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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 - PowerPoint PPT Presentation
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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 computer as we know it?
1999 Pentium IIIB
www.icknowledge.com
http://www.pbs.org/transistor/science/events/pointctrans.html
1947 First point contact transistor
by Bardeen and Brattain
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
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. Feyemann
D. Deutsch
P. Shor
Number of Atoms in a Useful SystemFrom R. Keyes, IBM J. Res. Develop (1988)
# atoms to store a bit # dopant atoms/bipolar transistor
EX: Quantum Parallelism
• Quantum:– Put all 7-bits into a superposition state– superposition allows quantum computer to
make calculations on all 128 possible numbers (27) in ONE iteration i.e. finishes in 1 second.
– Tremendous possibilities… imagine doing computations on even larger sample spaces all at the same time!!!
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
Qubits as binary Qudits• In multi-valued (MV) Quantum Computing (QC), the unit of memory
(information) is qudit.
• For instance, ternary logic values of 0, 1, and 2 are represented by a set of distinguishable different basis states of a qutrit.
• These states can be a photon’s polarizations or an elementary particle’s spins.
• After encoding these distinguishable quantities into multiple-valued values, qutrit states are represented by basis states |0>, |1> and |2> , respectively.
• A qubit, used in binary QC uses only two basis states, |0> and |1>
• Qubit and qutrit are then special cases of qudits A
B
AP
BQ
C CABR
Qudits• Qudits exist in a linear superposition of
states, and are characterized by a wave function .
• As an example (), it is possible to have light polarizations other than purely horizontal or vertical, such as slant 45 corresponding to the linear superposition of .
• In ternary logic, the notation for the superposition is , where , , and are complex numbers.
• These intermediate states cannot be distinguished, rather a measurement will yield that the qutrit is in one of the basis states, , , or .
• The probability that a measurement of a qutrit yields state is , state is , and state is .
• The sum of these probabilities is one. • The absolute values are required since, in
general, , and γ are complex quantities.
• Pairs of qutrits are capable of representing nine distinct states,, , , , , , , , and , as well as all possible superpositions of these states.
Quantum Logic
Circuits
Quantum Logic
50%
50%0
1
SpecchioSingle photon
Optical sensor
… strange behavior
1
1
0
0
Quantum Gate
0
1 0
1
NOT
1
1
0
0
notnot
Qubit
Qubit in a Ion Trap
Deterministic Turing Machine
Initial State Final State
Deterministic Turing Machine transits deterministically from initial to final state.
Probabilistic Turing Machine
P1
P2
P3
P4
P5
P6
P7
P8
P9
P = P2P7 + P3P8
Probabilities of final output states P
robabilistic output states
Quantum Computation
A1 A2
A3 A4
A = A1A2 + A3A4
P = |A1A2 + A3A4|2
= |A1A2 + A3A4|2
+2Re(A1A2A3A4)
Decoherence
A beam-splitter
0
1
0
1
%50
%50
The simplest explanation is that the beam-splitter acts as a classical coin-flip, randomly sending each photon one way or the other.
Quantum Interference
0
1
0
1 %100
The simplest explanation must be wrong, since it would predict a 50-50 distribution.
