Evolutionary approach to Quantum and Reversible Circuits synthesis

Transcription

1 Evolutionay appoach to Quantum and Revesible Cicuits synthesis Matin Lukac, Maek Pekowski, Hilton Goi, Mikhail Pivtoaiko +, Chung Hyo Yu, Kyusik Chung, Hyunkoo Jee, Byung-guk Kim, Yong-Duk Kim Depatment of Electical Engineeing and Compute Science, Koea Advanced Institute of Science and Technology, Guseong-dong,Yuseong-gu, Daejeon , Koea, +Depatment of Electical Engineeing, Potland State Univesity, Potland, Oegon, , USA Abstact: The pape discusses the evolutionay computation appoach to the poblem of optimal synthesis of Quantum and Revesible Logic cicuits. Ou appoach uses standad Genetic Algoithm (GA) and its elative powe as compaed to pevious appoaches comes fom the encoding and the fomulation of the cost and fitness functions fo quantum cicuits synthesis. We analyze new opeatos and thei ole in synthesis and optimization pocesses. Cost and fitness functions fo Revesible Cicuit synthesis ae intoduced as well as local optimizing tansfomations. It is also shown that ou appoach can be used altenatively fo synthesis of eithe evesible o quantum cicuits without a majo change in the algoithm. Results ae illustated on synthesized Magolus, Toffoli, Fedkin and othe gates and Entanglement Cicuits. This is fo the fist time that seveal vaiants of these gates have been automatically synthesized fom quantum pimitives. 1. Intoduction Quantum computing is a flouishing and vey attactive eseach aea [7, 46, 48]. Inheiting popeties fom Quantum Mechanics, it allows theoetically to build computes much moe efficient than the existing ones. Fo instance, cetain poblems non solvable in polynomial time in classical domain can be solved in polynomial time in quantum domain. Similaly, the complexity of othe poblems can be educed while tansfoming them into the Quantum domain [48]. Moeove, Quantum Cicuits (QC) have an advantage of being able to pefom massively paallel computations in one time step [7,46,48]. The motivation to develop automated CAD tools fo quantum cicuits becomes ecently quite high because accoding to the esults of [37,38,39,40,41] such computes can be physically build and the pogess of these ealizations is fast. It is aleady possible to pefom quantum-mechanically simple opeations with tapped ions o atoms. Simplified but complete quantum cicuits wee constucted using Nuclea Magnetic Resonance technology [48]. The state of the at in yea 2002 is a 7 qubit quantum compute [45]. Fo this size of compute the poblem of quantum cicuit synthesis and optimization is not tivial and cannot be solved by hand so some kind of design automation becomes necessay. While quantum mechanics and quantum computing ae established eseach aeas, systematic design methods of quantum computes, and especially logic cicuit design fo such computes still emain only at the beginning stages of exploing available possibilities. It can be compaed to the state of the at of logic design in 1940 s when

2 the fist standad binay computes wee built and the minimization algoithms like those fom Quine and Shestakov stated to appea. Cuently, the woks in quantum logic design ae on: designing new univesal gates and investigating thei popeties, ceating methods of composing gates to cicuits, and building elementay quantum cicuits fo pactical applications, fo instance aithmetics. New algebaic models fo quantum logic and cicuit design ae also investigated. The Compute Aided Design of Quantum Cicuits is even less developed and thee exist only few pogams to design such cicuits automatically o with a limited use intevention. This pape is elated to new appoaches fo CAD of Quantum Cicuits, the new eseach and development aea that we ty to establish by ou eseach [1,2,10,28,29,30,31,32,33,42,43,55,56,57,58]. So fa, analytic and seach appoaches have been used in quantum logic synthesis. They ae based on matix decomposition, local cicuit tansfomations, mapping fom vaious types of decision diagams, spectal appoaches, and on adaptations of seveal EXOR logic, Reed-Mulle, and othe classical combinational cicuit design methods. Ou appoach combines some of them but fundamentally belongs to the goup of Evolutionay Algoithms. The appoach has been also used to a simila poblem of Revesible Cicuit (RC) Logic synthesis; such cicuits can be ealized in CMOS, optical and nano-technologies [47]. Genetic Algoithms (GAs) ae one of the well-known Evolutionay Algoithm poblem solving appoaches to Soft Computing [9]. Thei use is vey popula in poblems with no identified stuctue and high level of noise, because: 1) a big poblem space can be seached, 2) the size of this space can be modeated by paametes, 3) a vaiety of new solutions can be poduced, and 4) with long enough time a cicuit can be obtained that is close to the optimal one. These advantages make GAs useful in the initial phases of eseach and investigations of the design poblem space. GAs ae vey good candidates to be used in logic synthesis of new types of cicuits, investigating the usefulness of new gate types and new cicuit stuctues. The special cases of Evolutionay Algoithms include: Genetic Algoithms, Evolutionay Pogamming, Evolution Stategies and Genetic Pogamming. So fa, to the best of ou knowledge, only two of the fou Evolutionay Algoithm types have been applied to the QC o RC synthesis. Genetic pogamming was used to synthesize EPR (Einstein-Podolsky-Rosen) pais of qubits [5], while a gowing numbe of woks uses a classical GA fo QC and RC synthesis. Fo instance, [1,2,3,6,7,8,54] used GA to evolve quantum and evesible gates and cicuits, like the telepotation cicuits. A evesible cicuit is one that has the same numbe of inputs and outputs and is a oneto-one mapping between vectos of input and output values. Such cicuits ae elated to quantum cicuits. An attempt at a geneal appoach to encode both Quantum and Revesible Cicuits was pesented in [1,2]. It is known that evey quantum cicuit is evesible [7,46,48], so the eseaches on classical binay evesible synthesis and quantum synthesis shae many ideas. The Revesible Logic (RL) cicuits [14,15,20] ae aleady technologically possible and have been implemented in CMOS technology [47]. Thus, some of ou esults below ae applicable also to such cicuits. As descibed late, most of gates with moe than one input used in QC ae deived and oiginate fom RC. Although this wok is concened with one appoach to automated synthesis, it is impotant to notice that a paallel eseach on new gates is also exploed by [10,20,21,22,23,24,25,26,27,28,29,50,51,52]. The seach of new Quantum gates (QG) and Revesible gates (RG) has two diffeent aspects: invention

3 and genealization. The invention of new gates is mainly aimed towad an optimization of a paticula design in a specific technology o towads inventing new univesal gates. The genealization appoach is the use of aleady known gates and exending them to new gates of cetain pefeable popeties. Usually thee ae also new paticula synthesis methods that come togethe with poposed new gates [30,31,32,33,34,55,57,58]. This pape is oganized as follows. Section 2 pesents the minimal backgound in quantum cicuit design necessay to undestand the pape. Examples ae used to illustate the most impotant notions. Section 3 descibes the geneal poblem of synthesis of quantum cicuits (called also quantum aays) fom pimitive quantum gates, especially using evolutionay algoithms. Section 4 pesents decomposition of gates to smalle pimitives elated to the cost of these gates. Moe ealistic cost function fo gates ae one of main innovations of this pape. Section 5 pesents ou entie minimization methodology fo quantum aays synthesis. The sixth section descibes local optimizing tansfomations used in the post-pocessing stage of GA. Section 7 pesents ou vaiant of GA and its settings fo this wok. Section 8 gives moe details on the most impotant aspect: the fitness function design fo GA and its ole. Section 9 pesents othe aspects of the Genetic Algoithm that we applied. Section 10 descibes expeimental esults and section 11 discusses issues and advantages of this appoach as well as ou cuent and futue eseach and open questions. Section 12 concludes the pape. 2. Fundamentals of Quantum Logic In quantum computation quantum bits (qubits) ae used instead of classical binay bits to epesent infomation. These infomation units ae deived fom the states of mico-paticles such as photons, electons o ions. These states ae the basis states (basis vectos, eigenstates) of the computational quantum system. Assume an electon with two possible spin otations: +1/2 and 1/2. Using Ket notation [12,13] these distinguishable states will be epesented as 0> and 1>, espectively. Each paticle in a quantum domain is epesented by a wave function descibing it as having both popeties of a wave and of a paticle as intoduced by [11,12,13,16]. Based on these popeties, quantum computation inheited the poweful concept of supeposition of states. Assume a paticle p 1 be epesented by a wave function ψ 1 = α 1 0> + β 1 1>. The coefficients α and β ae complex numbes called the eigenvalues. They must be in geneal complex because only having complex values in wave functions allows to eliminate themselves in ode to satisfy some expeimentally obseved popeties of quantum wold, such as fo instance in the Two-Slit expeiment [17]. It is not ou goal hee to explain such expeiments o fomulate fundaments of quantum mechanics. (The eade can find infomation in liteatue, especially [48]). Ou goal is only to intoduce the fomal calculus of quantum mechanics in ode to explain the basic concepts of ou CAD algoithms and especially the genetic algoithm fo quantum cicuit synthesis. Similaly as an enginee who has no undestanding of electonic cicuits, but knows only Boolean Algeba, is able to develop logic cicuits synthesis softwae, one with no undestanding of quantum phenomena and physics will be able to develop quantum CAD if he will only lean some fundamental quantum notation and associated algebaic popeties and tansfomations. To teach these to the Reade is one of the goals of this pape.

4 The intepetation of wavefunction ψ 1 is that α 1 2 is the pobability of the paticle being measued in state 0> and β 1 2 is the pobability of that paticle being measued in state 1>. Thus, the measuement o obsevation pocess tansfoms the quantum wold of complex wavefunctions to the maco-wold of events that occu with standad pobabilities. The supeposition of states is epesented by these facts: (1) each of these pobabilities can be non-null, (2) α β 1 2 =1, and (3) if anothe paticle p 2 with a wavefunction ψ 2 = α 2 0> + β 2 1> is added to the system, then the esulting wavefunction will be ψ 1 ψ 2 > = α 1 α 2 00> + α 1 β 2 01> + β 1 α 2 10> + β 1 β 2 11>. The system can be in any possible state of the wavefunction and will collapse to one of the eigenstates when measued [19]. The space of quantum computing is thus much lage than in classical computing, which causes that efficient algoithms fo synthesis and analysis of quantum cicuits ae moe difficult to develop. Howeve, cetain similaities exist which will be useful fo us to explain the synthesis issues, especially to the eades with a digital design engineeing backgound. The equations intoduced above ae the esult of the Konecke poduct [18] on matices of elementay quantum gates that opeate in paallel. The Konecke Poduct of Matices is defined as follows: a c b x d z x a y z = v x c z y v y v x b z x d z y ax v = az y cx v cz ay av cy cv bx bz dx dz by bv dy dv Mathematically, it is the Konecke Poduct opeation that allows the quantum logical system to gow dimensionally much faste than classical logics. Obseve that in a quantum system n qubits epesent a supeposition of 2 n states while in a classical system n bits epesent only 2 n distinct states. Opeations ove a set of qubits ae defined as matix opeations and map the physical states into the Hilbet space. [13,46,48]. The concept of Hilbet space will be used below only in the most elementay way necessay to undestand the calculations. States of the wave function ae eigenvalues of this space. Each matix-opeato epesents a modification to the complex coefficients of the wave function. The new coefficients esult in the modification of pobabilities to find the system in a paticula basic state. But we do not have to woy about classical pobabilities in this pape since the opeation of a quantum cicuit is puely deteministic, so we will always deal with eigenvalues (complex pobability) athe than standad pobabilities in cicuit design. Consequently a quantum gate will be a matix having fo input the vecto of complex coefficients of the wavefom and poducing a vecto of complex coefficients as the output. An illustative example can be seen in equation 1. a b α 0 aα 0 + bβ 1 = (1) c d β 1 cα 0 + dβ 1 whee a, b, c, and d ae complex coefficients of the matix indicating the (complex) pobability to tansit fom one state to anothe, and α 0>, β 1> ae the (complex) wavefunction coefficients to be popagated though the matix-opeato. Less fomally, we can think about this matix as a quantum equivalent of a classical gate such as AND which tansfoms its input states into output states. The designe of

