Search for Universal Ternary Quantum Gate Sets with Exact Minimum Costs

1

• Search for Universal Ternary Quantum Gate Sets
  with Exact Minimum Costs

Normen Giesecke, Dong Hwa Kim, Sazzad Hossain
and Marek Perkowski Department of Electrical
Engineering, Portland State University, FAB
160-05, 1900 SW Fourth Avenue, Portland, Oregon,
USA, E-mail mperkows_at_ee.pdx.edu Dept. of
Instrumentation and Control Engn., Hanbat
National University, 16-1 San Duckmyong-Dong
Yuseong-Gu, Daejon, Korea, 305-719.E-mail kimdh_at_h
anbat.ac.kr
2
Hierarchical Decomposition and Synthesis of
Ternary gates
Mozammel Khan, ISMVL 2007
Synthesis using Logic Blocks and Gates
Design of Logic Blocks and Gates
This paper
Ternary Quantum Multiplexers
Ternary Muthukrishnan-Stroud Gates
Soonchil Lee et al, MVL J 2006
Single-Qubit Rotation gates and 2-qubit
Interaction gates
3
Circuit Structures for Ternary Logic extend the
structures for binary logic
Binary Multi-Cube gate
Multivalued counterpart of the Multi-Cube gate
for any radix. FG is Feynman-Galois gate. Symbol
? stands for exor.
Kmap of function f(a,b,c,d) realized by the gate
F f(a,b,c,d) from above.
4
Toffoli-like 3-controlled gate structure for
Galois Field Sum of Product Circuits
5
A cascade of two 2-controlled Toffoli-like gates
for Modulo sum of minima type of circuits
6
Ternary Wave Cascade (Modsum of ternary Maitra
cascades)
Because these structures are used again and
again, it is definitely worthy to optimize their
components very well, even spending months of
computer time.
7
Quantum Reversible Cascades with Ternary Quantum
Multiplexers
Op. 4
A
A
Op. 5
Operations for a ternary system
Op. 6
0 Wire
1 Modulo Shift 1
0 1 2
Op. 1
Op. 7
2 Modulo Shift 2
B
B
01 Swap 0 1
Op. 2
Op. 8
02 Swap 0 2
12 Swap 1 2
Op. 3
Op. 9
Time

• Reversible cascades are used to represent logic
  gates. The gates themselves are realizable via
  quantum technologies.

• Reversible cascades are not schematics instead
  of being physical representations, they are
  chronological.
• Time flows from left to right
• Gates are not physical gates instead, they are
  electromagnetic pulses applied to some group of
  quantum particles that change their bit
  representation

• This means that there cannot be a feedback
  Gates cannot be controlled by previous states
  that have changed.

8
The Muthukrishnan-Stroud Gate
Operations for a ternary system
0 Wire
1 Modulo Shift 1
2 Modulo Shift 2
01 Swap 0 1
02 Swap 0 2
12 Swap 1 2
A
P
2
operation
B
Q
Two views of MS gate

• Multi-valued representation is based on the
  Muthukrishnan-Stroud gate
• It acts essentially as a multi-valued multiplexer
• There is one control line, one input line, and
  one output line
• When the control line qubit is at its highest
  order value (i.e., 2gt in a ternary system of
  0gt, 1gt, 2gt), it selects an operation to apply
  to the input line
• If the control line is at any other value, not
  the highest order value, the multiplexer acts as
  a quantum wire and passes the input directly to
  output

9
Quantum Reversible Cascades cont.
Operations for a ternary system
0 Wire
1 Modulo Shift 1
2 Modulo Shift 2
01 Swap 0 1
02 Swap 0 2
12 Swap 1 2

• Based on the Muthukrishnan-Stroud gate, we use a
  generalized multi-valued gate which we can
  implement via macros of the Muthukrishnan-Stroud
  gate
• It is similar to the Muthukrishnan-Stroud gate,
  except it can select different operations for
  different control line values, rather than a
  multiplexer that only operates when the control
  line is the highest value
• Ultimately we expect to see direct implementation
  of the generalized ternary gate (GTG)

