Advice Coins for Classical and Quantum Computation

Advice Coins for Classical and Quan tum Co mputation Scott Aaronson ∗ Andrew Druck er † Abstract W e study the power of classical and quan tum alg orithms equipped with nonuniform a dv ice, in the for m o f a coin whose bia s enco des useful information. This question takes on pa rticular impo rtance in the quan tum case, due to a surprising r esult that w e pro ve: a quantum finite automaton with just two states c an b e sensitive to arbitr arily smal l changes in a c oin ’s bias . This con tr a sts with classical probabilis tic finite automa ta , whose sensitiv ity to changes in a coin’s bias is bounded b y a classic 1 9 70 result of Hellman a nd Cov er. Despite this finding, we are able to bo und the p ow er of a dv ice co ins for space-b ounded clas- sical a nd quantum computation. W e define the cla sses BPPSP ACE / coin and BQPSP A CE / coin , of langua ges decidable by classica l and quantum polyno mial-space machines with advice co ins. Our main theorem is that both classe s coincide with PSP A CE / p oly . Proving this result turns o ut to require substantial machinery . W e use an a lgorithm due to Neff for finding ro ots of po lynomi- als in NC ; a result from algebr aic geometry that low er-b ounds the separ ation of a p olynomia l’s ro ots; and a res ult on fixed-p o int s of sup erop er ators due to Aaro nson and W atrous, o riginally prov ed in the cont ext of quantum computing with closed timelike curves. 1 In tro duction 1.1 The Distinguishing Problem The f undamental task of mathemat ical stati stics is to learn features of a random pro cess from empirical data generated by that pro cess. One of the simplest, y et most imp ortan t, examples concerns a coin with unkno wn bias. Sa y w e are giv en a c oin wh ich lands “ heads” with some unknown probabilit y q (called the bias ). In the distinguishing pr oblem , w e assume q is equal either to p or to p + ε , for some kno wn p, ε , and we w ant to decide which h olds. A traditional f o cus is the sample c omplexity of statistical learning pr o cedures. F or example, if p = 1 / 2, then t = Θ  log (1 /δ ) /ε 2  coin flips are necessary and sufficien t to su cceed with p robabilit y 1 − δ on the distingu ish ing pr oblem ab o v e. This assumes, ho wev er, that we are able to coun t the n um b er of heads seen, whic h requires log ( t ) b its of memory . F rom the p ersp ectiv e of computational efficiency , it is nat ural to wonder whether metho ds with a m uc h smalle r sp ace requiremen t are p ossible. This question was studied in a cla ssic 197 0 pap er by Hellma n and Co ve r [13]. They sho wed that any (cl assical, pr obabilistic) finite automaton th at distinguish es w ith b ounded error b et w een a coin of bias p and a co in of b ias p + ε , must ha v e Ω ( p (1 − p ) /ε ) s tates. 1 Their r esult holds with no r estriction on the n umber of co in flips p erformed b y the automat on. This mak es ∗ MIT. Email: aaronson@csail. mit.edu. This material is based up on work supp orted by th e N ational Science F ound ation under Grant N o. 0844626. Also supp orted by a DARP A YF A grant and a S loan F ello wship. † MIT. Email: adruck er@mit.edu . Supp orted by a DARP A YF A grant. 1 F or a formal statemen t, see Section 2.5. 1 the result esp ecially interesting, as it is not immed iately clear h o w sensitiv e su c h m achines can b e to small c hanges in the bias. Sev eral v ariations of the distinguishing problem for space-b oun ded automata w er e studied in related w orks by He llman [12] and C o ve r [10]. V ery recent ly , Brav erman, Rao, Raz, and Y ehu- da yoff [8] a nd Bro d y and V erbin [9] studied the pow er of restricted-width, r e ad-onc e br anching pr o gr ams for this problem. The distinguishing pr oblem is also closely relat ed to the appr oximate majority problem, in whic h giv en an n -bit string x , w e wan t to decide whether x has Hamming w eight less than (1 / 2 − ε ) n or more than ( 1 / 2 + ε ) n . A large b o dy of research has addr essed the abilit y of constan t-dep th circuits to solve the appr o ximate ma jorit y p r oblem and its v ari- an ts [1, 3, 4, 17, 20, 21]. 1.2 The Quan tum Case In this pap er, our fi rst con tribu tion is to inv estigate th e p o w er of quantum space-b oun ded algorithms to solv e th e distinguishing problem. W e pr o ve the sur p rising result that, in the absence of n oise, quan tum finite automata with a constan t num b er of states can b e sensitiv e to a rbitr arily smal l c hanges in bias: Theorem 1 (I nformal) F or any p ∈ [0 , 1] and ε > 0 , ther e is a quantum finite automa ton M p,ε with j ust two states (not c ounting the | Accept i and | Reject i states) that distinguishes a c oin of bias p fr om a c oin of bias p + ε ; the diffe r enc e in ac c eptanc e pr ob abilities b etwe en the two c ases is at le ast 0 . 01 . (This differ enc e c an b e amplifie d u si ng mor e states.) In other words, the low er b ound of Hellman and Co ver [13] has no analogue for quantum finite automata. T h e up shot is that we obtain a natural example of a task that a quant um fi nite automaton can s olve using arbitr arily fewer states than a probabilistic fin ite automaton, not merely exp onentia lly fewer states! Galv ao and Hardy [11] ga v e a related example, inv olving an automaton that mo ves con tinuously thr ough a field ϕ , a nd needs to decide wh ether an integral R 1 0 ϕ ( x ) dx is o dd or ev en, promised that it is an inte ger. Here, a quan tum au tomaton n eeds only a single qubit, whereas a classical automaton cannot guarantee success with any finite n umb er of bits. Nat urally , b oth our qu an tu m automaton and that of Galv ao and Hardy only work in the absence of noise. 1.3 Coins as Advice This u nexp ected p o w er of quan tu m finite automata in vites us to think fur ther ab out what sorts of statistica l learning are p ossib le using a small num b er of qubits. In particular, if sp ace-b ounded quan tum alg orithms can detect arbitrarily small changes in a coin’s bias, then could a p -biased coin b e an incredibly-p o werful information r esour c e for quan tum computation, if the b ias p wa s w ell-c hosen? A bias p ∈ (0 , 1) ca n b e viewed in it s binary expans ion p = 0 .p 1 p 2 . . . as an infinite sequence of bits; by flipping a p -biased coin, w e could hop e to access those bits, p erh aps to help us p erform computations. This idea can b e seen in “Buffon’s needle,” a prob ab ilistic exp eriment that in p rinciple allo ws one to calc ulate the digits of π to any desired accuracy . 2 It can also b e seen in the old sp eculation that computationally-useful information might somehow b e enco ded in dimensionless physical constan ts, suc h as the fin e-structure constan t α ≈ 0 . 