More experimental data
0
1
0
1
2
cos2
2
sin2
A new theory
0
1
0
1
2
cos2
2
sin2
12
10
2
i 1
2
e0
2
i i
The particle can exist in a linear combination or superposition of the two paths
12
)1e(i0
21e ii
Probability Amplitude and Measurement
0
1
0
1
2
0
If the photon is measured when it is in the state then we get with probability
10 10
2
1
2
00
12
1
2
0
Quantum Operations
The operations are induced by the apparatus linearly, that is, if
andthen
1
2
i0
2
11
2
10
2
i10 1010
12
10
2
i0
12
i0
2
11
12
i
2
10
2
1
2
i1010
Quantum Operations
Any linear operation that takes statessatisfying
and maps them to statessatisfying
must be UNITARY
12
1
2
0 10 10
10 '1
'0 1
2'1
2'0
Linear Algebra
10 10
0
0
1
1
1
0
1
010 1
0
0
1
corresponds to
corresponds tocorresponds to
Linear Algebra
2
i
2
12
1
2
i
corresponds to
corresponds to
ie0
01
Linear Algebra
0
corresponds to
0
1
2
i
2
12
1
2
i
ie0
01
2
i
2
12
1
2
i
Linear Algebra
I10
01uu
uu
uu
uuUU *
**
1110
0100t
11*01
1000
1110
0100
uu
uuU
is unitary if and only if
Abstraction
The two position states of a photon in a Mach-Zehnder apparatus is just one example of a quantum bit or qubitExcept when addressing a particular physical implementation, we will simply talk about “basis” states and and unitary operations like
and
0 1
H
H(a)
2
1
2
12
1
2
1
(b)
Im
++--
Re
2
1
2
1
2
1
2
1
|0>
|1>(c)
+
-2
1
2
1 (d)|0> |0>
|1> |1>
ie0
01
(a) (b)(c)
ie
|0>
|1>
(d)
-
+
cos
sin
cos
sin
An arrangement like
0
is represented with a network like
H H0
H 0(a)
(b)
H
+
-
2
1
2
1
2
1
2
1
+
-
+
-
2
1
2
1
2
1
2
1
+
-
+
cos
-sin
cos
sin
(b)
H
(a)H
+
-
2
1
2
1
2
1
2
1
+
-
+
-
2
1
2
1
2
1
2
1
+
-
+
-
2
1
2
1
2
1
2
1
+
-
+
-
2
1
2
1
2
1
2
1
+
-
|0>
|0>
|1>
|1>
|00>
|01>
|10>
|11>
|0>
|1>
|00>
|01>
|10>
|11>
|0>
|1>
More than one qubit
10 10
If we concatenate two qubits
11100100 11011000
10 10
we have a 2-qubit system with 4 basis states0000 0110 1001 1111
and we can also describe the state as
or by the vector
1
0
1
0
11
01
0
00
More than one qubit
In general we can have arbitrary superpositions
11011000 11100100
12
11
2
10
2
01
2
00
where there is no factorization into the tensor product of two independent qubits.These states are called entangled.
Measuring multi-qubit systems
If we measure both bits of
we get with probability
11011000 11100100
yx2
xy
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 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 vs. Quantum Circuits
ci in 1in 1i0i0
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
Quantum Circuit Characteristics
• Quantum Measurement– Measurement yields only one state X of the
superposed states – Measurement also makes X the new state and
so interferes with computational processes– X is determined with some probability, implying
uncertainty in the result– States cannot be copied (“cloned”), implying that
signal fanout is not permitted– Environmental interference can cause a
measurement-like state collapse (decoherence)
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
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
Quantum Gates
• One-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: 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
0 1
1 0
0 1
1 0
1 0
0 1
NOTNOT
=
Quantum Gates
• 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
Universal One-Input Gate Sets• 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
Quantum Gates
U Any state
2H H2
• Two-Input Gate: Controlled NOT (CNOT)
Quantum Gates
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 – Serves as a “non-demolition” measurement gate
x
y
x
x y
x
y
x
x y CNOT
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
x
y
x
x y
|00> |00>
|01> |01>
|10> |10>
|11> |11>
(a) (b)
(c) (d)
• 3-Input gate: Controlled CNOT (C2NOT or Toffoli gate)
Quantum Gates
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
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
(a) (b)
|000>
|000>
|001> |001>
|110>
|010>
|111>
|011>
(c)
|010>
|011>
|100>
|101>
|100>|101>|110>
|111>
• General controlled gates that control some 1-qubit unitary operation U are useful
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
• 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
Example 1: Simulation
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
?
Quantum Circuits
|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
Example 1: Simulation (contd.)
?
• Exercise: Simulate the two remaining cases
Quantum Circuits
Example 1: Algebraic analysis
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 Circuits
Example 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 Circuits
Example 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
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
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(x)
QC
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)
Quantum Gate Arrays
1-bit full adder
|0>
|0>
|x>
|y>
|c> |c>
|x>
|y>
|s>
|c’>
Let |c> = |1>, |x> = |0>, |y> = |1>
|0> |0> |1>
|1> |1> |0>
|s> = |0>, |c’> = |1>
Quantum Gate Arrays
It is possible to construct reversible quantum gates for any classical
computable function f with m input and k output bits.