5 quantum algoithms has to deal with standad pobabilities, but the designe of quantum cicuits, which is ou inteest hee, deals only with opeations in quantum wold because his input poblem is descibed in such a way. Assume j to be the squae oot of 1. Let us denote by U + a hemitian matix of matix U, which means the complex conjugate of the tansposed matix U (the matix U is fist tansposed and next each of its complex numbes is eplaced with its conjugate, thus a jb eplaces a +jb). We say that gate U is unitay when U*U + = I, whee I is an identity matix. It can be shown that because the pobabilities must be peseved at the output of the quantum gate, all matices epesenting quantum gates ae unitay. Thus evey quantum gate, block and the entie cicuit is descibed by a unitay matix. Evey quantum gate has theefoe exactly one invese matix quantum computing is evesible; quantum gate matices epesent logically evesible gates. Some of those gates ae exactly the same as in classical evesible computing, which allows to use some esults of binay evesible computing in quantum computing. While in geneal the coefficients in unitay matices of quantum cicuits ae complex numbes, thee ae some special and impotant gates fo which unitay matices ae just pemutation matices. (Let us ecall that a pemutation matix has exactly one 1 in evey ow and in evey column and all othe enties ae zeos it descibes theefoe an input-output pemutation of value vectos). The gates whose unitay matices ae pemutation matices ae called pemutation gates and they coespond to gates of classical evesible logic. Thee exist, howeve, othe impotant gates whose unitay matices ae not pemutation matices. Rotational opeatos (gates) and gates such as Hadamad gate (denoted by H) and Squae Root of Not Gate (denoted by V) belong to this second categoy which we will call tuly quantum pimitives. These gates ae esponsible fo supeposition, entanglement and all peculiaities of quantum computing, although they may constitute only a small faction of gates in a quantum cicuit. A Hadamad gate is an example of a gate that has a unitay matix which is not a pemutation matix. Hee ae some useful matices: Hadamad =, Pauli X = NOT =, Pauli Y = j j 1 j Pauli Z =, Phase =, V = NOT = j 2 j 1 j, 0 We will denote these gates by H, X, Y, Z, S and V, espectively. Only X is pemutative. It can be easily checked by the eade that multiplying the matix of gate V by itself poduces the matix of Pauli-X which is an invete o NOT gate. The eade can find inteesting identities by multiplying these matices by themselves and also by thei invese matices. Such identities in quantum logic ae used to simplify the quantum cicuits that come diectly fom evolutionay algoithms (section 6). They play a ole analogous to Boolean algeba identities such as De Mogan used in classical logic synthesis. Below we will pay special attention to cicuits composed of only pemutation gates (these ae the so-called pemutation cicuits, because thei unitay matices ae pemutation matices). An example of a pemutation gate is the Feynman gate (called also CNOT gate o Quantum XOR). This gate can be descibed by the following logic expessions: (A,B) => (P, Q) = (A, A B), see Figue 1a. This equation means that the

6 output bit P is just the same as its input A, while on the output of the second wie Q the opeation A B is pefomed. This opeation (EXOR of A and B) is a standad logic opeation. Sometimes notation A = P and B = Q is used (Figue 1c,d,e). (In othe papes notation A =A, and B =B is used to undelie the fact that the esult occus on the same quantum wie A o B). These expessions ae also elated to the quantum matix. Fo instance, the pemutation matix and the equations fom Figue 1c descibe the Feynman opeato fom Figue 1a,b. The caeful eade may veify that this is indeed a unitay matix and that it satisfies the definition of a pemutation matix as well. Obseve that each of its ows and columns coesponds to one of possible states of input and output values: 00, 01, 10, and 11. The binay states ae encoded to natual numbes as follows: 00=0, 01=1, 10=2, 11=3. We will use natual Figue 1: Feynman gates: (a) the cicuit in quantum aay notation, (b) the tansfomation between states executed by this gate, (c) the unitay matix with binay enumeation of ows and columns and the coesponding Boolean equation of outputs, (d) Feynman EXOR up cicuit scematics, (e) its unitay matix and Boolean equations numbes to addess ows and columns in matices. Let us obseve that when A = 0 and B = 0, then A =B =0. Thus, input combination 00 is tansfomed to output combination 00. This is epesented by a value of 1 at the intesection of ow 0 and column 0 of the matix. Similaly, input combination 11 is tansfomed to output combination 10, which is epesented by a value of 1 at the intesection of ow 3 and column 2 (Figue 1c). Anothe vaiant of Feynman gate is in Figue 1d,e.

7 Figue 2: Swap gate: (a) unitay matix and coesponding Boolean equations, (b) ealization using Feynman gates, (c) a schematic, (d) anothe ealization using two Feynman gates and one Feynman EXOR up gate. Figue 2 pesents anothe pemutation gate called a Swap gate. Obseve that this gate just swaps the wies A and B, since input state 01 is tansfomed to output combination 10, input vecto 10 is tansfomed to output vecto 01 and the emaining combinations ae unchanged (Figue 2a). Swap gate is not necessay in classical CMOS evesible computing, whee it is just a cossing of connecting wies that can be fo instance in two layes of metallization. Howeve, in quantum computing technology like NMR, evey unitay matix othe than identity is ealized by NMR electomagnetic pulses so its ealization has a cost. High cost of swap gates in quantum ealization is one of main diffeences between quantum and evesible cicuit synthesis (in evesible computing these gates ae fee). Thus quantum cicuit design coves some aspects of not only logic synthesis but also physical design (placement and outing) of standad CAD. Let us intoduce now few moe gates that we will use. Thee ae seveal 2*2 gates in binay evesible logic and they ae all linea. A linea gate is one that all its outputs ae linea functions of input vaiables. Let a, b, and c be the inputs and P, Q and R the outputs of a gate. Assuming 2*2 Feynman gate, P = a, Q = a b, when a = 0 then Q =b; when a = 1 then Q = b. (Sometimes the negation of a is also denoted by a ). With b=0 the 2*2 Feynman gate is used as a fan-out (o copying) gate. It can be shown that a swap gate can be built as a cascade of thee Feynman gates Figue 2 (the eade can check it by multiplying matices o by tansfoming Boolean equations of gates). Figue 2b shows ealization of Swap gate with thee Feynman gates, and Figue 2c its schematics. Obseve that thee is also anothe ealization of the Swap gate (Figue 2d). Cicuits fom Figue 2b and Figue 2d ae then called equivalent o tautological. Evey linea evesible function can be built by composing only 2*2 Feynman gates and invetes. Thee exist 8! = 40,320 3*3 evesible logic gates, some of them with inteesting popeties, but hee we ae not inteested in types of gates but in synthesis methods using abitay pemutation gates, so we will estict ouselves to only few gate types (many othe gate types have been defined in ou softwae). Thee exist two well-known univesal 3*3 evesible gates: Fedkin gate [14] and

8 Toffoli gate. Toffoli gate is also called 3*3 Feynman gate o Contolled-Contolled NOT (Figue 3). The 3*3 Toffoli gate is descibed by these equations: P = a, Q = b, R = ab c. Toffoli gate is an example of two-though gates, because two of its inputs ae given to the output. Similaly, the concept of k-though gates can be intoduced, as well as the concept of k*k Toffoli Gates. In geneal, fo a evesible gate with n inputs and n outputs, the matix is of size 2 n * 2 n. The 3*3 Fedkin gate (Figue 4) is descibed by these equations: P = a, Q = if a then c else b, R = if a then b else c. As we see, in tems of classical logic this gate is just two multiplexes contolled in a flipped (pemuted) way fom the same contol input a. The symbol notation fo a 3*3 Fedkin Gate is shown in Figue 4a. Qubit a is the contolling qubit. A classical schematics of this gate that uses multiplexes is shown in Figue 4b. As we see, the 3*3 Fedkin gates ae pemutation gates, they pemute the data inputs b,c of the multiplexes unde contol of the contol input a of these multiplexes that is also outputted fom the gate. The fact that output P eplicates input a is vey useful because it allows to avoid fanout gates fo the same level of the cicuit (let us ecall that fanout is not allowed in evesible logic). Copying of abitay signals is not allowed in quantum cicuits (no cloning pinciple) [48], so the eplication using Feynman gates can be applied only to basis states. Figue 4c pesents a Fedkin gate contolled with qubit c. In Figue 4d qubit b is the contolling qubit. Realization of Fedkin gates using Toffoli and Feynman gates is shown in Figue 5. Figue 3. Toffoli gates: (a) a schematic of Toffoli gate (EXOR down), (b) Toffoli gate (EXOR middle), (c) ealization of schematics fom Figue 3b using the Toffoli EXOR down and two Swap gates, (d) ealization of Toffoli EXOR middle with pemuted input ode

9 Fedkin gates ae examples of what we define hee as one-though gates, which means gates in which one input vaiable is also an output. Figue 4. Fedkin gates: (a) Schematics with contolled qubit a, (b) the classical schematics of this gate using multiplexes, (c) Fedkin gate contolled by qubit c (Fedkin Down), (d) Fedkin gate contolled by qubit b (Fedkin middle). Figue 6 pesents the well-known ealization of Toffoli gate fom [21]. Accoding to the figue in ight (Figue 6b), thee ae five 2-qubit pimitives. Let us a a a a b c Toffoli ac ba = b c ac ba ca ac Figue 5: Fedkin gate ealization using one Toffoli and two Feynman gates. (a) Toffoli shown schematically, (b) Toffoli with EXOR-down schematics a a a a b Toffoli b = b b c ab c c V V V + ab c Figue 6: Toffoli gate synthesized using only 2-qubit pimitives as poposed by Smolin. explain how the quantum pimitives coopeate to ceate coect gate behavio. The non-pemutation matices of quantum pimitives ae composed to ceate pemutation

10 behavio of the entie gate. Obseve that Contolled-V and Contolled-V + quantum pimitives ae used. V is the unitay matix called Squae-Root-of-Not, thus a seial connection of gates V ealizes an invete: V*V = NOT. Matix V + is the hemitian of V. As we emembe fom the definition of hemitian, we have that V * V + = I which is the identity matix (descibing a quantum wie). The Contolled-U gate (Figue 7) woks as follows: when the contolling signal a (uppe signal in gate Contolled-U) is 0, the output Q of the lowe gate epeats its input b. When the contolling signal a is 1, the output of the lowe gate is the opeato of this gate (matix U) applied to its input b (Figue 7). Theefoe, in case of the Contolled-V gate, the output is opeato V applied to the input. In case of Contolled-V + gate, the output is opeato V + applied to the input of the gate. The eade can veify in Figue 6b that when a=1, b=0, the opeation V * V + = I is executed in line c, thus the output of this gate epeats the value of c. When a = 1 and b = 1, the opeation V * V =NOT is executed in line c, thus the Figue 7. A geneal-pupose Contolled Gate. When a = 0 then Q = b, when a = 1, then Q = U(b). lowest output is a negation of input c. Which is the same as ab c executed in on the coesponding qubit in Toffoli (Figue 6a). Othe output values ae also the same as in the Toffoli gate equations. The eade is asked to analyze the opeation of this gate on the emaining input combinations to veify its coectness. Let us also emembe that Feynman gate has two vaiants. The EXOR gate can be on an uppe o on a lowe wie. Similaly, thee ae thee 3-qubit Toffoli gates, EXOR-up, EXOR-down and EXOR-middle (Figue 3). Obseve that these gates have all diffeent unitay matices so they ae fomally diffeent quantum gates. 3. Evolutionay Appoaches to Synthesis of Quantum Aays Thee ae basically two methods of designing and dawing quantum cicuits. In the fist method you daw a cicuit fom gates and you connect these gates by standad wies. This is like in classical cicuit design, but you gates ae evesible o quantum gates. The ules to design a evesible cicuit using this appoach ae the following: (1) no loops allowed in the cicuit and no loops intenal to gates, (2) fanout of evey gate is one (which means, evey output of a gate can be connected only to one input teminal). These ules peseve the evesible chaacteistic of gates thus the esulting cicuit is also completely evesible. Next, when you daw the cicuit, the gates ae placed on a 2-dimensional space and the connections between them ae outed. Evey cossing of two wies in the schematics is eplaced with the quantum Swap gate making the schematics plana, which means, no moe two wies intesect

