One-Way Quantum Computer Simulation

Transcription

1 One-Way Quantum Computer Smulaton Eesa Nkahd, Mahboobeh Houshmand, Morteza Saheb Zaman, Mehd Sedgh Quantum Desgn Automaton Lab Department of Computer Engneerng and Informaton Technology Amrkabr Unversty of Technology Tehran, Iran {e.nkahd, houshmand, szaman, Abstract In one-way quantum computaton (WQC) model, unversal quantum computatons are performed usng measurements to desgnated qubts n a hghly entangled state. The choces of bases for these measurements as well as the structure of the entanglements specfy a quantum algorthm. As scalable and relable quantum computers have not been mplemented yet, quantum computaton smulators are the only wdely avalable tools to desgn and test quantum algorthms. However, smulatng the quantum computatons on a standard classcal computer n most cases requres exponental memory and tme. In ths paper, a general drect smulator for WQC, called OWQS, s presented. Some technques such as qubt elmnaton, pattern reorderng and mplct smulaton of actons are used to consderably reduce the tme and memory needed for the smulatons. Moreover, our smulator s adjusted to smulate the measurement patterns wth a generalzed flow wthout calculatng the measurement probabltes whch s called extended one-way quantum computaton smulator (EOWQS). Expermental results valdate the feasblty of the proposed smulators and that OWQS and EOWQS are faster as compared wth the well-known quantum crcut smulators,.e., QuIDDPro and lbquantum for smulatng WQC model. Keywords-Quantum Computng, WQC, Smulaton INTRODUCTION Quantum nformaton [] s an nterdscplnary feld whch combnes quantum physcs, mathematcs and computer scence. Quantum computers have some advantages over the classcal ones, e.g., they gve dramatc speedups for tasks such as nteger factorzaton [3] and database search [4]. They use the quantum mechancal phenomena such as superposton, measurement and entanglement []. Novel deas have been ntroduced based on the use of measurement and entanglement to perform quantum computatons, whch are referred as measurement-based quantum computaton (MBQC) [5]. The computatonal resources n measurement-based quantum computng models can be characterzed by graph states. Graph Ths paper s an extended verson of the paper presented at Euromcro DSD conference [].

2 states are specal mult-qubt quantum states that can be shown by graphs n whch each node represents a qubt and each edge represents an entanglement between pars of qubts. Mult-qubt GHZ states wth applcatons n quantum communcaton, or cluster states of arbtrary dmensons are some examples of graph states [6]. Two basc models of MBQC are teleportaton quantum computaton (TQC) and one-way quantum computaton (WQC), frst proposed by Raussendorf and Bregel [7]. WQC has drawn researchers attentons, manly because t offers dfferent and promsng physcal realzatons of quantum computatons. In WQC, the quantum correlatons n an entangled state, called cluster state or graph states, are utlzed to perform quantum computatons by sngle-qubt measurements only. The unversalty of two-dmensonal cluster states has been proved [7]. The needed computatons, specfed n measurement patterns, are drven by rreversble projectve measurements and hence, the model s called one-way. As scalable and relable quantum computers have not been mplemented yet, quantum computaton smulators are the only wdely avalable tools to desgn and test quantum algorthms. The most mportant challenge of the classcal smulaton of quantum computaton models s the exponental tme and memory complexty and the proposed smulators attempt to reduce such complextes. Snce the number of qubts n a WQC measurement pattern s consderably more than the equvalent quantum crcut, ther smulaton s more complex and the methods [8-] proposed for smulatng the quantum crcut model cannot be drectly used for practcal smulaton of WQC, only by addng the capablty of applyng measurements n non-standard bass. The am of ths paper s to overcome the crcut model smulators lmtatons n smulaton of WQC patterns and present a general tool to drectly smulate the WQC model. Two man technques, namely qubt elmnaton and pattern reorderng, are proposed to consderably reduce the state complexty as well as the tme and memory needed for the smulatons of WQC measurement patterns. After usng these technques, we can utlze prevously proposed smulaton technques for the crcut model to acheve further optmzatons for smulatng the WQC model. Moreover, a proposed technque for mplct smulaton of actons can be appled due to the lmted number of basc actons n the WQC model. These three technques are used to present an array-based WQC smulator called OWQS. In addton, OWQS s adjusted to smulate the measurement patterns wth a generalzed flow [3] wthout calculatng the measurement probabltes whch s called extended one-way quantum computaton smulator (EOWQS). One of the applcatons of OWQS and EOWQS s the ablty to answer to the recognton problem as stated n [4]. Ths problem s for recognzng what untary a gven WQC pattern mplements. The paper s organzed as follows. In the next secton, the prelmnares are presented. In Secton 3, the related work s revewed. In Secton 4, the proposed approach s explaned. Expermental results are presented n Secton 5 and fnally Secton 6 concludes the paper wth suggestons for future research. PRELIMINARIES. Qubts, quantum states and gates In classcal bnary computaton, a bt assumes two dstnct values, and. Bts consttute the buldng blocks of the classcal nformaton theory. In an analogous manner, quantum bts or qubts are the fundamental unts of nformaton n quantum computng. A qubt s a unt vector n a two-dmensonal Hlbert space, H whose bass vectors are denoted as:

3 and. Unlke classcal bts, qubts can be n a superposton of and represented by where and are complex numbers such that. If a measurement n the standard bass{, }, s appled to the state, the outcome wll be () wth the probablty ( ), and the state mmedately after the measurement s ( ). The state of an n-qubt quantum system (quantum regster) s represented by a column vector n a n -dmensonal Hlbert space, H n as follows: n n n where represents tensor product operaton and.... n Entanglement s a unque quantum mechancal resource that plays a key role n many of the most nterestng applcatons of quantum computaton and quantum nformaton. A mult-qubt quantum state s sad to be entangled f t cannot be wrtten as the tensor product entangled quantum state. of two pure states. For example, the EPR par s an There are a number of models for the evoluton of quantum computaton. The man model to explore quantum computaton s the crcut model, based on untary evoluton of qubts by networks of gates. Every quantum gate s a lnear transformaton represented by a untary matrx. A matrx U s untary f UU I, where U s the conjugate transpose of the matrx U. Snce any untary operaton has an nverse, any quantum gate s reversble whch means that gven the state of a set of output qubts, t s possble to determne the state of ts correspondng set of nput qubts. Some useful sngle-qubt gates comprse Paul set whch are shown n the followng: I, X, Y, Z. Two other mportant sngle-qubt untares are Hadamard, H and Phase gates, P: H, P. If U s a gate that operates on a sngle qubt, then the controlled-u gate s a gate that operates on two qubts,.e., control and target qubts, and U s appled to the target qubt f the control qubt s and otherwse, leaves t unchanged. For example, controlled-z (CZ) and controlled-not (CNOT) gates perform the Z and X operators respectvely on the target qubt f the control qubt s and no acton s taken otherwse. The matrx representatons of the CZ and CNOT gates are []: 3

4 CZ, CNOT=. WQC model The necessary computatons n WQC are organzed as patterns. A WQC pattern [4, 5] s defned as P V, I, O, A where V s the set of qubts, I V the set of nput qubts, O V, the set of output qubts and A a fnte set of actons actng on V. The pattern s wrtten as a sequence of four dfferent types of actons. When performng a computaton, the actons are appled from rght to left. These actons are explaned n the followng. The preparaton acton N v prepares a qubt v nto state whch s appled to all of the non-nput qubts. Entanglement acton E uv entangles qubts u and v by applyng a CZ gate. The entanglement commands between the qubts can be represented by the edges n a graph, called the entanglement graph of the pattern. Sngle-qubt measurement acton M v measures the qubt v n the orthonormal bass of: where, e () s called the angle of measurement. The outcome of a measurement on a qubt v s denoted by s v ϵ. If the state of the qubt after the measurement s, then s, and f t s, then s. The measurement outcomes can be summed together modulo, whch are called sgnals. A measurement can depend on other ones through two sgnals s and t as follows: v t s s M M t () To calculate the sgnals, all of the measurement results that appear n the sgnals t and s need to be known. Ths means that all those measurements must be performed before the dependent measurement. s s Sngle-qubt Paul correcton actons X v and Z v apply the Paul X and Z gates to the qubt v, respectvely, f s= and do nothng f s=. The set of actons A must conform to the followng rules: (D) no acton depends on an outcome not yet measured. (D) no acton acts on a qubt already measured. (D) no acton acts on a qubt not yet prepared unless t s an nput qubt. (D3) a qubt v s measured f and only f v s not an output. v 4

5 A pattern s n the standard form [4, 5] f all of the entanglement operatons are at the begnnng of the pattern, followed by all of the measurement operatons and the correcton operatons are at the end of the pattern. The standard form s denoted by CME where C, M and E stand for correcton, measurement and entanglement operators, respectvely. The quantum computaton depth of a pattern or just quantum depth s the depth of the executon of the pattern that s due to the dependences of measurement and correcton commands on prevous measurement results. For example, the quantum depth of the standard s 3 pattern s 4 s s s s P {,,,5,, 5, X Z M M M M E } s 4 due to the dependences of the qubts Projectve measurement A set of mutually orthogonal projecton operators{ P, P,..., P } wth the followng propertes consttute projectve m measurements [6]: P P P P m P I When ths measurement s carred out n a system wth state, then the result s obtaned wth the probablty: prob P (3) and the state collapses to: P prob (4).4 Generalzed measurement The operators{ M, M,..., M } on a Hlbert space H are called generalzed measurement operators [6] f they satsfy: m m M M I (5) There are no condtons on M other than ths. When a generalzed measurement wth a set of measurement operators { M, M,..., M } s carred out n a system wth state, then the result s obtaned wth the probablty: and the state collapses to: m prob M M (6) M prob (7) 5

6 3 RELATED WORK In ths secton, a bref revew of quantum crcut model smulators as well as WQC smulators s presented. Vamontes et al. [9] defne a new graph-based data structure for smulatng quantum crcuts called quantum nformaton decson dagram (QuIDD). Ths data structure s used for the development of a quantum crcut smulator usng BDD operatons n the QuIDDPro software whch uses the BDD software package CUDD. The man motvaton of QuIDD s that vectors and matrces whch arse n quantum computng, exhbt repeated structures. QuIDD utlzes these smlartes n the vectors and matrces n order to reduce the memory and the run-tme needed for the smulatons. In [], a graph-based quantum crcut smulator based on quantum multple-valued decson dagram (QMDD) structure s developed. The QMDD structure explots the regular structure of the matrces whch represent quantum crcuts and gates. In the ffth verson of Feynman Program smulator [7], some features such as sngle-qubt measurements n the nonstandard bass, projectve and the generalzed measurements are added whch have been clamed to be useful for the MBQC smulaton. A class of quantum crcuts wth a restrcted gate lbrary can be effcently smulated n polynomal tme on classcal computers. The results of Gottesman-Knll Theorem [] and ts recent mprovement by Aaronson and Gottesman [8] apply only to crcuts wth stablzer gates CNOT, Hadamard, Phase, Paul gates and measurement of observables n the Paul group and stablzer nput states. Aaronson and Gottesman s algorthm to smulate crcuts wth these gates s called CHP, whose tme and space requrements scale only quadratcally wth the number of qubts [8]. In [9], CHP s mproved n such a way that t requres tme and space of O( nlog n) where n s the number of qubts by usng graph states to represent the system state, whch s called GraphSm. P-blocked smulaton s another technque whch has been proposed n the quantum crcut smulaton doman [3]. The basc dea of ths approach s decomposng the states nto smaller dstnct sub-states, whenever possble. The typcal way to pursue such separable states s employng entanglement metrcs and contrvng state representatons whose sze depend on these metrcs. Such representaton s called p-blocked f no subset of p+ qubts are entangled. In other words, the set of all qubts s parttoned nto k blocks B, B,,B k, where each block contans at most p entangled qubts. Therefore, the state complexty grows wth the number of entangled qubts nstead of the total number of qubts. Lbquantum [] s a C lbrary for the smulaton of quantum mechancs, wth a specal focus on quantum computng. It provdes mplementaton of quantum regsters, basc operatons for regster manpulaton such as the Hadamard gate or the CNOT gate, measurement n the standard bass, etc. In [], a tool was presented to smulate the crcut and one-way quantum computaton models n a parallel envronment provded by PC workstatons connected n a standard Ethernet network. The heart of the algorthm s the vector state transformaton by a untary matrx U. The most mportant challenge n ths approach s how to dvde the tasks nvolved n the smulaton among dfferent nodes whle an effcent mplementaton of measurements and other actons have not been addressed. In [], a WQC emulator, based on formal measurement calculus [4, 5] was presented. The approach focuses on WQC and attempts to perform a fathful emulaton of physcal systems. It ams to carry out every step descrbed by the theores of quantum mechancs and computatons. The man drawback of ths approach s that measurement and other actons have been mplemented as straghtforward matrx-vector multplcaton. 6