Uf
|x>
|y>
|x>
|y f(x)>
There exists a quantum gate array that implements the unitary transformation Uf : |x, y> |x, y f(x)>, where indicates bitwise xor.
Quantum Gate Arrays
The previous transformation Uf is reversible
Uf + = Uf Uf
+ Uf = Uf Uf = I
UfUf
|x>
|y>
|x> |x>
|y f(x)>
|y f(x) f(x)>
But |y f(x) f(x)> = |y>
Superposition of Quantum States
Consider the Tofolli Gate
|x>
|y>
|0>
|x>
|y>
|x> |y>
Apply T, the Tofolli transform, to the superposition of all inputs.
T(H|0> H|0> |0> ) = ½(|000> + |010> + |100> + |111>)
Quantum parallelism – Applying the Tofolli transform to a superposition of all of the input states produces a superposition of all of the states in the “truth table”
Superposition of Quantum States
BUT Only one of the superposed states can be extracted by measurement
T(H|0> H|0> |0> ) = ½(|000> + |010> + |100> + |111>)
0
=
1
|000> + |010> + |100> 0
|111> 1
Measurement of the output projects the superposition onto the set of states consistent with the result
Quantum ParallelismIn order to take advantage of quantum parallelism one must:
1. Transform the state in such a way as to amplify the values of interest – so that they have a higher probability of being selected during measurement
2. Find common properties of ALL the states of f(x)
Shor’s factoring algorithm
Grover’s unstructured search algorithm
Where to learn more
• Web Page of Marek Perkowski– class 572 - see description of student projects– Portland Quantum Logic Group
• Superposition property may be mathematically described using the Kronecker product (tensor product) operation
• The Kronecker product of two matrices is defined as follows:
dvdzcvcz
dydxcycx
bvbzavaz
bybxayax
vz
yxd
vz
yxc
vz
yxb
vz
yxa
vz
yx
dc
ba
Kronecker Product of Matrices
Tensor Products
• Similarly one can define tensor products for any size of matrices and in particularly for vectors representing superposed states.
• As an example, consider two qutrits with and . • When the two qutrits are considered to represent a
state, that state is the superposition of all possible combinations of the original qutrits, where:
2221201211
10020100
2121212121
212121212112
Superposition• The superposition property allows the qubit states to grow much faster
in dimension than classical bits, and the qudits faster than qubits. • In a classical system, n bits represent distinct states, whereas n qubits
correspond to a superposition of 2n states and n qutrits correspond to a superposition of 3n states.
• In the above formula some coefficient can be equal to zero, so there exists a constraint bounding the possible states in which the system can exist (entanglement).
• “Allowing d to be arbitrary enables a tradeoff between the number of qudits making up the quantum computer and the number of levels in each qudit”.
• Because in contemporary quantum technologies every qubit is costly, higher radices than 2 give an advantage of improved processing and storage power at the same realization cost.
• This is a strong argument for realization of multi-valued logic in quantum circuits.
• In addition to standard advantages of mv logic, quantum mv logic may be superior to binary one because of different nature of entanglement.
Quantum Notation
• An output of a gate is obtained by multiplying the unitary matrix of this gate by a vector of Hilbert space corresponding to this gate’s input state.
• (Unitary matrix U is one such that U . U+ = I, where U+ is a Hermitian matrix of U. A Hermitian U+ is a conjugate transpose matrix of U).
• A gate or a sub-circuit of a quantum circuit corresponds to a unitary matrix. • As shown below, the resultant unitary matrix of an arbitrary quantum circuit is created by
matrix multiplications or Kronecker multiplications of matrices of its composing sub-circuits. • Various quantum notations contribute to the difficulty in understanding the concepts of
quantum computing and creating efficient analysis, simulation, verification and synthesis algorithms for QC.
• Generally, however, we believe that once the minimal amount of formalism is understood, logic researchers can quickly contribute to new designs, since much can be learned from the history of Electronic Design Automation as well as from MV logic theory and design.
• The lessons learned there should be used to design efficient QDA tools for MV quantum computing.