11 in it. The schematics is thus ewitten to a quantum aay notation illustated below. It is elatively easy to tansfom a quantum aay to its coesponding unitay matix, as will be illustated in the sequel. Konecke and standad matix multiplications ae used in this tansfomation. A unitay matix of a paallel connection of gates A and B is the Konecke poduct A B of matices A and B. A unitay matix of a seial connection of gates A and B is the standad matix poduct A * B of matices A and B. Tansfomation of quantum aay to its unitay matix is done fo analysis o veification and is a pat of ou evolutionay algoithm. The appoaches that use this fist design method of evesible cicuits ae close to those of the classical digital CAD whee the phases of logic synthesis and physical (geometical) design ae sepaated. The second design method fo quantum cicuits is to synthesize diectly the quantum aay of a cicuit that was initially specified by a unitay matix. This is done without involving additional gaph-based o equation-based epesentations. The synthesis is done by one of two appoaches: (a) composing matices of elementay gates in seies o in paallel until the matix of the entie cicuit becomes the same as the specification matix, (b) decomposing the specification matix of the entie cicuit to paallel and seial connections of unitay matices until all matices coespond to matices of elementay gates diectly ealizable in the given technology. Fo simplification, fom now on we will talk intechangeably about a gate and its unitay matix. In anothe synthesis vaiant, called the appoximate synthesis, it is not equied that the cicuit specification matix and the matix of composed gates ae exactly the same. They can diffe by small allowed values o/and diffe in some matix coodinates only. The synthesis poblem of a quantum aay has theefoe a simple fomulation as a composition o decomposition poblem, but pactical ealization of synthesis algoithms fo quantum cicuits is extemely had. Obseve that even in the case of standad (ievesible) binay logic these composition and decomposition appoaches ae computationally vey difficult and it is only vey ecently that they ae becoming pactical in CAD softwae, because of thei high computational complexity. In the case of quantum logic the situation is much moe difficult, because of the fast incease of data sizes and because so fa the mathematics of desciption tansfomations is limited, heuistics ae not known and thee ae no countepats in quantum logic of such familia notions as Kanaugh Maps, pime implicants o eductions to coveing/coloing combinatoial appoaches. Theefoe most authos tuned to evolutionay algoithms as the fast pototyping methods fo quantum aays [1,2,3,5,6,8,54]. These appoaches seem to be good fo intoductoy investigations of the solution space and its popeties, with the hope that by analyzing solutions we will lean moe about the seach space and ultimately ceate moe efficient algoithms. As we will see below, this phenomenon actually happened in case of ou eseach. In the evolutionay computation appoach to quantum synthesis, two key elements stongly influence the convegence of the seach. These ae: (1) the evaluation function (called also the fitness function), and (2) the encoding of the individuals [9]. Genetic pogamming uses the encoding of data to tees so the opeations on them ae vey time-consuming [36]. Hee we popose a new way to calculate costs of gates and cicuits as a pat of the fitness function. Also poposed is a new method of QC and RL cicuit encoding in a GA. Since we use the same encoding

12 fo QC and RL and fo many vaiants of algoithms, design paametes and cost functions, we can easily compae espective synthesis esults unde stictly the same conditions. This has been not accomplished by pevious authos. The synthesis of quantum logic cicuits using evolutionay appoaches has two fundamental aspects. Fist, it is necessay to find the cicuit that eithe (A) exactly coesponds to the specification, o (B) diffes only slightly fom the specification. Case (A) is veified by a tautology of the specification function and solution function. In case of a tuly quantum cicuit this is done by a compaison of unitay matices. In case of pemutation functions this can be also done by compaing the tuth tables. Obseve that non-pemutation matices cannot be epesented by tuth tables which leaves the epesentation of unitay matices as the only canonical function epesentation. This epesentation is esponsible fo less efficient tautology veification duing fitness function calculations, which consideably slows down the softwae. Case (B) fo pemutation cicuits is veified by an incomplete tautology (tautology with accuacy to all combinations of input values and with abitay logic values fo don t cae combinations). In some applications such as obot contol o Machine Leaning it is sufficient that the specification and the solution ae close, like, fo instance, diffeing only in few input value combinations. Second fundamental aspect of quantum logic synthesis is that the size (cost) of the cicuit has to be minimal, in ode to allow the least expensive possible quantum hadwae implementation (like the minimum numbe of electomagnetic pulses in NMR technology). The cost diffes fo vaious technologies, but some appoximate costs functions will be fomulated hee that ae still moe accuate than those peviously poposed in the liteatue. Consequently, the fitness function of the GA should take into account both the above-mentioned aspects, which means the GA should be designed so as to minimize both the distance fom the specification and the final cicuit cost simultaneously. Sometimes we ae not inteested only in those solutions that meet the specification exactly. Usually we ae not inteested in the solution that would have the exact minimal cost. Multiple designs of a GA ae tested hee and thei esults ae compaed in ode to select the minimum cost solution. This appoach not only impoves the cost of any solution found, but also helps us to develop gadually bette geneations of ou evolutionay/algoithmic/heuistic pogamming tools fo QC CAD. In ou softwae the fitness function is optimized fo the minimal cost, speed of synthesis o pecision of synthesis. In case of designing a evesible cicuit, the fitness function can be also optimized fo the speed of synthesis, the cost of the cicuit o fo the total numbe of wies (insetion of a minimum numbe of constants). These two methods of optimization fo each technology and logic type allow to fomulate pecise constaints fo the synthesis method in use. In this pape we deive moe pecise cost functions fo gates to be used in the optimization algoithm. We will follow the footsteps of some pevious papes in quantum computing and we will ealize all gates fom 1* 1 gates (called also 1-qubit gates) and 2*2 gates (i.e. 2-qubit gates). Moeove, accoding to [21] we will assume that the cost of evey 2*2 gate is the same. Although ou methods ae geneal, to be moe specific we assume Nuclea Magnetic Resonance (NMR) quantum computes [4,49,50,51,52], fo which we appoximately assume costs of all 1-qubit gates to be 1, and 2-qubit gates to be 5. Thus, evey quantum gate (paticulaly, evey pemutation quantum gate) will be build fom only 1-qubit and 2-qubit pimitives and its cost

13 espectively calculated as the total cost of these pimitives used in the cicuit. Obviously, this appoach can be used fo both QC and RC, since in evolutionay appoaches we can always abitaily estict the gates used fo synthesis, and in case of RC only a subset of pemutation gates can be selected. 4. Costs of quantum gates. An impotant poblem, not discussed so fa by othe authos, is the selection of the cost function to evaluate the QC (o RC) designs. Although the detailed costs depend on any paticula ealization technology of quantum logic, so that the cost atios between fo instance Fedkin and Toffoli gates can diffe in NMR and ion tap ealizations, the assumptions used in seveal pevious papes [2,3,4,5,6,8]; that each gate costs the same, o that the cost of a gate is popotional to the numbe of inputs/outputs, ae both fa too appoximate. These kinds of cost functions do not povide any basis fo the synthesis optimization and as shown by us [1, 2] can lead to quantum sequences fo NMR that ae fa fom the minimum. In this section some of the well-known decompositions of known 3-qubit quantum pemutation gates, as well as the new gates, will be discussed to illustate this poblem. The cost of quantum gates in synthesis methods can be seen fom as many points of view as thee ae possible technologies. Pevious eseaches used specific cost to minimize the numbe of wies pe gate, the numbe of gates in the cicuit o othe task-specific citeia. Assume a andom function to be synthesized in QL and defined ove a set of basic vaiables a, b, c and d. This function will use 4 qubits and the unitay matix epesentation of the cicuit will be of size 2 4 * 2 4. The matix epesentation can be also called the evolution matix of the system [10] but this name has nothing to do with evolutionay synthesis algoithms. Defined as quantum evolutionay pocesses, thee is theoetically an infinity of possible quantum gates. In the pesent wok only well-known and widely used unitay quantum gates, as those listed above, will be used. The use of ou softwae can howeve vey easily extend the system by defining abitay gates as 256-chaacte symbols togethe with thei unitay matices and natual numbes fo costs. This applies also to multi-qubit gates and multi-valued quantum gates [55]. Based on NMR liteatue [49,50,51,52] we assume the 1-qubit gates to have a cost of 1 and the 2-qubit gates to have the cost of 5. This leads to 3-qubit gate costs of about 25. Fom this cost attibution it is evident that the optimization should be aimed towad the use of gates with as few qubits as possible. The cost of the cicuit is the total cost of gates used. Of couse, especially using GAs, the seach might not always be successful but a solution will have always two evaluation citeia: the final cicuit eo and the final cost calculated afte the optimizing local equivalence tansfomations being applied to the initial cicuit poduced by a GA. (Sometimes we use in GA moe pecise gate costs if they ae known fom the liteatue, but the costs given above will be used fo illustation puposes hee). In the ideal case the GA will find an optimal cicuit; i.e. the cicuit with the smallest possible cost and with the coect matix epesentation. Two wose altenatives ae: (1) a cicuit with a highe cost than the minimum but still with a coect matix, (2) a cicuit with not completely coect matix but with a smalle cost. Fo futue use let s efe to these thee goups of altenatives as C o (optimal cost), C a (aveage cost) and C w (wost cost), espectively. In the case of RL cicuit synthesis the diffeences in esults can occu on one moe level: as mentioned befoe, input constants insetion is allowed in RL synthesis. Howeve, we want to minimize the

14 numbe of such constants in ode to educe the cicuit s width. Once again, as will be shown below, this paamete can be also taken into account by ou softwae as pat of the cost function. Consequently, one can obtain a coect cicuit with a small cost of gates but with a highe numbe of wies. Cost of the esult in gates C g Function found Total Cost C t 1. Min C g YES Min C t 2. Min < C g < Max YES Min < C t < Max 3. C g Max YES Min < C t < Max 4. Min C g NO Min < C t < Max 5. Min < C g < Max NO Min < C t < Max 6. C g Max NO C t Max Table 1: Illustation of possible esults using a combination of cost function pe gate C g and a global evaluation cost function C t. The column in the middle indicates if the seached function was found o not found. Table 1 systematizes possible esults of a Quantum Cicuit synthesis. In the table possible seach solutions ae chaacteized with espect to the combinations of the two cost functions. The cost C g is the cost of the cicuit based on the sum of costs of all gates. C t is the total cost whee the coectness of the esult is also included, based on one of the peviously explained veification methods between the specification and the obtained cicuit desciptions. As will be seen, both of these two cost functions can be handled in ode to acceleate o modeate the speed of convegence of the entie synthesis pocess. A close look at the table will eveal thee categoies of esults as peviously descibed. These ae cicuits with small, aveage and high total costs, espectively. These types can be again sepaated in two, depending whethe the final solution was found o not. Goups 1, 2, and 3 ae fo exact design, goups 4, 5, and 6 fo appoximate. Goups 1 and 4 coespond to C 0, Goups 2 and 4 to C a, and goups 3 and 6 to C w. The categoy with the most cicuits is the one with an aveage cost, and as will be seen late it is this categoy that needs paticula attention while selecting and adapting a fitness function fo a synthesis poblem. Min and Max in the table ae some paametes set by the use and based on expeience. Now let us continue the discussion of the cicuit decomposition into pimitives. Let us conside one pactical example. The Toffoli o Fedkin gates intoduced in section 2 ae both univesal quantum logic gates that ae well-known. They have been built in seveal quantum and evesible technologies. The poblem is to find an optimal decomposition of the univesal gates into smalle pats, especially the diectly ealizable quantum pimitives such as Feynman, NOT o Contolled-V (C_V) gates. As mentioned ealie, the gates with one and two qubits have costs 1 and 5, espectively. Consequently the assumption of using only 1-qubit and 2-qubits gates will be obseved since only such gates ae diectly ealizable in quantum hadwae. Othe gates ae only abstact concepts useful in synthesis and not necessaily the best

