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From Quantum Gates to Quantum Learning:
recent research and open problems in quantum circuits
Marek A. Perkowski,
Portland Quantum Logic Group,Department of Electrical Engineering and Computer Science,Korea Advanced Institute of Science and Technology, and
Department of Electrical and Computer Engineering, Portland State University, USA.
The progress in classical computer technology has been dramatic
1999 Pentium IIIB
www.icknowledge.com
http://www.pbs.org/transistor/science/events/pointctrans.html
1947 First point contact transistor
by Bardeen and Brattain
Many researchers believe Many researchers believe an even greater revolution an even greater revolution is coming: quantum is coming: quantum computers computers
Nano-systemHow small is a nanometer?
• 1 meter
• 10 m
• 1 m
• 10 nm
• 1nanometer
• 0.1 nm
• 1 picometer
• 1 femtometer
– Size of red blood cell
– = a millionth of a meter
– Size of polio virus
– = a billionth of a meter
– Size of the hydrogen atom
– = a trillionth of a meter
– = 10 15 m, size of a proton
Number of Atoms in a Useful SystemFrom R. Keyes, IBM J. Res. Develop (1988)
# atoms to store a bit # dopant atoms/bipolar transistor
History• 1970s and 1980s, introduction of quantum
computers (Richard Feynmann, David Deutsch, and Paul Benioff)
• 1994, Peter Shor’s factoring algorithm• 1996, Lov Grover, searching algorithm• 1998, 1999, 2001 Isaac L. Chuang,
developed the world's first 2-qubit, 3-qubit, 5-qubit and 7-qubit quantum computer
People
Turing Machine …(1936)”
“… Quantum Circuits…(1985)”
First Ideas(1982)”
“…Factorization …(1997)”
A. TuringR. Feynmann
D. DeutschP. Shorr
Jiffy Quantum Theory
• Quantum nature: a combination of both.• In preparing the initial state: only one of the 2 states• On measurement: only one state found.• Probability: the state’s component in the mix• Both preparation and measurement in contact with a macro
system
|0>
|1>
|0> and |1>
Info unit: 1 bit. Physical system: 2 states
Qubit in a Ion Trap
Classical Versus
Quantum
• 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
Classical vs. Quantum Circuits
• 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)
Quantum Circuits are different
Decoherence
• 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
Classical versus Quantum Circuits
ci in 1in 1i0i 0
2n 1
• 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
More Quantum Circuit Characteristics
Classical vs. Quantum Circuits
Classical adder
cn–1
s0
s1
s2
s3
cn
a0
b0
a1
b1
a3
b3
a2
b2
Sum
Carry
Classical vs. Quantum Circuits
Quantum adder
Feynman gate
Reversible Circuits
Reversible Circuits
• Reversibility was studied around 1980 motivated by power minimization considerations
• Bennett, Toffoli et al. showed that any classical logic circuit C can be made reversible with modest overhead
……
n inputs
GenericBooleanCircuit
m outputs
f(i)i …
Reversible BooleanCircuit
…
f(i)
…
…
“Junk”i
“Junk”
• 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.
Reversible Circuits
ReversibleANDgate
a
b
f = ab c
a
b
c
a b c a b f0 0 0 0 0 00 0 1 0 0 10 1 0 0 1 00 1 1 0 1 11 0 0 1 0 01 0 1 1 0 11 1 0 1 1 11 1 1 1 1 0
• Reversible gate family [Toffoli 1980]
Reversible Circuits
(Toffoli gate)
• Every Boolean function has a reversible implementation using Toffoli gates.
• There is no universal reversible gate with fewer than three inputs
Quantum Gates
• One-Input gate: NOTOne-Input gate: NOT– Input state: c0 + c1
– Output state: c1 + c0
– Pure states are mapped thus: and
– Gate operator (matrix) is
– As expected:
0 1
1 0
0 1
1 0
1 0
0 1
NOT
NOTNOT
0 1
1 0
0
1
0
1
0
1
One-Input gate: “Square root of NOT”
– Some matrix elements are imaginary– Gate operator (matrix):
– We find:
so with probability |i/2|2 = 1/2
and with probability |1/ 2|2 = 1/2
Similarly, this gate randomizes input
– But concatenation of two gates eliminates the randomness!
