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.

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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
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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

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

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Qubit in a Ion Trap
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Classical Versus Quantum
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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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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)

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Decoherence
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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

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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

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Classical vs. Quantum Circuits
Classical adder
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Classical vs. Quantum Circuits
Quantum adder
Feynman gate
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Reversible Circuits
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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
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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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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!

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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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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

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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.

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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?

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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
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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
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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

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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

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Quantum Circuits Simulation
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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!

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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

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

U
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Simulation of Quantum Circuits
Simulation continued
1? 1? x?
1? 1? Vx?
1? 0?
1? 0? Vx?
1? 1?
1? 1? Ux?
?
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Analysis of Quantum Circuits based on Unitary
Matrices
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Algebraic Analysis of Quantum Circuits

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

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Quantum Circuits

• Example 1 (contd)

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Quantum Circuits

• Example 1 (contd)

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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

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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.)

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Quantum Circuits

• Half Adder Specific (reduced) design

x1?
x1?
CNOT
C2NOT (Toffoli)
x0?
sum
y?
y? ? carry
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1. Walsh Transform using classical logic circuits is
   expensive,
2. we need many adders and subtractors.

1. Walsh Transform using quantum logic circuits is
   NOT expensive,
2. we need only quantum Hadamard gates,
3. each gate costs only two pulses

1. We can go from standard space to spectral space,
   back to standard space and so on many times.
2. This would be very expensive in classical
   spectral-based logic circuits.
3. These methods can be used for filtering.

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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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Tremendous potential for truly innovative research
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• Potential new research areas in quantum circuits

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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.

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New Frontiers

• Quantum Computer

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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
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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.

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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?

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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.

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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.
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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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

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

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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.

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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.

76
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.

77
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.

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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
79
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.

80
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.

81
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.

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What to remember?

1. Types of quantum gates
2. Analysis of circuits based on various types of
   gates.
3. Six important states on the Bloch Sphere. How to
   use them?
4. Open Research areas in quantum circuits.
5. Simulation of quantum circuits (arrays)
6. Formal analysis of quantum circuits.
7. Decoherence.