15 gates fo any paticula quantum technology (like NMR in ou case). Figue 6 pesented the well-known ealization of Toffoli gate fom [21]. Accoding to Figue 6b thee ae five 2-qubit pimitives, so the cost is 5 * 5 = 25. Now that we leaned in section 2 how Toffoli gate woks intenally based on the opeation of its component quantum pimitives, we will ealize the Fedkin gate fom the Toffoli gate. Using a GA [1,2] o the synthesis method fom this pape, we can synthesize the Fedkin gate using two Feynman gates and one Toffoli gate as in Figue 5. The cost of this gate is 2* = 35. Substituting the Toffoli design fom Figue 6b to Figue 5 we obtain the cicuit fom Figue 8a. Now we can apply an obvious EXOR-based tansfomation to tansfom this cicuit to the cicuit fom Figue 8b. This is done by shifting the last gate at ight (Feynman with EXOR up) by one gate to left. The eade can veify that this tansfomation did not change logic functions ealized by any of the outputs. Obseve that a cascade of two 2*2 gates is anothe 2*2 gate, so by combining a Feynman with EXOR-up gate (cost of 5), followed by contolled-v gate (cost of 5) we obtain a new gate C_V with the cost of 5. Similaly gate C_V + with cost 5 is ceated. This way, a cicuit fom Figue 8c is obtained with the cost of 25. (This tansfomation is based on the method fom [21] and the details of cost calculation of C_V and C_V + ae not necessay hee). Thus, the cost of Toffoli gate is exactly the same as the cost of Fedkin gate, and not half of it, as peviously assumed and as may be suggested by classical binay equations of such gates. a) b) V V V + V V V + c) C_V V C_V + Figue 8: Example of educing cost in the Fedkin gate ealized with quantum pimitives. Gates C_V and C_V + in Figue 8c ae ceated by combining espective quantum pimitives fom Figue 8b which ae shown in dotted lines. Encouaged with the above obsevation that sequences of gates on the same quantum wies have the cost of only single gate on these wies, we used the same method to calculate costs of othe well-known gates. Let us fist investigate a function of thee majoities investigated by Mille [22,23,56]. This gate is descibed by equations: P = ab ac bc, Q = a b a c bc, P = ab ac b c. Whee a is a negation of vaiable a. Function P is a standad majoity and Q, R ae majoities on negated input aguments a and b, espectively [56]. We ealized this function with

16 quantum pimitives, found it useful in othe designs and thus wothy to be a standalone quantum gate. We call it the Mille gate [56]. As seen in Figue 9a, the Mille gate equies 4 Feynman gates and a Toffoli gate, which would suggest a cost of 4* = 45. Howeve, pefoming tansfomations as in Figue 9b, we obtain a solution with cost 35. Anothe solution obtained by the same method has cost 35 and is shown in Figue 9c. It is also based on simple EXOR tansfomation (x y) z = (x z) y applied to thee ightmost Feynman gates fom Figue 9a, with EXOR in the middle wie y. (Moe on optimizing equivalence-based tansfomations in section 6). Again, the Mille gate, based on its binay equations, looks initially much moe complicated than the Toffoli gate, but a close inspection using quantum logic pimitives poves that it is just slightly moe expensive. 5. Ou Entie minimization methodology of quantum aays synthesis. Based on examples fom section 4, let us obseve that a new pemutation quantum gate with equations: P = a, Q = a b, R = ab c can be ealized with cost 20. It is just like a Toffoli gate fom Figue 6b but without the last Feynman gate fom the ight. This is the cheapest quantum ealization known to us of a complete (univesal) gate fo 3-qubit pemutation gate (Figue 10). It is thus wothy futhe investigations. We found that the equation of this gate was known to Pees [35], but it has not been used in pactical designs of quantum o evesible cicuits. We popose hee to conside the Pees gate as a base of synthesis and tansfoms, similaly as 2-input NAND gates ae used in technology mapping phase of classical Boolean CAD. Obseve that assuming the availability of Pees gate, the algoithms fom liteatue (fo instance [25,26]) will not lead to a quantum aay with minimum quantum costs fomulated as above. When given the unitay matix of the Pees gate, these algoithms would etun the solution of cost 30 composed fom one Toffoli gate and one Feynman gate, which would lead to clealy non-minimal electomagnetic pulse sequence. Thus, impoved synthesis algoithms should be ceated to minimize the ealistic quantum gate costs intoduced in [21] and in this pape. Obseve please also, that if a post-pocessing stage wee added to the algoithms fom [22,23,24,25,26] (o to the esult of the GA fom section 7), then the optimum solution (of a single Pees gate and cost of 20) would be found fo this example. Theefoe ou entie synthesis and optimization appoach to quantum aays is the following. 1. Fist ceate the cicuit by some appoximate synthesis method (evolutionay algoithm in ou case), 2. Apply maco-geneation of complex gates to quantum pimitives, 3. Apply the optimizing equivalence-based tansfomations.

17 The application of this pocedue to ou example will have the following stages: (a) the initial solution is a Toffoli gate followed by a Feynman gate (with EXOR down) on quantum wies 1,2 (Figue 10b). (b) the maco-geneation of the Toffoli gate leads to a Pees gate followed by the Feynman gate (with EXOR down) on quantum wies 1,2. (c) the equivalence-based tansfomation (pesented below) will find a patten of two Feynman gates of the same type (EXOR down) in sequence, which is eplaced with two quantum wies, 1 and 2 (Figue 10c). (d) thus the two ight Feynman gates ae cancelled, the same way as two invetes in seies ae cancelled in standad logic synthesis. (e) the esultant cicuit, a final solution, will have just one Pees gate of cost 20 (Figue 10d). a) x b) y V V V + z V V V + c) V V V + Figue 9: Reducing the cost of ealization of Mille gate using quantum pimitives: Feynman EXOR up, Feynman EXOR down, Contolled-V (V down), and Contolled V + (V + down). As can be seen fom pevious examples, the gates matices have infinity of possible epesentations in QL. The above samples of synthesis ae situated in the C o categoy of Table 1, because if eventually a smalle cicuit of a Fedkin gate wee found, then the one pesented hee will emain vey close to the minimal cost. Consequently the goup that needs the closest inspection is the C a because it includes cicuits of geat inteest. As the numbe of cicuits is infinite, the cicuits in the last goup can be consideed as being too fa fom the seached solution. Howeve the cicuits in the goup having aveage cost (not too fa fom the solution and not too big) can be simplified to bette ones, and also they can aleady include pats of the optimal cicuits. 6. Local Tansfomations. The tansfomations ae gouped in 12 tansfomation sets. Thee ae the following sets: S1. 1-qubit tansfomations,

18 S2. 2-qubit tansfomations, S3. 3-qubit tansfomations, S4. 4-qubit tansfomations, S5. n-qubit tansfomations, S6. Tenay tansfomations, S7. Mixed binay/tenay tansfomations, S8. Maco-geneations, S9. Maco-cell ceations, S10. Pees Base tansfomations, S11. Toffoli Base tansfomations, S12. Contolled-V Base tansfomations, S13. Input/Output pemutting tansfomations. Many tansfomations ae shaed between sets. In addition, in each of the above base sets, thee ae subsets to be chosen fo any paticula un of the optimize pogam. Most of them ae taken fom [22-26,49,53] but some othe ae based on ou eseach o geneal quantum liteatue. Diffeent goups of tansfomations ae used in vaious stages of cicuit optimization. The 1-qubit tansfomations ae elated to 1-qubit gates Figue 10. Pees gate: (a) Pees gate as a block using Toffoli Base notation, (b) Solution to Pees gate in Toffoli Base with cost 30, (c) Result of maco-geneation of cicuit fom Figue 10b to Pees base, (d) Afte applying the optimizing tansfom that emoves a successive pai of Feynman gates. The cost is now 20. (Tables 2,3 and 4). They can pecede and also follow the 2-qubit, 3-qubit and othe tansfomations. The 2-qubit tansfomations ae fo 2-qubit cicuits o 2-qubit subcicuits of lage quantum cicuits, similaly the 3-qubit tansfomations ae fo 3- qubit cicuits o subcicuits of lage cicuits (Table 5). The n-qubit tansfomations ae geneal tansfomation pattens applicable to cicuits with moe than 3 qubits. They ae less computationally efficient and they use intenally tansfoms S1 S4. Tenay tansfomations (only 1-qubit, 2-qubit and 3-qubit) ae used fo tenay quantum logic synthesis and mixed tansfomations fo mixed binay/tenay quantum

19 logic synthesis [57]. Maco-geneations ae tansfomations that convet highe ode gates such as Fedkin, Magolus, DeVos, Kentopf o Pekowski gates to standad bases. Maco-cell ceations fom set S9 ae invese to those fom set S8. Thee ae thee standad bases of tansfomations: Toffoli Base, Pees Base, and Contolled-V Base. In Toffoli Base all pemutation gates ae conveted to X (i.e. NOT), 3-qubit Toffoli and 2-qubit Feynman gates. This is the standad synthesis base used by all othe authos in liteatue [22-26,48]. The Pees Base has been intoduced oiginally by us based on the obsevation of supeioity of this base in NMR ealizations (and pehaps othe ealizations as well). It includes only X, 3-qubit Pees and 2-qubit Feynman gates. Contolled-V Base is anothe new base that is vey useful to synthesize new low-cost pemutation gates fom tuly quantum pimitives of limited type [55,57]. This base includes Contolled-V, Contolled-V +, V, V +, X and Feynman gates. In all bases the gates like Feynman, Toffoli, Contolled-V ae stoed in all possible pemutations of quantum wies. Thus in 2-qubit base thee ae Feynman EXOR up and Feynman EXOR down gates and tansfomations espective to each of these gates. Fo simplification, in the tables below only some of the tansfomations ae shown, fo instance elated only to Feynman EXOR down o Toffoli EXOR down gates. Othe tansfomations ae completely analogous. The output pemutting tansfomations lead in pinciple to a cicuit that has an unitay matix which is diffeent fom the oiginal unitay matix. Obseve that each tansfomation can be applied fowad o backwad, so the softwae should have some mechanisms to avoid infinite loops of tansfomations. The matix of the new cicuit is the matix of the oiginal cicuit with pemuted output signals. In some applications the ode of output functions is not impotant, so if the cicuit is simplified by changing the output ode, the output pemutting tansfomation is applied. In addition to opeatos defined ealie, we define now the following opeatos: T 1 = 0 Y ( φ ) 0 jφ jπ 2, P ( φ ) = e I, 4 e φ φ = cos I j sin Y, 2 2 X ( φ ) = Z ( φ ) = φ cos I 2 φ cos I 2 j sin j sin φ X, 2 φ Z 2 X,Y,Z defined ealie ae Pauli spin matices and X(φ),Y(φ), Z(φ) ae the coesponding exponential matices, giving otations on the Bloch sphee [48]. P is a phase otation by φ/2 to help match identities automatically [53]. The tansfomation softwae opeates on sequences of symbols that epesent gates and thei pats. Symbol * is used to ceate sequences fom subsequences. This symbol thus sepaates two seially connected gates o blocks. Numeically, it coesponds to standad matix multiplication. Gate symbols within a paallel block may be sepaated by spaces, but it is necessay only if lack of space will lead to a confusion, othewise space symbol can be omitted. So fo instance symbols of macocells should be sepaated by spaces. Thee ae fou types of symbols: simple, otational, paameteized and contolled. Simple symbols ae just names (hee single chaactes, in softwae chaacte stings). The names of simple symbols (used also in othe types of symbols) ae the following: D standad contol point (a black dot in an aay), E contol with negated input, F contol with negated output, G contol

