Hypothesis elimination on a quantum computer

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1 Hypothesis elimination on a quantum computer arxiv:quant-ph/042025v 3 Dec 2004 Andrei N. Soklakov and Rüdiger Schack Department of athematics, Royal Holloway, University of London Egham, Surrey, TW20 0EX, UK 3 December 2004 Abstract Hypothesis elimination is a special case of Bayesian updating, where each piece of new data rules out a set of prior hypotheses. We describe how to use Grover s algorithm to perform hypothesis elimination for a class of probability distributions encoded on a register of qubits, and establish a lower bound on the required computational resources. Introduction In the standard setting of Bayesian inference one starts from a set of hypotheses H = {h} and a set of possible data D = {d}. Hypotheses and data are connected via conditional probabilities p(d h), known as the model. Given a prior probability distribution p(h), the updated probability p(h d) of the hypothesis h to be true given that the piece of data d was observed is obtained from Bayes s rule, [] p(h d) = p(d h)p(h). () p(d h)p(h) h H In this paper, we consider the problem of hypothesis elimination, which is a special case of Bayesian updating where the model is of the form { 0 if h is ruled out by d, p(d h) = otherwise. c h (2) The positive constant c h does not depend on d and is determined by normalization. We assume that there is a finite number, N, of hypotheses, which we label 0 to N, i.e., in Proceedings of the 7th International Conference on Quantum Communication, easurement and Computing (QCC 04), edited by Stephen. Barnett (AIP Press, elville, NY, 2004). a.soklakov@rhul.ac.uk r.schack@rhul.ac.uk

2 H = {0,,..., N }. Furthermore, we assume that the prior p(h) has been obtained by hypothesis elimination from an initial uniform prior distribution on H. There are several possible ways of encoding a probability distribution p(h) on a quantum register. Here we represent p(h) by the state ψ = h=0 p(h) h, (3) where h are computational basis states [2] of a register formed of log 2 N qubits. To be specific, we assume that the prior p(h) has been obtained by k hypothesis elimination steps (k > ). We thus assume that the prior is given in the form of a sequence of oracles o,..., o k, where { 0 if h is ruled by the data in step j, o j (h) = (4) otherwise. Likewise, we assume that the model Eq. (2) is given as an oracle o k, where o k (h) = c h p(d h). The sequence of oracles o j gives rise to a new sequence O,...,O k defined by { if i j : oi (h) =, O j (h) = (5) 0 otherwise. For each O j, we define the set of solutions, Ω j = {h : O j (h) = }, and denote by j the corresponding number of solutions, j = For j {k, k}, we now define the states h=0 Ψ j = j O j (h). (6) h Ω j h. (7) The state Ψ k encodes the prior p(h), and Ψ k encodes the posterior p(h d). 2 Quantum hypothesis elimination The problem of hypothesis elimination now takes the following form: Given a quantum register in the prior state Ψ k and given the oracles O,...,O k, transform the register state into the posterior state Ψ k. One can use Grover s algorithm [3] to solve this problem as follows. Define a quantum oracle corresponding to O k via Ô k h = ( ) O k(h) h. (8) There are standard techniques [2] to implement Ôk in the form of a quantum circuit. The Grover operator Ĝ(O k) associated with the oracle is then defined as Ĝ(O k ) = ( 2 Ψ 0 Ψ 0 l ) Ô k, (9) 2

3 where l is the identity operator and Ψ 0 = x (0) N is the equal superposition state. The posterior state Ψ k can now be prepared by repeated application of the Grover operator Ĝ(O k) to the equal superposition state Ψ 0. This requires (π/4) N/ k calls of the oracle Ôk. Notice that the hypothesis elimination algorithm outlined above makes no direct use of the prior state Ψ k. This raises the following question: Is it possible to reduce the number of Grover iterations (and therefore oracle calls) required to prepare Ψ k by starting from the prior state Ψ k instead of the equal superposition state Ψ 0? In other words, can one make use of the computational effort that went into preparing the prior state Ψ k in order to obtain the posterior state Ψ k more efficiently? As already suggested by the results in Ref. [4], the answer to this question is negative. Here we prove the following result. Consider the family of oracles O that consists of O k, O k and all possible combinations of O k and O k (see Eq (29) for the precise definition of this family). Now consider all possible algorithms that consist of applying the corresponding Grover operators x=0 A abc... =... Ĝ(O c)ĝ(o b)ĝ(o a), O a, O b, O c, O. () Then, in the limit of large N, any algorithm A abc... requires at least ( 2/8) N/ k oracle calls from the above family to convert Ψ k into Ψ k. In other words, making direct use of the prior state Ψ k does not improve the asymptotic cost of O( N/ k ) oracles calls required to prepare Ψ k. 3 Proof Consider an oracle O which accepts out of the total N hypotheses h: h=0 good h O(h) =. (2) We shall call such hypotheses good, as opposed to bad hypotheses that are rejected by the oracle. Using different notation [5] for the amplitudes of good and bad hypotheses, we have that after t consecutive applications of the Grover operator Ĝ(O) an arbitrary quantum state Ψ ini = gh ini h + b ini h h (3) is transformed into Ψ fin = Ĝ t (O) Ψ ini = good h bad h g fin h h + bad h b fin h h. (4) 3