• The operations used are multi-valued operators.
  The operators for a ternary system are listed

10
Muthukrishnan-Stroud Gate

• Internally, built from Interaction gates and
  rotations

11
What are the internals of the MS Gate?
Sequence of X, Y and Z rotations
Schematic view of Muthukrishnan-Stroud Gate as a
controlled sequance of rotations in X, Y and Z
axes by arbitrary angles
X rotation
Y rotation
Z rotation
12
Use of interaction gate
X rotation
Y rotation
Z rotation
General case
rotations
rotations
Z
rotations
Z
Special cases are cheaper
X rotation
Y rotation
Z
Z
13
General view of cascade for D-level circuits
rotations
rotations
rotations
rotations
rotations
Z
rotations
Z
Every multi-valued quantum multiplexer can be
build like this
14
First two structures based on cascaded quantum
multiplexers


15
One more structure based on cascaded quantum
multiplexers
16
Problem Formulation/Motivation

• The system in ternary logic
• A ternary output is specified
• Goals
• Find a (quasi)-minimum circuit in a form of a
  cascade, given input/output specification.
• Introduction of minimal number of ancilla bits
  (garbage/constant input)
• Gates
• Only Generalized Ternary Gates (GTG) in series
  are used for synthesis

17
Exhaustive Search Why?

• No experience and knowledge about the space to
  search. Nothing was published when this work
  started
• To get a feeling what GTG are capable for
• Straight forward process
• A breadth-first search seemed a good start

18
Breadth first search (BFS)

• BFS is a tree search algorithm used for searching
  a tree, tree structure, or graph.
• The breadth-first-search begins at the root node
  and explores all the neighboring nodes. Then for
  each of those nearest nodes, it explores their
  unexplored neighbor nodes, and so on, until it
  finds the goal.

• There are 216 different GTG realizations (63
  operation combinations)

19
Breadth first search (BFS) cont.

• 216 different GTG realizations

1.GTG - 1.GTG - 1.GTG
1.GTG - 1.GTG - 2.GTG

216.GTG - 216.GTG - 216.GTG
20
Exhaustive Search Example

• This example gives the implementation of a
  2-qudit ternary SWAP Gate
• The example begins on the first multiplexer and
  ensue the first truth table. Continuing that way,
  the last of the truth tables shows the results of
  the multiplication of the truth tables of the
  current multiplexer with the one before
• The third truth table shows the solution of the
  2-qudit ternary SWAP

21
Exhaustive Search Example cont.

• 2-qudit SWAP gate

22
Ternary Quantum Logic Exhaustive Search
23
Iterative deepening search
24
Iterative deepening search l 0
25
Iterative deepening search l 1
26
Iterative deepening search l 2
27
Iterative deepening search l 3
28
Iterative deepening search

• Number of nodes generated in a depth-limited
  search to depth d with branching factor b
• NDLS b0 b1 b2 bd-2 bd-1 bd
• Number of nodes generated in an iterative
  deepening search to depth d with branching factor
  b
• NIDS (d1)b0 d b1 (d-1)b2 3bd-2
  2bd-1 1bd
• For b 10, d 5,
• NDLS 1 10 100 1,000 10,000 100,000
  111,111
• NIDS 6 50 400 3,000 20,000 100,000
  123,456
• Overhead (123,456 - 111,111)/111,111 11

29
Properties of iterative deepening search

• Complete? Yes
• Time? (d1)b0 d b1 (d-1)b2 bd O(bd)
• Space? O(bd)
• Optimal? Yes, if step cost 1