007297 35253 77 th at c h aracterizes t he strength o f the 2 See h t t p://en.wikipedia.org/wiki/Buffon %27s needle 2 electromag netic in teraction. But lea ving aside the question of whic h biases p ∈ [0 , 1] can b e realized by actual ph ysical pro cesses, let u s assume that coins of any desired bias are a v ailable. W e ca n then ask: what computational p roblems can be sol v ed efficien tly using such coins? Th is question w as r aised to us by Erik Demaine (p ersonal comm u nication), and wa s initially motiv ated b y a problem in compu tational genetics. In the mo d el that w e u s e, a T urin g mac hine receiv es an input x and is giv en access to a sequence of b its d ra wn indep end ently f rom a n advic e c oin with some arb itrary bias p n ∈ [0 , 1], whic h may dep end on the input length n = | x | . The mac hine is supp osed to decide (with high success probabilit y) whether x is in some language L . W e allo w p n to dep end only on | x | , n ot on x itself, since otherw ise the bias could b e set to 0 or 1 dep end ing on wh ether x ∈ L , allo w ing membersh ip in L to b e decided trivially . W e l et B PPSP A CE / coin b e the class of languag es dec idable with b ound ed error by p olynomial-space algorithms w ith an advice coin. Similarly , BQPSP ACE / coin is the corresp onding class for p olynomial-space quan tum algorithms. W e imp ose no bou n d on th e runn in g time of these algorithms. It is n atural to compare these classes with the corresp ond ing classes BPPSP ACE / p oly an d BQPSP A CE / p oly , which consist of all languages decidable b y BPPSP ACE and B QPSP A CE mac h in es resp ectiv ely , with the h elp of an arbitrary advic e string w n ∈ { 0 , 1 } ∗ that can dep end only on the input length n = | x | . Compared to the standard advice classes, the str ength of the coin model is that an advice co in bias p n can b e an arbitrary real num b er, and so encod e infinitely m an y bits; the weakness is that this informatio n is only accessible indirectly thr ough the observ ed outcome s of coin flips. It is tempting to try to simulate an advice coin using a con ven tional advice string, whic h simply sp ecifies the coin’s bias to p oly ( n ) bits of precision. A t least in the classical case, the effect of “rounding” the bias can then b e b oun d ed b y the Hellman-Co ve r T heorem. Unfortu nately , that theorem (whose b ound is essentia lly tigh t) is not strong enough to mak e this work: if the bias p is extremely close to 0 or 1, th en a PSP AC E mac h in e r eally c an detect c h anges in p muc h smaller than 2 − poly ( n ) . T his means that upp er-b ounding the p o wer of advice coins is a non trivial problem ev en in the classica l case. In the qu an tu m case, the situation is ev en w ors e, since as men tioned earlier, the quantum analogue of the Hellman-Co ver Theorem is false. Despite these difficulties, we are able to sh o w strong limits on the p o wer of adv ice coins in b oth the cla ssical and quan tum cases. Our main theorem sa ys that PSP ACE machines can effectiv ely extract only p oly ( n ) bits of “useful information” fr om an advice coin: Theorem 2 (Main) B QPSP A CE / coin = BPPS P ACE / c oin = PSP ACE / p oly . The cont ainmen t PS P ACE / poly ⊆ B PPSP A CE / coin is easy . On the other hand, provi ng BPPSP A CE / coin ⊆ PS P ACE / poly app ears to b e no easier than the corresp onding quant um class con tainment. T o pro v e that BQPSP A CE / coin ⊆ PSP ACE / poly , w e will need to und erstand the b ehavio r of a space-b ound ed advice coin mac h in e M , as we vary th e coin bias p . By applying a theo rem of Aaronson and W atrous [2] (whic h w as origi nally dev elop ed to un derstand quan tum computing with closed timelik e curves), w e pro v e th e k ey prop ert y that, for eac h input x , the ac- c eptanc e pr ob ability a x ( p ) of M is a r ational function in p of de gr e e at mo st 2 poly ( n ) . It follo w s that a x ( p ) can “oscill ate” b et we en high and lo w v alues no more than 2 poly ( n ) times as w e v ary p . Using this fact, w e will sho w how to identify the “true” bias p ∗ to sufficien t precision with an advice string of p oly ( n ) bits. What make s this nontrivial is that, in our case, “sufficien t precision” sometimes means exp ( n ) bits! In other w ord s, the rational fun ctions a x ( p ) really c an b e sensitive 3 to doubly-exp onen tially-small c h anges to p . F ortunately , w e will sh ow that this d o es not h app en to o often, and can b e dealt with when it d o es. In ord er to manipulate coin biases to exp onen tially many bits of precision—and to in terpret our advice string—in p olynomial space, w e use tw o ma jor tools. The fir st is a sp ace-efficie n t algorithm for fin ding ro ots of univ ariate p olynomials, dev elop ed b y Neff [14] in th e 1990 s. Th e second is a lo we r b ound from algebraic geometry , on th e spacing b et w een consecutiv e ro ots of a p olynomial with b ounded in teger co efficients. Besides these t wo to ols, we will also need s pace-efficien t linear algebra algorithms d ue to Boro din, Co ok, and Pipp enger [7]. 2 Preliminaries W e assume familiarit y with basic n otions of quan tum computation. A detailed tr eatmen t of space- b ound ed qu antum T u r ing mac hin es was giv en by W atrous [22 ]. 2.1 Classical and Quan tum Space Complexit y In this p ap er, it will generally b e most con venien t to consider an asymmetric mo del , in w h ic h a mac hine M can acc ept only by halting and ente ring a sp ecial “Accept” state, but can reject simply b y nev er accepting. W e sa y that a language L is in the class BPPSP A CE / p oly if th ere exists a classical p robabilistic PSP ACE mac hin e M , as well as a collection { w n } n ≥ 1 of p olynomial-size advice strings, su c h that: (1) If x ∈ L , then Pr [ M ( x, w n ) accepts] ≥ 2 / 3. (2) If x / ∈ L , then Pr [ M ( x, w n ) accepts] ≤ 1 / 3. Note that we d o not require M to accept within any fixed time b ound . So for example, M could ha ve exp ected r unnin g time that is finite, y et doubly exp onentia l in n . The class BQPSP AC E / p oly is defi n ed similarly to the ab o ve, except that now M is a p olynomial- space quantum mac hine rather than a classical one. Also, w e assume that M has a designated accepting state, | Accept i . After eac h computational step, the algorithm is measured to determine whether it is in the | Accept i state, and if so, it halts. W atrous [22] prov ed the follo wing: Theorem 3 (W atrous [22]) BQPSP ACE / poly = BPPSP ACE / poly = PSP ACE / p oly . Note that W atrous stated his resu lt for u ni f orm complexit y c lasses, bu t th e p r o of carries o v er to the nonuniform case without c hange. 2.2 Sup erop erators and Linear Algebra W e will b e in terested in S -state quantum fi nite automata that can includ e non-unitary tr ansfor- mations suc h as m easuremen ts. Th e state of such an automato n need not b e a pur e state (that is, a u nit ve ctor in C S ), but can in ge neral b e a mixe d state (that is, a probab ility distrib ution o ve r suc h v ectors). Ev ery mixed state is uniquely r epresen ted by an S × S , Hermitian, trace-1 4 matrix ρ called th e density matrix . See Nielsen and Ch uang [16] for more ab out the densit y matrix formalism. One can transform a densit y matrix ρ using a sup er op er ator , whic h is an y op eration of the form E ( ρ ) = X j E j ρE † j , where the matrices E j ∈ C S × S satisfy P j E † j E j = I . 