20 with negated input and output, X Pauli-X, Y Pauli-Y, Z Pauli-Z, H Hadamad, S phase, T Π/8 (although Π/4 appeas in it). Symbols A, B and C ae auxiliay symbols that can match seveal gate symbol definitions. They do not coespond to any paticula gates but to goups of gates and ae useful to decease the set of ules and thus speed-up tansfomations. Othe simple symbols will be explained below. As we see, chaactes ae used hee not only fo gates but also fo pats of gates, such as D standad contol used in gates. Thus we can combine these symbols to ceate gate desciptions: DX (Feynman with EXOR down), XD (Feynman with EXOR up), DDX (Toffoli o Toffoli with EXOR down), DXD (Toffoli with EXOR in middle), XDD (Toffoli with EXOR up), and so on. Names of maco-cells ae fo instance: FE (Feyman), TOD (Toffoli with EXOR down), SW (swap), FRU (Fedkin contolled with uppe wie), MA (Magolus), KED (Kentopf with Shannon expansion in lowest wie), etc. Symbols like φ, θ, π denote angles and othe paametes. Paameteized symbols have the syntax: simple_name [paamete 1, paamete n ], whee paamete i ae paametes. Fo instance, X[4Π/8], Y[3Π/2], Z[φ], etc. Rotational symbols ae composed of the (simple) otation opeato symbol such as X, Y, Z, o P, and a numbe. X[φ i ], Y[φ i ], Z[φ i ], P[φ i ], whee φ i = 4Π/8 * i, i = 1,2,,7. We assume hee that all otational opeatos have peiod 4Π and equal identity when thei agument is 0, thus the choices fo φ I. In these opeatos the notation is like this, X = X[4Π/8 * ], = 1,2,,7, and so on fo Y, Z and P. Contolled symbols have the syntax simple_name [ sequence of simple,otational o paameteized symbols ] Example 1: D[X*X] is a symbol of a contolled gate that is ceated fom two subsequent Feynman gates. Obseve that DX*CX = D[X*X] = D[I] = I I, which means that two contolled-not gates in sequence ae eplaced by two paallel quantum wies denoted by I I. Example 2. D[S*T*H] is a sequence S*T*H contolled by single qubit. Example 3. DD[X1*Y2*Y2] is a sequence of otational gates contolled by a logic AND of two qubits (this is a genealization of a Toffoli gate). Obseve that contolled symbols ae ceated only as a tansitional step duing the optimization pocess such gates do not physically exist. Using the concept of contolled symbols, n-qubit cicuits can be optimized using 1-qubit tansfomations without duplicating all the 1-qubit identity ules. Similaly the paameteized and otational symbols allow the eduction and hieachization of the set of ules, which causes moe efficient and effective un of the optimization softwae. The simplified algoithm SA fo pefoming ule-based optimization of 3-qubit quantum aays is the following: 1. Apply all the 1-qubit tansfomations, until no moe applications of such ules becomes possible. 2. Apply all the 2-qubit tansfomations and 1-qubit tansfomations induced by them (fo instance using the contolled symbols). 3. Apply all the 3-qubit tansfomations until possible. 4. Iteate steps 1,2 and 3 until no changes in the cicuit. 5. Apply invese tansfomations that locally optimize the aay. 6. Repeat steps 1,2,3,4 until possible. 7. Apply invese tansfomations that do not wosen the cost of the aay. 8. Repeat steps 1,2,3,4 until possible.

21 Seveal simila vaiants of this heuistic algoithm can be ceated. In geneal, none of these vesions gives a waanty of the optimal o even sub-optimal solution, as known fom the theoy of Post/Makov algoithms. As an example, we pesent 1-qubit tansfomation algoithm A1q: A1. Combine modulo-8 the same types of otational opeatos P,X,Y,Z. Fo instance X2*X3 becomes X5, X2*X3*X3 becomes I, and Y3*Y3*Y3 becomes I*Y1=Y1. A2. Apply diectly applicable ules that do not include symbols A and B. A3. Iteate 1 and 2 until no moe changes possible. A4. Stating fom the left of the sequence, find a gouping patten such as A2 = - BC A5. Substitute symbols X,Y,Z fo A,B and C fom the patten. A6. Apply in fowad diections the standad simplifying tansfomations such as I*A=A A7. Repeat steps A1 to A6 until possible. Example 4. [53]. Given is an identity Y = - X*Y*X. Let us ty to veify this identity using algoithm A1q. We have theefoe to simplify the sequence X*Y*X. (A1) Thee ae no pattens of the same otational opeatos to combine, (A2) Thee ae no diectly applicable ules, (A4) We take patten -X*Y and match it with the ule A2 = C*B. This leads to Z2*X. (A1) no, (A2) no, (A3) no, (A4) we find patten A2 = CB which leads to Y*X*X. (A6) X*X is eplaced with I, Y*I is eplaced with Y. No futhe optimization steps ae possible, so the sequence was simplified to Y, poving that Y = -Y*Y*X. Example 5. Simplify HXH. (A2) Use ule H = X*Y1 twice. This leads to X*Y1*X*X*Y1. (A2) Use ule I=AA in evese diection. This leads to X*Y1*Y1. (A1) Y1*Y1 is eplaced by Y2. This leads to X*Y2. (A4) Use patten A=B*C2. This leads to Z. No futhe optimization steps ae possible. Thus we poved that HXH = Z. Moe optimization examples of algoithm SA will be given in the sequel. In ou uns of GA we look fo solutions with the accuacy of: (1) pemutation of quantum wies, (2) pemutation of inputs, (3) pemutation of outputs (tansfomation goup S13). In addition we can also geneate solution sets with accuacy of inveting input, output o input/output signals (the NPN classification equivalent cicuits). Theefoe, fo each unitay matix we geneate theefoe many logically equivalent solutions. We can geneate solution sets also fo the same set of Boolean functions, o fo the same NPN classification class. One inteesting aspect of such appoach is that one can ceate new local equivalence tansfomations fo cicuits in each of these classes. Finding these tansfomations and applying them exhaustively to paticulaly inteesting gates leads to levellized onion-like stuctues of gates, as the one shown in Figue 11. This Figue shows the layeed stuctue of gates ceated by adding only Feynman gates to a seed composed of othe gate types, in this case a Toffoli gate. Figue 12 shows the layeed stuctue with Pees gate as a seed, in which Toffoli, Fedkin and Mille gates ae ceated. These tansfomations ae used to find efficient ealizations of new gates fom known gate ealizations [55,57].

22 Figue 11. The layeed stuctue of gates. Fedkin fom Toffoli and Toffoli fom Fedkin. I1. I = A*A I2. I = -A*C2*B I3. I = HH I4. I = - H*Y3*X I5. I = P2*A2*A I6. I = P2*H*X2*Y1 I7. I = - P2*H*Y3*X2 I8. I = - P2*H*Z2*Y3 I9. I = P2*Y1*Z2*H I10. I = P2*Y2*Y I11. I = - P2*Y3*H*Z2 I12. I = - P2*Y3*X2*H I13. I = P2*Z2*Z I14. I = P3*Z3*S I15. I = P4*A4 I16. I = - Y*X2*Z I17. I = - Z*Y2*X I18. I = - Z*Y3*H A1. A = B*C2 A2. A = B2*C A3. A = B3*A*B3 A4. A = B3*C*B1 A5. A = - C*B2 A6. A = C1*B*C3 A7. A = - C2*B A8. A = C3*A*C3 A9. A = P2*A2 A1.1. A1 = - B*A1*C A1.2. A1 = - B*A3*B A1.3. A1 = - C*A3*C A1.4. A1 = P2*A3*A A2.1. A2.2. A2.3. A3.1. A3.2. A3.3. A4.1. A2 = - B*C A2 = CB A2 = - P2*A A3 = B*A3*C A3 = - C*A3*B A3 = - P2*A1*A A4 = P4 X1. X = - H*Y3 X2. X = S*X2*S X3. X = Y1*H X4. X = Y3*H*Y2 X5. X = Z1*X*Z1 X6. X = - Z3*Y*Z1 X1.1. X1.2. X2.1. X2.2. X2.3. X2.4. X2.5. X3.1. X3.2. X3.3. X3.4. X1 = H*Z1*H X1 = Z*X1*Y X2 = H*Z2*H X2 = P2*H*Y3 X2 = - S*X*S X2 = - Y3*H*Y X2 = Y3*Y*H X3 = H*Z3*H X3 = - S*H*S X3 = - Y*X1*Y X3 = - Z*X1*Z Table 2: 1-qubit tansfomations fo I, A and X goups.

23 Y1. Y = - H*X2*Y3 Y2. Y = H*Y3*Z2 Y3. Y = - H*Z2*Y1 Y4. Y = P2*Y2 Y5. Y = S*Y2*S Y6. Y = X1*Y*X1 Y7. Y = X2*Y3*H Y8. Y = X3*Y*X3 Y9. Y = - X3*Z*X1 Y10. Y = -Y1*X2*H Y11. Y = -Y3*H*X2 Y12. Y = - Y3*Z2*H Y13. Y = - Z1*X*Z3 Y13. Y = Z1*Y*Z1 Y14. Y = - Z2*H*Y3 Y15. Y = Z3*Y*Z3 Y1.1. Y1 = - H*Y*X2 Y1.2. Y1 = - H*Y2*X Y1.3. Y1 = - H*Y3*H Y1.4. Y1 = H*Z Y1.5. Y1 = P2*H*Z2 Y1.6. Y1 = P2*Y3*Y Y1.7. Y1 = X*H Y1.8. Y1 = Y*H*X2 Y1.9. Y1 = - Z*Y2*H Y1.10. Y1 = Z2*H*Y Y1.11. Y1 = - Z2*Y*H Y2.1. Y2.2. Y2.3. Y2.4. Y2.5. Y2 = - H*Y2*H Y2 = - H*Y3*Z Y2 = - P2*Y Y2 = - S*Y*S Y2 = - X*Y3*H Y3.1. Y3 = H*Z2*Y Y3.2. Y3 = - H*X Y3.3. Y3 = - P2*H*X2 Y3.4. Y3 = - P2*Y1*Y Y3.5. Y3 = X2*H*Y Y3.6. Y3 = Y*H*Z2 Y3.7. Y3 = Y*X2*H Y3.8. Y3 = - ZH Z1. Z = H*Y1 Z2. Z = P2*Z2 Z3. Z = S*S Z4. Z = - X1*Y*X3 Z5. Z = X1*Z*X1 Z6. Z = X3*Z*X3 Z7. Z = Y2*H*Y3 Z8. Z = - Y3*H Z1.1. Z1 = H*X1*H Z1.2. Z1 = P2*Z3*Z Z1.3. Z1 = - P3*S Z1.4. Z1 = - X*Z3*X Z1.5. Z1 = Y*Z1*X Z1.6. Z1 = - Y*Z3*Y Z2.1. Z2 = H*X2*H Z2.2. Z2 = H*Y3*Y Z2.3. Z2 = P2*Y3*H Z2.4. Z2 = - P2*Z Z2.5. Z2 = - P3*Z1*S Z2.6. Z2 = - Y*H*Y3 Z2.7. Z2 = Y1*Y*H Z3.1. Z3 = H*X3*H Z3.2. Z3 = - P1*Z*S Z3.3. Z3 = - P2*Z1*Z Z3.4. Z3 = - P3*Z2*S Z3.5. Z3 = - X*Z1*X Z3.6. Z3 = X*Z3*Y Z3.7. Z3 = - Y*Z1*Y Table 3: 1-qubit tansfomations fo Y and Z goups. S1. S = P1*Z1 S2. S = P3*Z3*Z S3. S = T*T S4. S = X1*S*Y1 S5. S = X2*S*Y2 S6. S = X3*S*Y3 S7. S = Y*S*X S8. S = - Y1*S*X3 S9. S = - Y2*S*X2 S10. S = - Y3*S*X1 H1. H = P2*Y1*Z2 H2. H = - P2*Y3*X2 H3. H = S*X1*S H4. H = X*Y1 H5. H = - X1*H*Z3 H6. H = - X2*H*Z2 H7. H = - X2*Y3*Y H8. H = - X3*H*Z1 H9. H = Y*X2*Y3 H10. H = - Y1*Y*X2 H11. H = - Y1*Z H12. H = Y2*H*Y2 H13. H = Y2*X*Y3 H14. H = Y3*H*Y3 H15. H = -Y3*X H16. H = Y3*Z*Y2 H17. H = Y3*Z2*Y H18. H = - Z*Y3 H19. H = - Z1*H*X3 H20. H = - Z2*H*X2 H21. H = - Z2*Y1*Y H22. H = - Z3*H*X1 X[Π] *Y[φ] = Y[- φ]*x[π] X[-Π]*Y[φ] = Y[- φ]*x[-π] X[φ] *Y[Π] = Y[Π] *X[-φ] X[φ]*Y[-Π] = Y[-Π]*X[- φ] X[Π/2]*Y[φ] = Z[φ] *X[Π/2] X[-Π/2]*Y[φ] = Z[-φ]*X[-Π/2] X[φ] *Y[Π/2] = Y[Π/2]*Z[φ] X[φ] *Y[-Π/2] = Y[-Π/2]*Z[-φ] X[3Π/2]*Y[φ] = Z[-φ] *X[3Π/2] X[-3Π/2]*Y[φ] = Z[φ] *X[-3Π/2] X[φ] *Y[3Π/2] = Y[3Π/2]*Z[-φ] X[φ] *Y[-3Π/2] = Y[-3Π/2]*Z[φ] Y[Π] *Z[φ] = Z[- φ]*y[π] Table 4: 1-qubit tansfomations fo S, H and paameteized otation goups.

