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[Quant-Ph] 8 Feb 2021 Ups That Suppose H W Rbesaeqatmplnmal Equivalent

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[Quant-Ph] 8 Feb 2021 Ups That Suppose H W Rbesaeqatmplnmal Equivalent A NEW QUANTUM ALGORITHM FOR THE HIDDEN SHIFT Zn PROBLEM IN 2t GERGELY CSAJI´ E¨otv¨os Lor´and University, 1117 Budapest, P´azm´any P´eter s´et´any 1/A Budapest, 1185, Hungary [email protected] February 9, 2021 Abstract In this paper we make a step towards a time and space efficient algorithm n for the hidden shift problem for groups of the form Zk . We give a solution to the case when k is a power of 2, which has polynomial running time in n, and only uses quadratic classical, and linear quantum space in n log(k). It can be a useful tool in the general case of the hidden shift and hidden subgroup problems too, since one of the main algorithms made to solve them can use this algorithm as a subroutine in its recursive steps, making it more efficient in some instances. 1 Introduction The hidden subgroup and hidden shift problems have been intensively studied by several authors since Shor’s discovery of efficient factoring and discrete log- arithm algorithms[7]. Many of the problems that have an exponentially faster quantum algorithm are instances of the first one and the latter is a closely re- lated problem, which is useful for example when we are dealing with the hidden subgroup problem in groups of the form G ⋊ Z2, G abelian. The hidden subgroup problem consists of a finite group G, a subgroup H G, ≤ a finite set S 0, 1 l, (this l is called the encoding length) and a function ⊂ { } f : G S such that f(x)= f(y) if and only if x and y are both elements of the → same left coset of H. We then say that f hides the subgroup H. Furthermore we suppose that f is efficiently computable or that we have an oracle Uf mapping the state x 0 to x f(x) . | i| i | i| i In the hidden shift problem we are given a finite group G, a finite set S 0, 1 l, arXiv:2102.04171v1 [quant-ph] 8 Feb 2021 ⊂{ } and two functions f ,f : G S, both injective and we know that there exists 0 1 → some s G such that f (x) = f (xs) for every x G. The task is to find s, ∈ 0 1 ∈ given oracle access to f0 and f1, which are unitary transforms Uf0 ,Uf1 , that map the state x 0 to x fi(x) . | i| i | i| i For example, if we have a finite abelian group G, then the hidden shift problem can be viewed as an instance of the hidden subgroup problem with G′ = G⋊Z2, and f : G′ S given with f(x, i)= fi(x). Then the function f hides the sub- → group H = (0, 0), (s, 1) , so if we find H, we can determine s too. Ettinger and { } Hoyer showed[2] that there is a reduction in the other direction as well, therefore the two problems are quantum polynomially equivalent. 1 There has been a massive amount of research regarding these problems, how- ever an efficient solution to the general case has not yet been discovered and is starting to seem impossible. But in some special cases we can at least give some algorithms, that are quite more efficient than the brute force search, which is generally the only classical solution. For example when G is abelian, then we can solve the hidden subgroup problem in polynomial time, but the hidden shift problem seems more difficult even in this case. The hidden shift problem, although not as general as the hidden subgroup prob- lem, is still very useful in a lot of areas. For example, as Regev showed[6], the case when G = Zn (which corresponds to the hidden subgroup problem in the dihedral group) is important, because an efficient fault tolerant algorithm for this would break a primitive from lattice based cryptography. Another relevant aspect for the hidden shift problem is that it is recursively used by an algorithm of Friedl et al.[3] which is one of the most important algorithms for the hidden subgroup problem (it can be used to solve the hidden shift problem as well), be- cause it works efficiently for a reasonably large class of groups, namely solvable groups with constant exponent and constant derived length. It introduces a new problem called Translating Coset, which is a generalization of both the hidden subgroup and the hidden shift problems and then the authors show that this Translating Coset in any finite solvable group G and for any normal subgroup N of G is reducible to the Translating Coset in G/N and in N. The hidden shift problem comes into the picture, because the algorithm used to solve