• Here we include the absolute minimum amount of formalism sufficient to start independent software development by people who have sufficient background in EDA tools and algorithms such as search or evolutionary programming
Quantum Circuits
• In terms of logic operations, anything that changes a vector of qudit states to another qudit vector satisfying measurement probability properties can be considered as a quantum operator (unitary matrix).
• These phenomena can be modeled using the analogy of a “quantum circuit” (called also quantum array).
• In a quantum circuit, a wire does not carry ternary values but corresponds to a 3-tuple of complex values, , , and γ.
• Quantum logic gates of the circuit map the complex values on their inputs to complex values on their outputs.
• As mentioned, operation of all quantum gates and their assemblies is described by matrix operations.
• Any quantum circuit is a composition of parallel and serial connections of blocks, from small to large.
Analysis of Quantum Circuits• Small blocks correspond to quantum gates that are easily directly
realizable (like Pauli rotations) or are very simple and require just few basic quantum operations such as Feynman gates or Stroud/Muthukrishnan gates.
• Serial connection of blocks corresponds to multiplication of their (unitary) matrices.
• Parallel connection corresponds to Kronecker multiplication of their matrices.
• So, theoretically, the analysis, simulation and verification are easy and can be based on matrix methods.
• Practically they are tough because the dimensions of the matrices grow exponentially.
• All these become much easier when one deals only with permutative matrices, which are equivalent to multi-output truth tables of completely specified functions. In such matrices there is exactly one “1” in every row and column.
• An active research area is to represent operations on unitary matrices (in particular, the permutation matrices) by new efficient data structures and algorithms.
Calculating output state of QC
• Typically the symbols |0> and |1> are not present in the matrix formulation of the equations, only the probability amplitudes (i.e. and ) are included; however, there are kept in Equation (1) for illustrative purposes.
(1)
| 0 | 0 |1
|1 | 0 |1
a b a b
c d c d
• Because the qubit probabilities must be preserved at the output of the quantum gate, all matrices representing them are unitary.
• An important unitary matrix property is that of a full rank. • This property implies that quantum gate matrix rows and columns are
orthonormal. • Therefore, past results from spectral methods for classic digital logic
are directly applicable to quantum logic synthesis. • Furthermore, since quantum logic gates are represented using unitary
orthonormal matrices, they represent logically reversible gates. • These observations mean that the single input/output quantum logic
gates as represented in Equation (1) are rotation matrices characterized by some particular rotation angle , where, for example, a = cos, b = sin, c = -sin and d = cos.
• With this viewpoint, it can be seen that there are, in fact, an infinite number of single input/output qubit gates.
Rotation Gates• However, three elementary gates can be used to generate any rotation [7]. • These are the R, S, and T gates described in matrix notation by
(1a)
cos sin 1 0 0
sin cos 0 0
j
j j
eR S T
e e
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
XOR logic synthesis is useful for QC
• In this example, the input is |10> = ((0)|0>+(1)|1>) ((1)|0>+(0)|1>), and the input vector is represented by the coefficients shown in parentheses.
• It is a significant fact that the unitary gates described by Equations (1) and (2) can realize any quantum logic function (including standard binary).
• There are several strong similarities of quantum logic to classic digital circuit design using AND/XOR logic.
• Our research group has been heavily involved in AND/XOR logic circuit design as well as related algebraic and spectral methods for several years.
• We found these experiences very useful in quantum
circuit design.
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.
1
j
Figure 1. Bloch Sphere with 6 values shown
1
j
• The identity matrix and three Pauli matrices:
• form a basis for the 2x2 density matrices. • So every density matrix can be written as p = ½ (I + ax X + ay Y + az Z). • We associate with every 1-qubit state p = ½ (I + ax X + ay Y + az Z) the
vector (ax, ay, az). If p = | | for a state | = e j (cos ( / 2) |0 + e j sin ( / 2) |1 ).
• Then the corresponding vector is (ax, ay, az) = (sin cos , sin sin , cos ).
• It can be easily derived that the vectors (ax, ay, az) satisfy |ax|2 + |ay|2 + |az|2 = 1, which means that all pure states are located on the surface of the Bloch Sphere.