3 W e will often find it more con v enien t to work with a “v ectorized” represen tation of mixed s tates and su p erop erators. Giv en a densit y matrix ρ ∈ C S × S , let vec ( ρ ) b e a v ector in C S 2 con taining the S 2 en tries of ρ . Similarly , giv en a sup erop erator E , let mat ( E ) ∈ C S 2 × S 2 denote the mat rix that describ es the action of E on v ectorized mixed s tates, i.e., that satisfies mat ( E ) · v ec ( ρ ) = vec ( E ( ρ )) . W e will need a th eorem due to Aa ronson and W atrous [2], whic h giv es us co n structiv e acce ss to the fixe d-p oints of sup erop erators. Theorem 4 (Aa ronson-W atrous [2]) L et E ( ρ ) b e a sup er op er ator on an S - dimensional system. Then ther e exists a se c ond sup er op er ator E fix ( ρ ) on the same syst em, such that: (i) E fix ( ρ ) is a fixe d-p oint of E for every mixe d state ρ : that is, E ( E fix ( ρ )) = E ( ρ ) . (ii) Eve ry mixe d state ρ that is a fixe d-p oint of E is also a fixe d-p oint o f E fix . (iii) Given the entries of mat ( E ) , the entries o f mat ( E fix ) c an b e c ompute d in p olylog( S ) sp ac e. The follo wing fact, which w e call the “Leaky S ubspace Lemma,”will pla y an imp ortant role in our analysis of quantum fi nite au tomata. In tuitive ly it sa ys that, if rep eatedly ap p lying a linear transformation A to a v ector y “leaks” y into the span of another v ector x , then there is a un iform lo we r b oun d on th e rate at wh ic h the leaking happ ens. Lemma 5 ( Leaky Subs pace Lemma) L et A ∈ C n × n and x ∈ C n . Supp ose that for al l ve ctors y in some c omp act set U ⊂ C n , ther e exists a p ositive inte ger k suc h that x † A k y 6 = 0 . Then inf y ∈ U max k ∈ [ n ]    x † A k y    > 0 . Pro of. It su ffices to prov e the follo wing claim: for al l y ∈ U , ther e exists a k ∈ [ n ] such that x † A k y 6 = 0 . F or giv en this claim, Lemma 5 follo ws by the fact that f ( y ) := max k ∈ [ n ]   x † A k y   is a con tinuous p ositiv e function on a compact set U . W e n o w pro ve the claim. Let V t b e the v ector space sp anned b y  Ay , A 2 y , . . . , A t y  , let V := S t> 0 V t , and let d = dim V . T hen clearly d ≤ n and d im ( V t − 1 ) ≤ d im ( V t ) ≤ d im ( V t − 1 ) + 1 for all t . No w supp ose dim ( V t ) = dim ( V t − 1 ) for some t . Then it must b e p ossib le to wr ite A t y as a linear combinatio n of Ay , . . . , A t − 1 y : A t y = c 1 Ay + · · · + c t − 1 A t − 1 y . 3 This condition is necessary and sufficient to ensure that E ( ρ ) is a mixed state, for every mixed state ρ . 5 But this means th at ev ery higher iterate ( A t +1 y , A t +2 y , etc.) is also expressible as a linear com bi- nation of the low er iterates: for example, A t +1 y = c 1 A 2 y + · · · + c t − 1 A t y . Therefore d = dim ( V t − 1 ). Th e conclusion is that B :=  Ay , A 2 y , . . . , A d y  is a basis for V . But then, if there exists a p ositiv e integ er k suc h that v † A k w 6 = 0, then there must also b e a k ≤ d such that x † A k y 6 = 0, b y the fact that B is a basis. This pro v es the claim. 2.3 Coin-Flipping Finite Automata It will often b e con v enient to u se the language of fin ite automata rather than that of T u ring mac hines. W e mo del a coin-flipping quantum finite automaton as a p air of su p erop erators E 0 , E 1 . Sa y that a coin has bias p if it lands heads with indep endent p robabilit y p ev ery time it is flipp ed. (A coin here is j u st a 0 / 1-v alued random v ariable, with “h eads” meaning a 1 outcome.) Let $ p denote a coin with bias p . When the automat on is giv en $ p , its state ev olv es according to the sup erop erator E p := p E 1 + (1 − p ) E 0 . In our mod el, the sup erop erators E 0 , E 1 b oth i ncorp orate a “measur emen t step” in whic h the a u- tomaton c hec ks wh ether it is in a designated b asis state | Accept i , and if so, halts and accepts. F ormally , this is represen ted by a pro jectiv e measuremen t with observ ables { Γ Acc , I − Γ Acc } , where Γ Acc := | Accept i h Accept | . 2.4 Advice Coin Complexit y Classes Giv en a T ur ing mac hin e M , let M ( x, $ p ) denote M giv en inp ut x together with the abilit y to fl ip $ p at an y time step. Th en BPPS P ACE / co in , or BPPSP ACE with an advic e c oin , is d efined as the class of languages L for w hic h there exists a PSP ACE m achine M , as w ell as a sequence of real n um b ers { p n } n ≥ 1 with p n ∈ [0 , 1], su c h th at for all inputs x ∈ { 0 , 1 } n : (1) If x ∈ L , then M ( x, $ p n ) accepts w ith probabilit y at least 2 / 3 ov er the coin flips. (2) If x / ∈ L , then M ( x, $ p n ) accepts w ith probabilit y at most 1 / 3 ov er the coin flips. Note that there is no requ iremen t for M to halt after at most expon entiall y many steps, or ev en to halt with probabilit y 1; also, M ma y “reject” its input by lo oping forever. This makes our main r esult, whic h b oun ds the co mputational p o wer o f adv ice coins, a str on ger statemen t. Also note that M has no source of rand omness other than the coin $ p n . Ho w ever, this is not a serious restriction, since M can easily use $ p n to generate un biased random bits if needed, by u sing the “v on Neumann tric k.” Let q ( n ) b e a p olynomial space b ound. Then w e mo d el a q ( n )-space qu antum T uring mac hin e M with an advice coin as a 2 q ( n ) -state automaton, with state space {| y i} y ∈{ 0 , 1 } q ( n ) and in itial state   0 q ( n )  . Giv en advice coin $ p , the m ac hine’s state evo lv es according to the sup erop erator E p = p E 1 + (1 − p ) E 0 , where E 0 , E 1 dep end on x and n . The individual en tries of the matrix represent ations of E 0 , E 1 are required to b e computable in p olynomial space. The machine M has a designated | Accept i state. In vecto rized notation, w e let v Acc := v ec ( | Accept i h Accept | ). Since | Acc ept i is a computational basis state, v Acc has a single co ord inate 6 with v alue 1 and is 0 elsewhere. As in Section 2.3, the mac hin e measur es after eac h computation step to determin e w h ether it is in the | Accept i state. W e let ρ t denote the alg orithm’s state after t steps, and let v t := ve c ( ρ t ). If w e p erform a standard-basis m easuremen t a fter t steps, then the probabilit y a x,t ( p ) of seeing | Accept i is giv en b y a x,t ( p ) = h Accept | ρ t | Accept i = v † Acc v t . Note that a x,t ( p ) is nondecreasing in t . Let a x ( p ) := lim t →∞ a x,t ( p ). Th en B QPSP A CE / coin is the class of languages L for whic h there exists a BQPSP ACE m achine M , as w ell as a sequence of advice coin biases { p n } n ≥ 1 , suc h that for all x ∈ { 0 , 1 } n : (1) If x ∈ L , then a x ( p n ) ≥ 2 / 3 . (2) If x / ∈ L , then a x ( p n ) ≤ 1 / 3 . 