7 In [], t s shown that WQC patterns on non-unversal one-dmensonal cluster states can be smulated on classc computers effcently,.e., n polynomal tme n the number of qubts. However, as these WQC patterns are not unversal, a general smulator proposed for smulatng one-way quantum computers cannot be of polynomal complexty. A WQC smulaton algorthm by contractng tensor networks was proposed by Markov and Sh [3] and t was shown that WQC can be effcently smulated f ts underlyng graph has a small tree-wdth. 4 PROPOSED APPROACH In ths secton, practcal algorthms for smulaton of WQC patterns on conventonal computers usng state vector representaton are descrbed. The proposed technques can decrease the state space as well as tme and memory complextes of smulaton. The extent of reducton n the complextes depends on the structure of the underlyng graph of entanglements and the measurement dependences. The structure of OWQS smulator s shown n Fgure. Frst, t takes a pattern n the standard form as well as the state of the nput qubts. Then, the pattern actons are reordered by an algorthm called, PROA. Subsequently, the actons are appled n the new order by the smulator core. Fnally, the state of the output qubts s reported. 4. State representaton State vector representaton s used to represent the system state. Intally, all of the qubt states are separate (except the nput qubts whch can be entangled), therefore the separate states (called sub-states) can be saved n dstnct vectors whle the system s total state s composed of these sub-states. Two dfferent sub-states are combned only when the entanglement operaton s appled to the correspondng qubts n these sub-states. Thus, n an n-qubt system, the memory complexty of m storng the state space s O, where m s the number of qubts n the largest sub-state. The sub-state of an m-qubt system s saved n an array of sze m of complex numbers. For example, the representaton of a three-qubt state s shown n Eq. 8.,,,,-, --,, - T (8) Fgure.Structure of OWQS smulator where E, M and C stand for entanglement, measurement and correcton, respectvely. Pattern ReOrderng Algorthm 7

8 4. Qubt elmnaton In WQC, sngle-qubt projectve measurements are used and the measurement matrces{ P, P} are defned as: P e e (9) e P e () However, n ths paper a new generalzed measurement bass s appled n order to mplement qubt elmnaton technque easer as explaned n the followng. Suppose that we are to perform a measurement on qubt v n an n-qubt state. can be dvded nto two parts based on or parts of the qubt v whch are defned by and, respectvely as: n n b b b b b b, n, n v,, n, n v, where and are complex numbers and b b... b, n, n, s the bnary expanson of number. After each measurement, the state of the measured qubt v s not mportant for further steps of smulaton and can be removed from the state space. Ths s because n a WQC pattern each non-output qubt can be measured only once and after that no actons are allowed to apply on ths qubt and fnally only the state of output qubts determnes fnal state of system. Therefore, the dmensonalty of the n- qubt state reduces from n to n-. In order to elmnate qubts after the measurement, we transform the known mentoned measurement bass M ( ) to the standard bass as explaned n the followng. Every sngle-qubt measurement can be assocated wth a unt vector on the Bloch sphere whch corresponds to ts + egenstate and can be parameterzed by the co-lattude θ and longtude φ of ths vector, wrtten as a par of angles (θ, φ) (Fgure ) [4]. The measurement bass n WQC, that s corresponds to the angles,. In order to transform ths measurement bass to the standard bass, t s necessary to make the measurement vector concde wth the north pole of the Bloch sphere. Therefore, frst, the measurement vector s rotated around the Z-axs by an angle by applyng R z e to the qubt. Ths acton makes the vector concde wth. Then by applyng a Hadamard gate to the qubt, the measurement vector wll concde wth the north pole of the Bloch sphere and then we can measure ths qubt n the standard bass. Fgure 3 llustrates these steps. 8

9 Fgure. Sngle-qubt measurement can be represented by the par of angles (θ, φ) on the Bloch sphere [4]. Therefore, the new generalzed measurement bass s ntroduced as follows: e M. H. m e j, j{,} () M H m e e '.. j, j {,} () Applyng the new generalzed measurement on a qubt can be done n three steps:. The measurement operators M and M wth respect to α are determned by usng Eq.s and.. The probablty of the measurement result s calculated by usng Eq The state after the measurement s computed from Eq. 7. It can be readly verfed that M M P ; {,} : e e M M. P e e and therefore: e M M. P e e e prob M M P (3) z z z α y y y x a) Vector assocated wth M() x b) Vector after applyng RZ(-) x c) Vector of part (b) after applyng H Fgure 3. Steps to transform measurement bass to the standard bass 9