• When there many identical quantum circuits working together they are described by density matrices and the (mixed) states may lay inside the sphere, not on the surface
10
01Z,
0i
i0Y,
01
10X,
10
01I
One way to realize multi-valued logic using binary quantum computing.
• Figure shows the location of 6 points, that may correspond to values of some multi-valued algebras.
• For binary logic we use |0 and |1.
• For quaternary logic we use |0, |1, |0+|1, and |0-|1.
• For 6-valued logic we may use additionally |0 + j |1 and |0 - j|1.
• A rotation or a combination of rotations leads from one value to any other value.
1
j
Important quantum gates
• Because global phase does not count, the T gate can be also written as follows:
T (/8) = .
• H denotes the important Hadamard gate:
4/0
01je
11
11
2
1H
• The Hadamard and the /8 gate can be used to approximate any given single-qubit unitary operation with arbitrary accuracy.
• On the Bloch sphere, T and HTH are rotations by an angle /4 radians around the z- and x-axes, respectively.
• The composition of these two operations gives a rotation by an angle , which is defined by cos/2 = cos2/8, around an axis n, which is defined by n = (cos /8, sin /8, cos/ 8).
• Since is irrational, any rotation around the -axis can be build, to arbitrary precision, from T and HTH.
• Furthermore, since for arbitrary H R n () H = R m () with m = (cos/8, - sin/8, cos/8 ) not collinear n, there are angles , , such that any given U can be written U = Rn() Rm () R n().
• It can be also shown that any given 2-qubit gate can be composed from CNOT and a single qubit gate.
• Similarly other universal sets of 1-qubit gates can be found and illustrated using Bloch Sphere.
• This sphere is also useful to find operator identities (quantum generalizations of rules like “ Not (Not B) = B “ ) which play fundamental role in quantum circuit optimization.
• Study of universality and power as well as quantum realization costs of these gates are still active research areas.
• More study should be devoted to multi-valued Bloch Sphere, operators in it and their transformations and realization.
• Above we showed how multiple-valued logic can be encoded in binary quantum computing.
• Quaternary logic requires two binary measurements (readings).
• The first reading distinguishes states |0 and |1, and the second reading uses additional rotation gates to distinguish between states |0+|1, and |0-|1.
• It can be shown that the logic with 2n values requires n readings.
• Another approach to multi-valued quantum circuits requires measurements with more than two basis states.
• Also, new gates should be defined as well as the synthesis methods for these gates.
• While several books and numerous papers have been published on binary quantum circuits [45,72,103] not much information on their multi-valued counterparts is available. In their pioneering paper,
• Muthukrishnan and Stroud [68] developed in 2000 multi-valued logic for multi-level quantum computing systems and showed their realizability in linear ion trap devices.
• However, no experimental data are known so far. In addition, this approach generates circuits that are too large and no procedure was proposed to minimize them.
• In 2002, Brylinski and Brylinski [13] discussed the universality of n-qudit gates without giving any design algorithms.
• Since 2001, PQLG group [2-5,51-55] proposed Galois Field approach to multi-valued quantum logic synthesis in several regular structures.
• They used gates were ternary counterparts of classical binary Feynman and Toffoli gates. De Vos [23] proposed two ternary 1*1 gates and two ternary 2*2 gates, but again no synthesis method was proposed.
• In 2002, Perkowski, Al-Rabadi, and Kerntopf [75] proposed a 2*2 Generalized Ternary Gate (GTG gate) based on the ternary conditional gate [68] and ternary shift gates [52-54] and showed the realization of ternary Toffoli gate using GTG gates. This work introduced for the first time the practical realizability of Galois Field circuits in realizable multi-valued quantum technology.
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.
Quantum Circuit Simulation
• Simulation of quantum circuits plays absolutely fundamental role in many areas of quantum physics and engineering.
• Similarly as in classical circuits design, simulation is used to verify correctness of the design, analyze its properties and find some interesting aspects that cannot be found by “hand and pencil” methods.
• It is amazing that the first quantum algorithms were invented without quantum simulators, but now the researchers routinely use quantum simulators to help them with inventions and verify their design guesses.
