Alternative Coins for Quantum Random Walk Search Optimized for a Hypercube

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1 Journal of Quantum Informaton Scence, 215, 5, 6-15 Publshed Onlne March 215 n ScRes. Alternatve Cons for Quantum Random Walk Search Optmzed for a Hypercube Hrsto Tonchev Department of Physcs, Sofa Unversty, Sofa, Bulgara Emal: h_tonchev@phys.un-sofa.bg Receved 21 September 214; accepted 24 March 215; publshed 25 March 215 Copyrght 215 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract The present paper s focused on non-unform quantum cons for the quantum random walk search algorthm. Ths s an alternatve to the modfcaton of the shft operator, whch dvdes the search space nto two parts. Ths method changes the quantum cons, whle the shft operator remans unchanged and sustans the hypercube topology. The results dscussed n ths paper are obtaned by both theoretcal calculatons and numercal smulatons. Keywords Quantum Informaton, Quantum Random, Quantum Random Walk Search 1. Introducton The search algorthms for unstructured databases are wdely used n statstcal data processng for searchng the maxmum or mnmum element or an element correspondng to specfc crtera. Effectve search algorthms can provde a soluton for one of the Non-Determnstc Polynomal Tme Complete (NPTC) problems, from whch a soluton can be found to any NPTC problem by an algorthm wth polynomal complexty. These are the reasons for the great nterest n the quantum search algorthms and ther expermental mplementaton. The frst quantum search algorthm for unstructured databases s created by Grover [1] and s based on a quantum Fourer transformaton. Quantum search on a two-qubt cavty QED database has been done by Yamaguch et al. [2]. Quantum search on a three-qubt database wth NMR has been done by Vandersypen et al. [3]. Grover s algorthm cannot be used wthout knowng the exact number of solutons. To fnd the exact number of elements and satsfy the search crtera, the quantum countng algorthm should be used. There are already many classcal random walk algorthms that perform much better n ther tasks than determnstc algorthms. Two classes of such algorthms are Las Vegas algorthms (whch always end wth a correct result when used for a fnte tme) and Monte Carlo algorthms (whch depend on random nput and mght pro- How to cte ths paper: Tonchev, H. (215) Alternatve Cons for Quantum Random Walk Search Optmzed for a Hypercube. Journal of Quantum Informaton Scence, 5,