24 R2.1. R2.2. R2.3. R2.4. R2.8. R2.9. R2.12. R2.13. R2.15. R2.18. R2.19. R2.27. R2.29. R2.35. R2.43. R2.44. (a) (b) IX*DX = DX * IX DX*XD = XD * SW DX*XD*DX = SW DX*XX = XI*DX DA*DB = D[A*B] DZ = ZD HH*DX*HH = XD IH*DZ*IH = DX DX*XI*DX = XX DX*YI*DX = YX DX*ZI*DX = ZI ZI DX*IX*DX = IX DX*IY*DX = ZY DX*IZ*DX = ZZ Z[θ] I *DX = DX* Z[θ] I I X[θ] *DX = DX* I X[θ] R3.11. R3.12. R3.21. R3.22. R3.24. R3.25. R3.28. R3.43. R3.44. R3.48. R3.49. R3.54. R3.55. R3.57. R3.58. R3.59. R4.1. R4.2. R4.3. DDA*DDB = DD[A*B] DIX*DDX = DDX * DIX DDX*DXI*DDX = DXI *DIX DXI * DDX = DDX * DXI * DIX IXI*DDX*IXI = DDX*DIX DDX*XII*DDX = XII*IDX DXI*DIX*DDX = DDX*DXI IXD*DDX = DDX * DXD * DDX * IXD IIX*DDX = DDX * IIX DDX*DXD*DDX = IDX*DXD*IDX IXI*DIX*DDX = DDX*IXI IDX*DIX*DXI = DXI*IDX IDX*DXI*IDX = DXI*DIX XII * DDX = DDX * X DX DXI*DDX*DXI = DDX*DIX DXI*IDX*DXI = IDX*DIX DIIX*IDDX = IDDX* DIIX XIID*IDDX = IDDX*XDII*XIID DXII*IDDX = IDDX*DIDX*DXII (c) Table 5: Examples of 2-qubit and 3-qubit tansfomations: (a) 2-qubit tansfomations, (b) 3-qubit tansfomations; obseve a space between X and DX in ule R3.57 that signifies that D contol the lowe X, (c) 4-qubit tansfomations Figue 12: Intenal pat of a layeed stuctue with Pees gate as a seed.

25 7. Genetic Algoithm fo Quantum Logic Synthesis Now that we undestand all the basic peequisites fo ou appoach to quantum aay synthesis and we have a geneal plan fo it, let us finally discuss the Genetic Algoithm itself. Similaities with genetic infomation and its tansmission in Natue ae applied and constitute the coe of the evolutionay appoach. The application of GA is well known in poblems with high noise because thei convegence can be modeated accoding to the fitness function. GAs ae widely used optimization algoithms, howeve only few authos pesent thei application pogams fo QC synthesis [1,2,3,6]. Geneal steps of using ou GA vaiant fo QC (and RL) ae the following: 1. Initiate a andom population of individuals (hee each individual is a Quantum o Revesible cicuit). An individual is an encoding of a chomosome of a cicuit fo fitness function evaluation. The size of the population is geneally between 50 and 500 individuals. A too small population will yield a too fast convegence while a too big population will uselessly seach a too lage poblem space. 2. Steps descibing the seach using the GA ae called geneations. The seach stops when a condition is attained, geneally this is a esult of having fitness geate than a limit, o that cetain numbe of geneations has been attained. 3. Pick n pseudo-andomly selected individuals. The ules of selecting individuals can speed up the convegence of the seach, howeve if the selection opeato is too geedy (selecting only the best individuals) the algoithm can get stuck in a local minimum. We expeimented with seveal ules, some pesented below. They ae based on fitness function. 4. Fo all the individuals of the new population apply two genetic opeatos: a. Cossove opeato allows to ecombine two individuals by exchanging pats of thei espective chomosomes. This opeation is the main powe of the seach with GAs. b. Mutation opeato intoducing noise into the GA in ode to avoid local minima. Geneally applied with a vey small pobability [0.01, 0.1] (compaed to the one of Cossove [0.3, 0.85]) modify andomly a chomosome of a andomly selected individual. 5. Replace the old geneation of individuals by the new one. 6. If solution is found then exit, else go to step 2. All individuals in the GA ae quantum cicuits patitioned into paallel blocks. Ou epesentation allows descibing any Quantum o Revesible cicuit [1,2]. This patitioning of the cicuit was in ou case induced fom the epesentation of any QC such as one in Figue 14. Each block has as many inputs and outputs as the quantum aay width (fo instance thee in Figues 3 6). The chomosome of each individual is a sting of chaactes with two types of tags. To indicate to the algoithm whee a paallel block begins and ends, a special tag diffeent than any gate was equied. This special tag is named hee R-tag o. Gates tags ae chaactes fom the alphabet and in ou examples tags ae the fist lette(s) of gates names. An example of a chomosome can be seen on Figue 13. The cicuit coesponding to the paent 1

26 fom Figue 13 can be seen on Figue 14. In this paticula encoding each space (empty wie o a gate) is epesented by a tag. Ou poblem-specific encoding was applied to allow the definition of as simple genetic opeatos as possible. The advantage of these stings is that they allow encoding an abitay QL o RL cicuit without any additional paametes known fom pevious eseach [5,6]. And only the possibility to move gates, emove and add them to the chomosome consequently make it possible to constuct and abitaily modify a cicuit. To speed-up the seach the cossove opeato has only one estiction. Both cossove candidates ae esticted to have the same numbe of wies. This consideably simplifies this opeation because it allows to avoid the case when the two cicuits ae cut in incompatible locations. Consequently, fo any cicuit with a numbe of wies highe than one, the cossove is done only between paallel blocks. This assues that all childen ae well fomed and eliminates also the necessity of a epai algoithm. In the case of individuals having only one wie in the cicuit, the cossove is executed at a andom location in the chomosome. The mutation is moe delicate. Its esults should be: adding a gate, eplacing a gate by anothe one, o a emoval of a gate. All these opeations can be easily implemented except fo the eplacement opeato. Assume the Hadamad gate fom cicuit in Figue 14 is to be changed to a Feynman gate. It is a paticula gate whee we need to ceate a new paallel block because we want to peseve the emaining of the cuent cicuit and also not to ceate a block with moe wies than the cicuit actually has. In the emaining cases: (1) the emoving of a gate can easily be seen as eplacing it by wies, (2) the adding of a gate implies a ceation of a new paallel block. The esults of genetic opeatos ae shown in Figue 13. In the middle ae two paents which yield two childen on the left side as a esult of a cossove. On the ight side ae the esults of mutation (emoval of a gate, adding a new gate and changing of a gate, espectively) on paent 2. As can be seen, the opeation of adding a gate esults in a ceation of a new paallel block, but as will be shown late this opeation is not equivalent to the opeation of adding one block. Child 1 5 I SW I I H T H Child 2 5 F C H I T I SW I I Paent 1: cicuit fom Fig 14 5 I SW I I H I T I SW I I Paent 2 5 F C H T H Cossove Mutation 5 F C H I I I H R G 5 F C F I I H T H A G 5 F I I T I I H T H C G Figue 13: Chomosome epesenting the cicuit on figue 14 and genetic opeations as example. On the left is a esult of cossing-ove and on the ight is ae esults of mutation Removal of a gate (R G ), Adding of a gate (A G ) and Replacing of a gate (C G ).

27 The evaluation of a QC is pat of fitness function calculation. In the context of ou encoding it is done in two steps. Fist fo each paallel block a matix epesenting the pat of the cicuit is calculated. Then each such a unitay matix is multiplied with its neighbo so as at the end a matix epesenting the whole cicuit is obtained. An example of this calculation can be seen in Figue 15. Inside of each paallel block only Konecke matix poducts ae used, while standad matix multiplications ae used to multiply these blocks. In the case of ou softwae vaiant fo RC synthesis the evaluation is simila to the one used in QC but no matix opeations ae equied and tuth tables of (in)complete multi-output functions ae used instead. As mentioned ealie, constants insetion is possible in RC logic synthesis. Consequently a function defined ove 3 wies can be seached on a cicuit with 4, 5 o even moe wies. The evaluation of a RC cicuit consists in finding logical equation fo each wie and then testing all input/output combinations on the seached wies. In this case the constants ae altenatively tested fo each wie. As can be seen, the length of the chomosome is popotional to the size of the cicuit. Also the time equied to calculate each cicuit is popotional to this size. A cicuit with seven gates will have seven blocks if each gate cannot be connected in paallel with anothe one. In such case, the chomosome with seven blocks encoding a cicuit fo 3 qubits can encode maximally 7*3 = 21 one-qubit gates. 8. The Fitness Function The fitness function used was multiple times modified in ode to obseve impovements o changes in the seach fo the new optimized cicuits. Fo dydactic easons we will explain some of these vaiants. The fitness function evaluates each individual and assigns to it a fitness value, epesenting the quality of the encoded cicuit, and the encoding is a diect mapping fom genotype to phenotype. The oiginal GA was defined as GA with Dawinian leaning. It means, the fitness function evaluates the genotype. In Baldwinian GA the genotype is conveted to the phenotype and the fitness function evaluates the phenotype. The objective of ou expeiments was consequently to modify the fitness function in ode to obseve the impact on the evolution of the desied solution. The fitness function does not modify the cicuits but only biases the selection opeato. This mechanism, accoding to the fitness function value of each cicuit, will be able to select bette o wose cicuits. Fo example, a fitness function weighted too stongly towads the length of a cicuit will pefe cicuits with highe length, even if they ae not coect. This kind of flexibility allows the GA to exploe egions of the poblem space that ae inaccessible to classical computational methods without an exteme time and computational esouces consumption. The equation (7.1) epesents the basic fitness function. F 1 = 1 + Eo (7.1) whee Eo is the evaluation of the coectness of the cicuit. Although we found this fitness function vey useful in space exploation, it is howeve too geneic and does not take into account the cost of the cicuit. An impoved function is shown in equation 7.2. As can be seen in the equation, two paametes ae sufficient to ceate a moe sophisticated fitness function

28 F 1 Min _ cost = (7.2) 1+ Eo Cost whee the fist ight hand element is the evaluation of the coectness of the solution fom equation (7.1) and the next one is an additional constaint focing the selection opeato to select cicuits with a smalle numbe of gates o with gates having a smalle total cost. In the case of the QC seach the eo evaluation is based on the compaison of the esultant matix with the unitay matix of the evaluated cicuit. The equation (7.3) descibes the Eo calculation Eo = O S (7.3) i j ij ij whee O xy is an element fom the expected synthesized unitay matix and S xy is an element fom the matix of the cuent cicuit (fo speed a tuth table is used in case of RL). The eo geneated by this function is exact because it eflects any diffeences in the whole matix of the cicuit. In the case of an exact match, this eo substituted in the equation (7.2) gives a fitness of 1. As can be seen in Table 6, the eo has always value 128 in the denominato, since thee ae 64 (2 3 x 2 3 ) units in the evaluation matix of a 3-qubit gate and each of them is a complex numbe. The epesentation of complex numbes in computes is done via a definition of complex numbes as one numbe fo the eal pat and one fo the imaginay. Consequently one can scale the eo eithe vesus the complete set of eal and imaginay pats of the matix coefficients eithe vesus the numbe of coefficients in the matix of the esulting cicuit. Figue 14: Tansfomation of a QC fom the encoded chomosome (on the left) to a final quantum cicuit notation epesentation of this QC (on the ight). Hee S is a Swap gate, H is a Hadamad gate and W is a wie. In the middle thee is one CCNOT (Toffoli) gate. Although useful in seveal uns of ou pogam, the equation (7.2) has still a slight impefection. In the case one wants to measue the fitness in the inteval [0, 1] the second element of the equation (7.2) will in some maginal cases output a fitness of 0 while the coect esult was found. Consequently a modification of the fitness function is equied. Equation (7.4) shows one possible fom of a function poducing fitness in the inteval [0, 1].