30
Summary of algorithms
31
Multiplexer Implementation

• Multiplexer implementation for two variables is
  in fact straightforward
• Here we also introduce the idea of mirroring
• After a constant input line performed its
  operation, it can be reused.
• But for before it needs to be reset
• Mirroring serves this purpose well, at the cost
  of some additional gates.
• By introducing N additional gates, where N is the
  number of gates required for implementation, an
  inverse set of gates can be implemented to
  realize the original set of inputs on the output
• Notice that each and every operation (both swap
  and shift operations) have a conservative map
  or inverse operation

32
Inverse Gates

• Realization of the ternary Toffoli Gate as an
  example for mirroring

33
Limitations on the Goal Function

• Because the operation of a GTG gives outputs that
  are always conservative, the goal function must
  be conservative with respect to the input
  variable
• Functions that are NOT balanced cannot be
  directly implemented they can, however, be
  implemented if we introduce an ancilla bit
• An ancilla bit is simply an input line that is a
  known constant e.g. 0gt
• Also referred to as garbage input. Unless
  restored using the property of reversibility, it
  will result in a garbage output
• Formula to calculate the number of balanced
  functions for a given radix and number of qudits
  (pradix nnumber of qudits)

Balanced function
0
1
2
A\B
0
1
2
0
1
2
0
1
2
0
1
2
Unbalanced function
A\B
0
1
2
0
0
0
0
0
1
2
1
0
2
1
2
34
Implementation with more Input Variables

• In the previous examples, the input was two
  variables.
• Here we see an example of a 3-variable problem,
  the ternary Toffoli Gate
• The Realization uses MS Gates and needs the
  minimum cost of 4 single qudit operations.
• The Toffoli gate is a balanced gate and therefore
  no ancilla bit is needed.

A
B
C
Karnaugh map and realization of the ternary
Toffoli gate
35
Results The MIN and MAX Gate

• The following two gates are the MIN and MAX gates
• They can be used to build up a PLA like structure
  (using Mod-Sum)
• Their drawback is the required ancilla qudit, but
  contemporary circuit CAD systems may be reused to
  start building quantum circuits out of MIN/MAX
  gates

36
Results The Feynman Gate

• The Feynman Gate was found to be universal to
  construct complete quantum circuits.
• There is a second version, which is called
  ternary Feynman Galois gate
• Their realizations using GTG are shown below on
  the right-hand side

2-qudit ternary Feynman (Galois) gate
A
RA
A
R
B
S
B
1
2
37
Results 2-qudit SWAP Gate

• The SWAP gate exchanges a pair of inputs to the
  output.
• It has no counterpart in the classical binary
  logic because the crossing of electrical wires,
  for instance within 2 layers of metallization, is
  applied wherever it is needed and no special gate
  is required for this action.
• There are no real wires and thus a copying or
  cloning gate is required to perform this

2-qudit ternary SWAP gate Symbol (a),
Input/Output table (b), Realization (c)
B
A
B
A
1
0
0
0
0
A
A
2
1
0
0
1
2
0
0
2
0
1
0
1
02
2
1
1
1
1
B
B
2
1
1
2
1
12
0
2
2
0
01
1
2
2
1
2
2
2
2
(a) (b)
(c)
38
Results 2-qudit Inverse SWAP Gate

• Similar to the SWAP gate is the Inverse SWAP gate
  that we proposed
• The pairs of inputs and outputs are also
  exchanged but in addition the order of the output
  is flipped around.
• It is expected that it is universal as the
  2-qudit SWAP gate.

NEW 2-qudit ternary Inverse SWAP gate Symbol
(a), Input/Output table (b), Realization (c)
B
A
B
A
A
B
2
0
0
2
2
A
A
1
1
0
2
1
2
0
2
0
0
1
2
1
01
Flipping
1
1
1
1
Swapping
B
2
1
1
0
B
12
1
0
2
0
2
02
2
1
2
0
1
2
2
0
0
(a) (b)
(c)
39
Results Ternary Toffoli Gate

• Toffoli is viewed as universal, and thus another
  important gate.
• Its realization using GTG is possible without an
  ancilla qudit.
• From the Toffoli gate, which is a 2 -
  Controlled-Not, it is possible to build up an
  n-qudit Controlled-Not.
• The realization requires only 4 segments and 4
  single quditoperations. It seems to be the best
  realization found so far, compared to the
  literature
• No mirroring is needed.