2 duce an ncorrect result). Las Vegas algorthms are wdely used n felds lke artfcal ntellgence [4], bology [5] and others. Monte Carlo algorthms are used n mathematcs, condensed matter physcs [6]-[8] and others. Another type of quantum algorthms are the ones based on the quantum random walk; they are analogous to the classcal random walk. There are two types of those algorthms: contnuous tme evoluton random walk algorthms (CTRWA) and dscrete tme random walk algorthms (DTRWA). The CTRWA were frst ntroduced by Farh and Gutmann [9]. They have showed that CTRWA propagate exponentally faster through graphs [1] and can solve any black box problem lke searchng exponentally faster than any classcal algorthm [11]. Chlds has shown that the contnuous tme quantum random walk search algorthms (CTRWS) can fnd an element n a graph wth dmenson over 4D faster than Grover s search algorthm [12]. DTRWA have been frst proposed by Aharonov et al. [13]. Examples for DTRWA algorthms are the quantum random walk algorthms for element dstncton [14] and the quantum random walk search algorthms [15]. Dscrete tme random walk search algorthms (DTRWSA) have been frst created by Shenv et al. [15] and are denoted as SKW. The orgnal SKW search algorthm can fnd an element wth probablty less than 1/2. Hen has proposed a faster DTRWSA, but to be effectve, the ntal state of the algorthm should take nto account whch elements are to be searched [16]. Potocek et al. have shown that f the searched space s dvded nto two parts, the probablty to fnd a soluton can be ncreased close to 1, wth large enough searched space [17]. They also have demonstrated that the probablty of fndng a soluton can be ncreased f the shft operator s modfed to dvde the searched space. Tuls has shown that DTRWSA s faster than Grover s search when the searched space s two-dmensonal [18]. Grover s search, CTRWSA and DTRWSA dffer conceptually n terms of workng prncple. Ths s the reason for ther dfferent advantages and dsadvantages. Grover s search algorthm and DTRWSA can be modfed to fnd a soluton wth probablty close to one. Long has shown that Grover s search can be modfed so that the probablty of successfully fndng a soluton wth t to be exactly equal to one [19], whch means that the algorthm evolves from a quantum probablstc to a quantum determnstc method. Potocek et al. have shown that a probablty to fnd soluton close to one can be obtaned n DTRWSA by two dfferent methods [17]. Grover s search algorthm needs only one oracle call for each teraton of the algorthm and less number of qubts (as much as needed to store the searched space); let ths number be denoted as n. DTRWSA needs more qubts: O(n), and two oracle calls for each teraton [2]. DTRWSA s much better than Grover s algorthm, when there s the need to search n a regster of two or more dmensons [18] [21]. The present paper s organzed as follows. In Secton 2, the dscrete quantum random walk s revewed, and the quantum random walk on a lne s shown as an example. In Secton 3, the quantum random walk on a hypercube and the quantum random walk search on a hypercube are revewed. In Secton 4, a new alternatve way s demonstrated for the method shown n [17] for dvdng the searched space of the algorthm n two, whle sustanng the hypercube topology and effectvely dvdng the searched space by usng cons, whch unequally dstrbute the probablty of transton to adjacent nodes. In Secton 4.1, Householder reflecton s revewed. In Secton 4.2, the general form of specalzed cons s shown; whereas ther use s dscussed n Secton 4.3. In Secton 4.4, some examples of such cons and quantum crcut for ther expermental mplementaton n quantum random walk search algorthm are shown. The results of numercal smulatons wth the cons are also gven. Secton 5 s the concluson of the artcle. 2. Classcal and Dscrete Quantum Random Walk, Quantum Random Walk on Lne The classc random walk on a lne starts at an ntal state and at every step, a con s tossed. There s a dfferent probablty of the possble outcomes of the toss. The sum of these probabltes s equal to one. Each of the possble outcomes of the toss s assocated wth a dstnct drecton, and the drectons depend on the structure of the graph whch s traveled over. For example, for a lne the drectons are left and rght. For a square grd, the drectons are left, rght, up and down. The partcle moves one step n the correspondng drecton, accordng to the result of the con toss. The quantum random walk algorthm s the quantum analogue of the classc random walk algorthm. H C s the Hlbert space of the quantum con (con space) and H S s the Hlbert space of the nodes of the structure of the graph. Agan, each step of the algorthm (whch s descrbed by the operator U) has two parts. Frst s the con toss. The con flp s defned by the untary operator of the con C, whch acts n the con space H C. The con C S operator acts upon the Hlbert space H H and s denoted by C = C I. The result of the acton of the con operator upon the con s a chral state [22]. Ths s an analogue of the classc probablty. As the chral state s a quantum state, t can be n a quantum superposton of drectons. Accordng to the toss outcome, the state of 7

3 the system s changng. The exact change of the state depends on the structure. The quantum operator whch represents ths structure s a permutaton matrx whch performs controlled shft, dependng on the state of the C S con, and s denoted by S. The operator S acts upon the space H H. Summarly, each step of the quantum random walk can be wrtten as: U = SC. (1) An example for a shft operator s S L correspondng to a quantum random walk on a lne [23]: 1 d = ( ) d SL = d, x 1 d, x, (2) where x s the poston of the partcle on the lne. The values of d ( or 1) correspond to left and rght drectons. For the con, a Hadamard matrx can be used: C L = (3) Summarly, a DQRW step on lne can be wrtten as UL = SL L L. The classc random walk spreads as a bnomal dstrbuton after each step. In DQRW, after each step of the algorthm, quantum nterference occurs when more than one possble path exsts to reach the respectve poston. The nterference can be constructve or destructve, whch leads to very dfferent dstrbuton compared to the classc random walk on a lne. The varance n the number of steps t between the classc random walk and DQRW s very dfferent. DQRW spreads as O(t); n comparson, the classc random walk spreads as O( t ) [23]. If the quantum random walk s measured at each con flp, or after the end of each step, t wll revert to the classc random walk [15]. 3. Quantum Random Walk and Search on a Hypercube The hypercube s a graph wth N = 2 n nodes and n edges between nodes. Each one of the nodes wll be denoted by a n-bt strng x. Two nodes of a hypercube, x 1 and x 2 are connected only f the modulus of hammng C log( n) weght of ther dfference s equal to one: x1 x2 = 1. That s why the Hlbert space of the con s H = H, N n N C the Hlbert space of the nodes s H = H and the Hlbert space of the random walk s H = H H [15]. The shft operator for the hypercube S C s: S = d, x e d, x, C n 1 d d= x where d s the drecton of the moton and e d s the d-th bass vector of the Hypercube. The Grover con G s frequently chosen for a con for quantum random walks on a hypercube. Ths con s nvarant to all permutatons of the n edge drectons, so t sustans the permutaton symmetres of the hypercube. c c C = G = I + 2 s s, (5) c where I s dentty operator, s s an equal weght superposton of the states of all drectons. To make a quantum random walk search algorthm, a quantum oracle should be appled that marks the wanted element by applyng a con upon t. The oracle does ths by usng the functon f ( x ), whch s used to determne whch con would be appled: C or C 1, 1 x = xt f ( x) =. (6) x xt Summarly, the operator of the con becomes: ( ) C = C I + C C x x (7) 1 t t, where C 1 can be almost any untary operator but most often t s taken C1 = I. The reason for ths s the faster spread through the graph and the smplcty n expermental realzaton. Summarly, the random walk search teraton can be wrtten as: (4) 8