2.5 The Hellman-Co ver Theorem In 19 70 Hell man and Co v er [13] pro v ed the follo wing imp ortant result (for conv enience, we sta te only a sp ecial case). Theorem 6 (H ellman-Co v er Theorem [13 ]) L et $ p b e a c oin with bias p , and let M ($ p ) b e a pr ob abilistic finite a utomaton that takes as input an infinite se quenc e of indep e ndent flips of $ p , and c an ‘ halt and ac c ept’ or ‘halt and r eje ct’ at any time step. L et a t ( p ) b e the pr ob ability that M ($ p ) has ac c epte d after t c oin flips, and let a ( p ) = lim t →∞ a t ( p ) . Supp ose that a ( p ) ≤ 1 / 3 and a ( p + ε ) ≥ 2 / 3 , for some p and ε > 0 . Then M must h ave Ω ( p (1 − p ) /ε ) states. Let us make tw o remarks ab out Th eorem 6. First, the theorem is easily seen to b e essen tially tigh t: for an y p a nd ε > 0, o ne can constru ct a fi nite automaton with O ( p (1 − p ) /ε ) states such that a ( p + ε ) − a ( p ) = Ω (1). T o d o so, lab el the automaton’s states by intege rs in {− K , . . . , K } , for some K = O ( p (1 − p ) /ε ). Let the initial state b e 0. When ever a heads is seen, incremen t the state by 1 with probability 1 − p and otherwise do n othing; w henev er a tails is seen, d ecremen t the state by 1 with probabilit y p and otherwise do nothing. If K is ev er reac hed, then halt and accept (i.e., guess that the bias is p + ε ); if − K is ev er r eac hed, then halt and reject (i.e., guess that the bias is p ). Second, Hellman and Co ver actually prov ed a stronger result. Supp ose w e consider the relaxed mo del in w hic h the finite automato n M nev er needs to halt, and one defines a ( p ) to b e the fraction of time that M sp ends in a designated s u bset of ‘Accepting’ states in the limit of infi nitely many coin flips (this limit exists with p robabilit y 1). Th en the lo w er b ound Ω ( p (1 − p ) /ε ) on the n umb er of states still holds. W e will hav e more to sa y ab out fi nite automata that “accept in th e limit” in Section 5. 2.6 F acts A b out Polynomials W e now collect some u s eful facts ab out p olynomials and r ational functions, and ab out sm all-space algorithms for ro ot-finding and linear algebra. First w e will need the follo wing f act, whic h follo w s easily from L’Hˆ opital’s Rule. 7 Prop osition 7 Whenever the limit e xi sts, lim z → 0 c 0 + c 1 z + · · · + c m z m d 0 + d 1 z + · · · + d m z m = c k d k , wher e k i s the smal lest inte ger su c h that d k 6 = 0 . The next tw o facts are muc h less elemen tary . Firs t, we state a b oun d on the minimum sp acing b et w een zeros, for a low-degree p olynomial with intege r co efficien ts. Theorem 8 ([5, p. 359, Corollary 10.22]) L et P ( x ) b e a de gr e e- d u ni v ariate p olynomial, with inte ger c o efficients of bitlength at most τ . If z, z ′ ∈ C ar e distinct r o ots of P , then   z − z ′   ≥ 2 − O ( d log d + τ d ) . In p articular, if P is of de gr e e at most 2 poly ( n ) , and has inte ger c o efficients with absolute values b ounde d by 2 poly ( n ) , then | z − z ′ | ≥ 2 − 2 p oly( n ) . W e will need to lo cate the zeros of u niv ariate p olynomials to high precision u s ing a small amount of m emory . F ortunately , a b eautiful algorithm of Neff [14] f rom the 1990s (imp r o ved by Neff an d Reif [15] and by Pan [18]) provides exactly what we n eed. Theorem 9 ([14, 15, 18]) Ther e exists an algorithm that (i) T akes as input a triple ( P , i, j ) , wher e P is a de gr e e- d univariate p olynomial with r ational 4 c o efficie nts whose numer ators and denominators ar e b ounde d in absolute va lue by 2 m . (ii) Outputs the i th most sig ni fic ant bits of the r e al and imagina ry p arts of the binary exp ansion of the j th zer o of P (in some or der indep endent o f i , p ossibly with r ep e titions). (iii) Use s O (p olylog ( d + i + m )) sp ac e. W e will also need to in vert n × n matrices using p olylog ( n ) space. W e can do so usin g an algorithm of Boro d in, Co ok, and Pipp enger [7] (whic h w as also used for a similar app lication b y Aaronson and W atrous [2]). Theorem 10 (Boro din et al. [7, Corollary 4.4]) Ther e exists an algorithm tha t (i) T akes as input an n × n matrix A = A ( p ) , whose entries ar e r ational functions in p of de gr e e p oly ( n ) , with the c o efficients sp e cifie d to poly ( n ) bits of pr e cision. (ii) Computes det ( A ) (and as a c onse quenc e, also the ( i, j ) e ntry of A − 1 for any given c o or dinates ( i, j ) , assuming th at A is inve rtible). (iii) Use s p oly ( n ) time and p olylog ( n ) sp ac e. Note that the algorithms of [7, 14, 15, 18] are all stated as p ar al lel ( NC ) algorithms. Ho wev er, an y parallel algorithm can b e con verted into a space-efficien t algorithm, usin g a standard redu ction due to Boro din [6]. 4 Neff ’s original algor ithm assumes polyn omials with inte ger co efficients; the result for rational coefficients follo ws easily b y clearing denominators. 8 1                           0 Figure 1: A quan tum fin ite automaton that distinguish es a p = 1 / 2 coin f rom a p = 1 / 2 + ε coin, essen tially by u sing a qubit as an analog coun ter. 3 Quan tum Mec hanics Nu llifies the Hellman-Co v er Theorem W e now sho w that the quan tum analogue of the Hellman-Co v er Th eorem (Theorem 6) is false. Indeed, we w ill sho w that for any fi xed ε > 0, there exists a quantum fin ite automaton with only 2 states that can distin gu ish a coin with bias 1 / 2 from a coin with bias 1 / 2 + ε , with b ound ed probabilit y of error indep endent of ε . F ur th ermore, this automaton is ev en a halting automaton, whic h halts with p robabilit y 1 and ent ers either an | Accept i or a | Reject i state. The k ey idea is that, in this setting, a single qubit can b e used as an “analog coun ter,” in a w a y that a classical prob ab ilistic bit cannot. Admittedly , o ur r esult w ould fai l were the qubit sub ject to noise or decoherence, as it wo uld b e in a realistic physical situation. Let ρ 0 b e the designated starting state of the automaton, and let ρ 1 , ρ 2 , . . . , b e d efined as ρ t +1 = E p ρ t , with notation as in Section 2.4. Let a ( p ) := lim n →∞ h Accept | E n p ( ρ 0 ) | Accept i b e the limiting probabilit y of acceptance. This limit exists, as argued in S ection 2.4. W e no w pro v e Theorem 1, which w e restate for con venience. Fix p ∈ [0 , 1] and ε > 0 . Then ther e e xists a quantum finite automat on M with two states (not c ounting the | Accept i and | Reject i states), such that a ( p + ε ) − a ( p ) ≥ β for some c onstant β indep endent o f ε . (F or e xample, β = 0 . 0117 works.) Pro of of Theorem 1. The state of M will b elong to the Hilb ert sp ace s panned b y {| 0 i , | 1 i , | Accept i , | Reject i} . The initial state is | 0 i . Let U ( θ ) :=  cos θ − sin θ sin θ cos θ  9 b e a unitary transformation that rotates coun terclo ckwise by θ , in the “coun ter su bspace” spanned b y | 0 i and | 1 i . Also, let A and B b e p ositive in tegers to b e sp ecified later. Then the finite automaton M ru ns the follo wing pro cedure: (1) If a 1 bit is en coun tered (i.e., the coin land s heads), apply U ( ε (1 − p ) / A ). (2) If a 0 bit is en coun tered (i.e., the coin land s tails), apply U ( − εp/ A ) . (3) With p robabilit y α := ε 2 /B , “measure” (that is, mo ve all p robabilit y mass in | 0 i to | Reject i and all probab ility mass in | 1 i to | Accept i ); otherwise do n othing. W e n o w analyze the b eha v ior of M . F or simplicit y , let us first consider steps (1) and (2) only . In this ca se, w e can think o f M as ta king a random wa lk in the space o f p ossible angles b et wee n | 0 i and | 1 i . I n particular, after t steps, M ’s state will ha v e the form cos θ t | 0 i + sin θ t | 1 i , for some angle θ t ∈ R . (As w e follo w the wal k, we sim p ly let θ t increase or decrease without b oun d , rather than confining it to a range of size 2 π .) Supp ose th e coin’s bias is p . Th en after t steps, E [ θ t ] = pt · ε A (1 − p ) + (1 − p ) t ·  − ε A p  = 0 . On the other h and, supp ose the bias is q = p + ε . T hen E [ θ t ] = q t · ε A (1 − p ) + (1 − q ) t ·  − ε A p  = ε A t · [ q (1 − p ) − p (1 − q )] = ε 2 t A . So in particular, if t = K/ε 2 for so me constant K , then E [ θ t ] = K/ A . Ho wev er, we also need to understand the v ariance of the angle, V ar [ θ t ]. If the bias is p , then by the indep endence o f the coin flips, V ar [ θ t ] = t · V ar [ θ 1 ] = t ·  p  ε A (1 − p )  2 + (1 − p )  ε A p  2  ≤ ε 2 t A 2 , and lik ewise if the bias is q = p + ε . If t = K/ε 2 , this implies that V ar [ θ t ] ≤ K/ A 2 in b oth cases. W e n o w incorp orate step (3). Let T b e the n u m b er of steps b efore M halts (that is, b efore its state gets m easur ed). Then clearly Pr [ T = t ] = α (1 − α ) t . Also, let u := K/ε 2 for some K to b e 10 sp ecified later. Then if the bias is p , we can upp er-b oun d M ’s acceptance pr ob ab ility a ( p ) as a ( p ) = ∞ X t =1 Pr [ T = t ] · E  sin 2 θ t | t  ≤ Pr [ T > u ] + u X t =1 Pr [ T = t ] · E  sin 2 θ t | t  ≤ Pr [ T > u ] + u X t =1 Pr [ T = t ] · E  θ 2 t | t  ≤ (1 − α ) u + E  θ 2 u | u  ≤  1 − ε 2 B  B /ε 2 · K/B + ε 2 u A 2 ≤ e − K/B + K A 2 . Here the third li ne uses sin x ≤ x , while t he fourth lin e uses th e fact that E  θ 2 t  is nondecreasing for an unbiased random walk. So long as A 2 ≥ B , we can min im ize the fi nal expression by setting K := B ln  A 2 /B  , in whic h case we ha ve a ( p ) ≤ B A 2  1 + ln  A 2 B  . On the other han d , sup p ose the b ias is p + ε . Set v := L/ε 2 where L := π A/ 4. T hen for all t ≤ v , w e ha ve Pr [ | θ t | > π / 2 | t ] ≤ Pr      θ t − ε 2 t A     > π 2 − ε 2 t A | t  < ε 2 t/ A 2 ( π / 2 − ε 2 t/ A ) 2 ≤ ε 2 v / A 2 ( π / 2 − ε 2 v / A ) 2 = 4 π A where the second line uses Ch ebyshev’s inequalit y . Also, let ∆ t := θ t − ε 2 t/ A . T hen for all t ≤ v w e ha ve E  θ 2 t | t  = E "  ε 2 t A + ∆ t  2 | t # = ε 4 t 2 A 2 + E  ∆ 2 t | t  + 2 ε 2 t A E [∆ t | t ] ≥ ε 4 t 2 A 2 . 11 Putting the pieces together, w e can lo wer-b ound a ( p + ε ) as a ( p + ε ) = ∞ X t =1 Pr [ T = t ] · E  sin 2 θ t | t  ≥ v X t =1 Pr [ T = t ] · E  sin 2 θ t | t  ≥ v X t =1 Pr [ T = t ] · Pr [ | θ t | ≤ π / 2 | t ] · E  θ 2 t / 3 | t  ≥ v X t =1 α (1 − α ) t ·  1 − 4 π A  · ε 4 t 2 3 A 2 =  1 − 4 π A  ε 4 α 3 A 2 L/ε 2 X t =1  1 − ε 2 B  t t 2 ≥  1 − 4 π A  ε 4 α 3 A 2 e L/B L/ε 2 X t =1 t 2 ≥  1 − 4 π A  ε 6 3 A 2 B e L/B ·  L/ε 2  3 6 =  1 − 4 π A  L 3 18 A 2 B e L/B =  1 − 4 π A  π 3 A 1152 B e π A/ 4 B . Here th e third line u ses th e fact that sin 2 x ≥ x 2 / 3 for all | x | ≤ π / 2. If w e now c ho ose (for example) A = 10000 and B = 7500, then we h av e a ( p ) ≤ 0 . 0008 and a ( p + ε ) ≥ 0 . 0125, whence a ( p + ε ) − a ( p ) ≥ 0 . 01 17 . W e can strengthen Theorem 1 to ensure that a ( p ) ≤ δ and a ( p + ε ) ≥ 1 − δ for any desired error probabilit y δ > 0. W e sim p ly use standard amplification, whic h increases the n u m b er of states in M to O (p oly (1 /δ ) ) (or equiv alen tly , the num b er of qubits to O (log(1 /δ )) ). 4 Upp er-Bounding the Po w er of Advice Coins In this section w e pro v e Th eorem 2, th at BQPSP A CE / coin = B PPSP A CE / coin = PS P ACE / poly . W e start with th e easy half: Prop osition 11 PSP ACE / poly ⊆ BPPSP ACE / co in ⊆ B QPSP A CE / coin . Pro of. Giv en a p olynomial-size advice string w n ∈ { 0 , 1 } s ( n ) , we enco de w n in to th e first s ( n ) bits of th e b inary expans ion of an advice bias p n ∈ [0 , 1]. Then b y flippin g the coin $ p n sufficien tly man y times ( O  2 2 s ( n )  trials suffice) and tall ying the fractio n of heads, a T ur ing mac h ine can reco v er w n with high success probability . Counting out the desired num b er of trials and determining the fraction of heads seen c an b e d on e in space O  log  2 2 s ( n )  = O ( s ( n )) = O (p oly ( n )). Thus we can simulat e a PSP ACE / p oly mac hin e with a BPPSP A CE / coin mac h ine. 12 The rest of th e section is dev oted to sho wing that B QPSP A CE / coin ⊆ PSP ACE / p oly . First w e giv e some lemmas ab out quan tu m p olynomial-space advice coin algorithms. Let M b e such an algorithm. S upp ose M u ses s ( n ) = p oly ( n ) qubits of m emory , and has S = 2 s ( n ) states. Let E 0 , E 1 , E p b e the sup erop erators for M as describ ed in S ection 2.4. Recalling the vec torized notation from Section 2.4 , let B p := mat ( E p ). Let ρ x,t ( p ) b e the state of M after t coin fl ips steps on input x and coin bias p , and let v x,t ( p ) := v ec ( ρ x,t ( p )). Let a x,t ( p ) := v † Acc v x,t ( p ) b e the pr obabilit y that M is in th e | Accept i state, if measured after t steps. Le t a x ( p ) := lim t →∞ a x,t ( p ). As discus s ed in Section 2.4, the q u an tities a x,t ( p ) are nondecreasing in t , so the limit a x ( p ) is we ll-defined. W e no w sh o w that—e xcept possib ly at a fi n ite num b er of v alues— a x ( p ) is actually a r ational function of p , w hose degree is at most the num b er of s tates. Lemma 12 Ther e exist p olynomials Q ( p ) and R ( p ) 6 = 0 , of de gr e e at most S 2 = 2 poly ( n ) in p , such that a x ( p ) = Q ( p ) R ( p ) holds whenever R ( p ) 6 = 0 . M or e over, Q and R have r ational c o efficients that ar e c omputable in p oly ( n ) sp ac e given x ∈ { 0 , 1 } n and the index of the d esir e d c o efficient. Pro of. T hroughout, w e suppress the dep endence on x for con v enience, so that a ( p ) = lim t →∞ a t ( p ) is simply the limiting acceptance probability of a fin ite automaton M ($ p ) giv en a coin with b ias p . F ollo win g Aaronson and W atrous [2], for z ∈ (0 , 1) define the matrix Λ z ,p ∈ C S 2 × S 2 b y Λ z ,p := z [ I − (1 − z ) B p ] − 1 . The matrix I − (1 − z ) B p is inv ertible, since z > 0 and all eigen v alues of B p ha ve absolute v alue at most 1. 5 Using Cramer’s ru le, w e can represent eac h e n tr y of Λ z ,p in the form f ( z ,p ) g ( z ,p ) , wh ere f and g are biv ariate p olynomials of degree at most S 2 in b oth z and p , and g ( z , p ) is not identi cally zero. Note that b y collecting terms, we can write f ( z , p ) = c 0 ( p ) + c 1 ( p ) z + · · · + c S 2 ( p ) z S 2 g ( z , p ) = d 0 ( p ) + d 1 ( p ) z + · · · + d S 2 ( p ) z S 2 , for some co efficien ts c 0 , . . . , c S 2 and d 0 , . . . , d S 2 . No w let Λ p := lim z → 0 Λ z ,p . (1) Aaronson and W atrous [2] sho w ed th at Λ p is precisely the matrix represen tation mat ( E fix ) of the sup erop erator E fix asso ciated to E := E p b y Theorem 4. Thus we hav e B p (Λ p v ) = Λ p v for all v ∈ C S 2 . 5 F or the latter fact, see [19] and [2, p. 1 0, fo otnote 1]. 