10 By usng ths new bass, the measured qubt collapses to the state or nstead of other qubts reman at the same states as for the measurement n the or, respectvely and the bass. As a result, to elmnate measured qubt from the state space, smply half of the state wth zero ampltudes s removed. Furthermore, whle the tradtonal measurement operator has no zero element, the new proposed measurement matrx has only two non-zero elements, and ths feature s utlzed n mplct smulaton of acton n order to reduce the needed computatons. Nevertheless, one can elmnate the measured qubt wthout usng ths new measurement operator but wth more computatonal effort. 4.3 Pattern reorderng Consder a pattern wth sze n n the standard form. All of the entanglement operatons are performed at the begnnng of the pattern. Ths leads to a state wth sze n. However, we propose an approach to keep the sze of the states as small as possble by reorderng the measurement actons and qubt elmnaton after each measurement to manage the state space as well as tme and memory complextes. Proposton: Before measurng a qubt v (wth resolved dependences) t s only needed to perform the entanglement operatons to ts adjacent qubts. Proof. Wthout loss of generalty, assume that the nput pattern s n the standard form. The followng relatons hold: a) The measurements on dsjont qubts can commute: v u u v M M M M ; u v b) The entanglements on qubts (,j) and (u,v) commute: (4) E E E E (5) j uv uv j c) The entanglement and the measurement operatons on dsjont qubts commute: M E ;, k j E jm j k (6) k where k s the set of qubts acted by M whch does not contan and j. Consder a subset of CME operatons C M M M E E E o k v t s vp q and a target qubt v where the symbols as x represent the sets of qubts acted by M and E whch do not nclude v. It should be mentoned that the measurements on the qubts on whch v depends are assumed to be already performed. If we show that the measurement on v can be carred out only after performng the E actons on v, the proposton s proved. Usng Eq. 5, all of the v entanglements can be moved to the rght of the pattern. In ths case, the pattern s as follows: C M M M E E E o k v t s q vp Now by usng Eq. 4, the pattern can be wrtten as follows: Fnally usng Eq. 6, we wll have: C M M M E E E o k t v s q vp C M M E E M E (7) o k t s q v vp

11 Therefore, usng the above proposton, each qubt can be chosen n an approprate order for measurement. Then, for the selected qubt, the entanglement operatons wth ts adjacent qubts have to be performed, and fnally the qubt can be measured. It s worth mentonng that n a pattern n the standard form, all of the correcton commands are on the left of the pattern and reman unchanged. Ths s the bass of the pattern reorderng algorthm. Snce the qubts can be elmnated after each measurement, the complexty of the state space s kept small. Fndng the best orderng of the measurements can be expressed as the followng: Fgure 4 shows a graph G that represents the entanglement operatons for the pattern of SWAP gate. Each vertex shows a qubt and the edges correspond to the entanglement operatons. q6 and q8 are the output qubts whch are not measured and the black vertces represent the qubts whch have already been measured. The qubts whch are ready to be measured are shown by whte vertces. Fgure. 4. SWAP entanglement graph. Each vertex represents a qubt. The black vertces and the whte non-boxed vertces represent the measured qubts and the qubts ready to be measured, respectvely. The output qubts whch are not measured, are represented by the boxed vertces. In Table I, an optmzed order of qubts to be measured for the graph n Fgure 4 s shown as an example. At stage, each qubt s n a separate set. Then q s chosen for measurement, and therefore, the entanglement E(q,q3) must be performed. To perform the entanglement, the sub-states contanng the two qubts are tensored n a two-qubt state. After that, q s measured and elmnated. The sub-states n the last column of the table represent the subsets after performng the measurements. At stage, q3 s chosen for measurement. Then, E(q3,q4) and E(q3,q) must be performed. Therefore, the sub-states contanng q3, q and q4 are tensored. q3 s measured and s then elmnated. At stage 3, q s chosen and then E(q,q5) s performed. q s measured and then s deleted subsequently. The other qubts are chosen n a smlar way, and fnally, sub-state {q6, q8} shows the output state. Ths example ndcates how the maxmum sze of state can be reduced from 8 to 3. In the general case, dependent measurement cannot be performed as long as ther dependences are not resolved. Therefore, we defne two measurement lsts: ready lst and dependent lst. Ready lst conssts of the qubts wth resolved dependences and the qubts wth dependent measurements are mantaned n the dependent lst. When all of the dependences of a qubt have been removed, t s moved nto the ready lst. To choose the best order of measurements, a heurstc approach s used. The frst crteron s the sze of the produced subset for measurng a qubt v (MS(v)) Hgher prorty s assgned to the smaller measurement state-spaces. The second crteron to be consdered for each qubt v s the number of connectons that t has to the output qubts (OS(v)). Less number of connectons leads to a hgher prorty snce the output qubts are never measured and wll reman n the sub-state untl the