i / 1/ 2 1/ 1/ 2
1/ 1/ 2 i / 1/ 2
1
2
i 1
1 i
1
2
i 1
1 i
1
0
1
2
i
1
1
2
i 1
1 i
i 1
1 i
0 i
i 0
NOTNOT
Quantum Gates
• One-Input gate: Hadamard
– Maps 1/ 2 + 1/ 2 and 1/ 2 – 1/ 2 . – Ignoring the normalization factor 1/ 2, we can write
xx (-1)xx xx – ––xx
• One-Input gate: Phase shift
1
2
1 1
1 1
H
1 0
0 e i
• Requirement:
• Hadamard and phase-shift gates form a universal gate set
• Example: The following circuit generates = cos 0 + ei sin 1 up to a global factor
Universal One-Input Quantum Gate Sets
U Any state
2H H2
Quantum XOR gate• Called also Feynman gate or Controlled Not gate. • This gate allows inputs of |00> and |01> to appear
unchanged at the outputs, but interchanges the pairs |10> and |11>.
• For example, consider the quantum XOR gate’s operation for an input |10>.
1
0
0
0
0
1
0
0
0100
1000
0010
0001|00> |00>
|01> |01>
|10> |10>
|11> |11>
x
y
x
x y CNOT
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
– CNOT maps x xxand x xNOT x x xx looks like cloning, but it’s not.
– These mappings are valid only for the pure states and
x
y
x
x y
Quantum XOR gate
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 1 0
b
c
b
ab c
a a
|000>
|000>
|001> |001>
|110>
|010>
|111>
|011>
|010>
|011>
|100>
|101>
|100>|101>|110>
|111>
3-Input gate: 3-Input gate: Controlled Controlled CNOT CNOT (C2NOT or (C2NOT or Toffoli gate)Toffoli gate)
• General controlled gates that control some 1-qubit unitary operation U are useful
Controlled Quantum GatesControlled Quantum Gates
U
u00 u01
u10 u11
C(U)
U
C2(U)
U
U
etc.
Universal Gate Sets• To implement any unitary operation on n qubits
exactly requires an infinite number of gate types• The (infinite) set of all 2-input gates is universal
– Any n-qubit unitary operation can be implemented using (n34n) gates [Reck et al. 1994]
• CNOT and the (infinite) set of all 1-qubit gates is universal
Quantum Gates
Discrete Universal Gate Sets• The error on implementing U by V is defined as
• If U can be implemented by K gates, we can simulate U with a total error less than with a gate overhead that is polynomial in log(K/)
• A discrete set of gate types G is universal, if we can approximate any U to within any > 0 using a sequence of gates from G
Quantum Gates
E(U,V ) max
(U V )
Discrete Universal Gate Set• Example 1: Four-member “standard” gate set
Quantum Gates
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
1
2
1 1
1 1
H
1 0
0 i
S /8
1 0
0 e i / 4
CNOT Hadamard Phase /8 (T) gate
• Example 2: {CNOT, Hadamard, Phase, Toffoli}
Quantum Circuits
Simulation
• A quantum (combinational) circuit is a sequence of quantum gates, linked by “wires”
• The circuit has fixed “width” corresponding to the number of qubits being processed
• Logic design (classical and quantum) attempts to find circuit structures for needed operations that are– Functionally correct– Independent of physical technology– Low-cost, e.g., use the minimum number of qubits or gates
• Quantum logic design is not well developed!
Quantum Circuits
• Ad hoc designs known for many specific functions and gates• Example 1 illustrating a theorem by [Barenco et al. 1995]: Any
C2(U) gate can be built from CNOTs, C(V), and C(V†) gates, where V2 = U
Quantum Circuits
V V† V
=
U
Simulation of Quantum CircuitsSimulation of Quantum Circuits
|0
|1
|x
|0
|1
|x
|0
|1
|xV V† V
=
U
|0
|1
V|x
|0
|1
|0
|1
|x
|0
|1
|0
|1
|x
?
|1
|1
|x
|1
|1
|x
|1
|1
U|xV V† V
=
U
|1
|1
V|x
|1
|0
|1
|0
V|x
|1
|1
|1
|1
U|x
Simulation continued
?
Simulation of Quantum CircuitsSimulation of Quantum Circuits
Analysis of Quantum
Circuits based on Unitary Matrices
Algebraic Analysis of Quantum Circuits
U4U2 U3U1 U5U0
V V† V
=
U
?x1
x2
x3
• Is U0(x1, x2, x3) = U5U4U3U2U1(x1, x2, x3)
= (x1, x2, x1x2 U (x3) ) ?