13 No w, the en tries of Λ z ,p are biv ariate ratio nal functions, whic h ha ve absolute v alue at most 1 for all z , p . Thus the limit in equation (1) must exist, and t he co effient s c k , d k can b e computed in p olynomial s p ace using Theorem 10. W e claim that ev ery en try of Λ p can b e represen ted as a rational function of p of degree at most S 2 (a repr esen tation v alid for all but fin itely man y p ), and that the co efficients of this rational function are computable in p olynomial space. T o see this, fix some i, j ∈ [ S ], and let (Λ p ) ij denote the ( i, j ) th en try of Λ p . By the ab o v e, (Λ p ) ij has the form (Λ p ) ij = lim z → 0 f ( z , p ) g ( z , p ) = lim z → 0 c 0 ( p ) + c 1 ( p ) z + · · · + c S 2 ( p ) z S 2 d 0 ( p ) + d 1 ( p ) z + · · · + d S 2 ( p ) z S 2 . By Prop osition 7, th e ab o ve limit (wh enev er it exists) equals c k ( p ) /d k ( p ), where k is the sm allest in teger such that d k ( p ) 6 = 0. No w let k ∗ b e the smallest in teger such that d k ∗ is not the iden tically- zero p olynomial. Then d k ∗ ( p ) has only finitely many zeros. It follo ws that (Λ p ) ij = c k ∗ ( p ) /d k ∗ ( p ) except when d k ∗ ( p ) = 0, whic h is what we wan ted to show. That th e coefficients are rational and computable in p olynomial space follo ws b y construction: we can loop through all k until w e find k ∗ as ab ov e, and then compu te the co efficien ts of c k ∗ ( p ) and d k ∗ ( p ). Finally , w e claim that we can write A ’s limiting acceptance probability a ( p ) as a ( p ) = v † Acc Λ p v 0 , (2) where v 0 is the v ectorize d initial s tate of A (indep enden t of p ). It will follo w from equation (2) that a ( p ) has the d esir ed rational-function represen tation, sin ce t he map Λ p → v † Acc Λ p v 0 is linear in the entries of Λ p and can b e p erf orm ed in p olynomial space. T o establish equation (2 ), consider the T aylo r series expansion for Λ z ,p , Λ z ,p = X t ≥ 0 z (1 − z ) t B t p , v alid for z ∈ (0 , 1) (see [2] for details). Th e equalit y X t ≥ 0 z (1 − z ) t = 1 , z ∈ (0 , 1) , implies that v † Acc Λ z ,p v 0 is a weig h ted av erage of t he t -step acceptance probabilities a t ( p ), for t ∈ { 0 , 1 , 2 , . . . } . Letting z → 0, the w eigh t on eac h individual step approac hes 0. Since lim t →∞ a t ( p ) = a ( p ), we obtain equation (2). The next lemma lets us “pat c h up” the finitely man y singularities, and sho w that a x ( p ) is a rational fun ction in the en tire op en interv al (0 , 1). 6 Lemma 13 a x ( p ) is c ontinuous for al l p ∈ ( 0 , 1) . Pro of. O nce again w e suppr ess the dep en dence on x , so that a ( p ) = lim t →∞ a t ( p ) is j ust the limiting acceptance p r obabilit y of a finite automaton M ($ p ). T o sho w that a ( p ) is contin uous on (0 , 1) , it suffi ces to sh o w that a ( p ) is co n tinuous on ev ery closed subinterv al [ p 1 , p 2 ] su c h th at 0 < p 1 < p 2 < 1. W e will pro v e this by proving the follo w in g claim: 6 Note that th ere could still b e singularities at p = 0 and p = 1, and this is n ot just an artifact of th e proof ! F or example, consider a finite automaton that accepts when and only when it sees ‘heads.’ The acceptance probability of such an automaton satisfies a ( 0) = 0, but a ( p ) = 1 for all p ∈ (0 , 1]. 14 (*) F or every subinterval [ p 1 , p 2 ] and eve ry δ > 0 , ther e exists a time t (not dep ending on p ) such that a t ( p ) ≥ a ( p ) − δ for al l p ∈ [ p 1 , p 2 ]. Claim (*) implies that a ( p ) can b e un iformly appro ximated by cont in u ous fun ctions on [ p 1 , p 2 ], and hence is contin uous itself on [ p 1 , p 2 ]. W e now prov e claim (*). First, call a mixed state ρ de ad for bias p if M ($ p ) halts with probabilit y 0 when run with ρ as its initial state . No w , the sup erop erator applied b y M ($ p ) at eac h time step is E p = p E 1 + (1 − p ) E 0 . Th is m eans that ρ is dead for any bias p ∈ (0 , 1) , if and only if ρ is dead for b ias p = 1 / 2. So w e can simp ly refer to suc h a ρ as de ad , with no dep enden ce on p . Recall that B p := mat ( E p ). O bserve that ρ is dead if and only if v † Acc B t 1 / 2 v ec ( ρ ) = 0 for all t ≥ 0. In particular, it follo ws that there exists a “dead subspace” D of C S , suc h that a pure state | ψ i is dead if and only if | ψ i ∈ D . (A mixed state ρ = P i p i | ψ i i h ψ i | is d ead if and only if | ψ i i ∈ D for all i s u c h th at p i > 0 .) By its definition, D is orthogonal to the | Acc ept i state. Define the “liv e su b space,” L , to b e the orthogonal complemen t of | Accept i and D . Let P b e the pro jector onto L , and let v Live := vec ( P ). Also, recalli ng that v 0 is the vecto r ized initial state of M , let g t ( p ) := v † Live B t p v 0 b e the probability that M ($ p ) is “still aliv e” if measured after t steps—i.e., th at M has neither accepted nor entered the dead subsp ace. Clearly a ( p ) ≤ a t ( p ) + g t ( p ). Th us, to prov e claim (*), it suffices to pr o ve that for all δ > 0, there exists a t (not dep ending on p ) suc h that g t ( p ) ≤ δ for all p ∈ [ p 1 , p 2 ]. First, let U b e the set of a ll ρ supp orted only on the live s ubspace L , and notice that U is compact. Therefore, by Lemma 5 (the “Leaky Su bspace Lemma”), there exists a constan t c 1 > 0 such th at, for all ρ ∈ U ,  v † Acc + v † Dead  B S 2 p 1 v ec ( ρ ) ≥ c 1 and hence v † Live B S 2 p 1 v ec ( ρ ) ≤ 1 − c 1 . Lik ewise, there exists a c 2 > 0 suc h that, for all ρ ∈ U , v † Live B S 2 p 2 v ec ( ρ ) ≤ 1 − c 2 . Let c := min { c 1 , c 2 } . T h en by con v exit y , for all p ∈ [ p 1 , p 2 ] and all ρ ∈ U , we ha ve v † Live B S 2 p v ec ( ρ ) ≤ 1 − c, and hence v † Live B S 2 t p v ec ( ρ ) ≤ (1 − c ) t for all t ≥ 0. Th is means that, to ensu re th at g t ( p ) ≤ δ for all p ∈ [ p 1 , p 2 ] simulta neously , we just need to c h o ose t large enough that ( 1 − c ) t/S 2 ≤ δ . This pro ves claim (*). 15 W e are no w ready to complete th e p ro of of Theorem 2. Let L b e a language in BQPSP ACE / co in , whic h is decided b y the quan tum p olynomial-space advice -coin mac hine M ( x , $ p ) on advice coin biases { p n } n ≥ 1 . W e w ill sho w that L ∈ BQPSP ACE / p oly = PSP ACE / p oly . It ma y not b e p ossible to p erfectly sp ecify the bias p n using p oly ( n ) bits of advice. Instead, w e use our advice string to simula te acce ss to a second bias r n that is “almost as go o d” as p n . This is ac hiev ed b y the follo win g lemma. Lemma 14 Fixing L, M , { p n } as ab ove, ther e exists a classic al p olynomial-sp ac e algorithm R , as wel l as a family { w n } n ≥ 1 of p olynomial-size advic e strings, for which the fol lowing holds. Given an index i ≤ 2 poly ( n ) , the c omputation R ( w n , i ) outputs the i th bit of a r e al numb er r n ∈ (0 , 1) , such that for al l x ∈ { 0 , 1 } n , (i) If x ∈ L , then Pr [ M ( x, $ r n ) ac c epts ] ≥ 3 / 5 . (ii) If x / ∈ L , then Pr [ M ( x, $ r n ) ac c epts ] ≤ 2 / 5 . (iii) The binary exp ansion of r n is identic al ly zer o, for su fficiently lar ge indic es j ≥ h ( n ) = 2 poly ( n ) . Once Lemma 14 is p ro ved, sho win g the c on tainment L ∈ B QPSP A CE / p oly is easy . Firs t, w e claim that using the advice f amily { w n } , w e can sim ulate access to r n -biased coin flips, as follo w s. Let r n = 0 .b 1 b 2 . . . denote the b in ary expansion of r n . given w n j := 0 while j < h ( n ) let z j ∈ { 0 , 1 } be random b j := R ( w n , j ) if z j < b j then output 1 else if z j > b j then output 0 else j := j + 1 output 0 Observe that this algorithm, which ru ns in p olynomial s pace, outputs 1 if an d only if 0 .z 1 z 2 . . . z h ( n ) < 0 .b 1 b 2 . . . b h ( n ) = r n , and this o ccurs w ith probabilit y r n . Thus we can simula te r n -biased coin fl ips as claimed. W e define a BQPSP A CE / p oly mac hine M ′ that take s { w n } fr om Lemma 14 as its