• Quantum simulators are used to simulate a good circuit and a circuit with inserted faults, for test generation and fault localization.
• Moreover, because the search-based synthesis methods for quantum circuits such as exhaustive search, genetic algorithms, genetic programming, simulated annealing or heuristic search do not use deeper knowledge of circuit structure and properties, simulation is the only way (to be used as a part of the fitness function) to direct the search towards a circuit that satisfies the given requirements.
• The results of the simulation are compared with the circuit specification many times in the loop of the search program.
• The same is true for quantum fault simulation. • As we see, in all these applications the simulation of quantum circuits must be very
fast and the computer memory should be large. • On the other hand, matrix operations on unitary matrices are slow, thus new methods
and representations should be found to allow for very fast and low in memory usage simulation.
• This is attempted to be achieved by two fundamental methods: – (1) acceleration of standard operations by using special hardware emulators, parallel
computers or processor networks [71,73],– (2) creating new advanced data structures to represent quantum data more efficiently
using standard computers.
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.
Multi-valued Quantum Circuit Synthesis
• Let us first briefly summarize current results in binary quantum circuit synthesis.
• This is the most advanced research area and there are two gate models for synthesis (especially for permutative circuits):– (1) The first gate model assumes that only gates with limited
number of inputs can be used (for instance universal Toffoli3 gate that operates on three qubits; P=a, Q=B, R=abc).
– 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.
Multi-valued Quantum Circuit Synthesis
• (2) The second gate model assumes that for any given number of qubits N for which a function is realized, there exist a Toffoli gate ToffoliN (or a similar universal gate in which one data qubit is controlled by more than 2 control qubits) that operates on N qubits.
• We will call it the unlimited qubit gate model. • In the first model it was proved by Shende et al that every
N-qubit reversible function which is represented by an even number of cycles, is realizable without constant wires (ancilla bits) and every N-qubit function that is represented by an odd number of cycles is realizable with N+1 wires (one ancilla bit).
• (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.
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.
• 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.
What can we do?
Quantum Computers
• Our community should should develop a systematic methodology and CAD tools for synthesizing, verifying, testing and simulating of quantum computers.
• These methods and tools will be counterparts of what exists now in binary CMOS.
• Development of these tools will require understanding of real quantum circuit technology.
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,
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.
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.
Four Synthesis Models
• There exist the following synthesis models, both for binary and multiple-valued logic:
1. limited qubit gate model and full reversible function (way a). Usually zero or one ancilla bits are expected.
2. unlimited qubit gates and full reversible function (way a). Usually zero or one ancilla bits are expected.
3. limited qubit gates and single output function (way b). Usually at most M ancilla bits are expected.
4. unlimited qubit gates and irreversible input function (way b). Usually at most M ancilla bits are expected.
• 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.
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.
• 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].
Research Challenges
• Recent works suggest that many uniform general methods can be created to realize various multiple-valued logics that will use generalized rotations with respect to 3 orthogonal basis axes, rotations by angles 2/k, where k>1 is a natural number.
• In general, rotations with respect to any axis n can be used, but using some of Z, X, and Y simplifies gates.
• Every existing algorithm for binary quantum circuit design can be extended to its various multiple-valued quantum counterparts, but these generalizations are not trivial and algorithms that use these gates are numerically very challenging.
• These problems form then a good base for new research by people who understand search-based EDA algorithms and multiple-valued logic.
Figure 2. 3*3 Toffoli gate
• Figure 2 presents a standard binary reversible Toffoli gate.
• Its ternary counterpart has Galois Field 2 operations of multiplication and addition replaced with Galois Field(3) operations.
A
B
AP
BQ
C CABR
• Observe that the internal structure of this gate is complex when using quantum realizable gates (Figure 3). The Controlled-V gate works like this: when the control (top) signal is |0>, the data input is forwarded to output with no change. When the control signal is |1> the operation of the lower box (so-called V) is executed. In our case this is a square-root-of-NOT operation. Thus if two Controlled-V gates in series are controlled by the same signal A, if A=1 then their qubit data line is a negation. Two such gates in series serve then as a controlled-NOT or Feynman gate. Also, the operation of V and V+ annihilate ( V V+ = I ) . The reader can simulate “by hand” the circuit from Figure 3a to see that it truly realizes the Toffoli3 gate. Let us observe that the circuit from Figure 3a can be redrawn to one from Figure 3b. This circuit emphasizes that both CNOT, CV and C V+ are Controlled-Gates that leave data signal unchanged when the control is |0> and apply its internal transformation (the symbol of this transformation is in the input to multiplexer) when the control is |1>.