4 Let ḡ ini and b ini be the averages of the initial amplitudes corresponding to the good and the bad hypotheses respectively: ḡ ini = good h and similarly for the final amplitudes ḡ fin = Let us also define g ini h good h g ini h, bini = g fin h, bfin = bad h bad h b ini h, (5) b fin h, (6) = gini h ḡini, b ini h = bini h b ini. (7) In other words, gh ini and bini h define the features of the initial amplitude functions gini h and b ini h relative to their averages ḡ ini and b ini. Biham et. al. [4] have shown that the change of the amplitudes is essentially determined by the change of the averages: gh fin b fin h = ḡ fin + gh ini = b fin + ( ) t b ini h, (8) where the averages ḡ fin and b fin are given as follows. Define ( ω = arccos 2 ), (9) N α = b ini 2 + ḡ ini 2, (20) ( ) ḡ ini φ = arctan. (2) bini The averages are given by ḡ fin = α sin(ωt + φ), bfin = α cos(ωt + φ). (22) Let us also define the separation of the averages ini = ḡ ini b ini, fin = ḡ fin b fin. (23) Equations (8) imply that after t applications of the Grover operator the values of individual amplitudes gh fin of accepted hypotheses do not change relative to their average ḡfin g fin h ḡfin = g ini h = gini h ḡini. (24) The same is true for the rejected hypotheses if b ini = 0 or t is even. This observation suggests that the action of any algorithm of the type A aabbcc... =...Ĝ2 (O c )Ĝ2 (O b )Ĝ2 (O a ) (25) 4

5 can be analyzed by looking at the changes of the average amplitudes of the accepted and rejected hypotheses relative to various oracles Ôa, Ôb,.... Before we proceed with this analysis let us first clarify what kind of oracles are relevant to this problem. Let f be a real-valued step function which takes only three values, f if O k (h) =, f(h) = f 2 if O k (h) =, but O k (h) = 0, (26) f 3 if O k (h) = 0. Let us also require that h=0 f2 (h) =, and introduce a quantum state Ψ(f) that is defined by f in a natural way, Ψ(f) = h=0 f(h) h. (27) Evidently, both Ψ k and Ψ k can be written in this way. It follows from the above discussion that the action of the operator Ĝ2 (O k ) on the state (27) can be completely described by the changes of f, f 2 and f 3. oreover Ĝ2 (O k ) preserves the value of δ k = f 2 f 3. Similarly, the action of Ĝ2 (O k ) on (27) can be completely described by the changes of f, f 2 and f 3, and it preserves the value of δ k = f f 2. In general, for any oracle O, the corresponding operator Ĝ2 (O) preserves the amplitude differences between any two hypotheses for as long as either both hypotheses are accepted or both are rejected by O. However, if f good and f bad denote the amplitudes of an accepted and a rejected hypothesis, respectively, then the difference δ = f good f bad is changed by an amount which satisfies the inequality 4 2, (28) where is the number of accepted hypotheses with respect to O (see the Appendix). Using this inequality the action of any algorithm of the type (25) can be analyzed by calculating how individual changes of δ k and δ k accumulate during the action of the algorithm. In order to convert Ψ k into Ψ k the net result of such changes must be sufficient to increase δ k from 0 to / k and decrease δ k from / k to 0. It follows that all oracles that are relevant for this task can be obtained from O k and O k. Since each oracle is completely characterized by the set of acceptable hypotheses, the relevant family of oracles generated by O k and O k can be written out as oracles that correspond to the sets Ω k, Ω k, Ω k Ω k, and the complementary Ω k, Ωk, Ω k Ω k. (29) Let us consider the first three oracles from the family defined by the sets (29), namely the oracles that accept hypotheses from the sets Ω k, Ω k, Ω k Ω k. (30) 5