12 end of the smulaton. Let S(v) be the set of qubts whch are dependent on the measurement result of qubt v. The sze of S(v) can be consdered as the thrd crteron (SS(v)). Hgher prorty s assgned to the qubts whose sze of S s larger. Although selectng qubts n separate sub-states leads to some sub-states wth sub-optmal sze, when these sub-states need to be merged, a large sub-state wll be produced. In order to prevent ths, qubts whch are connected to prevously created substates are selected wth a hgher prorty. Ths s the fourth crteron that s controlled by flag n Eq. 8 whch assgns a cost to each unmeasured qubt. * MS( v) * OS( v) * SS( v) flag Cost( v) = *(( MS( v) )* ) * OS( v) * SS( v) flag where α, β, γ and δ are postve real numbers that control the mportance of the frst, second, thrd and fourth crtera, respectvely and flag s set to f qubt v s connected to any prevously selected qubts and set to otherwse. In each step, the Measurement state-space of all qubts s generated, the qubt wth the smallest cost s chosen, and then the sub-states are updated. PROA conssts of two nested loops. The number of teratons n the outer loop s equal to the number of qubts whch are measured,.e., K V O. In each teraton, one qubt s chosen to be measured. In the nner loop, for each canddate qubt v, the measurement state-space, whch wll be created n the case of choosng ths qubt, s produced wth a complexty proportonal to the number of the pattern s qubts and the adjacent qubts to the qubt v. The number of the adjacent qubts to each qubt s E V on average, where E s the number of edges n the entanglement graph. Therefore, the tme complexty to create measurement state-space for each canddate qubt s E V. V E on average. In teraton, the ready qubts to be measured s equal to K- n the worst case. Therefore, the total number of teratons s equal to: and the complexty of PROA algorthm s of K K K K K ( K( K )) O ( * E ) O ( K ( K ) E ) O( EK ) (9) PROA ams to order actons n such a way that the problem state space s kept as small as possble. Whenever the quantum depth of the patterns s greater, PROA s more lmted because of more measurement dependences. Moreover, the entanglements of a selected qubt to ts adjacent ones should be appled frst and so a hgher graph degree leads to larger substates. The output qubts are not measured and hence are not removed. Therefore, the number of adjacent qubts to each qubt and whether they are output qubts or not, affect PROA. The extent of the state complexty reducton cannot be exactly formulated but t s ntatvely affected by the degree of the entanglement graph, the depth of the pattern and the number of output qubts. The smaller degree of entanglement graph, the less number of output qubts and the smaller depth of the pattern helps PROA reduce the state complexty more. Moreover, the values of α, β, γ and δ need to be approprately set to utlze these characterstcs, as explaned n Secton 5. In an n-qubt system, the state space complexty of the problem s O( m ), where m s the number of qubts n the largest sub-state. In the worst case, m s equal to n and n the best case for a pattern wth a connected entanglement graph, m s equal (8)

13 to O, where O s the number of output qubts. The extent of reducton for each measurement pattern s shown n Table III, Secton Implct smulaton of actons In WQC, actons are lmted to CZ, X and Z gates as well as sngle-qubt measurements. CZ, X and Z gates can be smulated wth respect to ther behavor, mplctly. Moreover, sngle-qubt measurement of a qubt n an n-qubt system leads to a measurement matrx wth some regularty whch s descrbed n the followng. Ths allows us to measure the specfed qubt wth no need to construct the measurement matrx and explctly perform the matrx-vector multplcaton. Table I An example of an optmzed order of measurements Stage No. Measurement Measurement orders state-space Sub-states - - {},{},{3},{4},{5},{6},{7},{8} {,3} {},{3},{4},{5},{6},{7},{8} 3 {3,,4} {,4},{5},{6},{7},{8} 3 {,4,5} {4,5},{6},{7},{8} 4 4 {4,5,7} {5,7},{6},{8} 5 5 {5,6,7} {6,7},{8} 6 7 {7,6,8} {6,8} 4.4. Implct smulaton of entanglement, X and Z actons In ths secton, the smulaton of CZ, X and Z gates are explaned. The basc dea s to smulate these gates wth respect to ther behavor nstead of usng the conventonal matrx-vector multplcaton, as explaned n the followng. If the qubts u and v on whch the CZ gate s to be appled, are n dfferent sub-states, frst the tensor product of the two correspondng sub-states are computed and then the CZ gate s appled. Let be an n-qubt state that conssts of the qubts u and v as follows: n n n n b b b b b b b b b b b b, n, n u v,, n, n u v,, n, n u v,, n, n u v, where,, and are complex numbers and, n, n..., b b b s the bnary expanson of. In order to apply CZ, t s suffcent to change nto - where u v. Therefore, CZ can be done n n- steps. The pseudo code of ths mplementaton s shown n Fgure 5, whch takes then-qubt state state as well as u and v as the parameters. (In the code, the sze of state s denoted by sze (state).) As a result, one CZ gate s appled to the u th and v th qubts where v u n and λindex can be computed by only shft operatons. 3

14 ENTANGLEMENT (state, u, v) for to sze( state ) 4 do j mod u λindex [j/ v- ]* v + v- + (j mod v- ) + [/ u- ] * u + u- state[λindex] state[λindex] * (-) end for Fgure 5. Pseudo code of entanglement acton Let be an n-qubt state that conssts of the qubt v as follows: n n b b b b b b, n, n v,, n, n v, In order to apply X and Z gates on the qubt v, t s suffcent to exchange wth for X and change nto whch can be appled both n n- steps. for Z 4.4. Implct smulaton of measurement Suppose that the measurement acton M(α) s to be appled on the qubt vn an n-qubt state: n n b b b b b b, n, n v,, n, n v, Frst, the measurement operator M wth respect to α s determned. In order to compute the state after the measurement, M s calculated as follows: M I... I M I... I () n-v v Therefore, the straghtforward computaton of the state after the measurement usng matrx-vector multplcaton s of n n O *. In the followng, the proposed approach to perform ths s explaned. The measurement operator M has the followng property: Each column and row of M ncludes only the values of M n the correspondng column and row as well as some other zero elements wth certan regulartes. In other words, there are only two non-zero elements n each column and row of M. For example: m m m m m m M I M m m m m m m 4

15 M I M I m m m m m m m m m m m m m m m m Therefore, we only need to save the elements of M (only four complex numbers) nstead of mantanng the numbers. Moreover, the mplct and n-place matrx-vector multplcaton M can be appled. Implct matrx-vector multplcaton n complex M refers to applyng M wthout constructng matrx M and passng up the zero elements durng the multplcaton. In-place matrx-vector multplcaton refers to savng the results of the multplcaton n the orgnal vector wthout usng any extra vectors. Let M be a matrx and V a n -element column vector. Normally, applyng M*V needs to save the results n the new n -element column vector V. However n-place matrx-vector multplcaton can be used snce there s only one non-zero element n each column of M. Therefore, each element of V s used one tme. Thus, we can use ths element and then replace t wth new one wthout savng the results n a new vector. These technques are explaned n the followng example. be a three-qubt state and Let 3 3 n n M v be a measurement acton on the second qubt (v=). Assumng that the measurement outcome s zero, the measured state collapses to: e e M prob prob e prob e 3 3 e e e 3 3e 3 5