• Observe that the internal structure of this gate is complex when using quantum realizable gates (Figure 3).
• The Controlled-V gate works like this: when the control (top) signal is |0>, the data input is forwarded to output with no change.
• When the control signal is |1> the operation of the lower box (so-called V) is executed.
• In our case this is a square-root-of-NOT operation. • Thus if two Controlled-V gates in series are controlled by the same signal A,
if A=1 then their qubit data line is a negation.• Two such gates in series serve then as a controlled-NOT or Feynman gate.
Also, the operation of V and V+ annihilate ( V V+ = I ) . • The reader can simulate “by hand” the circuit from Figure 3a to see that it
truly realizes the Toffoli3 gate. • Let us observe that the circuit from Figure 3a can be redrawn to one from
Figure 3b. • This circuit emphasizes that both CNOT, CV and C V+ are Controlled-Gates
that leave data signal unchanged when the control is |0> and apply its internal transformation (the symbol of this transformation is in the input to multiplexer) when the control is |1>.
• 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.
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
A
B
C V V V+
P
R
S
A
B
C
P
R
S
(a)
(b)V V V+
+1 +1
•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.
Figure 4: Conceptual ternary multiplexer
op = Logical Operations:
+0, +1, +2 represent Galois Addition of constants 0, 1, and 2, respectively
01, 02, 12 represent logical replacement i.e. a 01 operation will replace 0->1, 1->0, and 2->2
•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
Other problems in MV QC(1)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.
(2)Development of methods based on unitary matrix decomposition, group theory, Lie groups and Clifford algebras,
(3)Methods for incompletely specified functions, to be used in machine learning and data mining,
(4)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),
(5)Efficient methods for local optimization of quantum circuits on many levels of description,
(6)High-level quantum hardware description languages that will play in QDA a role similar to VHDL and Verilog in EDA,
(7)Development of formalisms and synthesis methods for sequential circuits.
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.
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 from section 2 of this paper.
• The name NP means non-deterministically polynomial, because there are no deterministic algorithms to solve NP problems in polynomial time (w.r.t the size of the problem).
• Any problem in the NP-complete class can be transformed into any other problem in this NP-complete class using polynomial number of steps.
• The quantum search algorithms can be used to solve the ”constraint satisfaction problems” into which all other NP-complete algorithms can be reduced [17].
• In a constraint satisfaction problems (SAT is the simplest example, graph coloring is another one) we deal with multi-valued variables and constraint rules on value relations between values of subsets of variables (relations like, “two adjacent nodes in a graph should have different colors”).
• In other words, one has to find assignment of values b to all a variables so that all constraints are satisfied.
• All such problems can be reduced theoretically to SAT, but this is not necessarily the best way to solve them.
• On a classical computer O(ba) assignments must be searched before finding a valid solution, if any.
• Using heuristics, or domain-dependent knowledge of a particular problem’s structure, the search can be dramatically speeded up to O(bka), where k1 and is problem dependent.
• Grover’s quantum search algorithm for structured problems further reduces the number of states searched to O(bak/2), which means a polynomial speedup over classical algorithms.
• This may be enough to solve many currently intractable problem instances [58].
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.
• 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.
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.
Research Challenges in QCI
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.
Open Problems in Reversible Non-quantum Circuits
• Reversible adiabatic gates and circuits.– Are they different than in quantum? Why?
• Are garbage and ancilla bits important?
• Realization of adiabatic reversible circuits and synthesis for them.
• Applications in practical architectures for low power.
• Applications in practical architectures for standard computing (FFT, butterflies, DSP circuits).
• Testing and Test localization.
• Realization in DNA and other nano-technologies (fluidic)
• Optical realizations.
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