3-qudit ternary Toffoli gate (2-Controlled-NOT)
Symbol (a), Realization (b)
(a)
(b)
40
Some New Gates Invented by Exhaustive Search

• Using the exhaustive program I found the
  following
• 1. all 2-qudit gates can be realized within 4
  segments (4 quantum multiplexers).
• 2. 1680 out of the 19683 2-qudit gates need no
  additional ancilla qudit to be realized, the
  rest do
• 3. the number of single qudit operations at the
  multiplexers is not higher than 6 for all of the
  2-qudit gates
• 4. The exhaustive algorithm produced a library
  where the realization of all 2-qudit gates, their
  structure and single qudit operations are stored.
  This data can be used for a CAD system for
  quantum logic circuits.

41
Gates used in GA

• Not all 216 Generalized Ternary Gates (GTG) were
  used
• Yen et al. showed that 12 Generalized Ternary
  Gates (GTG) out of the 216 GTG are universal and
  sufficient to realize quantum gates
• The Genetic algorithm used only those 12 GTG, and
  the single qudit operations (1,2,01,02,12)

42
What was invented 2-qudit Feynman

• The solutions found by the GA have higher cost

2-qudit ternary Feynman gate (Controlled-NOT)
A
R
R
A
B
S
B
2
2
2-qudit ternary Feynman (Galois) gate
43
Results 2-qudits SWAP

• The nature of the GA can be seen again. The
  solution that were found are not optimal
• The found result can be minimized.

2-qudit ternary SWAP gate Symbol (a) and
Realization found by the GA (b)
(a)
(b)
1
1
1
2
A
B
1
2
12
1
1
1
B
A
1
1
1
1
1
1
44
Results 2-qudits Inverse SWAP

• The GA found a realization for the new proposed
  Inverse SWAP gate

2-qudit ternary Inverse SWAP gate Symbol (a) and
GA realization (b)
(a)
(b)
1
2
2
A
B
2
1
1
1
1
2
2
2
2
2
1
2
B
A
2
2
02
2
2
1
2
2
2
2
1
1
1
AE
AC
N
G
Q
L

N
T
R
P

W
V
I
P
I
P
T
Genotype pppAEppVppWppACppNppTppRppPppppIppPppG
ppNppIppLppQppPppTp
45
Results cont.

• The 3-qudit SWAP gate was not possible to find
  with the exhaustive search and therefore
  indicates the ability of the GA
• The 3-qudit SWAP exchanges the 3 input to the
  output
• There are Ns Number of SWAP gates for Nq qudits

3-qudit ternary SWAP gate Realization (a) and
Symbol (b)
(a)
(b)
1
B
A
2
1
1
02
1
C
B
1
12
1
1
1
2
01
02
1
1
C
A
12
1
01
46
Improvements on the GA

• The GA is restricted to an small number of the
  216 different GTGs
• Therefore analyze the GTGs in the 2-qudit library
  and use those for the GA
• Automation of the GA
• e.g. If diversity of the population goes down
• Change of the mutation ratio (erasure/addition/fli
  pping)
• or increase the mutation probability

47
Conclusion
48
Exhaustive Search

• Benefits
• Toffoli Gate is realized in 4 GTGs
• An algorithmic method was given to implement
  ternary quantum logic gates using the principles
  of MS gates and GTG
• Exhaustive search for 2-variable goal functions
  results in maximum of 4 levels of multiplexer,
  and one ancilla bit.
• Realizations of well known universal quantum
  gates for 2- and 3-qudit were found and verified.
• Formula to calculate the number of balanced
  functions for a given radix and number of qudits
  was presented.
• Results for all 2-qudit quantum gates are now
  available.
• The gates discovered in this thesis can be used
  as building block in higher-level synthesis
  methods, as presented in the literature.
• Drawbacks
• Limitations with respect to number of levels and
  qudits are given.