16 and after the elmnaton of the measured qubt, the fnal state wll be: 3 () As a result, we need to move through half of the rows whch only consst of m and m and then mplctly multply these rows by usng only two multplcatons. Each element of s used only once for computng, and so we can replace the old elements by the new ones (n-place matrx-vector multplcaton). Computng the probablty of the measurement s done mplctly too, as shown n Fgure 6. Therefore, the measurement s performed by consumng constant sze of extra memory and the tme complexty of the algorthm for complex-number multplcatons s O n. The pseudo code of ths measurement operaton s shown n Fgure 6 whch takes a measurement angle α, an n-qubt state and qubt number v as the nput parameters. In ths pseudo code, pj are the elements of the former projectve measurement matrx p P p p p and mj and m' j are the elements of the new measurement matrces M and M respectvely, and rand(,) returns a random number between zero and one. 4.5 Extended OWQS In ths secton, an accelerated extenson of OWQS for a specfc class of patterns s proposed. A pattern s correct,.e., t mplements a determnstc untary f t s strongly determnstc. In patterns whch have such ths property, each branch occurs wth the same probablty prob =.5 and all of the branches mplement an dentcal untary (up to a global phase). Therefore, there s no need to compute the probabltes of the measurements. Browne et al. [3] present a necessary and suffcent condton for strong unform determnsm based on the geometry of the entanglement graph called generalzed flow (gflow). Moreover, n [5] an algorthm s presented for fndng a gflow n the patterns n polynomal tme. If a pattern has a gflow, the probablty of measurement results s.5; otherwse, the pattern s reported as an ncorrect one. Furthermore, one may preselect the measurement results arbtrarly or smulate only the postve branch of the computatons, n whch all of the measurement outcomes are pre-selected to be zero. Therefore, the dependences n the pattern are elmnated durng the smulaton and consequently, applyng PROA algorthm potentally can lead to better results. Ths s because the results of all of the measurements are determned and therefore there s no dependent lst and all of the qubts are ready for measurement from the begnnng of smulaton. Ths approach, whch s called EOWQS, leads to a sgnfcant mprovement n memory and run-tme. The gflow algorthm [5] s ncorporated nto EOWQS and s appled before smulatng each pattern as a preprocess operaton. 6

17 MEASUREMENT(state, α, v) Create M and P matrces wth respect to α //computng the measurement probablty of zero for to sze(state) do f (/ v- mod ) = end for else v then temp p state [ ] p state [ ] then * [ v temp p state ] p * state [ ] prob prob conj ( state [ ]* temp) //computng the state after measurement f (rand (,) prob) //f the measurement result s zero. then a b m prob m prob else b a m //f the measurement result s one. prob m prob for to sze( state ) do j v v v mod ( ) [ ] state a state j b state j v [ ] [ ] [ ] end for Fgure 6. Pseudo code for smulaton of the measurement acton 4.6 Effcent smulaton For a cluster state of sze N*M whch s shown n Fgure 7, our approach leads to m=n+, where N qubts on the rght column of the cluster state are output qubts. To ths end, the qubts for measurement are chosen column by column from left to rght. In each column, qubts may be selected ether from top to bottom or vce versa. Therefore, for a cluster state wth N O (log ), the proposed approach s of polynomal complexty where n s the number of qubts n the cluster state. In n comparson to [], the state space complexty of smulaton of a cluster state *M (D cluster state) s O(). 7

18 Fgure 7. An N*M cluster state. N qubts n the boxes on the rght most column are outputs. 4.7 A smple example In ths secton, the smulaton of CNOT pattern wth nput state qq s elaborated. The CNOT pattern s as follows [4]: CNOT,,3,4,,,,4, X Z Z M M E E E () OWQS takes ths pattern as nput and then the pattern s reordered by PROA algorthm as follows: CNOT,,3,4,,,,4, X Z Z M E E M E (3) The new reordered pattern s smulated by the Smulator Core of OWQS from rght to left. The steps of applyng the actons are llustrated n Fgure 8. The ntal state s shown n part (a). Note that total system state conssts of four dstnct sub-states. Then E 3 s appled. As a result, sub-states contanng q and q3 are tensored (part (b)). q s measured n the next step and then elmnated (part (c)). The other actons are performed n a smlar way. Fnally, the sub-state n part (f) shows the output state as q4q. 5 EXPERIMENTAL RESULTS OWQS was mplemented n C++ and was run on a workstaton wth 4GB RAM and Core Duo.6 GHz CPU. Some known quantum crcuts were decomposed nto CZ and J ( ) gates. Then we appled the approach presented n [6, 7] n order to extract the correspondng pattern after performng the proposed optmzatons whch nclude standardzaton, sgnal shftng and Paul smplfcatons. In Table II, the run-tme results obtaned from our smulaton algorthms are shown and compared wth those produced by Emulator [], QuIDDPro [9], and lbquantum []. The numbers n the parentheses n the frst column ndcate the number of qubts used n the quantum crcut. Each pattern s smulated by all of the smulators wth the same nput states and the outputs are verfed. We used a large sample of vald random nput states n order to have a far comparson. The table shows that the space complexty of the proposed approaches s consderably reduced by utlzng the qubt elmnaton and pattern 8

