Superiority of exact quantum automata for promise problems

Sup eriorit y of exact quan tum automata for promise problems Andris Am bainis ⋆ and Abuzer Y ak aryılmaz ⋆⋆ Universit y of Latvia, F aculty o f Computing, Raina bulv. 19, Riga, L V-1586, Latvia { ambainis ,abuzer } @lu.l v Octob er 26, 20 18 Abstract. In this n ote, w e presen t a n infinite family of p romise p rob- lems whic h can be solv ed exactly by just tuning transition amplitudes of a tw o-state quantum finite automata op erating in realtime mo de, whereas the size of t he corresp onding classical automata gro w without b ound. Keywords: exact quantum computation, promise problems, succinct- ness, quantum finite automaton, classical finite aut omaton 1 In tro duction The exact qu an tum computation has b een widely examined for p artial (promise) and total fu n ctions (e.g. [BH97 , BV97, BBC + 98, BCdWZ99, Kla00, BdW03, MNYW05, FI09, YFSA10]). O n the other hand, in au- tomata theory , only t w o r esults ha ve b een obtained: (i) Klauc k [Kla00] has shown that realtime quan tum finite automata (QF As) cannot b e more concise than realtime d eterministic finite au - tomata (DF As) 1 in case of language recognition, (ii) Murak ami et. al. [MNYW05] hav e sh o wn th at there is a promise prob- lem solv able by quan tum pushd o w n automata but not by an y deter- ministic push do w n a utomata. ⋆ Ambainis was supp orted b y ESF pro ject 1DP/1.1.1.2.0 /09/APIA/VIAA/044, FP7 Marie Curie International Reintegration Grant PIRG02 -GA-2007-224886 and FP7 FET-Op en pro ject QCS. ⋆⋆ Y ak aryılmaz was partially supp orted by the Scientific and T echnological Research Council of T urkey (T ¨ UB ˙ IT AK ) with gran t 108E142 and FP7 FET-Op en pro ject QCS. 1 The pro of wa s basically giv en for Kondacs-W atrous realtime QF A model [KW97] but it can b e extended for any mo del of realtime QF As including the most general ones [Cia0 1 , BMP03, Hir10, YS11 ]. In this note, w e consider succinctness of realtime QF As for promise problems. W e p r esen t an infinite family of promise p roblems which can b e solv ed exactly b y just tuning transition amp litudes of a t wo -state rtQ F As, whereas the size of the corresp ondin g classical automata gro w without b ound . 2 Bac kground Throughout the pap er, (i) Σ denotes the in put alphab et not cont aining left- and righ t-end mark- ers ( ¢ and $, resp ectiv ely) and ˜ Σ = Σ ∪ { ¢ , $ } , (ii) ε is the empty string, (iii) w i is the i th sym b ol of a giv en string w , and (iv) ˜ w represents the string ¢ w $, for w ∈ Σ ∗ . Moreo ve r, all mac hines p resen ted in the p ap er op erate in r ealtime mo de. That is, the inpu t head mo ves one square to the right in eac h step and the computation stops after reading $. A promise problem is a pair A = ( A y e s , A no ), w h ere A y e s , A no ⊆ Σ ∗ and A y e s ∩ A no = ∅ [W at09 ]. A pr omise problem A = ( A y e s , A no ) is solv ed exactly by a machine M if eac h strin g in A y e s (resp., A no ) is accepte d (resp., r ejected) exactly by M . Note that, if A y e s = A no , this is the same as the r ecognitio n of a language ( A y e s ). W e giv e our quantum r esult w ith the most restricted of the kno wn QF A mo del, i.e. Mo or e - Crutchfield quantum finite automaton (MCQF A) [MC00], (see [YS11] for the definition of the most general Q F A mo del). A MCQF A is a 5-tuple M = ( Q, Σ , { U σ | σ ∈ ˜ Σ } , q 1 , Q a ) , where Q is the set of states, q 1 is the initial state, Q a ⊆ Q and is the set of accepting states, and U σ ’s are un itary op erators. The computation of a MCQF A on a giv en input string w ∈ Σ ∗ can b e traced by a | Q | - dimensional vecto r. Th is v ector is in itially set to | v 0 i = (1 0 · · · 0) T and ev olves according to | v i i = U ˜ w i | v i − 1 i , 1 ≤ i ≤ | ˜ w | . A t the en d of the computation, w is accepted (resp ., rejected) with p rob- abilit y || P a v | ˜ w | || 2 (resp., || P r v | ˜ w | || 2 ), where P a = P q ∈ Q a | q ih q | and P r = I − P a . If we replace th e unitary op eration with a zero-one left sto c hastic op erator, w e obtains a realtime DF A (which we call simply a DF A). 