A Fourier‐based quantum computational learning algorithm with similarities to classical basis function networks is developed. Instead of a Gaussian basis, the quantum algorithm uses a discrete Fourier basis with the output being a linear combination of the basis. A set of examples is considered as a quantum system that undergoes unitary transformations to produce learning. The main result of the work is a quantum computational learning algorithm that is unique among quantum algorithms as it does not assume a priori knowledge of a function f.
A Fourier‐based quantum computational learning algorithm with similarities to classical basis function networks is developed. Instead of a Gaussian basis, the quantum algorithm uses a discrete Fourier basis with the output being a linear combination of the basis. A set of examples is considered as a quantum system that undergoes unitary transformations to produce learning. The main result of the work is a quantum computational learning algorithm that is unique among quantum algorithms as it does not assume a priori knowledge of a function f.
An interesting classical result due to Jackson allows polynomial-time learning of the function class DNF using membership queries. Since in most practical learning situations access to a membership oracle is unrealistic, this paper explores the possibility that quantum computation might allow a learning algorithm for DNF that relies only on example queries. A natural extension of Fourier-based learning into the quantum domain is presented. The algorithm requires only an example oracle, and it runs in O(sqrt(2^n)) time, a result that appears to be classically impossible. The algorithm is unique among quantum algorithms in that it does not assume a priori knowledge of a function and does not operate on a superposition that includes all possible states.
Implementing general functions of operators is a powerful tool in quantum computation. It can be used as the basis for a variety of quantum algorithms including matrix inversion, real and imaginary-time evolution, and matrix powers. Quantum signal processing is the state of the art for this aim, assuming that the operator to be transformed is given as a block of a unitary matrix acting on an enlarged Hilbert space. Here we present an algorithm for Hermitian-operator function design from an oracle given by the unitary evolution with respect to that operator at a fixed time. Our algorithm implements a Fourier approximation of the target function based on the iteration of a basic sequence of single-qubit gates, for which we prove the expressibility. In addition, we present an efficient classical algorithm for calculating its parameters from the Fourier series coefficients. Our algorithm uses only one qubit ancilla regardless the degree of the approximating series. This contrasts with previous proposals, which required an ancillary register of size growing with the expansion degree. Our methods are compatible with Trotterised Hamiltonian simulations schemes and hybrid digital-analog approaches.
The application of the quantum Fourier transform (QFT) within the field of quantum computation has been manifold. Shor’s algorithm, phase estimation and computing discrete logarithms are but a few classic examples of its use. These initial blueprints for quantum algorithms have sparked a cascade of tantalizing solutions to problems considered to be intractable on a classical computer. Therefore, two main threads of research have unfolded. First, novel applications and algorithms involving the QFT are continually being developed. Second, improvements in the algorithmic complexity of the QFT are also a sought after commodity. In this work, we review the structure of the QFT and its implementation. In order to put these concepts in their proper perspective, we provide a brief overview of quantum computation. Finally, we provide a permutation structure for putting the QFT within the context of universal computation.
Linear threshold functions are fw,θ(x)=sign(⟨x,w⟩−θ), where the weight vector w∈Rn is a unit vector, θ∈R is a threshold, and typically x∈Rn or x∈{−1,1}n. When θ=0, the LTF is called homogeneous, and we write fw:=fw,0. Such functions are among the most important objects in machine learning, since they serve to linearly discriminate positive and negative examples. We give three positive results about learning LTFs:
1. Suppose we can make real-domain queries, meaning we can compute fw,θ(x) at any x∈Rn of our choice. We give a quantum algorithm that learns fw,θ up to Euclidean error ε using O(log(n/ε)) membership queries and O(n) other gates. Then we have also learned fw,θ up to error O(ε) when x is Gaussian. Classical algorithms need Ω(nlog(1/ε)) queries.
2. A homogeneous LTF fw on domain {−1,1}n where w has only k nonzero entries of the same value, is the Majority function on the support of w. Belovs gave a bounded-error quantum algorithm that identifies the hidden support exactly (and hence learns fw) using O(k1/4) queries. We give an exponential improvement, using O(logk) queries.
3. Suppose we have a unitary U that can produce (discretized) quantum examples under Gaussian measure, corresponding to ∫xγn(x)∣x⟩∣fw(x)⟩dx. This is a weaker access model than membership queries. We give a quantum algorithm based on the efficient Hermite transform of Jain et al. to learn homogeneous LTFs fw with error ε under the Gaussian distribution, using O(n1/4/ε) applications of U and U† and O(n2/ε4) other gates.