19 reorderng technques. The speed-ups of OWQS and EOWQS compared wth the QuIDDPro are shown n Fgure 9 for the subset of benchmarks that the QuIDDPro can successfully smulates. In [], the space complexty of smulatng an n-qubt pattern s O( n ), due to the n -element projector P, and the patterns wth at most 3 qubts can be smulated n ths way. As all of the qubts n the standard WQC patterns are entangled at the begnnng of smulaton, n our benchmarks the lbquantum fals to smulate patterns wth more than 3 qubts. The complexty of the proposed approach s O( m ), where m s the number of qubts n the largest sub-state durng the smulaton. sub-state q T sub-state q T sub-state 3 q T 3 sub-state 4 q T 4 a) Intal state wth state as nput. sub-state q T sub-state qq T 3 sub-state 3 q T 4 b) System state after applyng E 3 sub-state q T sub-state q T 3 sub-state 3 q T c) System state after applyng 4 result. M wth a zero measurement sub-state q q q T 4 3 d) Applyng E 3 E 34 sub-state qq T e) System state after applyng 4 result. M 3 wth a one measurement Fgure 8. Steps of smulaton of CNOT gate sub-state qq T 4 f) Fnal state after applyng 3 X Z Z. 4 4 Table II Smulaton run-tme of WQC patterns by OWQS and EOWQS compared wth Emulator [], lbquantum [] and QuIDDPro [9] #qubts n Gate Emulator(sec) Lbquantum(sec) QuIDDPro(sec) OWQS(sec) EOWQS(sec) pattern (n) CNOT () Toffol (3) 7 N/A Not-applcable: the smulator exts wth an error mostly caused by memory lmtaton 9

20 Fredkn (3) 45 N/A N/A GHZGate () 39 N/A N/A GHZGate (3) 45 N/A N/A GHZGate (5) 49 N/A N/A QFT (3) 3 N/A QFT (5) 9 N/A N/A QFT (8) 4 N/A N/A > hour QFT (9) 36 N/A N/A N/A QFT () 38 N/A N/A N/A _49 (4) [8] 88 N/A N/A nth_prme4_nc (4) [8] N/A N/A hwb4 (4) [8] 66 N/A N/A ham7 (7) [8] 9 N/A N/A Shor code (9) 3 N/A N/A Toffol_Starcase (3) [9] 97 N/A N/A Toffol_Starcase (7) [9] 9 N/A N/A Toffol_Starcase () [9] 6 N/A N/A rc_adder4 (6) [3] 5 N/A N/A rc_adder5 () 88 N/A N/A rd73 () [8] 8 N/A N/A rd84 (5) [8] 33 N/A N/A N/A sym () [8] 6 N/A N/A N/A gf_4 () [8] 4 N/A N/A N/A N/A 3.66 As stated n Secton 4.3 the result of the PROA algorthm s hghly affected by the values of α, β and γ n Eq. 8. The results reported n ths secton are obtaned as descrbed n the followng emprcal approach. Intally α, β, γ and δ are set to O, O,.5 and O respectvely. Then n O teratons, α s decreased by one and fnally the value that produces the best result n all of the teratons s used. It should be noted that PROA s a polynomal-tme algorthm and ts run-tme s neglgble as compared wth the man smulatons run-tme. Furthermore RPOA s appled for each pattern only once and can be consdered as a preprocess operaton. As shown n Table III, ths approach leads to the mnmum possble state-space n many cases,.e., m O (the numbers ndcated by*).

21 Table III Smulaton state-space of OWQS and EOWQS (O( m )). * shows PROA leads to the best possble results of m,.e., O Gate N m(owqs) m (EOWQS) CNOT () 4 3* 3* Toffol (3) 7 4* 4* Fredkn (3) 45 4* 4* GHZGate (5) 49 6* 6* QFT (3) 3 4* 4* QFT (5) 9 6* 6* QFT (8) 4 9* 9* QFT (9) 36 * * QFT () 38 * * 4_49 (4) [8] nth_prme4_nc (4) [8] 5* 5* hwb4 (4) [8] 66 5* 5* ham7 (7) [8] Shor code (9) 3 4 Toffol_Starcase (3) [9] Toffol_Starcase (7) [9] Toffol_Starcase () [9] rc_adder4 (6) [3] 5 7* rc_adder5 () 88 5 * rd73 () [8] 8 rd84 (5) [8] 33 6sym () [8] 6 6 gf_4 () [8] Due to the elmnaton of the qubt dependences, EOWQS usually leads to a smaller state space than OWQS. Moreover, due to the lack of need for calculatng probabltes, the smulaton run-tme and space complexty of EOWQS s reduced.

22 Speed-up OWQS EOWQS Benchmarks Fgure 9. Speed-ups of OWQS and EOWQS compared wth QuIDDPro 5. Results analyses and comparsons wth prevous quantum crcut model smulators As Table II and Fgure 9 ndcate, the methods proposed for smulatng the quantum crcut model cannot be drectly used for smulaton of WQC n practce, as the number of qubts n a WQC pattern s consderably more than the equvalent quantum crcut. Ths paper overcomes ths problem, manly by proposng pattern reorderng and qubt elmnaton technques. The proposed smulators do not use the prevous quantum crcut model deas. However, after usng pattern reorderng and qubt elmnaton technques, other prevously proposed crcut model smulaton technques can be exploted for obtanng further mprovements. The p-blocked smulaton technque decomposes states nto smaller states durng the smulaton. For example, assume that two qubts are entangled and hence they compose a sngle two-qubt entangled state. Applyng some gates to them may change the state of the qubts nto non-entangled ones. Therefore, they can be decomposed nto two sngle-qubt nonentangled states. However, our smulators smply mantan qubts n the separated states at the begnnng of smulaton and do not decompose them durng the smulaton. The p-blocked technque can be used along wth our proposed smulators for obtanng further mprovements by decomposng each sub-state nto the smaller ones whenever possble durng the smulaton. The extent of the mprovement nduced by the p-blocked smulaton hghly depends on the state of the nput qubts and the appled gates (or the graph structure of the WQC pattern) where these determne the number of entangled qubts durng the smulaton. QuIDDPro and QMDD are powerful graph-based crcut model smulators. The dea of usng graph structure n ths scope was born out of the smlarty and repeated structure of matrces and vectors n quantum computng. Dependng on the extent of regularty and smlarty n the state vectors and matrces (gates), the state complexty vares from O() to O( n ) whle the worst case does not usually happen. However, our smulator smply uses an array structure to store states and matrces. Addng the graph structure to our smulators can also mprove the results. Snce the graph-based smulators utlze smlarty and repeated structure of matrces and vectors, ther effectveness hghly depends on the state of nput qubts and appled gates (or graph structure of WQC pattern). The nput states whch lead to more smlarty n the system state are preferred by