3 The main results Let A k y e s = { a i 2 k | i is a nonn egativ e even in teger } and A k no = { a i 2 k | i is a p ositiv e o dd int eger } b e tw o u nary languages, where k is a p ositiv e in teger. W e will sho w that a t wo -state MCQF A can solv e p romise problem A k = ( A k y e s , A k no ), but any DF A (and so an y PF A) must hav e at least 2 N states to solv e the same pr oblem exactly . Theorem 1. Pr omise pr oblem A k = ( A k y e s , A k no ) c an b e solve d by a two- state M CQF A M k exactly. Pr o of. W e w ill u se a wel l-kno wn tec hniqu e giv en in [AF98]. Let N = 2 k and M k = ( Q, Σ , { U σ | σ ∈ ˜ Σ } , q 1 , Q a ), where Q = { q 1 , q 2 } , Σ = { a } , Q a = { q 1 } , U ¢ = U $ = I , and U a is a rotation in | q 1 i - | q 2 i plane with angle θ = π 2 N , i.e., U a =  cos θ − s in θ sin θ cos θ  . The computation b egins with | q 1 i and after reading eac h blo c k of N a ’s, the follo win g pattern is follo wed by M k : | q 1 i a N − → | q 2 i a N − → −| q 1 i a N − → −| q 2 i a N − → | q 1 i a N − → · · · . Therefore, it is ob vious that M k solv es promise problem A k exactly . Lemma 1. Any DF A solving A k = ( A k y e s , A k no ) exactly must have at le ast 2 k +1 states. Pr o of. Let N = 2 k and D b e a m -state DF A solving A k exactly . W e s ho w that m cannot b e less than 2 N . Since b oth A k y e s and A k no con tain infinitely m an y u nary strings, there m ust b e a c hain of t states, sa y s 0 , . . . , s t − 1 suc h that, for sufficien tly long strings, D en ters this c h ain in wh ic h D transmits f rom s i to s ( i +1 mo d t ) when r eading an a , where 0 ≤ i ≤ t − 1 and 0 < t ≤ m . Without lose of generalit y , we assume th at D accepts the input if it is in s 0 b efore r eading $. Thus, D rejects the in put if it is in s ( N mod t ) b efore reading $. Let S a b e the set of { s ( i 2 N mo d t ) | i ≥ 0 } . Th en, D accepts the inp u t if it is in one of the states in S a b efore reading $. Note that s ( N mod t ) / ∈ S a . Let d = gcd( t, 2 N ), t ′ = t d , and S ′ b e the set { s id | 0 ≤ i < t ′ } . Since S a ⊆ S ′ and | S ′ | = t ′ , w e can easily follo w S a = S ′ if we sho w | S a | ≥ t ′ . Firstly , we show that eac h i satisfying ( i 2 N ≡ 0 mo d t ) m ust be a m ultiple of t ′ : F or suc h an i , there exists a j su c h that i 2 N = j t . By dividing b oth sides with t = dt ′ , w e get i t ′ 2 N d = j . This implies that i m ust b e a multiple of t ′ since left side m ust b e an int eger and g cd ( t ′ , 2 N ) = 1. Secondly , we sho w that there is no i 1 and i 2 , i.e. t ′ > i 1 > i 2 ≥ 0, such that ( i 1 2 N ≡ i 2 2 N mo d t ). If so, w e h a ve ( i 1 2 N − i 2 2 N ≡ 0 mo d t ) and then (( i 1 − i 2 )2 N ≡ 0 mo d t ). This imp lies th at ( i 1 − i 2 ) must b e a m ultiple of t ′ . Th is is a contradict ion. Th us, for eac h i ∈ { 0 , . . . , t ′ − 1 } , w e obtain a d ifferen t v alue of ( i 2 N mo d t ) and so | S a | con tains at least t ′ elemen ts. If gcd( t, N ) = d , then s ( N mod t ) also b ecomes a mem b er of S a . Th er e- fore, gcd( t, N ) must b e differen t than gcd( t, 2 N ). T his can only b e p ossible whenev er t is a multiple of 2 N . Therefore, m cannot b e less th an 2 N . Since a 2 k +1 -state DF A solving p r omise problem A k exactly can b e constructed in a straigh tforwa rd w a y , w e obtain the follo w ing theorem. Theorem 2. The minimal D F A solving the pr omise pr oblem A k = ( A k y e s , A k no ) exactly has 2 k +1 states. 4 Concluding remarks In this paper, we identify a case in whic h the sup eriorit y of qu an tum computation to classical one cannot b e b ou n ded. F or this purp ose, we use an infi nite family of tw o unary disj oint languages con taining the strings of the form ( a 2 n ) ∗ and a n ( a 2 n ) ∗ , resp ectiv ely , wh er e n is a p o wer of 2. What happ ens if n is not an exact p ow er of 2? F or quantum case, w e can still solve th e same problem with 2 states. On the other hand, for the classical case, th e minimum n umber of states is determined by the b iggest f actor of the num b er, which is a p o wer of 2. Let k , l > 0. 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