49
Genetic Algorithm

• Benefits
• A realization for a 3-qudit SWAP gate was found
• A second algorithmic method was given to
  implement ternary quantum logic gates using the
  principles of MS gates and 12 GTGs
• It supports the search for quantum gates where
  the exhaustive search is not applicable anymore
• Serves as a foundation for future research
• Drawbacks
• There is no guarantee to find a solution
• If a solution was found it may not need to be
  minimal with respect to the number of levels and
  single qudit operations

50
In Conclusion

• Presented today were two software programs for
  logic synthesis for quantum realizable gates
• Exhaustive Search
• Genetic Algorithm
• We believe now that the best method is combining
  Iterative Deepening Depth First with A Algorithm
  and recognizing easy functions on lower levels
  of the tree.

51
References

• Ch. H. Bennett and R. Landauer, "The Fundamental
  Limits of Computation", Scientific American, July
  1985, pp. 38-46.
• R. Landauer, "Irreversibility and heat generation
  in the computational process" I.B.M. Journal of
  Research and Development, 5 (1961), pp. 183-191.
• A. Muthukrishnan and C R. Stroud, Jr.,
  Multivalued Logic Gates for Quantum
  Computation, Physical Review A, vol. 62, no. 5,
  2000, pp. 052309/1-8
• Edward Fredkin, A physicists Model of
  Computation, Proceedings of the XXVIth RENCONTRE
  DE MORIOND, 1991 Savoie, France
• http//www.waters.com
• http//aemc.jpl.nasa.gov/activities/mms.cfm

52
Outline

• Introduction
• Why Quantum Logic?
• Reversible Logic
• A Brief Background
• Quantum Logic Gate Synthesis Method
• Exhaustive Search
• Comparison to GA
• Conclusion

53
Why Quantum Computing?

• Moores law will reach fundamental limits within
  the coming future
• Transistor size approaching single atom
• Power density problem
• Quantum phenomenon (tunneling, etc.)
• Many other issues..
• Computationally, quantum computing is
  exponentially more powerful
• Due to quantum phenomenon, for N ternary qudits
  (a quantum bit with three states), 3N states
  can be computed simultaneously.

54
Reversible Logic and Quantum Computing

• How does reversible logic relate?
• In addition to being a method of power reduction,
  reversibility is an intrinsic property of quantum
  computing.
• What is Reversible Logic?
• Logic where no information is lost between
  inputand output.
• Given an output, a the single distinct input can
  be derived.
• A special case is the permutative logic where
  the outputs are simply some permutation of the
  inputs.

55
What is Reversible Logic?

• Logic where no information is lost between input
  and output.
• Given an output, a the single distinct input can
  be derived.
• A special case is the permutative logic where
  the outputs are simply some permutation of the
  inputs.

• Reversible Logic
• Example Permutative logic

Reversible Logic
Input
Output
A
B
A
B
0
0
0
0
0
1
0
1
1
0
1
1
1
1
1
0

• Non-Reversible Logic
• Example Standard AND/OR/EXOR Logic

Non-Reversible Logic
Input
Output
A
B
R
0
0
0
1
0
?
Can you give example of reversible logic that is
not permutative? This would require different
numbe of input and output signals, we discussed
the interaction gate in my class.
1
0
1
1
1
56
What is Reversible Logic?
Reversible Logic

• Reversible Logic
• Example Interaction Gate (2-input, 4 output)
• One can derive the input by knowing the output

Input
Output
A
B
AB
AB
AB
AB
0
0
0
0
0
0
0
1
0
1
0
0
1
0
0
0
1
0
1
1
1
0
0
1

• Permutative Logic
• Example Feynman Gate
• One-to-One mapping between input and output(so
  called bijectiv function)