23 graph-based smulators. The effectveness of lbquantum also depends on the state of nput qubts because t benefts from the sparse structure for storng state vectors. The GraphSm and the CHP smulators have both been proposed for a specal sub-class of quantum crcuts,.e., stablzer crcuts whose smulaton can be done n polynomal tme accordng to Gottesman-Knll theorem. These smulators can be appled to WQC patterns only when the measurements are done n X, Y and Z bases and the nput states are stablzer states. Otherwse, these smulators are unable to smulate the patterns. However, our proposed smulators are general n the sense that they accept any knd of patterns wth any measurement angles and arbtrary nput states. As Table IV shows, the GraphSm smulator can be appled on a lmted sub-set of WQC patterns of Table II and as expected, leads to smaller runtmes for these patterns as compared wth the proposed smulators. It s suggested to have GraphSm and the proposed smulators n a comprehensve smulaton tool and smulate the stablzer crcuts usng GraphSm. Our smulators do not use any of the above factors or any other factors makng the performance dependent on the nput state and hence they do not depend on the state of nput qubts. The smulaton results also confrm ths statement. In order to verfy the dependences of the prevously proposed quantum crcut and the proposed smulators on the nput data, Table V compares the run-tme of these approaches for dfferent types of nput data,.e., when the nput qubts are n the bass ({, }) or states. As ths table ndcates, the run-tme of OWQS and EOWQS do not depend on the state of nput qubts, whle the run-tmes of QuIDDPro and lbquantum do. Table IV Smulaton run-tme of OWQS and EOWQS compared wth the GraphSm smulator No. of qubts n Gate GraphSm(sec) OWQS (sec) EOWQS (sec) pattern (n) CNOT SWAP GHZGate () GHZGate (3) GHZGate (5) Table V Smulaton run-tmes of WQC patterns by OWQS and EOWQS compared wth lbquantum and QuIDDPro [9] for the standard bass ({, }) and nput state of each qubt. #qubts Gate n pattern (n) CNOT () 4 Toffol (3) 7 Input Lbquantum QuIDDPro OWQS Emulator(sec) state (sec) (sec) (sec) EOWQS(sec) bass bass N/A N/A

24 Fredkn (3) 45 GHZGate () 39 GHZGate (3) 45 GHZGate (5) 49 QFT (3) 3 QFT (5) 9 QFT (8) 4 QFT (9) 36 QFT () 38 4_49 (4) [8] 88 bass N/A N/A N/A N/A bass N/A N/A N/A N/A N/A bass N/A N/A N/A N/A N/A bass N/A N/A N/A N/A N/A bass N/A N/A bass N/A N/A N/A N/A > hour.6.53 bass N/A N/A > hour N/A N/A > hour bass N/A N/A > hour.34.3 N/A N/A N/A bass N/A N/A N/A.5.47 N/A N/A N/A bass N/A N/A N/A N/A nth_prme4_nc (4) [8] bass N/A N/A N/A N/A hwb4 (4) [8] 66 ham7 (7) [8] 9 Shor code (9) 3 bass N/A N/A N/A N/A bass N/A N/A N/A N/A N/A bass N/A N/A N/A N/A Toffol_Starcase (3) [9] 97 bass N/A N/A N/A N/A > hour..73 4

25 Toffol_Starcase (7) [9] Toffol_Starcase () [9] rc_adder4 (6) [3] bass N/A N/A N/A N/A > hour bass N/A N/A N/A N/A N/A bass N/A N/A N/A N/A N/A rc_adder5 () 88 rd73 () [8] 8 rd84 (5) [8] 33 6sym () [8] 6 gf_4 () [8] 4 bass N/A N/A N/A N/A N/A bass N/A N/A N/A N/A N/A bass N/A N/A N/A N/A N/A N/A bass N/A N/A > hour N/A N/A N/A.7.98 bass N/A N/A N/A N/A 3.6 N/A N/A N/A N/A CONCLUSIONAND FUTURE WORKS WQC has drawn consderable attentons, manly because t offers dfferent physcal realzatons of the quantum computatons. However, to the best of our knowledge no practcal tool has been proposed for the smulaton of ths model. It should be mentoned that the number of qubts n the WQC pattern s consderably more than the equvalent quantum crcut whch makes ther smulaton more complex. Therefore, conventonal crcut model smulators such as QuIDDPro cannot be drectly used for the smulaton of WQC. In ths paper, a practcal approach to smulatng WQC patterns on the classcal computers, called OWQS, was proposed and then was extended n a way that t can reduce run-tme and potentally space complexty of smulaton by explotng the concept of gflow. Two man technques, qubt elmnaton and pattern reorderng were presented to consderably reduce the state complexty as well as the tme and memory needed for the smulatons of WQC patterns. After usng these technques one can utlze prevously proposed crcut model smulaton deas to acheve further optmzatons. Usng graph structure to represent system states [9] nstead of arrays s underway. REFERENCES [] E. Nkahd, M. Houshmand, M.Saheb. Zaman, M. Sedgh, OWQS: One-Way Quantum Computaton Smulator, 5th Euromcro Conference on Dgtal System Desgn (DSD),, pp

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