29 F 1 = (7.4) Min _ cost Eo + Cost Howeve, this equation assumes that Min_Cost is the cheapest possible cicuit. In the case a cheape coect cicuit is found the fitness will be of couse smalle than 1. This fitness function is centeed in the point of the optimal cicuit (Min_Cost and coect esult) similaly to a Gaussian cuve. It is clea fom this example that the Min_Cost paamete should be also set to 1 which leads to the next vesion of the fitness function fom equation (7.5): F = 1 (7.5) Eo + Cost This equation then should be only maximized because the value of 1 is pactically impossible to find. It is subsumed in the Cost paamete. The value of used gates vaies as peviously intoduced in this pape. Consequently a cicuit ealized with only 1-qubit gates will always have highe fitness than one ealized with gates of two and moe qubits if fo each moe-than-one-qubit gate a cheape vaiant with only 1-qubit gates is found. Impotant point to notice is the numeical non-compatibility of the Eo and Length paametes. The Eo is in the ange [0, 2 1+(2*n) ] while the ange of the Length paamete is in [Min, ]. This ill-scaled atio would in consequence make the selection pessue mainly with espect to the length of the cicuit and not to its coectness. As a solution to this poblem, the final vaiant of the fitness function scales these two vaiables as shown in equation (7.6): F Eo 1 = α ( 1 ) + β Max _ eo Cost (7.6) whee α and β ae scaling paametes that allow to modify the selection pessue eithe towads cheape o moe coect cicuits. Anothe possible appoach induced fom the pevious fitness function was the intoduction of the Paeto optimality (PO) fitness function [9]. PO is used fo multiple objectives optimization poblems such as hee the Cost and the Eo. A GA using PO uses a diffeent evaluation fo the fitness of each individual. This appoach is the following: - Each individual is compaed with all othes and is assigned with a ank. - The ank of an individual depends on the numbe of wins while compaed to all othe individuals. - The compaison is made on all paametes. Example: an individual with the highest ank will have all of its paametes highe (fo maximization) o lowe (fo minimization) than most of all othe individuals.

30 Such a compaison foces the selection pocess of the GA to epoduce mainly the globally optimal individuals. Similaly to equation (7.5) one needs to scale paametes of compaison when they ae not equivalent. Figue 15: Examples of Konecke poduct, and of Matix poduct * on a sample of a cicuit. 9. Othe paametes of GA In the following discussion of ou esults, Stochastic Univesal Sampling (SUS) was used as the selection opeato, although we expeimented also with Roulette Wheel and othe selection appoaches. A decision fo using SUS instead of Roulette Wheel (RW) was implied fom the following demonstation of high nonlineaity in cicuit design. Consequently an opeato allowing a selection of less locally optimal cicuits such as SUS was needed in ode to obtain good expeimental esults. Fitness (6) Cicuit Eo (3) Cost (blocks) ΣC g α = 0.9 β = / , 3, 4 18/128 12, 18, , 0.778, , 6 4/128 30, , (cicuit completed) 0/ Table 6: Results of sequential evaluation of Fedkin gate fom Figue 5a. An implication of the pevious pat of this pape, one of the goals of the pesented wok was to einvent gates such as those in Figues 2, 3 and 4, o to synthesize smalle ones with the same functionality. Consequently, the analysis of a cicuit can give moe pecise explanation of possible non-monotonicity in the

31 synthesis pocess evaluation. Fo this pupose let s have a close look at the cicuit fom Figue 8a. This cicuit has seven gates. The measuement of Eo as fom equation (7.3) and fitness fom equation (7.6) ae ecoded in columns of Table 6. A close look at the above table will eveal non-monotonicity in the evaluation of this cicuit. The fist tap fo GA is between the fist gate and the adding of the second. Because both the eo and the cost incease, the fitness is significantly educed and ceates a elatively big local minimum. Consequently this step is difficult fo any automated synthesis without heuistics. Two next smalle poblems ae that between steps 2, 3, 4 and 5, 6 the eo emains constant while the cost of the cicuit is linealy gowing with each segment. A monotonic gowth of the cost can be obseved in the thid column. It is impotant to notice that all gates have a cost including the wie gate (identity gate). Such a setting allows the algoithm to avoid some local minima and constucting a cicuit with only empty wies. Fom the above example an impotant modification to the algoithm can be deived and is discussed below. A solution to this non-monotonicity poblem can be a modification of the above pesented genetic opeatos. In this wok an extension of the mutation opeato was ceated. Its esult is the addition of a complete paallel block to the cicuit. Also the mutation is tested fo each gate of the cicuit because multiple mutations can occu on one chomosome. Such a bitwise mutation o noise intoduction into the cicuit can avoid such local minima as those in Table 6. The modification of the opeato is equied to oveide such taps as in Table 6. Moeove, as aleady mentioned, in each of the goups of steps {2, 3, 4} and {5, 6} the cost inceases while the fitness deceases. It is possible to use a fitness function such as (7.1), howeve then the solution can have an abitay length and no paametes foce the selection of a shote cicuit. The esults and thei compaison is descibed in the next section. 10. Expeimental esults As was shown ealie, seveal well-known gates can be synthesized in diffeent mannes depending on the cost function selection. Thee exist aleady a vaiety of implementations of univesal gates, especially Fedkin o Toffoli. Ou goal, howeve, is to obtain the gates of the smallest possible cost fo ealistic physical ealizations. In section 2 of this pape few vaiations of Toffoli and Fedkin gates wee given and we discussed possible minimization of thei cost by ealizing them with smalle, tuly quantum, pimitives. The two impotant popeties allowing the minimization opeation ae the commutability and concatenation popeties of gates in QL. The fist popety, whethe ode of two subsequent gates can be changed, is puely mathematical and can be studied in any liteatue on Quantum Computing. In geneal AB BA, but some opeatos (gates) do commute. This can be used in synthesis, fo instance as pat of local optimization algoithms fom section 6. The second popety is moe poblematic because thee ae no any theoetical basis allowing o not this opeation to be applied. It is only a heuistic useful to calculate appoximate cost functions of composed gates [21] and the issue of concatenation should be futhe investigated. All uns of the GA have these geneic settings. The mutation pobability was in the ange of [0.01,0.1], the cossove pobability was in inteval [0.75, 0.9], and the size of the population was [100, 200] individuals. All uns wee done on a Pentium 500Mhz with 64 megs of am. We eiteate again the non-monotonicity issue discussed peviously. Because of two contay flows of values in the evaluation function (gowing a cost and

32 educing an eo) the modifications to the fitness function and selection opeato needed to be done. The fist possible and aleady intoduced modification is the scaling of the fitness function with the coefficients α and β. Such a use of scaling coefficients allows to eliminate the too small negative modifications due to the linealy inceasing cost of the cicuit. It is advised and expeimentally poven in following esults that the α coefficient should be in the inteval of 0.9 to 0.99 while β should be a complement to it; 1 - α. Similaly these coefficients should be used in a PO selection opeato to scale the anking of individuals. Figue 16: Vaious ealizations of Toffoli gate obtained fom GA befoe optimizing tansfomations. Befoe compaing available esults one needs to define the input gates fo the GA. In this case a distinction between a complete stating set and a biased stating set has to be made. A complete stating set of gates is one containing all available gates fo synthesis, excluding the esulting gate. In a complete stating set of gates the following gates wee available: Wie, Hadamad, Pauli X (NOT), Pauli Y, Pauli Z, Phase, V, Feynman, Swap, Contolled-V (C-V), Contolled-V-Hemitian (C-V + ), Fedkin, Toffoli, and Magolus. Fom these the esulting one was emoved. A biased stating set of gates is one based on some constaints, fo instance only 2-qubit gates ae used, o only an abitay subset of gates is available. In figues epesenting esults of the seach following lettes epesent these gates: Z Pauli Z, V Figue 17: Results of synthesis of Fedkin gate. a) is esult of application of fitness 7.1 and a complete set and b) the esult of fitness 7.6 and a complete set. contolled V, V + - contolled V +. Let us compae esults of diffeent uns using vaious fitness functions. Figue 16 pesents the esults of seach fo the Toffoli gate. The fist cicuit (Figue 16a) is

33 the esult of a un whee fitness fom equation 7.1 was used and a complete stating set of gates. As can be seen, this esult is vey expensive because includes thee Fedkin gates and fou Feynman gates. This cicuit can be simplified by emoving two of the thee subsequent identical Feynman gates based on the fact that Feynman gate is its own invese. The next one, Figue 16b, is a esult of the same setting as with the pevious one but the stating set was a biased one. The available gates in this biased set wee only the ones with numbe of inputs smalle o equal to 2 qubits. Similaly to the pevious cicuit, the goup of gates in dash-squaes can be concatenated in ode to educe the cost of the cicuit. Moeove two pais of consecutive identical Feynman gates (in dotted goups) can be emoved. Thus, the second fom left dotted goup is emoved entiely and the fist fom ight dotted goup is eplaced by a single Feynman gate. The ightmost Feynman gate can be moved befoe the last Contolled-V gate and Figue 18. Simplification of Fedkin gate based on local tansfomations: (a) the gate found by the GA, (c) esult of maco-geneation to Toffoli Base, (d) cicuit fom Fig.18b ewitten to show left-side pattens, (d) esult of maco-cell ceation, (e) final cicuit afte pemutative Swap emovals, (f) anothe esult of 2-qubit tansfomations in Toffoli Base applied to the initial cicuit. added to the dotted ectangle. Although this cicuit is longe than the fist one its cost is the same. It is composed only fom 2-qubit o 1-qubit gates. The fist cicuit has cost 55 (15*3 + 5*2) afte minimization and the second cicuit also afte minimization has cost of 45 (9*5). The two cicuits below in Figue 16 wee found using the impoved fitness function fom equation 7.6. The cicuit fom Fig. 16c was found using a complete stating set and the one fom Figue 16c using the biased set simila to the one used fo the cicuit fom Figue 16b. The majo impovement using the equation 7.6 as a fitness function is that the esults ae epoducible, because they ae diven as aleady mentioned by two opposite flows of evaluation. Obseve that by flipping ove two Feynman gates fom Figue 16c and emoving the swap gates a solution with one Fedkin and two Feynman is used, which is close to a known ealization of Toffoli gate fom Fedkin gate. The only diffeence ae two Pauli-Z otation gates, which can be emoved, as analysis shows. Inteestingly, ou softwae einvented also the famous cicuit of Smolin, since the sequence of two Pauli-X (NOT gates) can be emoved, and the sequence of two Contolled-V is equivalent to

