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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: [email protected] * Dept. of Instrumentation and Control Engn., Hanbat National University, 16-1 San Duckmyong-Dong Yuseong-Gu, Daejon, Korea, 305-719.E-mail: [email protected] Hierarchical Decomposition and Synthesis of Ternary gates Synthesis using Logic Blocks and Gates Design of Logic Blocks and Gates Mozammel Khan, ISMVL 2007 This paper Ternary Quantum Multiplexers Ternary Muthukrishnan-Stroud Gates Single-Qubit Rotation gates and 2-qubit Interaction gates Soonchil Lee et al, MVL J 2006 Circuit Structures for Ternary Logic extend the structures for binary logic a b c d 0 e a FG ab c FG cd -1 FG -1 FG Galois Product FG (Galois Product)-1 b c d 0 F = f (a,b,c,d)e Binary Multi-Cube gate Multivalued counterpart of the Multi-Cube gate for any radix. FG is Feynman-Galois gate. CD AB 00 00 0 01 0 11 0 10 0 a 01 11 10 0 0 0 1 1 1 Symbol stands for exor. 1 (ab)(cd) = acadbcbd Kmap of function f(a,b,c,d) realized by the gate F = f(a,b,c,d) from above. Toffoli-like 3-controlled gate structure for Galois Field Sum of Product Circuits a b a b c c 0 0 d Galois Product Galois Product x x FG xd (Galois Product)-1 (Galois Product)-1 0 0 A cascade of two 2-controlled Toffoli-like gates for Modulo sum of minima type of circuits a b a b c c 0 d min-1 min +1 0 min-1 min (01) 0 Ternary Wave Cascade (Modsum of ternary Maitra cascades) a a a b b b 0 c F 0 -1 min min 0 -1 max max c c 0 d max-1 max modsum c min-1 min modsum Because these structures are used again and again, it is definitely worthy to optimize their components very well, even spending months of computer time. 0 0 0 Quantum Reversible Cascades with Ternary Quantum Multiplexers Op. 4 A A’ Op. 5 Operations for a ternary system Op. 6 B Op. 1 0 Op. 7 Op. 2 1 Op. 8 Op. 3 2 Op. 9 B’ +0 +1 +2 01 02 12 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. Wire Modulo Shift +1 Modulo Shift +2 Swap 0 1 Swap 0 2 Swap 1 2 The Muthukrishnan-Stroud Gate A’=A A Operations for a ternary system +0 +1 +2 01 02 12 wire B’ B Wire Modulo Shift +1 Modulo Shift +2 Swap 0 1 Swap 0 2 Swap 1 2 wire operation Two views of MS gate A B 2 operation P Q 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., |2> in a ternary system of |0>, |1>, |2>), 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 Quantum Reversible Cascades cont. A’=A A Operation 1 B Operation 2 Operation 3 Time +0 +1 +2 01 02 12 Wire Modulo Shift +1 Modulo Shift +2 Swap 0 1 Swap 0 2 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 B’ Operations for a ternary system 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 Muthukrishnan-Stroud Gate Internally, built from Interaction gates and rotations 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 Use of interaction gate X rotation Y rotation Z rotation General case rotations Z rotations Z rotations Special cases are cheaper X rotation Z Y rotation Z General view of cascade for Dlevel circuits rotations rotations rotations Z rotations rotations Z rotations Every multi-valued quantum multiplexer can be build like this First two structures based on cascaded quantum multiplexers A P f3 f0 B Q f1 f4 f2 f5 f3 A P f4 f5 f0 f6 f1 f7 f2 f8 B Q One more structure based on cascaded quantum multiplexers A fd PA fe B ff fa PB fb Z fa11 |0> PZ fc fb fc fg fj fm fq fh fk fn fr fi fl fp fs PR 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 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 Breadth first search (BFS) BFS is a tree search algorithm used for searching a tree, tree structure, or graph. 1 2 5 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) 3 4 6 7 8 9 10 11 Breadth first search (BFS) cont. 216 different GTG realizations A’ A |0> +0 +0 12 +0 +0 12 +0 +1 12 1. GTG Gate 2. GTG Gate 216. GTG Gate A B |0> A B +0 +0 +0 +0 +0 +0 +0 +0 +0 A B |0> R R 1.GTG - 1.GTG - 1.GTG A B +0 +0 +0 +0 +0 +0 +0 +0 +1 R 1.GTG - 1.GTG - 2.GTG … A B |0> A B 12 12 12 12 12 12 12 12 12 R 216.GTG - 216.GTG - 216.GTG Exhaustive Search: Example A B B A 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 Exhaustive Search: Example cont. 2-qudit SWAP gate +1 +2 A B B +2 02 +1 12 A 01 A 0 0 0 1 1 1 2 2 2 A B 0 1 2 0 1 2 0 1 2 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 2 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 2 1 0 0 0 0 0 0 0 1 0 2 2 0 0 0 0 0 0 0 0 1 A 0 0 0 1 1 1 2 2 2 A B 0 1 2 0 1 2 0 1 2 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 2 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 2 1 0 0 1 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 1 A 0 0 0 1 1 1 2 2 2 A B 0 1 2 0 1 2 0 1 2 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 1 0 2 0 0 0 1 0 0 0 0 0 0 2 1 0 0 0 0 0 1 0 0 0 2 2 0 0 0 0 0 0 0 0 1 Ternary Quantum Logic: Exhaustive Search Iterative deepening search Iterative deepening search l =0 Iterative deepening search l =1 Iterative deepening search l =2 Iterative deepening search