4 c c ( t t ) U = SC = U 2S s s x x H. Tonchev. (8) The quantum crcut of the random walk search algorthm s shown n Fgure 1 [15]. Ths crcut does not actually depend on the shape of the searched graph. When the graph s dfferent, the shft operator S has to be changed. Summarly, the steps of the algorthm are [2]: 1) Intalzng the startng state of the con and node regster n an equal weght superposton. Ths can be done by applyng Hadamard gate on each qubt of the state ; 2) Applyng quantum random walk search teraton t = π 2 2 n tmes [15]. The steps of the quantum random walk search teraton are: a) Applyng a quantum oracle; b) Applyng an approprate con dependng on the state of the control regster; c) Applyng the quantum oracle; d) Applyng the shft operator. Due to the symmetry of the hypercube, ts nodes can always be re-labeled n such way that the marked node x t becomes node x t = [17]. The poston of x t = and the fact that the ntal state s an equal weght superposton allows to project the quantum random walk on hypercube onto a quantum random walk on a lne, as shown n Fgure 2. The basc states of the collapsed random walk on a hypercube are: x! ( n x 1! ) = Rx, dx,, (9) n! x= xxd= ( x 1! ) ( n x)! = Lx, dx,. (1) n! x= xxd= The shft operator n ths collapsed random walk bass R, L becomes: n 1 S= Rx, Lx, Lx, + 1 Rx,. (11) x= The quantum random walk on a lne strongly depends on the poston. In the bass R, L, the Grover con becomes: ( ωx) sn ( ωx) ( ω ) cos( ω ) where cos( ω x) = 1 2xnand sn ( x) ( 2 n) x( n x) n cos C = x x x= sn, (12) x x ω =. The perturbed con becomes C = C 2 R, R,. 4. Algorthm wth Cons from Generalzed Householder Reflecton Potocek et al. have shown n [17] that f the regster s dvded nto two subspaces for even and for odd elements by the shft operator, they can both evolve separately. Thus, the probablty to fnd a soluton ncreases twofold. In ths chapter t wll be demonstrated that the same result can also be obtaned by usng approprate cons Generalzed Householder Reflecton The generalzed Householder reflecton ( ; ) expresson: M χϕ, s wdely used n quantum nformaton and t s gven by the ( ) ( ) M χϕ ; = I + e ϕ 1 χ χ (13) where χ s a normalzed N-dmensonal vector, generally complex, φ s the phase, I s the dentty operator. In the orgnal SKW algorthm [15] for searchng a markng con, the operator (C 1 ) wth a mnus sgn s used. It can be vewed as M ( χϕ ; ) wth a phase equal to zero. For a walkng con (C ), the Grover s con s used whch s also a Householder reflecton when χ s an equal-weght superposton of all basc states and the phase s equal to π. 9

5 Fgure 1. Quantum crcut for random walk search algorthm. The box marked as T s the random walk search teraton and shoud be repeated t tmes. The value of t s shown n Secton 3. Fgure 2. Projectng random walk search algorthm on a hypercube to a random walk on a lne. The marked state s shown wth orange. ( χπ) ( χ) M ; = M = I 2 χ χ (14) In [16], the case s dscussed when C s agan an equal superposton vector, but the phase s random. In ths paper, only Householder reflecton wth phase equal to π wll be dscussed, when χ s dfferent from the one used n SKW General Form of the Cons Here we wll vew the case when C s a standard Grover con, as t s n the orgnal SKW search algorthm: C = I 2 s c s c, (15) s c 1 1 N where j s the j-th bass vector (,1, ) = j, (16) N j= j =, and For the markng con C 1, an arbtrary Householder reflecton s taken wth a phase π: here a j s real and a c s s the equal weght superposton vector. C = I 2 χ χ, (17) N 1 a j χ1 = j, (18) a j= N 1 2 aj j= a =, (19) 1