Permutative Logic
Input
Output

• Non-Reversible Logic
• Example Standard AND/OR/EXOR Logic
• It is not possible to derive the input only from
  the output

Non-Reversible Logic
Input
Output
?
57
Quantum Logic Synthesis

• Logic synthesis for quantum computing can be
  divided into
• two main categories
• Synthesis using Purely Quantum Gates
• Takes into account the effects of quantum
  phenomenon such as superposition, entanglement,
  etc.
• It is due to quantum phenomenon that for N
  qudits, 3N states can be computed simultaneously
  in case of ternary quantum circuits.
• Synthesis of Permutative functions, Binary or
  Multiple-valued
• This area has stronger ties with existing logic
  synthesis methods, as it deals solely with basic
  quantum states 0gt and 1gt (for binary).
• Ultimately, the Hilbert space transformations and
  quantum logical manipulations from purely quantum
  gate logic must be related to some basic state
  forms for data input and output.
• Binary logic is a solution, but because of the
  nature of quantum technology, it is possible to
  directly realize gates that are characterized in
  multi-valued logic.
• Permutative functions are similar to an identity
  matrix, where the rows are permutated

58
Quantum Logic Synthesis

• Logic synthesis for quantum computing can be
  divided into
• two main categories
• Synthesis using Purely Quantum Gates
• Synthesis of Permutative functions, Binary,
  Ternary or Multiple-valued

• Takes into account the effects of quantum
  phenomenon such as superposition, entanglement,
  etc.
• It is due to quantum phenomenon that for N
  qubits, 2N states can be computed simultaneously
  in case of binary quantum circuits.

• This area has stronger ties with existing logic
  synthesis methods, as it deals solely with basic
  quantum states 0gt, 1gt.
• Ultimately, the Hilbert space transformations and
  quantum logical manipulations from purely quantum
  gate logic must be related to some basic state
  forms for data input and output.
• Binary logic is a solution, but because of the
  nature of quantum technology, it is possible to
  directly realize gates that are characterized in
  multi-valued logic.

59
Ternary Quantum Logic Synthesis using a Genetic
Algorithm
Additional Slides in case of questions
60
Genetic Algorithm Why?

• Exhaustive Search was a good starting point for
  synthesis
• However, the limits were the amount of cascaded
  multiplexers, the number of qudits and the
  exhaustive time
• Quantum gates with more than 2-qudits inputs are
  possible e.g. 3-qudit SWAP

61
Definition of Genetic Algorithm

• GA is another search technique used in various
  fields e.g.
• Mobile communications infrastructure optimization
• Electronic circuit design and
• many more
• To find approximate solutions to optimization and
  search problems
• GA is one of the evolutionary algorithms and is
  based on biological evolution theory.
• It is implemented as a computer simulation in
  which a population of abstract representations
  (called chromosomes) of candidate solutions
  (called individuals) to an optimization problem
  evolves toward better solutions.
• Individuals can be represented as strings of 0s
  and 1s or strings of characters

011001101010
ACGHIKL
62
Pseudo Code of GA

• The Pseudo code shows the general structure of a
  GA with
• After the initialization and first evaluation
  begins the life cycle

• 01 t ? 0
• 02 initialize(P(t)) / initial population /
• 03 evaluate(P(t)) / evaluate population /
• 04 while (not termination-condition) do
  /begin of the life cycle/
• 05 t ? t 1
• 06 (t) ? select(P(t - 1)) / selection
  operator /
• 07 (t) ? recombine((t)) / crossover operator
  /
• 08 P(t) ? mutate((t)) / mutation operator /
• 09 evaluate(P(t)) / evaluate fitness /
• 10 end while

