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arXiv:0806.3362 · Quantum Information and Computation

Quantum algorithms for shifted subset problems

Ashley Montanaro

Published May 1, 2009• 4 citations
arXivHTMLPDF

Abstract

We consider a recently proposed generalisation of the abelian hidden subgroup problem: the {\em shifted subset problem}. The problem is to determine a subset $S$ of some abelian group, given access to quantum states of the form $\ket{S+x}$, for some unknown shift $x$. We give quantum algorithms to find Hamming spheres and other subsets of the boolean cube $\{0,1\}^n$. The algorithms have time complexity polynomial in $n$ and give rise to exponential separations from classical computation.

DOI
10.26421/qic9.5-6-10

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We consider a recently proposed generalisation of the abelian hidden subgroup problem: the {\em shifted subset problem}. The problem is to determine a subset SSS of some abelian group, given access to quantum states of the form ∣S+x⟩\ket{S+x}∣S+x⟩, for some unknown shift xxx. We give quantum algorithms to find Hamming spheres and other subsets of the boolean cube {0,1}n\{0,1\}^n{0,1}n. The algorithms have time complexity polynomial in nnn and give rise to exponential separations from classical computation.
Attempts to separate the power of classical and quantum models of computation have a long history. The ultimate goal are exponential separations for computational (promise) problems, however, these do not come a dime a dozen: while there were some early successes in the form of hidden subgroup problems for abelian groups–which generalizes Shor’s factoring algorithm perhaps most faithfully–only for a handful of non-abelian groups efficient quantum algorithms were found. Recently, problems have gotten increased attention that seek to identify hidden sub-structures of other combinatorial and algebraic objects besides groups, notably the quantum algorithms by Childs, Schulman, and Vazirani for shifted subset problems for spheres over a finite field. In this paper we provide new examples for exponential separations by considering hidden shift problems that are defined for several classes of highly non-linear Boolean functions. These so-called bent functions arise in cryptography, where their property of having perfectly flat Fourier spectra on the Boolean hypercube gives them resilience against certain types of attack. We present quantum algorithms that solve the hidden shift problems for several well-known classes of bent functions in polynomial time with a constant number of queries, while the classical query complexity is shown to be exponential. Our approach uses a technique that exploits the duality between bent functions and their Fourier transforms. We also give a quantum algorithm that solves the hidden shift problem for quadratic forms that might be of independent interest. This algorithm can also be used to find approximating polynomials and shifts for functions with large Gowers U3 norm. 1 1
We examine the application of quantum algorithms to the non-Abelian hidden subgroup problem and focus on the dihedral hidden subgroup problem (DHSP). Ettinger and Høyer have recently discovered an algorithm which, although efficient in the number of operations of the quantum computation, requires classical post-processing which grows exponentially with the size of the input. We first show that the DHSP can be reduced to another problem: how to efficiently estimate an unknown shift k , in a one-to-one map f : M → M , given two oracles which provide x → f ( x ), and x → f ( x ⊕ k ), with k and f unknown. We devise an algorithm which uses amplitude amplification to obtain an estimate for the unknown shift k in a number (( M ) 1/2 ) oracle calls.
Consider the following generalized hidden shift problem: given a function f on {0,...,M-1} x Z_N satisfying f(b,x)=f(b+1,x+s) for b=0,1,...,M-2, find the unknown shift s in Z_N. For M=N, this problem is an instance of the abelian hidden subgroup problem, which can be solved efficiently on a quantum computer, whereas for M=2, it is equivalent to the dihedral hidden subgroup problem, for which no efficient algorithm is known. For any fixed positive epsilon, we give an efficient (i.e., poly(log N)) quantum algorithm for this problem provided M > N^epsilon. The algorithm is based on the "pretty good measurement" and uses H. Lenstra's (classical) algorithm for integer programming as a subroutine.
We study quantum algorithms for the hidden subgroup problem (HSP). The HSP provides a unified framework for many important problems, including factoring integers, graph isomorphism problem, etc. It has been known that the problem for abelian groups can be efficiently solved. However, for non-abelian groups, it remains widely open. In this dissertation we study the problem for some specific non-abelian groups, called q-hedral groups. We start with a quantum algorithm, proposed by Moore et al. (2003), for finding some hidden conjugate subgroups of affine groups, a special case of q-hedral groups for large q. We then adapt their algorithm to construct an algorithm for solving the hidden conjugate problem for q-hedral groups for large q and point out a difficulty in complexity analysis that involves certain exponential sums. We discuss certain bounds on these sums and present some computational data on their values to make a conjecture. For small q, Regev (2003) gave an efficient algorithm for solving the HSP for dihedral groups (q = 2) using an oracle that solves a certain subset sum problem (SSP). A crucial step of his algorithm relies on the probability that elements in a cyclic group ZN can be expressed as sums of random elements in the group. We generalize this problem to any finite group and prove the following theorem: for any finite group G and r ≥ 32 log2 |G| + 5, if r elements from G are chosen randomly, then the 2r possible sums (or products) of the r elements will cover all the elements in G with high probability. Also, by making a further generalization of the SSP, we show that Regev's algorithm can be generalized for q-hedral groups for any fixed q. We also study this generalized subset sum problem. In addition to the algorithm developing for small q, we give an efficient quantum algorithm for solving the coset problem and an efficient quantum algorithm for solving the hidden conjugate problem. Furthermore, we also give a survey of an algorithm for finding the largest normal subgroup in a hidden subgroup of any finite group, introduced by Hallgren et al. (2000) and provide a simplified and unified proof.
Attempts to separate the power of classical and quantum models of computation have a long history. The ultimate goal is to find exponential separations for computational problems. However, such separations do not come a dime a dozen: while there were some early successes in the form of hidden subgroup problems for abelian groups--which generalize Shor's factoring algorithm perhaps most faithfully--only for a handful of non-abelian groups efficient quantum algorithms were found. Recently, problems have gotten increased attention that seek to identify hidden sub-structures of other combinatorial and algebraic objects besides groups. In this paper we provide new examples for exponential separations by considering hidden shift problems that are defined for several classes of highly non-linear Boolean functions. These so-called bent functions arise in cryptography, where their property of having perfectly flat Fourier spectra on the Boolean hypercube gives them resilience against certain types of attack. We present new quantum algorithms that solve the hidden shift problems for several well-known classes of bent functions in polynomial time and with a constant number of queries, while the classical query complexity is shown to be exponential. Our approach uses a technique that exploits the duality between bent functions and their Fourier transforms.
The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms are special cases of the HSP. In this paper we show that there exist a new efficient quantum algorithm for the HSP on groups ZN⋊Zqs\Z_{N}\rtimes\Z_{q^s}ZN​⋊Zqs​ where NNN is an integer with a special prime factorization, qqq prime number and sss any positive integer.