6 The con and the random walk step are unchanged, as n the standard SKW search algorthm: C = C I, (2) 2 n U = SC. (21) The perturbed random walk con s: C = C I C C = C 2 s s x x, c c ( ) υ υ ( χ χ ) c c 2 ( χ1 χ1 ) t t n U = SC = U S s s x x. (23) Ths form s too general, so n the next subsecton some examples for cons wll be dscussed Algorthm The steps of ths mplementaton of QRWS are the same as n the SKW search algorthm. The quantum crcut of ths algorthm s almost the same as n SKW and s shown n Fgure 1, the dfference beng that the markng con s dfferent and an addtonal qubt s taken whch does not need to be measured at the end of the algorthm Examples for Some Good Cons Some examples of cons sutable for a random walk search are proposed n ths secton. These examples are probably not only the useful ones but also can be performed relatvely easly n experments. A Householder reflecton can easly be done wth an N-pod system. For smplcty, a hypercube wth dmenson 2K, nstead of a hypercube wth dmenson N wll be revewed. N = 2K = 2 n. From here on, y wll denote arbtrary values, and y may or may not be equal to y j at j. Also, x s an arbtrary value, the modulus of whch s larger than the modulus of y at any. The number of y as altogether s n 1. One case of asymmetrcal cons s when ar = x, a r= y, where y < x, s an nteger and [, n 2], and r can be or n 1. These cons are desgned for random walk search on a hypercube. They have an asymmetrcal shape whch effectvely leads to dvson of the searched space to two (Fgure 4). In the frst searched subspace, the con marks the element marked by the oracle. In the second searched subspace, the markng con effectvely marks one of the adjacent nodes. The matrx chosen to be used n the markng con defnes whch of the nodes s marked. The dvson of the searched space n two requres an addtonal qubt (the number of states of the regster pror to the dvson s 2K) n order to perform the search and to have a probablty of fndng a soluton above 8%. Here are two examples of such cons, dependng on the way of doublng the number of states by addng a qubt: The frst type of such con s when a = n 1 x, a ( n 1) = y, where y < x, s an nteger and [, n 2] : K = t t (22) a = x + y (24) Ths result can easly be explaned when a n = 1, a n=, α = 1. The cons C 1, C, G (see Equaton (5)) dffer from each other only by the sgn of the components of ther matrces, so they mark the same edges wth the same ampltudes. The con C marks all edges connected wth the states wth a plus sgn. The con G marks all edges connected wth the marked state wth a mnus sgn. The con C 1 marks all edges except the last one wth a plus sgn, and the last one wth a mnus sgn (Fgure 3). The frst hypercube wth sze (K) for quantum random walk search s obtaned from the marked state n such way as not to nclude the state marked by the con wth a mnus sgn (Fgure 4). On the other hand, the node whch s marked wth a mnus sgn partakes n a hypercube wth sze (K) so that t does not nclude the marked state (Fgure 4). Ths s second hypercube n the searched space dvded nto two. The number of steps needed depends on the exact values of x and y. The quantum crcut needed for those types of cons s shown n Fgure 5. 11