63
Selection methods Roulette Wheel

• Three fitness proportional selection methods are
  implemented
• The individuals get a fitness value and upon this
  they get a larger or smaller section on the
  roulette wheel
• A random number is generated (the ball on the
  roulette wheel) and the section the is hit by the
  ball is chosen for recombination

64
Selection method Stochastic Universal Sampling

• Second selection method is SUS
• It is similar to the Roulette Wheel. Every
  individuals gets a section on the roulette wheel
  related to their fitness
• Another wheel is laid above the Roulette Wheel
  and it is turned around by a random value
• The individuals selected by the second wheel are
  chosen for recombination

65
Selection method Tournament Selection

• The last implemented selection method is the
  Tournament Selection
• Individuals are chosen randomly for a tournament
  (with k individuals) and the one with the highest
  fitness of the Tournament is chosen for
  recombination

Population
k Tournament individuals
Individual with highest fitness
S1
S4
S1
S8
S7
S3
S3
S3
S6
S6
S5
S2

• If k is chosen to big ? high selection pressure
• ? good individuals are preferred to much

66
Implemented crossover methods

• Crossover is the primary operator in the GA
• New Individuals are produced out of selected
  parents
• Fragments of chromosomes are exchanged and thus
  information is exchanged between potential
  solutions
• The location were the crossover is applied is
  chosen randomly.
• Two methods are implemented
• 1- point crossover
• 2- point crossover

67
Two Mutation methods

• The secondary operator is the mutation. It
  inserts new or lost information into the
  population. It is performed seldom otherwise the
  GA degenerated to a complete random search.
• Three method are implemented

68
Extra Slide Structure of an Ion Trap
ION TRAP realization
End Cap
End Cap
Ion Injection

Detection
Ring electrode
V
2)
1)
Ions, or charged atomic particles, can be
confined and suspended in free space using
electromagnetic fields. Qubits are stored in
stable electronic states of each ion, and quantum
information can be processed and transferred
through the collective quantized motion of the
ions in the trap (interacting through the Coulomb
force). Lasers are applied to induce coupling
between the qubit states (for single qubit
operations) or coupling between the internal
qubit states and the external motional states
(for entanglement between qubits). The
fundamental operations of a quantum computer have
been demonstrated experimentally with high
accuracy (or "high fidelity" in quantum computing
language) in trapped ion systems, and a strategy
has been developed for scaling the system to
arbitrarily large number of qubits by shuttling
ions in an array of ion traps. This makes trapped
ion system one of the most promising
architectures for a scalable, universal quantum
information processor.
The principle of the trap is to store the ions in
a device consisting of a ring electrode and two
end cap electrodes. The ions are stabilized in
the trap by applying a RF voltage on the ring
electrode. For maximum efficiency, the ions must
be focused near the centre where the trapping
fields are closest to the ideal and the least
distorted - maximizing resolution and
sensitivity. This is achieved by introducing a
damping gas (99.998 helium) that collisionally
cools injected ions, damping down their
oscillations until they stabilize.
Make it to3 or 4 slides, letters are too small
here
2)http//aemc.jpl.nasa.gov/activities/mms.cfm
1)http//www.waters.com
69
Extra Slide How does Logic Loss introduce Power
Loss?

• In the Billiard Ball Model of reversible
  computing, logic operations are represented by
  collisions between billiard balls.

Billiard Ball Model
reversible

• Suppose we have two billiard balls with some
  velocity vectors that will collide, as shown.

• At some given time later, knowing their positions
  and velocities, one can derive the original state
  of the system. This is an example of reversible
  logic.

irreversible

• In contemporary irreversible logic, some
  information is lost, preventing the reversibility
  of the system. This also results in a loss of
  energy to the system.