Translating Zn Coset uses an algorithm for the hidden shift problem in p as a subroutine. So more efficient algorithms for the hidden shift problem in these cases or efficient Zn algorithms for a larger class of groups, for example pt would mean a more efficient solution in the general case too, the latter because then the algorithm could save t 1 recursive calls. Our paper adresses this problem and gives an − Zn efficient solution to the case when G = 2t . Generally there are two main algorithms for the hidden shift problem. The first [4] (√log G ) one is Kuperberg’s , which solves the problem in 2O | | time and space. It is currently considered the best algorithm for the general case of the prob- lem. Although Kuperberg’s algorithm used exponential space, Regev made a modified version, which uses a pipeline of routines, each waiting until it gets enough objects from the previous one and reduces the space complexity to only polynomial in log G . | | The other is the previously mentioned FIMMS algorithm of Friedl et al.[3] Its √log(s) (e) r running time is ((log G +e+2 )O log(1/ε)) (see Ref. [3] Theorem 4.13), | | if G has a subnormal series of length r, where each factor is either elementary abelian of prime exponent bounded by e or is an abelian group of order at most s and ε is the allowed error probability. So it works efficiently in solvable groups with constant exponent and constant derived lenght. The space complexity is exponential, because the algorithm has to have all the states it needs at the beginning, because making further copies would violate 2 the no cloning theorem. In this paper we give a new algorithm, that solves the hidden shift problem in Zn 2t . It combines some ideas from previous algorithms to achieve both polyno- mial space complexity and polynomial running time in n. So it is faster than Kuperberg’s algorithm in some cases, for example when t is constant, or is asymptotically small compared to n. The FIMSS algorithm has similar running time, but its exponential space complexity is a lot worse than our quadratic space need. So ours is the first algorithm for the hidden shift problem in groups Zn of the form k , that has both polynomial running time in n and polynomial Zn space complexity (apart from Simon’s algorithm for the case G = 2 ). We hope its ideas can be extended to make more general algorithms with similar efficiency. Oddly enough, space complexity hasn’t had too much attention in the litera- ture of quantum computing, despite the fact that even if we can build quantum computers, that can work like the mathematical model we use to describe them, they probably won’t have too much space to operate with, especially in the be- ginning, because the more qubits we have, the more difficult to maintain the quantum states of qubits and prevent errors, decoherence and unwanted enta- glement, so it’s important to make algorithms, that will be able to run with these limitations. Now we state our main result: Theorem 1. There is a quantum algorithm, that solves the hidden shift problem n 3 (t+2) 1 in Z t with success probability 1 ε (for any ε> 0) in (t (n+1) l) log(ε− ) 2 − O time and (tn2 + l) classical and (tn + l) quantum space. O O The structure of this paper is the following. In Section 2 we describe the new algorithm in details. In Section 3 we analyse the algorithm as well as prove its correctness and clarify some of the methods mentioned in Section 2. In the end of Section 3 we give a modified version of our algorithm that runs with coset states as input. Zn It turns out that the FIMSS algorithm can be tailored for the special case 2t to achieve very close running time to our method’s, although the space complexity remains exponential in t. We outline such a version of the FIMSS in Section 4. 2 The new algorithm In this section we present the algorithm for solving the hidden shift problem Zn Zn ⋊ Z in 2t or equivalently the hidden subgroup problem in 2t 2, with hidden n subgroup H = (0, 0), (s, 1) and f : Z t ⋊ Z S, f(x, i) = fi(x). It has { } 2 2 → some of its main ideas based on Kuperberg’s subexponential-time algorithm[4] for solving the dihedral hidden subgroup problem in Z2t ⋊ Z2. Although it becomes exponential in t, it is only polynomial in n, so in the case when t is 3 constant or n is asymptotically large compared to t, it is more efficient than Kuperberg’s general algorithm for this problem. Algorithm: Input: N,n Z, where N =2t.
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