7 (a) (b) (c) Fgure 3. Dfference between uses of a markng con n: (a) SKW search algorthm on a hypercube; (b) Standard walk on hypercube of Grover Con wth no marked state; (c) Implementaton of walk wth ge- M χϕ ;. In general case con s asymmetrc. Here angle ϕ s neralzed Householder reflecton con ( ) equal to π, and χ s descrbed n the text. Green denotes a mnus sgn, cyan denotes a plus sgn, orange denotes a marked state, and n black s the state before the con s appled. Fgure 4. The smplest case s when ar = x, a r= y, where y < x, where r can be any number n the nterval [, n 1]. Those cons are desgned for random walk search on a hypercube. They have an asymmetrcal shape, whch leads to effectve dvson of the searched space nto two hypercubes. Green denotes a mnus sgn, cyan denotes a plus sgn, orange denotes a state not marked wth the markng con, but marked by the asymmetry of the markng con, and n black s marked the state where the con for unmarked state s appled. In the frst hypercube s the state marked wth the markng con (ts nodes are denoted by unprmed numbers). The second hypercube contans the state marked by the assymetry of the markng con (ts nodes beng denoted by prmed numbers); the second hypercube does not contan nodes from the frst one. Double prmed boxes show that the whole hypercube can be revewed as a hypercube wth dmenson reduced by one. The fgure s drawn as a cube for smplcty and easer understandng. C log( n) Smulatons are made wth Hlbert space of the con H = H. The smulatons are made by two qubt cons because of the absence of enough computatonal power to make smulatons for cons wth more qubts. Values a n = 1, a n=, α = 1 are good for explanaton of the workng of the algorthm. For two qudt cons, when they are used, the probablty for fndng a soluton at the 6-th teraton s.678, and 9 teratons are needed to obtan the maxmal probablty.859. Wth two-qubt cons, a n = 58 and a n= 18 the algorthm needs 6 random walk steps. The result of the numercal smulaton wth searched element 4 s shown n Fgure 6. Smulatons demonstrate that these cons can also be used when there s more than one marked state. 12

8 Fgure 5. Quantum crcut for cons when an = x, a n= y, where y < x. As n Fgure 1 for the SKW, the box marked by T s the random walk search teraton and should be repeated. For the number of tmes, see text. Fgure 6. Result of smulatng a quantum crcut wth a con wth a K = 58 and a K= 18 and 6 random walk steps. The searched element n the smulaton s 4 and the sze of the node regster s 3 qubts. It has been obtaned by numercal smulatons that a hgher probablty of fndng the searched element s acheved when a n = 58 and a n= 18 s used. Another type of such con s the case when a = x, a 1 = y, s an nteger and [ 1, n 1 ], where y < x. An example for ths s the con wth a = 1 and a 1 =. The quantum crcut for those types of cons s shown n Fgure 7. The algorthm needs 6 random walk steps, when the regster of the con conssts of two con qubts and a = 58and a 1 = 18. The result of the smulaton n ths case wth searched element 2 s shown n Fgure 8. Smulatons demonstrate that these cons can also be used when there s more than one marked state. When a = 1 and a 1 =, the algorthm also needs 9 teratons to obtan ts maxmal probablty.859. A hgher probablty of obtanng the searched element s acheved when a = 58and a 1 = 18, by analogy wth a n = 58 and a n= 18. Wth all other cases havng ths structure of the con, when ar = x, a r= y, s an nteger and [ 1, r) ( rn, 1], where y < x, r can be any number n the nterval [, n 1], the quantum crcut for obtanng the result wll be more complcated or addtonal classcal processng s needed. Numercal smulatons show that at least when the sze of the node regster N = 16, there are also other cons wth dfferent shape of the vector χ whch can be used effcently. Examples for such cons are when ( ) w a = 1 n+ 1, (25) ( 1 ) w a = n+, (26) 13

9 Fgure 7. Quantum crcut for cons when a1 = x, a = 1 y, where y < x. The box marked by T s the random walk search teraton and should be repeated. For the number of tmes, see the text. and when Fgure 8. Result of the smulaton of the quantum crcut wth a con wth a 1 = 58 and a 1 = 18 and 6 random walk steps. The searched element n the smulaton s 2 and sze of the node regster s 3 qubts. a a = 1 w, (27) w =. (28) An example for such cons s when the formula (28) s used wth the quantum crcut shown n Fgure 5. The probablty for fndng a soluton s approxmately.77, wth w = 3. Another example of search con s when the formula (26) s used wth the quantum crcut shown n Fgure 7. The probablty for fndng a soluton s approxmately.77, wth w = 3. The number of steps needed for the cons showed n ths secton s obtaned by numercal smulatons and has not been found emprcally yet. 5. Conclusons A dscrete quantum random walk search algorthm optmzed for hypercube s dscussed. A new alternatve DTRWS method for dvdng the searched space n two s presented. The searched space s dvded effectvely nto two by usng asymmetrc cons, whch dstrbute the probablty of shftng nto neghborng nodes non-unformly. The advantage of ths method s that t preserves the topology of the hypercube and does not dvde t by modfyng the shft operator. The cons are obtaned by usng Householder reflecton, whch can be easly performed n experments by usng N-pod systems. 14

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