Quantum CircuitsExample 1 (contd);
U1 I1 C(V)
1 0
0 1
1 0 0 0
0 1 0 0
0 0 v00 v01
0 0 v10 v11
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 v00 v01 0 0 0 0
0 0 v10 v11 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 v00 v01
0 0 0 0 0 0 v10 v11
Quantum CircuitsExample 1 (contd);
U2 U4 CNOT(x1, x2 ) I1
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
1 0
0 1
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
Quantum CircuitsExample 1 (contd);
– U5 is the same as U1 but has x1and x2 permuted (tricky!)– It remains to evaluate the product of five 8 x 8 matrices
U5U4U3U2U1 using the fact that VV† = I and VV = U
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 v00 v01 0 0
0 0 0 0 v10 v11 0 0
0 0 0 0 0 0 v00 v01
0 0 0 0 0 0 v10 v11
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 v00 v10 0 0 0 0
0 0 v01 v11 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 v00 v10
0 0 0 0 0 0 v01 v11
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 v00 v01 0 0 0 0
0 0 v10 v11 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 v00 v01
0 0 0 0 0 0 v10 v11
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 v00v00 v10v10 v00v01 v10v11
0 0 0 0 0 0 v 01̀ v00 v11v10 v01v01 v11v11
U0
Quantum Circuits
• Implementing a Half Adder– Problem: Implement the classical functions
sum = x1 x0 and carry = x1x0
• Generic design:|x1
Uadd|x0|y1|y0
|x1|x0|y1 carry|y0 sum
Quantum Circuits
• Half Adder: Generic design (contd.)
UADD
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
Quantum Circuits
• Half Adder: Specific (reduced) design
|x1
|x0
|y
|x1
|y carry
sum C2NOT (Toffoli)
CNOT
1. Walsh Transform using classical logic circuits is expensive,
2. we need many adders and subtractors.
1. Walsh Transform using quantum logic circuits is NOTNOT 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.
Walsh Transform for two binary-input many-valued variables
+
-
+
-
+
-
+
-
You need a butterfly
•H
Classical logic Quantum logic
•H
Butterfly is created automatically by tensor product corresponding to superposition
•minterms
Variable 1 Variable 1
Tremendous potential for
truly innovative research
Potential new Potential new research areas research areas
in quantum in quantum circuitscircuits
Research Potential
• Our community should develop a systematic methodology and CAD tools for synthesizing, verifying, testing and simulating of quantum computers.
• These methods and tools will be counterparts of what exists now in binary CMOS.– Re-use spectral approaches, DDs, XOR logic, etc.
• Development of these tools will require understanding of real quantum circuit technology.
New Frontiers
Quantum Computer Quantum Mechanics
Quantum Logic
Quantum Computers
Quantum Programming
Open Problems in Quantum Circuits
• Synthesis of binary quantum cascades with no garbage or small garbage – (Maslov, Dueck, Miller, Kerntopf,
Perkowski, Khlopotine, Mishchenko, Curtis, Khan, Jha and Agrawal, Hayes, Markov)
• 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
Open Problems in Quantum Circuits
• What is the Fault Model for quantum circuits?– Technology dependent?
• Formal Verification of quantum circuits
• Test Generation for quantum circuits
• Fault Localization of quantum circuits
• Synthesis of testable quantum circuits
• Synthesis of fault-tolerant, error correcting quantum circuits.
Open Problems in Quantum Circuits
• What are universal gates?
• How to calculate costs of elementary gates for each quantum technology such as NMR or ion trap?
• What are the gates that can be truly realized in a quantum technology?
• What are the synthesis, analysis and test methods for sequential quantum circuits?
Open Problems in Quantum Circuits
• Invent new quantum algorithms.
• What are the principles to create quantum algorithms
• The nature of entanglement.
• Quantum computer architectures.
• Quantum formalisms. (Clifford algebras).
• Quantum Logic.
Example 1:Example 1: First method to realize First method to realize MV quantum circuitsMV quantum circuits
• 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:
2221201211
10020100
2121212121
212121212112
MV Tensor Products
This approach to multi-valued quantum circuits requires measurements with more than two basis states. Also, new gates should be defined as well as the synthesis methods for these gates.
Quantum MV Superposition• The superposition property allows the qubit states to grow
much faster in dimension than classical bits, and the qudits faster than qubits.
• In a classical system, n bits represent distinct states, whereas n qubits correspond to a superposition of 2n states and n qutrits correspond to a superposition of 3n states.
• Because in contemporary quantum technologies every qubit is costly, higher radices than 2 give an advantage of improved processing and storage power at the same realization cost.
• This is a strong argument for realization of multi-valued logic in quantum circuits.
• In addition to standard advantages of mv logic, quantum mv logic may be superior to binary one because of different nature of entanglement.
1
j
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.