6 Oracles that correspond to the complementary sets in (29) can be analyzed in a completely analogous manner. The oracles corresponding to the sets (30) are O k, O k and Ok k, where { if h Ok (h) k Ωk = Ω k (3) 0 otherwise. Using the inequality (28) we see that, regardless of its position in the algorithm, the operator Ĝ2 (O k ) changes δ k by at most d = 4 2/ k without changing δ k. Similarly, the operator Ĝ2 (O k ) changes δ k by at most d 2 = 4 2/ k without affecting the value of δ k. The Grover operator with the combined oracle Ĝ(Ok k ) changes both δ k and δ k by an equal amount that does not exceed d 3 = 4 2/ N ( k k ). Let N(O) be the number of times that the oracle O is called within the algorithm. We would like to find a lower bound on the total number of oracle calls, N total = N(O k )+ N(O k ) + N(O k k ), that is needed by the algorithm to convert Ψ k into Ψ k. Let x, y and z be the number of times that the operators Ĝ2 (O k ), Ĝ 2 (O k ) and Ĝ2 (O k k ) appear in the algorithm. Then N total is bounded from below by the minimal value of 2(x + y + z) subject to the constraints xd + zd 3 U, yd 2 + zd 3 V, (32) where U = / k and V = / k are the required changes of δ k and δ k respectively. This is a simple linear optimization problem that should be considered in the nonnegative x, y, z octant. Keeping k and k constant we obtain that in the limit of large N the value of N total is approaching ( 2/4) N/ k. For any algorithm A abc... the action of the corresponding sequence of Grover operators can be rewritten in the form similar to that in Eq. (25). Using the definition of the Grover operator, we have Ĝ(O an )...Ĝ(O a 3 )Ĝ(O a 2 )Ĝ(O a ) = (ÂÔa n )...(ÂÔa 3 )(ÂÔa 2 )(ÂÔa ), (33) where  = 2 Ψ 0 Ψ 0 l. Using the fact that (Ôk) 2 = l and Â2 = l we obtain (ÂÔa n )...(ÂÔa 3 )(ÂÔa 2 )(ÂÔa ) = ÂÔa n...âôa 3 ÂÔa 2 (ÂÔa ) (ÂÔa ) 2 = ÂÔa n...âôa 3 Â(Ôa a 2 Â)(ÂÔa ) 2, (34) where Ôa a 2 = Ôa Ô a2. Denoting Ôa a 2...a j = j p= Ôa p we proceed ÂÔa n... ÂÔa 3 Â(Ôa a 2 Â)(ÂÔa ) 2 =...ÂÔa 3 Â(Ôa a 2 Â) (Ôa a 2 Â) 2 (ÂÔa ) 2 =...ÂÔa 4 (ÂÔa a 2 a 3 )(Ôa a 2 Â) 2 (ÂÔa ) 2 = ˆRĜ±2 (O a...a n )...Ĝ2 (O a a 2 a 3 )Ĝ 2 (O a a 2 )Ĝ2 (O a ), (35) where the + and signs are chosen for odd and even values of n, respectively, O a...a j denote classical oracles that correspond to the quantum oracle Ôa...a j, and ˆR is a residual operator { Ĝ(Oa...a ˆR = n ) if n is even, (36) Ĝ (O a...a n ) otherwise. 6

7 Since all oracles, Ô a...a j, belong to the family associated with the sets (29), the above arguments allow us to derive a bound on the minimum number of oracle calls from this family that are required to convert Ψ k to Ψ k. Indeed, transformations between Eqs. (33) and (36) at most double the number of oracle calls that are used by the original algorithm. To be more precise, if n is the number of oracle calls used by the original algorithm (see the left-hand side of Eq. (33)), then the equivalent modified algorithm, defined by the right-hand side of Eq. (35), requires at most 2n+ oracle calls. It remains to note that after the application of the residual operator that concludes the modified algorithm one has to arrive at the target state Ψ k, or, which is equivalent, the algorithm Ĝ ±2 (O a...a n )...Ĝ2 (O a a 2 a 3 )Ĝ 2 (O a a 2 )Ĝ2 (O a ) (37) must prepare the state ψ k = G ± (O a...a n ) Ψ k. In the limit of large N the state ψ k coincides with Ψ k. Using an analysis analogous to that of algorithm (25) we therefore conclude that, in the limit of large N, the algorithm (37) requires at least ( 2/4) N/ k oracle calls to convert Ψ k into ψ k. The original algorithm, therefore, will need, asymptotically, at least ( 2/8) N/ k oracle calls to convert Ψ k into Ψ k. Appendix Using the notation of Eqs. (2 24), the inequality (28) can be written as fin ini 4 2, (38) where the number of iterations is t = 2. By definition, we have N fin ini = α sin(2ω + ξ) sin ξ N = α 2 sin ω cos(ω + ξ) N 4α sin ω. (39) 2 Since 0 ω π and therefore sin ω 2 = cosω 2 = N, (40) fin ini 4α. (4) A bound on α can be obtained using the fact that, for any a, a 2,...,a R such that k= (a k) 2 =, we have ( ) max a k =. (42) k= 7

8 This can be easily shown using the method of Lagrange multipliers. Using (42) we obtain It then follows that maxḡ =, and max b = α < max b 2 + maxḡ 2 = Combining this bound with Eq. (4) we obtain Eq. (38) as intended. References. (43) 2. (44) [] J.. Bernardo and A. F.. Smith, Bayesian Theory (Wiley, Chichester, England, 994). [2]. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). [3] L. K. Grover, Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett. 79, 325 (997). [4] E. Biham, O. Biham, D. Biron,. Grassl, and D. A. Lidar, Grover s Quantum Search Algorithm for an Arbitrary Initial Amplitude Distribution, Phys. Rev. A 60, 2742 (999). [5] A. N. Soklakov and R. Schack, Efficient state preparation for a register of quantum bits, quant-ph/

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