l =3 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 = (d+1)b0 + d b^1 + (d-1)b^2 + … + 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% Properties of iterative deepening search Complete? Yes Time? (d+1)b0 + d b1 + (d-1)b2 + … + bd = O(bd) Space? O(bd) Optimal? Yes, if step cost = 1 Summary of algorithms 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 Inverse Gates Realization of the ternary Toffoli Gate as an example for mirroring: Mirroring gates A R=A B S=B |0> |0> +1 +1 +2 +2 Z C Q where Z{+1,+2,01,02,12} Operations for a ternary system Conservative or Inverse Operations +0 +1 +2 01 02 12 +0 +2 +1 01 02 12 Wire Modulo Shift +1 Modulo Shift +2 Swap 0 1 Swap 0 2 Swap 1 2 Wire Modulo Shift +2 Modulo Shift +1 Swap 0 1 Swap 0 2 Swap 1 2 Limitations on the Goal Function Balanced function A\B 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1 Unbalanced function A\B 0 1 2 0 0 0 0 1 0 1 2 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 2 0 2 1 n ( p )! A [( p n 1 )! )] p An ancilla bit is simply an input line that is a known constant e.g. “|0>” 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 (p=radix; n=number of qudits) Implementation with more Input Variables A RA B SB C NOT C , if A 2 B 2 Q C otherwise In the previous examples, the input was two variables. Here we see an example of a 3variable 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. Karnaugh map and realization of the ternary Toffoli gate: AB\C 00 01 02 10 11 12 20 21 22 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 0 2 1 0 2 1 0 2 0 A B A B C C +2 01 +1 01 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 2-qudit ternary MAX gate A B A MAX A B B B MAX (A,B) |0> A +1 12 +1 +1 |0> +1 R 12 2-qudit ternary MIN gate A B |0> A MIN A A B B B MIN (A,B) +1 |0> 02 01 +2 12 R +2 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 gate (Controlled-NOT) A B R NOT B, if A 2 S B otherwise A R B NOT B, if A 2 S B otherwise +1 2-qudit ternary Feynman (Galois) gate A A R B A 3 B B R=A +1 +2 S= A B 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) A B B A (a) A 0 0 0 1 1 1 2 2 2 B 0 1 2 0 1 2 0 1 2 A’ B’ 0 0 1 0 2 0 0 1 1 1 2 1 0 2 1 2 2 2 (b) +1 A B A’ +2 +2 02 +1 12 01 (c) B’ 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. B B A (a) B 0 1 2 0 1 2 0 1 2 A’ B’ 0 0 1 0 2 0 0 1 1 1 2 1 0 2 1 2 2 2 (b) Flipping A A 0 0 0 1 1 1 2 2 2 Swapping NEW 2-qudit ternary Inverse SWAP gate; Symbol (a), Input/Output table (b), Realization (c) A’ B’ 2 2 1 2 0 2 2 1 1 1 0 1 2 0 1 0 0 0 +2 A A‘ +1 01 B +1 12 +2 02 (c) B‘ 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 qudit operations. 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 R B S C NOT C , if A 2 and B 2 Q C otherwise A B A‘ B‘ C C‘ +2 (a) +1 01 (b) 01 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. 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) What was invented – 2-qudit Feynman The solutions found by the GA have higher cost 2-qudit ternary Feynman gate (Controlled-NOT) A B R NOT B, if A 2 S B otherwise A R B S +2 +2 2-qudit ternary Feynman (Galois) gate A R A 02 02 R B S A 3 B B +2 +2 S +1 +2 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) A B B A (b) +1 12 A +1 B +1 +1 +2 +1 +1 +2 B +1 +1 +1 +1 A +1 +1 +1 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) A B B A (b) +1 A +2 B 02 ^ A E +2 V +2 +2 +2 +1 +1 +2 +2 +1 +1 W A C +2 N +2 +2 T R +2 +2 +2 +2 +1 +2 +1 P ] +1 P G N A +1 +1 I +2 +2 I L Q P B T Genotype: p^ppAEppVppWppACppNppTppRppPpp]ppIppPppGppNppIppLppQppPppTp 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 N S Nq 1 3-qudit ternary SWAP gate; Realization (a) and Symbol (b) A B B C C A (a) (b) +1 +2 A +1 B +1 B 02 +1 12 +1 +1 +1 +2 01 C +1 +1 C +1 02 +1 A 12 01 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/flipping) or increase the mutation probability Conclusion 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 3qudit 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. 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 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. 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 physicist’s 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 Outline Introduction Why Quantum Logic? Reversible Logic A Brief Background Quantum Logic Gate Synthesis Method Exhaustive Search Comparison to GA Conclusion Why Quantum Computing? Moore’s 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), 3^N states can be computed simultaneously. 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 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. 