advice. Given an inpu t x ∈ { 0 , 1 } n , the mac hine M ′ sim u lates M ( x, $ r n ), by generati ng r n -biased coin flips using the metho d d escrib ed ab o v e. Then M ′ is a BQPS P ACE / poly a lgorithm for L b y parts (i) a nd (ii) of Lemma 14, alb eit with error b ounds (2 / 5 , 3 / 5) . The error b oun ds can b e b o osted to (1 / 3 , 2 / 3) b y ru nning sev eral indep en den t trials. S o L ∈ BQPSP A CE / p oly = PSP A CE / poly , completing the pro of of Th eorem 2 . Pro of of Lemma 14. Fix an input length n > 0, and let p ∗ := p n . F or x ∈ { 0 , 1 } n , recall that a x ( p ) denotes the acceptance prob ab ility o f M ( x, $ p ). W e are inte rested in the wa y a x ( p ) oscillate s as we v ary p . Defin e a tr ansition p air to b e an ordered pair ( x, p ) ∈ { 0 , 1 } n × ( 0 , 1) su c h that a x ( p ) ∈ { 2 / 5 , 3 / 5 } . It will b e also b e con venien t to define a larger set of p otential tr ansition 16 p airs , denoted P ⊆ { 0 , 1 } n × [ 0 , 1), that contai ns the transition pairs; the b enefit of considering this larger set is that its elemen ts will b e easier to enumerate. W e d efer the precise defin ition of P . The advice string w n will simply sp ecify the numb er of distinct p otential tr ansition p airs ( y , p ) such that p ≤ p ∗ . W e first give a high-lev el ps eu do co de description of th e algorithm R ; after pro ving that parts (i) and (ii) of Lemma 14 are met b y th e algorithm, w e will fill in the algorithmic details to sho w th at the pseudo co de can b e implemen ted in PSP A CE , and that w e can satisfy part (iii) of the Lemm a. The pseudo co de for R is as follo ws: given ( w n , i ) for all ( y , p ) ∈ P s := 0 for all ( z , q ) ∈ P if q ≤ p then s := s + 1 next ( z , q ) if s = w n then let r n := p + ε (fo r some small ε = 2 − 2 p oly( n ) ) output the i th bit of r n end if next ( y , p ) W e no w pr o ve that parts (i) and (ii) of Lemma 14 are satisfied. W e call p ∈ [0 , 1) a tr ansition value if ( y , p ) is a transition pair f or some y ∈ { 0 , 1 } n , and w e call p a p otential tr ansition value if ( y , p ) ∈ P for some y ∈ { 0 , 1 } n . Th en by defin ition of w n , the v alue r n pro du ced ab o v e is equal to p 0 + ε , where p 0 ∈ [0 , 1) is th e largest p oten tial tran s ition v alue less than or equal to p ∗ . (Note that 0 will alw a ys b e a p otentia l transition v alue, so this is well-defined.) When w e define P , w e will argue that any distinct p oten tial transition v alues p 1 , p 2 satisfy min {| p 1 − p 2 | , 1 − p 2 } ≥ 2 − 2 p oly( n ) . (3) It follo ws that if ε = 2 − 2 p oly( n ) is su itably small, then r n < 1, and there is no p oten tial transition v alue lying in the range ( p 0 , r n ]. Also, there are n o p oten tial transition v alues in the interv al ( p 0 , p ∗ ). No w fix any x ∈ { 0 , 1 } n ∩ L . Since M is a BQPSP ACE / co in mac hine for L with bias p ∗ , w e ha ve a x ( p ∗ ) ≥ 2 / 3. If a x ( r n ) < 3 / 5, then Lemma 13 implies that th ere must b e a transition v alue in the op en in terv al b et ween p ∗ and r n . But there are no suc h transition v alues. Thus a x ( r n ) ≥ 3 / 5 . Similarly , if x ∈ { 0 , 1 } n \ L , then a x ( r n ) ≤ 2 / 5 . This establishes p arts (i) and (ii) of Lemma 14. No w w e formally defin e the p otenti al transition p airs P . W e include (0 n , 0) in P , guaran teeing that 0 is a p oten tial transition v alue as r equired. No w recall, by Lemma 12, that for eac h x ∈ { 0 , 1 } n , the acceptance probabilit y a x ( p ) is a rational fu nction Q x ( p ) /R x ( p ) of degree 2 poly ( n ) , for all b ut fi n itely many p ∈ (0 , 1). Th erefore, the fun ction ( a x ( p ) − 3 / 5) ( a x ( p ) − 2 / 5) also h as a rational-function repr esen tation: U x ( p ) V x ( p ) =  a x ( p ) − 3 5   a x ( p ) − 2 5  , v alid for all but finitely man y p . W e will include in P all pairs ( x, p ) for whic h U x ( p ) = 0. It follo ws from Lemmas 12 and 13 that P con tains all transition p airs, as desired. 17     ! Figure 2: Graph ical d epiction of the pro of of T heorem 2, that BQPSP A CE / coin = PSP ACE / p oly . F or eac h in put x ∈ { 0 , 1 } n , the acceptance probability of the BQPSP ACE / coin mac hine is a rational function a x ( p ) of the coin bias p , with degree at most 2 poly ( n ) . Su c h a function can cross the a x ( p ) = 2 / 5 or a x ( p ) = 3 / 5 lines at most 2 poly ( n ) times. So even consider in g al l 2 n inputs x , there can b e at most 2 poly ( n ) crossings in total . It follo ws that, if w e w ant to sp ecify whether a x ( p ) < 2 / 5 or a x ( p ) > 3 / 5 for all 2 n inputs x sim ultaneously , it suffices to giv e only p oly ( n ) bits of information ab ou t p (for example, the total num b er of crossings to the left of p ). W e can no w establish equation (3). Fix an y distinct p oten tial transition v alues p 1 < p 2 in [0 , 1). Since p 2 6 = 0 is a p oten tial transition v alue, there is some x 2 suc h that ( x 2 , p 2 ) ∈ P . If p 1 = 0, then p 1 , p 2 are distinct ro ots of th e p olynomial pU x 2 ( p ), when ce | p 1 − p 2 | ≥ 2 − 2 p oly( n ) b y Theorem 8. Similarly , if p 1 > 0, then ( x 1 , p 1 ) ∈ P for some x 1 . W e observe that p 1 , p 2 are co mmon ro ots of U x 1 ( p ) U x 2 ( p ), fr om w h ic h it again foll o ws that | p 1 − p 2 | ≥ 2 − 2 p oly( n ) . Finally , 1 − p 2 ≥ 2 − 2 p oly( n ) follo ws since 1 and p 2 are distinct r o ots of (1 − p ) U x 2 ( p ). T h us equation (3) holds. Next we sho w that the pseudo co de can b e imp lemented in PS P ACE . Observe first that the degrees of U x , V x are 2 poly ( n ) , w ith rational co efficien ts ha v in g n u merator and denominator b ounded b y 2 poly ( n ) . Moreo ve r, the co efficient s of U x , V x are co mputable in PSP ACE from the coefficien ts of Q x , R x , and these coefficien ts are themselv es PSP ACE -co m putable. T o loop o v er the elemen ts of P as in the for-lo ops of the pseud o co de, w e can p erform an outer lo op o v er y ∈ { 0 , 1 } n and an inner lo op o ver the zeros of U y . These zeros are indexed by Neff ’s algorithm (T h eorem 9) and can b e lo op ed o v er with that indexing. T he algorithm of Theorem 9 ma y return dup licate ro ots, b ut these can b e iden tified and remo v ed by comparing eac h ro ot in tur n to all previously visited ro ots. F or e ac h pair of d istinct zeros of U y differ in their b in ary expansion to a su fficien tly large 2 poly ( n ) n um b er of bits (b y Theorem 8), and Theorem 9 allo w s us to compare suc h bits in p olynomial sp ace. Similarly , if ( y , p ) , ( z , q ) ∈ P then we can determine in PSP ACE whether q ≤ p , as requ ir ed. The only remaining implement ation step is to pr o duce the v alue r n in PSP A CE , in suc h a wa y that part (iii) of Lemma 14 is satisfied. Giv en the v alue p c hosen by the in ner lo op, and the index i ≤ 2 poly ( n ) , w e need to pr o duce the i th bit of a v alue r n ∈  p, p + 2 − 2 p oly( n )  , such that the binary expansion of r n is iden tically zero for sufficien tly large j ≥ h ( n ) = 2 poly ( n ) . But this is easily d one, since we can compu te an y desired j th bit of p , for j ≤ 2 poly ( n ) , in p olynomial sp ace. 18 5 Distinguishing Problems for Finite Automata The distinguishing pr oblem , as