70
Extra Slide Bloch Sphere

• Dirac Notation Quantum logic states are often
  represented in Dirac notation
• i.e., A0gt B1gt C2gt
• where quantum states 0gt, 1gt and 2gt are
  representative of superpositional states as
  weighted by A, B and C, such that a2, b2 and
  c2 are the probabilities of measurement of
  basic quantum state 0gt, 1gt or 2gt (and a2
  b2 c2 1).
• A multi-valued Bloch sphere can be described as a
  Bloch sphere for which more than two states have
  been defined as additional logic states, such as
  given by Figure b.
• It is important to understand that the number of
  values of the logic is formally related to the
  measurement process and not to what happens in
  Hilbert space.
• Similarly as one can create a base of measurement
  of 0gt and 1gt, a base of 0gt, 1gt and 2gt can be
  created by measuring at angles 120 apart. This
  may be difficult to achieve for particular
  technologies, but is possible in principle and
  does not contradict principles of quantum
  mechanics.

71
Structure of an Ion Trap

• The principle of the trap is to store the ions in
  a device consisting of a ring electrode and two
  end cap electrodes.
• The ions are stabilized in the trap by applying a
  RF voltage on the ring electrode.
• For maximum efficiency, the ions must be focused
  near the centre where the trapping fields are
  closest to the ideal and the least distorted -
  maximizing resolution and sensitivity.
• This is achieved by introducing a damping gas
  (99.998 helium) that collisionally cools
  injected ions, damping down their oscillations
  until they stabilize.

ION TRAP realization
End Cap
End Cap
Ion Injection

Detection
Ring electrode
V
2)
1)
72
Structure of an Ion Trap

• Ions, or charged atomic particles, can be
  confined and suspended in free space using
  electromagnetic fields.
• Qubits are stored in stable electronic states
  of each ion, and quantum information can be
  processed and transferred through the collective
  quantized motion of the ions in the trap
  (interacting through the Coulomb force).
• Lasers are applied to induce coupling between
  the qubit states (for single qubit operations) or
  coupling between the internal qubit states and
  the external motional states (for entanglement
  between qubits).
• The fundamental operations of a quantum
  computer have been demonstrated experimentally
  with high accuracy (or "high fidelity" in quantum
  computing language) in trapped ion systems, and a
  strategy has been developed for scaling the
  system to arbitrarily large number of qubits by
  shuttling ions in an array of ion traps.
• This makes trapped ion system one of the most
  promising architectures for a scalable, universal
  quantum information processor.

1)http//www.waters.com
2)http//aemc.jpl.nasa.gov/activities/mms.cfm
73
Conclusions on GA

• A second algorithmic method was given to
  implement ternary quantum logic gates using
  principle MS gates, GTG and single qudit
  operations
• Limitations with respect to the small amount of
  principle gates are given.
• The GA showed that solutions on 3 qudits (3-qudit
  SWAP) can be realized.
• Realizations for some universal well known
  quantum gates for 2- and 3 qudits were presented

74
Conclusion on Exhaustive Search

• An algorithmic method was given to implement
  ternary quantum logic gates using the principles
  of MS gates and GTG
• Limitations with respect to number of levels and
  qudits are given.
• Exhaustive search for 2-variable goal functions
  results in maximum of 4 levels of multiplexer,
  and one ancilla bit.
• Realizations of well known universal quantum
  gates for 2- and 3-qudit were found and verified.
• Formula to calculate the number of balanced
  functions for a given radix and number of qudits
  was presented.
• The gates discovered in this thesis can be used
  as building block in higher-level synthesis
  methods, as presented in the literature.

75
References

• Ch. H. Bennett and R. Landauer, "The Fundamental
  Limits of Computation", Scientific American, July
  1985, pp. 38-46.
• R. Landauer, "Irreversibility and heat generation
  in the computational process" I.B.M. Journal of
  Research and Development, 5 (1961), pp. 183-191.
• A. Muthukrishnan and C R. Stroud, Jr.,
  Multivalued Logic Gates for Quantum
  Computation, Physical Review A, vol. 62, no. 5,
  2000, pp. 052309/1-8

reversibility
Reversibility and thermodynamic
Multi-valued quantum gates in ion trap technology