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 Input A 0 0 1 1 Output A’ 0 0 1 1 B 0 1 0 1 B’ 0 1 1 0 Non-Reversible Logic Input A 0 0 1 1 B 0 1 0 1 Output ? Example: Permutative logic Non-Reversible Logic R 0 1 Reversible Logic Example: Standard AND/OR/EXOR Logic 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. What is Reversible Logic? Reversible Logic Input A 0 0 1 1 AB A’B AB’ 0 0 0 0 1 0 0 0 1 1 0 0 B 0 1 0 1 Permutative Logic Input A 0 0 1 1 Output A 0 0 1 1 B 0 1 0 1 Output ? Example: Feynman Gate One-to-One mapping between input and output (so called bijectiv function) Non-Reversible Logic R 0 1 Example: Interaction Gate (2-input, 4 output) One can derive the input by knowing the output Permutative Logic Non-Reversible Logic Reversible Logic AB 0 0 0 1 A’ B’ 0 0 0 1 1 1 1 0 B 0 1 0 1 Input Output Example: Standard AND/OR/EXOR Logic It is not possible to derive the input only from the output 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, 3^N 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 |0> and |1> (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 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 qubits, 2^N states can be computed simultaneously in case of binary quantum circuits. Synthesis of Permutative functions, Binary, Ternary or Multiple-valued This area has stronger ties with existing logic synthesis methods, as it deals solely with basic quantum states |0>, |1>. 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. Additional Slides in case of questions Ternary Quantum Logic: Synthesis using a Genetic Algorithm 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 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 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: 02: 03: 04: 05: 06: 07: 08: 09: 10: t ← 0; initialize(P(t)); evaluate(P(t)); /* initial population */ /* evaluate population */ while (not termination-condition) do /*begin of the life cycle*/ t ← t + 1; (t) ← select(P(t − 1)); /* selection operator */ (t) ← recombine((t)); /* crossover operator */ P(t) ← mutate((t)); /* mutation operator */ evaluate(P(t)); /* evaluate fitness */ end while 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 S7: 13% S8: 8% S1: 23% S6: 14% S5: 12% S4: 16% S3: S2: 5% 9% 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 S8: 8% S7: 13% S1: 23% S6: 14% S5: 12% S4: 16% S3: S2: 5% 9% S7: 13% S8: 8% S1: 23% S6: 14% S5: 12% S4: 16% S3: S2: 5% 9% 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 S8 S1 S4 S7 S6 S5 S2 S3 k Tournament individuals Individual with highest fitness S1 S6 S3 If k is chosen to big high selection pressure “good” individuals are preferred to much S3 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 0 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 1 0 0 0 1 1 1 1 1 0 1 2- point crossover 0 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 1 1 1 10 0 0 1 01 0 1 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: Flip mutation Erasure of a gene Addition of a gene 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 0 0 Extra Slide – Structure of an Ion Trap ION TRAP realization End Cap End Cap Detection Ion Injection Ring electrode V 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 ~ 1) 2) 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 Extra Slide – How does Logic Loss introduce Power Loss? Billiard Ball Model In the “Billiard Ball Model” of reversible computing, logic operations are represented by collisions between billiard balls. 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. 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. reversible irreversible Extra Slide – Bloch Sphere z z |0> |0> 120° 180° |1> 120° y 180° y 120° x x |2> |1> Dirac Notation Quantum logic states are often represented in Dirac notation: i.e., A|0> + B|1> + C|2> where quantum states |0>, |1> and |2> are representative of superpositional states as weighted by A, B and C, such that |a|2, |b|2 and |c|2 are the probabilities of measurement of basic quantum state |0>, |1> or |2> (and |a|2 + |b|2 + |c|2 = 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 |0> and |1>, a base of |0>, |1> and |2> 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. 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. ION TRAP realization End Cap End Cap Detection Ion Injection Ring electrode V ~ 1) •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. 2) •This is achieved by introducing a damping gas (99.998% helium) that collisionally cools injected ions, damping down their oscillations until they stabilize. • Ions, or charged atomic particles, can be confined and suspended in free space using electromagnetic fields. Structure of an Ion Trap • 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 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 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 3qudit 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. References Ch. H. Bennett and R. Landauer, "The Fundamental Limits of Computation", Scientific American, July 1985, pp. 38-46. reversibility R. Landauer, "Irreversibility and heat generation in the computational process"; I.B.M. Journal of Research and Development, 5 (1961), pp. 183191. Reversibility and thermodynamic 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 Multi-valued quantum gates in ion trap technology