describ ed in Section 1, is a natural problem with which to in v estigate the p o wer of restricted mo dels of computation. The b asic task is to distinguish a coin of bias p from a coin of b ias p + ε , usin g a fi nite automaton with a b ounded num b er of states. Sev eral v ariations of this problem hav e b een explored [13, 10], which mo dify either the mo del of compu tation or the mo de of acceptance. A basic question to explore in ea c h case is whether the distinguishin g task can b e solv ed b y a finite automaton whose n u m b er of states is indep endent of the v alue ε (for fixed p , sa y). V ariations of interest in clude: (1) Classic al vs. quantum finite automata. W e show ed in Section 3 that, in some cases, quantum finite automata can solve the distinguish ing p roblem where classical ones cannot. (2) ε -dep endent vs. ε -indep endent automata. Can a single automaton M distin gu ish $ p from $ p + ε for ev ery ε > 0, or is a different automaton M ε required for d ifferent ε ? (3) Bias 0 vs. bias 1 / 2 . Is the setting p = 0 easier th an the setting p = 1 / 2? (4) Time-dep endent vs. time-indep endent automata. An alternativ e, “non uniform” mo del of finite automata allo ws their state-transition f u nction to dep end on the curr ent time step t ≥ 0, as w ell a s on the current state and the cu rrent b it b eing read. This dep end ence on t can b e arbitrary; the transition function is not requ ired to b e computable giv en t . (5) A c c eptanc e by halting vs. 1 -side d ac c eptanc e vs. ac c eptanc e in the limit. Ho w do es the finite automaton register its final decision? A first p ossibilit y is th at the automato n halts and en ters an | Accept i s tate if it thinks the bias is p + ε , or halt s and enters a | Reject i sta te if it thinks the bias is p . A second p ossibilit y , whic h corresp ond s to the model considered for most of this pap er, is that the automaton halts and ent ers an | Accept i state if it thin ks the bias is p + ε , but can r eject b y simply n ev er halting. A third p ossibility is that the automaton never n eeds to halt. In this third mo del, w e d esignate some sub set of the states as “accepting states,” and let a t b e the pr ob ab ility th at the automaton would b e found in an accepting state, w ere it measured at the t th time step. Then the a utomaton is said t o ac c ept in the limit if lim inf t →∞ ( a 1 + . . . + a t ) /t ≥ 2 / 3, and to r eje ct in the limit if lim s up t →∞ ( a 1 + · · · + a t ) /t ≤ 1 / 3. The automaton solv es t he d istinguishing p roblem if it accepts in the limit on a coin of bias p + ε , and rejects in the limit on a coin of bias p . F or almost ev ery p ossible com bination of th e ab o ve, we can determine whether the distinguish in g problem can b e solve d by an automaton w hose num b er of states is ind ep endent of ε , by u sing the results and tec hniqu es of [13, 10] as w ell as the p resen t pap er. Th e situation is summarized in the follo wing t wo tables. 19 Classical case Coin distinguishing task 1 2 vs. 1 2 + ε 0 vs. ε T yp e of automaton Halt 1-Sided Limit Halt 1-Sided Limit Fixed No No No No Y es (easy) Y es ε -dep endent No No No [13] Y es (easy) Y es Y es Time-dep end ent No Y es [10] Y es No Y es Y es ε ,time-dep endent Y es [10] Y es Y es Y es Y es Y es Quan tum case Coin distinguishing task 1 2 vs. 1 2 + ε 0 vs. ε T yp e of automaton Halt 1-Sided Limit Halt 1-Sided Limit Fixed No No (here) ? No Y es Y es ε -dep endent Y es (here) Y es Y es Y es Y es Y es Time-dep end ent No Y es Y es No (easy) Y es Y es ε ,time-dep endent Y es Y es Y es Y es Y es Y es Let us br iefly discuss the p ossib ilit y an d imp ossib ilit y r esults. (1) Hellman and C ov er [13] sho w ed that a classical fi nite automaton needs Ω (1 /ε ) states to distinguish p = 1 / 2 from p = 1 / 2 + ε , ev en if the transition probabilities can dep end on ε and the automaton only needs to succeed in the limit. (2) By con trast, T heorem 1 sho ws that an ε -dep end en t quan tu m finite automaton with only two states can distin gu ish p = 1 / 2 from p = 1 / 2 + ε for an y ε > 0, even if the automaton needs to halt. (3) Cov er [10] ga v e a constru ction of a 4-state time- dep endent (but ε -indep endent) classical finite automaton that distinguishes p = 1 / 2 from p = 1 / 2 + ε , f or an y ε > 0, in the limit of infinitely man y coin flip s. Th is automaton can eve n b e m ade to halt in the case p = 1 / 2 + ε . (4) It is easy to mo dif y Co ver’s construction to get, for any fixe d ε > 0, a time-dep endent, 2-state finite automaton that distinguish es p = 1 / 2 from p = 1 / 2 + ε with h igh probabilit y and that halts. Ind eed, we simply need to lo ok for a ru n of 1 /ε consecutiv e heads, rep eating this 2 1 /ε times b efore halting. If suc h a run is foun d, then we guess p = 1 / 2 + ε ; otherw ise w e guess p = 1 / 2. (5) If w e merely wan t to distinguish p = 0 from p = ε , then ev en simpler constru ctions suffice. With an ε -dep end en t finite automaton, at eve ry time step w e flip the co in with probabilit y 1 − ε ; otherwise we halt and guess p = 0. If the coin ev er land s heads, then w e h alt an d output p = ε . Indeed, ev en an ε -indep endent finite automato n can d istinguish p = 0 fr om p = ε in the 1-sided m o del, b y flipping the coin o ver and o ver, and accepting if the coin ev er lands heads. (6) It is not hard to sho w that ev en a time-dep endent, q u antum fi nite automaton cannot solv e the d istinguishing problem, ev en for p = 0 v er s us p = ε , provided that (i) the automato n has t o halt when ou tp utting its answ er, and (ii) the s ame automato n has to work for every 20 ε . T he argumen t is simp le: give n a candidate automaton M , ke ep decreasing ε > 0 unti l M halts, with high pr ob ab ility , b efore observing a single hea ds. This m ust b e p ossible, sin ce ev en if p = 0 (i .e., the coin ne v er lands heads), M still needs to halt with high probabilit y . Th us, w e can simply w ait for M to halt with high probabilit y—sa y , after t coi n flip s—and then set ε ≪ 1 /t . On ce we h a ve done this, w e h a ve found a v alue of ε such that M cannot distinguish p = 0 fr om p = ε , s in ce in b oth cases M sees only tails with high pr obabilit y . 6 Op en Problems (1) Ou r advice-coin computational mo del can b e generalized significan tly , as follo ws. Let BQPSP A CE / dice ( m, k ) b e the class of languages decidable by a B QPSP A CE mac hine that can samp le fr om m distributions D 1 , . . . , D m , eac h of whic h ta kes v alues in { 1 , . . . , k } (t h us, these are “ k -sided dice”). Note that BQ PSP ACE / co in = BQPSP ACE / dice (1 , 2). W e conjecture that BQPSP A CE / dice (1 , p oly ( n )) = BQPSP A CE / dice ( p oly ( n ) , 2) = PSP ACE / p oly . F ur thermore, we are hop eful that the tec hn iques of this p ap er can shed ligh t on this and similar questions. 7 (2) Not all com binations of mo del features in Section 5 are well- understo o d. In particula r, can w e distinguish a c oin with b ias p = 1 / 2 from a c oin with bias p = 1 / 2 + ε usin g a quan tum finite automaton, not dep end en t on ε , that only needs to su cceed in the limit? 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