Example 2:Example 2: Second Method to Second Method to realize MV quantum circuits.realize MV quantum circuits.
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 - j|1.
• A rotation or a combination of rotations leads from one value to any other value.
1
j
Second Method to realize MV quantum circuits (cont).
• Above we showed how multiple-valued logic can be encoded in binary quantum computing.
• Quaternary logic requires two binary measurements (readings).
• The first reading distinguishes states |0 and |1, and the second reading uses additional rotation gates to distinguish between states |0+|1, and |0-|1.
• It can be shown that the logic with 2n values requires n readings.
Example 3:Example 3: Quantum Circuit Quantum Circuit SimulationSimulation
• Simulation of quantum circuits plays absolutely fundamental role in many areas of quantum physics and engineering.
• Simulation is used to:– verify correctness of the design, – analyze its properties and – find some interesting aspects that cannot be found by “hand and
pencil” methods. – Fault simulation– Evolutionary algorithms
• Researchers routinely use quantum simulators to help them with inventions and verify their design guesses.
Fast simulation is extremely important
• Matrix methods are slow. • Acceleration is attempted to be achieved
by two fundamental methods: – (1) acceleration of standard operations by
using special hardware emulators, parallel computers or processor networks,
– (2) creating new advanced data structures to represent quantum data more efficiently using standard computers.
Example 4:Example 4: Quantum Decision Quantum Decision DiagramsDiagrams
• New data structures, such as QUIDDs [Viamontes, Markov, Hayes] allow for implicit parallelism when executing Kronecker multiplications on them.
• QUIDDs are based on ADDs and MTBDDs, • so hopefully in future other decision diagrams may be used
to represent quantum circuits. • It is also expected that basic software engines used
successfully in classical CAD (such as for instance SAT or ATPG methods) may be used to deal with quantum circuits.
• Also, the fast simulators based on new types of decision diagrams should be in future parallelized and possibly accelerated in FPGA-based boards.
• Even before quantum computers will be available, their emulations on standard computers and ASIC/FPGA may prove useful to solve some practical problems.
Example 5Example 5:: Testing and Testing and diagnosis of quantum circuitsdiagnosis of quantum circuits
• Patel, Markov, and Hayes showed that reversible circuits are much better testable than irreversible circuits.
– This is because every test covers half faults and every fault is covered by half tests.
– The reversible circuits are then “transparent” to faults, making them well observable and controllable.
• We showed that fault localization in reversible circuits is easier.
• We presented preliminary results on testing binary quantum circuits and on fault localization of quantum circuits.
Testing Quantum Circuits (1)
• The good circuit is simulated. • Next every possible quantum fault is inserted (our
fault model is inserting arbitrary matrix in place of fault, this allows to simulate many different types of faults) and the circuit with fault is simulated in Hilbert space (no measurement).
• All possible measurement values are calculated with their probabilities.
• The comparison of a measurement from the unitary matrix of a correct circuit and a circuit with fault determines which input combinations (tests) give different measurements.
• In some cases the circuit is modified for multi-valued realization in order to distinguish the values.
Testing Quantum Circuits (2)• Observe that in contrast to standard testing
and reversible circuits testing, there are three types of faults in quantum domain: – (1) faults that can be detected deterministically, – (2) faults that cannot be detected (like global
phase faults), and – (3) faults that can be detected by repeated
application of tests, possibly with special measuring gates.
• These faults can be detected only with certain probability.
• Thus, quantum testing is probabilistic testing.
Research Challenges in Quantum Test
• Open problems include basically everything: – fault models, – fault simulation, – test generation, – test minimization, – fault coverage,– fault localization using probabilistic adaptive
trees.
Quantum Computational Intelligence (QCI)• The two most famous quantum
algorithms to date were created by Peter Shor and Lov Grover.
• Shor’s algorithm is for factoring integers:– It produces an exponential
computational speedup over classical algorithms
– It can break the RSA encryption techniques.
• Grover’s algorithm searches an unordered list of data, to find a particular item. – It has a provable quadratic speedup
over the best classical algorithm. – It is like looking for name of a person
in yellow pages knowing only his telephone number.
Research Challenges in Quantum Algorithms for Computational Intelligence
• “How these algorithms can be used in the field of Computational Intelligence?”.
• Quantum computing is in every particular instance at least as powerful as standard computing.
• It is therefore very reasonable to look for quantum counterparts to all concepts created in past in:– algorithm design, – Artificial Intelligence, – Machine Learning, – Computational Intelligence,– Soft Computing.
Future Applications in Structured Search
• Grover algorithm for searching an unstructured database started many practical applications because of the generality of its main idea – phase amplification.
