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Annual Symposium on Foundations of Computer Science

An improved quantum Fourier transform algorithm and applications

Lisa Hales, Sean Hallgren

Published Nov 7, 2002• 164 citations
Publisher

Abstract

We give an algorithm for approximating the quantum Fourier transform over an arbitrary Z/sub p/ which requires only O(n log n) steps where n=log p to achieve an approximation to within an arbitrary inverse polynomial in n. This improves the method of A.Y. Kitaev (1995) which requires time quadratic in n. This algorithm also leads to a general and efficient Fourier sampling technique which improves upon the quantum Fourier sampling lemma of L. Hales and S. Hallgren (1997). As an application of this technique, we give a quantum algorithm which finds the period of an arbitrary periodic function, i.e. a function which may be many-to-one within each period. We show that this algorithm is efficient (polylogarithmic in the period of the function) for a large class of periodic functions. Moreover, using standard quantum lower-bound techniques, we show that this characterization is right. That is, this is the maximal class of periodic functions with an efficient quantum period-finding algorithm.

DOI
10.1109/sfcs.2000.892139

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We give an algorithm for approximating the quantum Fourier transform over an arbitrary Z/sub p/ which requires only O(n log n) steps where n=log p to achieve an approximation to within an arbitrary inverse polynomial in n. This improves the method of A.Y. Kitaev (1995) which requires time quadratic in n. This algorithm also leads to a general and efficient Fourier sampling technique which improves upon the quantum Fourier sampling lemma of L. Hales and S. Hallgren (1997). As an application of this technique, we give a quantum algorithm which finds the period of an arbitrary periodic function, i.e. a function which may be many-to-one within each period. We show that this algorithm is efficient (polylogarithmic in the period of the function) for a large class of periodic functions. Moreover, using standard quantum lower-bound techniques, we show that this characterization is right. That is, this is the maximal class of periodic functions with an efficient quantum period-finding algorithm.
We isolate and generalize a technique implicit in many quantum algorithms, including Shor's algorithms for factoring and discrete log.In particular, we show that the distribution sampled after a Fourier transform over Zp can be efficiently approximated by transforming over Z, for any q in a large range.Our result places no restrictions on the superposition to be transformed, generalizing previous applications.In addition, our proof easily generalizes to multi-dimensional transforms for any constant number of dimensions.
The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor's celebrated factoring and discrete log algorithms are a special case. We begin by addressing various computational issues surrounding the QFT and give improved parallel circuits for both the QFT over a power of 2 and the QFT over an arbitrary cyclic group. These circuits are based on new insight into the relationship between the discrete Fourier transform over different cyclic groups. We then exploit this insight to extend the class of hidden subgroup problems with efficient quantum solutions. First we relax the condition that the underlying hidden subgroup function be distinct on distinct cosets of the subgroup in question and show that this relaxation can be solved whenever G is a finitely-generated abelian group. We then extend this reasoning to the hidden cyclic subgroup problem over the reals, showing how to efficiently generate the bits of the period of any sufficiently piecewise-continuous function on R. Finally, we show that this problem of period-finding over R, viewed as an oracle promise problem, is strictly harder than its integral counterpart. In particular, period-finding over R lies outside the complexity class MA, a class which contains period-finding over the integers.
We isolate and generalize a technique implicit in many quantum algorithms, including Shor's algorithms for factoring and discrete log. In particular, we show that the distribution sampled after a Fourier transform over Zp{\mathbb Z}_pZp​ can be efficiently approximated by transforming over Zq{\mathbb Z}_qZq​ for any q in a large range. Our result places no restrictions on the superposition to be transformed, generalizing the result implicit in Shor which applies only to periodic superpositions. In addition, our proof easily generalizes to multi-dimensional transforms for any constant number of dimensions.
We investigate and improve algorithms for computing the approximate quantum Fourier transform (QFT ) and simulating a sparse Hamiltonian. We present an improved near-linear time algorithm for the approximate QFT modulo 2™. We give a circuit for the arbitrary modulus version that matches the best currently-known bound. We then relate the difficulty of the arbitrary modulus QFT to classical integer multiplication and division. We also investigate the problem of simulating an n-qubit sparse Hamiltonian. As input we receive the system's start state, a black box which, via queries, provides the non-zero entries of each row of the Hamiltonian, and a time t. We present an improved polynomial-time quantum algorithm for computing an approximation of the state of the system at time t. Prior to this, we provide an introduction to quantum computing, and survey some quantum algorithms that are used in our improved algorithms.
The Discrete Fourier Transform (DFT) is central to the analysis of uniformly sampled signals, yet many practical applications involve non-uniform sampling, requiring the Non-Uniform Discrete Fourier Transform (NUDFT). While quantum algorithms for the standard DFT are well established, a corresponding framework for the non-uniform case remains underdeveloped. This work introduces a quantum algorithm for the Non-Uniform Quantum Fourier Transform (NUQFT) based on a low-rank factorization of the NUDFT matrix. The factorization is translated into an explicit quantum construction using block encodings, Quantum Signal Processing, and the Linear Combination of Unitaries framework, yielding an εεε-accurate block encoding of the NUDFT matrix with controlled approximation error from both classical truncation and quantum implementation. Under standard oracle access assumptions for non-uniform sampling points, we derive explicit, non-asymptotic gate-level resource estimates. The resulting complexity scales polylogarithmically with target precision, quadratically with the number of qubits through the quantum Fourier transform, and logarithmically with a geometry-dependent conditioning parameter induced by the non-uniform grid. This establishes a concrete and resource-efficient quantum analogue of the NUDFT and provides a foundation for quantum algorithms on irregularly sampled data.
How powerful are Quantum Computers? Despite the prevailing belief that Quantum Computers are more powerful than their classical counterparts, this remains a conjecture backed by little formal evidence. Shor's famous factoring algorithm [Shor97] gives an example of a problem that can be solved efficiently on a quantum computer with no known efficient classical algorithm. Factoring, however, is unlikely to be NP-Hard, meaning that few unexpected formal consequences would arise, should such a classical algorithm be discovered. Could it then be the case that any quantum algorithm can be simulated efficiently classically? Likewise, could it be the case that Quantum Computers can quickly solve problems much harder than factoring? If so, where does this power come from, and what classical computational resources do we need to solve the hardest problems for which there exist efficient quantum algorithms? We make progress toward understanding these questions through studying the relationship between classical nondeterminism and quantum computing. In particular, is there a problem that can be solved efficiently on a Quantum Computer that cannot be efficiently solved using nondeterminism? In this thesis we address this problem from the perspective of sampling problems. Namely, we give evidence that approximately sampling the Quantum Fourier Transform of an efficiently computable function, while easy quantumly, is hard for any classical machine in the Polynomial Time Hierarchy. In particular, we prove the existence of a class of distributions that can be sampled efficiently by a Quantum Computer, that likely cannot be approximately sampled in randomized polynomial time with an oracle for the Polynomial Time Hierarchy. Our work complements and generalizes the evidence given in Aaronson and Arkhipov's work [AA2013] where a different distribution with the same computational properties was given. Our result is more general than theirs, but requires a more powerful quantum sampler.