
  0?1?, and 0?-1?.
• It can be shown that the logic with 2n values
  requires n readings.

85
Example 3 Quantum Circuit Simulation

• Simulation of quantum circuits plays absolutely
  fundamental role in many areas of quantum physics
  and engineering.
• Simulation is used to
• verify correctness of the design,
• analyze its properties and
• find some interesting aspects that cannot be
  found by hand and pencil methods.
• Fault simulation
• Evolutionary algorithms
• Researchers routinely use quantum simulators to
  help them with inventions and verify their design
  guesses.

86
Fast simulation is extremely important

• Matrix methods are slow.
• Acceleration is attempted to be achieved by two
  fundamental methods
• (1) acceleration of standard operations by using
  special hardware emulators, parallel computers or
  processor networks,
• (2) creating new advanced data structures to
  represent quantum data more efficiently using
  standard computers.

87
Example 4 Quantum Decision Diagrams

• New data structures, such as QUIDDs Viamontes,
  Markov, Hayes allow for implicit parallelism
  when executing Kronecker multiplications on them.
  
• QUIDDs are based on ADDs and MTBDDs,
• so hopefully in future other decision diagrams
  may be used to represent quantum circuits.
• It is also expected that basic software engines
  used successfully in classical CAD (such as for
  instance SAT or ATPG methods) may be used to deal
  with quantum circuits.
• Also, the fast simulators based on new types of
  decision diagrams should be in future
  parallelized and possibly accelerated in
  FPGA-based boards.
• Even before quantum computers will be available,
  their emulations on standard computers and
  ASIC/FPGA may prove useful to solve some
  practical problems.

88
Example 5 Testing and diagnosis of quantum
circuits

• Patel, Markov, and Hayes showed that reversible
  circuits are much better testable than
  irreversible circuits.
• This is because every test covers half faults and
  every fault is covered by half tests.
• The reversible circuits are then transparent to
  faults, making them well observable and
  controllable.
• We showed that fault localization in reversible
  circuits is easier.
• We presented preliminary results on testing
  binary quantum circuits and on fault localization
  of quantum circuits.

89
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.

90
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.

91
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.

92
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.

93
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.

94
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.

95
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.

96
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.

97
?????????????
Importance of intelligent learning

• ???
• ?????????????
• ??????????????

98
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.

99
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.

100
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.

101
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.

102
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
103
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.

104
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.

105
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.

106
(No Transcript)
107

• Additional slides for questions

108
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.

109
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).

110

• (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.

111
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.

112

• 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.

113
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.

114
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.

115

• 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.

116
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.

117

• 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.

118
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.

119
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.

120

• 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.

121

• 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.

122

• 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.

123

• 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.

124

• 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

125
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.

126

• 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 problems structure, the search
  can be dramatically speeded up to O(bka), where
  k?1 and is problem dependent.
• Grovers 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.

127
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
128
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
129
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.

130
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.

131
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.

132
Quantum Notation

• As shown, the resultant unitary matrix of an
  arbitrary quantum circuit is created by matrix
  multiplications or Kronecker multiplications of
  matrices of its composing sub-circuits.
• Various quantum notations contribute to the
  difficulty in understanding the concepts of
  quantum computing.
• Generally, however, we believe that once the
  minimal amount of formalism is understood, logic
  researchers can quickly contribute to new
  designs.
• Much can be learned from the history of
  Electronic Design Automation as well as from MV
  logic theory and design.
• The lessons learned there should be used to
  design efficient QDA tools for MV quantum
  computing.
• Here we include the absolute minimum amount of
  formalism sufficient to start independent
  software development by people who have
  sufficient background in EDA tools and algorithms
  such as search or evolutionary programming