34 Feynman gate. Afte these tansfomations, ou gate is composed fom the same basic quantum pimitives as the Smolin s solution, but in a diffeent ode. The analysis of unitay matices shows howeve that the cicuits ae equivalent. These examples show that not only can ou softwae einvent the ealization of the known gates but it can also ceate simila minimum cost ealizations of othe gates, as will be pesented below. We wee not able to find a bette solution to Toffoli and Fedkin gates fom quantum pimitives, since pehaps they do not exist. Figue 17 pesents two of many found ealizations of the Fedkin gate. Again, obseve that the consecutive Pauli-Z gates can be cancelled since this gate is its own invese (a standad local tansfomation). Next, two Feynman gates can be flipped ove and coesponding swap gates emoved, leading to the known (minimal) ealization Fedkin gate using one Toffoli and two Feynman gates. Figue 18a shows step-by-step simplification of the cicuit fom Figue 17b afte emoval of Pauli-Z gates. Applying macogeneation of gates to Toffoli Base the cicuit fom Figue 18b is ceated which is the same as one fom Figue 18c, in which gates wee diffeently gouped to satisfy the left-hand-side pattens of ules. Then maco-cell ceation leads to the schematics fom Figue 18d. Finally, pemutative tansfoms fom set S13 that emove Swap gates lead to the classical ealization of Fedkin gate fom Figue 18e. Obseve that the same solution is found when 2-qubit tansfomations ae applied to Figue 18a (leading to the cicuit fom Figue 18f) and next the maco-cell geneation is applied to the aay fom Figue 18f. The cicuit fom Figue 17a is the esult of using the equation 7.1 and a complete stating set. Fig. 17b is the esult of fitness equation 7.6 and the same stating set as in Fig. 17a. The ealization of the Magolus gate fom Figue 19a is elegant and new. By applying the swap-elated tansfomations of the Fedkin gates (Figue 19b, Figue 19c) two cicuits with two Fedkin gates each ae ceated that ae Figue 19. Magolus gate: (a) Oiginal gate obtained diectly fom GA, (b) Removal of the ight Swap gate pemuted the ode of output wies Q and R, (c) emoval of the fist Swap gate pemuted output wies P and R. not known fom the liteatue. The 3-qubit Kentopf gate family includes gates with one Shannon expansion (of the fom v u+vx, whee v,u,x {a,b,c,a,b,c }and the expession cannot be simplified) and two Davio expansions (these expansions have the fom v ux, whee v,u,x {a,b,c,a,b,c }and the expession cannot be

35 simplified). Figue 20a pesents a gate fom this family in Toffoli Base, and Figue 20b shows a gate fom the Magolus gate family (NPN class), also ealized in Toffoli Base. Based on these and othe esults, we can claim that ou pogam invented seveal ealizations of basic gates that have been not ceated yet by humans and ae pehaps patentable. Figue 20. Kentopf-like and Magolus-like gates in Toffoli Base afte optimizing tansfomations. These ae the best epoted esults fo such gates. Figue 21. Entanglement and telepotation cicuits fom the liteatue found also by ou pogam. Figue 21 shows the entanglement and telepotation cicuits found also by the GP appoach fom [5]. Finally, Figue 22 illustates the using of tansfomations to pove that Fedkin gate is its own invese. Tansfomations of this type lead often to significant eductions of long aays found intially by the GA. One of the stiking obsevations of the obtained esults is the espective pesence in Fedkin o Toffoli gates in the synthesis of each of them. As can be seen in the esults of synthesis using the complete stating set, the Fedkin gate is pesent in all Toffoli implementations and vice vesa. It can be concluded that thee is a local

36 minimum that ou algoithm was unable to oveide. Oveiding this local minimum can be done using only basic quantum pimitives such as contolled-v. Figue 22. Using simplifying tansfomations to pove that Fedkin gate is its own invese. An inteesting diffeence between ou esults of synthesis of Fedkin o Toffoli gate and those shown in Figues 5 and 6, is that ou algoithm found cicuits with geneally slightly highe cost (in the case of impoved fitness function). To undestand this point one needs to look at the evolution of the seach duing a un of a GA. Figue 23 shows the ecoding of one un seaching fo a gate using the fitness function 7.1. Fo illustation the fitness 7.6 is also dawn. The figue shows the best esult of each hunded geneations. Fist cuve Eo shows the evolution of the scaled eo as used in the fitness function. Second cuve is the cost (1/Cost) and it is to be maximized to incease the global fitness function. The lage a cicuit the moe the cost is educed. The last thee cuves show the values fo fitness fom equation 7.1 and 7.6. Fitness function 7.6 is shown by two cuves; fist with paametes α = 0.99 and β = 0.01 then with α = 0.9 and β = 0.1. Moeove fo moe claity all fitness functions ae mapped on the seconday y-axis on the left. Since thee ae many solutions to a gate at a highe cost, it is moe likely that GA finds those solutions, but hopefully unning the local optimizing tansfomations on each of these non-minimal solutions leads to the same optimal solution, as was the case with the solutions discussed above. Figue 23 illustates the poblem of fitness function used in quantum logic synthesis. While both vaiants of the fitness function ae globally stable o stuck in a local minimum, the paamete having the geatest vaiation is the cost of the cicuit. This is because, as mentioned befoe, an infinity of cicuits exist theoetically in QL fo a cetain unitay matix. This is illustated by the fact that while the eo is constant all along the un, the cost oscillates. Moeove, once the coect cicuit is found, the fitness function 7.1 is at its maximum but the fitness function 7.6 is elatively smalle because the cicuit is lage than the local minima shown in the gaph. Fo this eason while using the fitness function 7.6, once a coect cicuit was found its fitness was set up to 1 and the un was ove. The pesented cuves captue also diffeent levels of detail while seaching fo a coect cicuit. The Fitness

37 function (fitness-7.6 (0.9)) captues stongly the size of the cicuit but the final jump to the coect solution is vey lage in the value of fitness ( 0.1). Such a fitness function can be used to minimize the size constaints. Howeve, the best fitness function to captue global popeties duing a seach fo a cicuit is the fitness function (fitness-7.6 (0.99)), because on one hand it is mainly influenced by the coectness of the cicuit and on the othe hand it takes into account the size of it. Consequently, compaing it to the fitness function (fitness-7.1) one can obseve Figue 23: An example of evolution of the fitness function and its elated paametes duing a un. Fo moe pecision the two fitness function cuves ae shown on a seconday y-axis on the left. simila esult but with small oscillations in the fitness function (fitness-7.6 (0.99)). An impotant point to be discussed is the impovement of the esults obtained with the heeby poposed cost functions and fitness functions vaiants. Intuitively one could think that the cost of gates educing the global cost of the solution will only impove the esults. Howeve, as can be seen in Table 6 and Figue 23 the easoning is not so simple. The duation of uns using nomal fitness function as in equation 7.1 has geneated solutions in less than 2000 geneations. These esults wee not optimal; eithe too long and too expensive, o not coect at all. On the othe hand the paamete that foces the selection opeato to pick the individuals with smalle gate costs dives the evolutionay pocess deepe into the poblem space. As the consequence, the seach time using the impoved fitness function inceases. Also, when using the fitness function 7.6, the paametes α and β can be set to values in the inteval [0, 1]. Ultimately when using α = 1 and β = 1 - α, this leads to simila esults as using the fitness 7.1. The poblem emains the same: What ae the coect paametes fo an optimal elation between the eo evaluation and the cost of the cicuit in ode to find the optimal epesentation of the seached function?

38 Poblem seached Solution found F7.1/F7.6 Time of seach in numbe of geneations F7.1/F7.6 Toffoli YES/YES <2000 / <50000 Fedkin YES/YES <1000 / <75000 Magolus YES/NO <1000 / < Table 7: Compaison of Fitness Function vaiants Table 7 pesents a summay of some esults in QC seach. The fist column is the function seached, the second column shows if the solution was found using eithe the non-optimized fitness function (F7.1), o the optimized one (F7.6). The last column shows, fo each cicuit, the time in geneations necessay to aive at a solution. Each un was stopped afte geneations in the case no solution was found peviously in the un. The above pesented agument that the impoved fitness function inceases the equied time to find a solution can be agued hee. As can be seen in two cases out of the thee poblems, a significant time incease was obseved. The pesented esults ae statistical aveages ove five uns fo each poblem. Out of them only the best solutions ae pesented in Figues 16, 17, and 19. Two aguments suppot the use of the impoved fitness function. Fist, the solution was found in all cases and it was found even when using a biased set. This implies that the cost function allows the GA to exploe pats of the poblem space that wee not accessible to the GA which used the nomal fitness function, i.e. without the cost of gates. Also all esults fom Table 7 ae fom uns using the biased stating set of gates. The following obsevations can be made when using the GA poposed hee with vaiable length of chomosomes fo quantum logic synthesis. - No diect elation between the eo and the cost values was obseved. In geneal this means that thee is almost an infinity of solutions to a given poblem and the numbe of these solutions gows with inceasing size of the cicuit. A solution of non-minimal length can be found easie because of the vaiable length of chomosomes and next tansfomed to a shote solution. It should be futhe investigated if this a bette stategy than esticting a size to given value. When the length is shot, such as 5, the second stategy woks, but fo longe cicuits the vaiable length of chomosomes seem to be moe poweful. - When dawing the fitness of the best individual each (let s say 100 th ) geneation we obseve that thee is no tendency of optimization (convegence). This confims ou pevious conclusion that the space of the fitness function is non-monotonic with lot of flat space (local optima) and some occasional peaks of good solutions. - The best type of Mutation is Bitwise, and the best type of Replication is SUS. The bitwise mutation opeato seems to be moe appopiate fo such a high noise space as the one exploed hee. Also, this obsevation is based on an aveage measuement of fitness taken fom the best individuals fom each 100 th geneation but this appoach does not necessay finds bette solutions. - Best type of GA obseved fom all pevious esults is Baldwinian, using SUS and bitwise mutation opeato.

39 To illustate these mentioned poblems and conclusions we set up a small compaative expeiment. In this expeiment, thee diffeent paameteized GA wee used. Two types of compaative expeiments ae set up and ae used to exploe the Figue 24: Results of the compaative tests with all 32 possible vaiations of the GA. Results ae all nomalized ove 20 uns fo each. The X axis epesents each of the 32 possible combinations of the 5 concened paametes. Y axis is the fitness function as fom equation 7.6. fitness landscape. In the fist case all thee configuations ae using the same value of paametes (mutation, cossove, etc.) and five diffeent paametic settings ae modified. Fo this a binay encoding of each GA type was used. Each GA has a binay signatue such as o coesponding to a paticula configuation. Each digit in the binay signatue epesents if a cetain featue is used o not. The five positions in the signatue coespond espectively to the following options: fitness type, theshold, eplication type, mutation type and Baldwinian. Fitness type means that if it is set to zeo, the fitness will be calculated individually and if it is set to one the fitness will be calculated as a shaed fitness in a goup of 20 individuals. Theshold set to one means only individuals having fitness highe than theshold (0.6) will be used fo eplication. Replication type set to zeo means the GA will be using the Roulette Wheel type of selection and if it is set to one the stochastic univesal sampling is used. The mutation type set to zeo contols the GA to use nomal mutation (once pe individual based on the mutation pobability) while set to 1 selects the bitwise mutation (applied to each element of the chomosome). And finally Baldwinian bit indicates if the GA is using fitness calculated diectly fom the chomosome o calculated fom a minimized chomosome. The esults ae pesented on Figue 24. As can be seen the GA behaves diffeently fo diffeent paametes. In the case whee no fitness = 1 is attained the GA was not always successful in finding the solution, while in the opposite case the GA found a good solution in at least in half