• Grover himself extended his algorithm for the structured search problem, one of the main tough research issues in AI, with a multitude of important and practical applications, including in EDA.
• Many interesting papers about quantum search using problem structure were written by Hogg and collaborators.
• Boyer developed bound for quantum searching algorithms. • The class of NP-complete problems includes:
– graph coloring, – satisfiability, – planning, – set covering, – combinatorial optimization, – tautology verification – and many other problems
that are useful for instance to solve the synthesis and optimization problems.
Generalizations of gates, circuits and automata
• Because gates, the basic concept of quantum computing, are a powerful generalization of gates in standard computing, researchers are systematically generalizing all the fundamental concepts of computing to involve quantum concepts in one way or another.
• And thus; – a quantum circuit is a generalization of a combinational
Boolean circuit, – Quantum Automata (various formalizations) generalize
Finite State Machines, – Quantum Turing machine generalizes Turing Machines and
Probabilistic Turing Machines, – and so on.
From CI to QCI• The same tendency is seen in Computational Intelligence. • Its concepts and algorithms are being generalized to those of
the Quantum Computational Intelligence (QCI). • And thus;
– Quantum Neural Networks, – Quantum Associative Memories, – Quantum Bayesian Nets, – Quantum Games, – Quantum Markets, – Quantum Agents, – Quantum Formulas, – Quantum Fuzzy Networks, – Quantum Spectral Transforms and Networks, – Quantum Evolutionary Algorithms, – Quantum Braitenberg Vehicles,– and many others have been created and are actively investigated both theoretically, using
software simulators, hardware emulators and in real quantum circuits.
理论计算机科学中的几个问题
应明生
清华大学计算机科学与技术系智能技术与系统国家重点实验室
Importance of intelligent learning
Research Challenges in NP problems
• Because laws of quantum mechanics proved useful to improve algorithmic performance of some NP problems, there is a high probability that more problems will find efficient solutions in quantum domain.
Quantum-Neural Algorithms:
• Quantum Associative Memories of Ventura and Martinez, • Competitive Learning in Quantum System by Ventura and
Perus. • While neural net processes real values, quantum NN
processes complex values. • It includes therefore standard NN and binary computers as
special cases• Thanks to superposition and entanglement can do much
more. • Weights that are complex values will allow to express much
more and higher order information. • Totally new algorithms can be invented for learning and using
such nets. • QuAM is analogous to a linear associative memory but all
neurons are quantum mechanical gates.
Research Challenges in QCI
• There are dual influences of CI and quantum computing. – 1. The quantum ideas can be used to create
powerful quantum-inspired algorithms to solve many types of problems in EDA, QDA and robotics.
– 2. The ideas and algorithms from many classical computer science areas can be now used in quantum domain or transformed and extended to quantum domain.
• Very little operational software packages that use these ideas.
Quantum Computational Intelligence
• Quantum Neural Nets
• Quantum Associative Memories
• Quantum Inspired Genetic Algorithms
• Learning by synthesis of quantum circuits
• Other models of learning based on quantum concepts.
• Quantum Braitenberg Vehicles.
In 2020 quantum computing will be In 2020 quantum computing will be a realitya reality
• As a community, we have a unique chance to work on the forefront of the future dominating technology.
• Logic design community did not have this opportunity in the past.
Quantum Circuit Design
And Technology
Mathematics and logic
Quantum Design Automation
Quantum Information and Quantum Computational Intelligence
Conclusions (1) • Emerging new area of Quantum Design Automation
(QDA). • Similarly as in design automation, there will appear
sub-areas of:– high level quantum synthesis, – logic level quantum synthesis, – quantum test, – quantum verification, – quantum simulation, – quantum software-hardware co-design, – quantum physical design, – automatic learning from examples, – data mining, – and so on.
Conclusions (2) • At the moment, even a single paper has been not
published in many of these areas• But surely they will appear in the forthcoming 10
years.
• We outlined some subjective choice of recent papers as a potential base of future research in QDA.
• Conventional logic synthesis, test and machine learning methods, for both binary and multiple-valued logic, form a powerful base of new approaches in quantum engineering.
Conclusions (3)
• Similarly the data structures like decision diagrams or fundamental algorithms such as satisfiability or reachability analysis continue to have their role.
• Because of high numerical demands of quantum logic there exist even higher expectations on these methods.
• Growing mutual influence of QDA and QCI, leading in long term to their unification.
My husband has not taken his decision yet, he is not sure if he should work on quantum
computing
What to remember?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.
