We present two new results about exact learning by quantum computers. First, we show how to exactly learn a k-Fourier-sparse n-bit Boolean function from O(k^{1.5}(log k)^2) uniform quantum examples for that function. This improves over the bound of Theta~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Our main tool is an improvement of Chang’s lemma for sparse Boolean functions. Second, we show that if a concept class {C} can be exactly learned using Q quantum membership queries, then it can also be learned using O ({Q^2}/{log Q} * log|C|) classical membership queries. This improves the previous-best simulation result (Servedio-Gortler, SICOMP'04) by a log Q-factor.
We present two new results about exact learning by quantum computers. First, we show how to exactly learn a k-Fourier-sparse n-bit Boolean function from O(k^{1.5}(log k)^2) uniform quantum examples for that function. This improves over the bound of Theta~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Our main tool is an improvement of Chang’s lemma for sparse Boolean functions. Second, we show that if a concept class {C} can be exactly learned using Q quantum membership queries, then it can also be learned using O ({Q^2}/{log Q} * log|C|) classical membership queries. This improves the previous-best simulation result (Servedio-Gortler, SICOMP'04) by a log Q-factor.
We study exact learning with membership queries for concept classes C⊆{0,1}N, focusing on the relationships among their deterministic, randomized, and quantum query complexities, denoted D(C), R(C), and Q(C), respectively. The two canonical quantum speedups in this model are witnessed by Grover search and Bernstein-Vazirani, leading to the longstanding conjecture R(C)=O(Q(C)2+Q(C)logN). We first refute this conjecture by constructing concept classes C and C′ satisfying R(C)=Ω(logQ(C)Q(C)3logN)andD(C′)=Ω(Q(C′)3logN). The first bound matches the upper bound of Arunachalam et al.~[Quantum'21] up to constant factors, while the second matches the upper bound of Servedio and Gortler~[SICOMP'04]. In particular, this shows that the saving in the randomized upper bound of Arunachalam et al. fundamentally relies on randomness. Apart from characterizing the optimal relationship between classical and quantum query complexity, our results are the first to show that quantum speedups for learning can go beyond the Grover and Bernstein-Vazirani paradigms.
We present two new results about exact learning by quantum computers. First, we show how to exactly learn a k -Fourier-sparse n -bit Boolean function from O(k1.5(logk)2) uniform quantum examples for that function. This improves over the bound of Θ~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Additionally, we provide a possible direction to improve our O~(k1.5) upper bound by proving an improvement of Chang's lemma for k -Fourier-sparse Boolean functions. Second, we show that if a concept class C can be exactly learned using Q quantum membership queries, then it can also be learned using O(Q2logQlog|C|) classical membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP'04) by a logQ -factor.
We study the number of uniform quantum examples needed to exactly learn a Boolean function with at most k nonzero Fourier coefficients. We give a learner with near-optimal O(k) sample complexity independent of the input length n. The learner uses only separate measurements of individual examples. This improves the previous O(k3/2log2k) upper bound (Arunachalam, Chakraborty, Lee, Paraashar, and de Wolf, 2021). The main ingredient is a candidate set formed by symmetric differences of subsets from independent groups of Fourier samples. A multilinear majority polynomial combines unbiased random approximations into an accurate witness whose Fourier support lies in this set. Granularity then forces the set to contain the entire Fourier support, including subsets never directly sampled. For comparison, we derive the tight classical sample complexity Θ(nk) by combining known sparse-recovery and lower-bound results.
This thesis studies strengths and weaknesses of quantum computers. In the first part we present three contributions to quantum algorithms. 1) consider a search space of N elements. One of these elements is "marked" and our goal is to find this. We describe a quantum algorithm to solve this problem using essentially sqrt{N} queries and other operations, improving over the gate count of Grover's algorithm. 2) We give a succinct characterization of quantum algorithms in terms of polynomials, and develop a new technique for showing upper and lower bounds on quantum query complexity based on this. 3) One generic technique used to compute the minimum of a given function is "gradient descent". We present a quantum gradient-calculation algorithm and a gradient-optimization algorithm that is quadratically faster than classical algorithms. In the second part of this thesis we look at quantum learning theory. 1) We survey quantum learning theory, describing the main results known for three models of learning, using classical as well as quantum data: exact learning from membership queries, the probably approximately correct (PAC) learning model, and the agnostic learning model. 2) We then consider if a quantum learner can PAC-learn a given concept class from fewer quantum examples. We give a negative answer by showing that quantum examples are not more powerful than classical random examples, both for PAC learning and for agnostic learning.
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.
In learning theory, the VC dimension of a concept class C is the most common way to measure its “richness.” A fundamental result says that the number of examples needed to learn an unknown target concept c∈C under an unknown distribution D, is tightly determined by the VC dimension d of the concept class C. Specifically, in the PAC model
Θ(dϵ+log(1/δ)ϵ)
examples are necessary and sufficient for a learner to output, with probability 1−δ, a hypothesis h that is ϵ-close to the target concept c (measured under D). In the related agnostic model, where the samples need not come from a c∈C, we know that
Θ(dϵ2+log(1/δ)ϵ2)
examples are necessary and sufficient to output an hypothesis h∈C whose error is at most ϵ worse than the error of the best concept in C. Here we analyze quantum sample complexity, where each example is a coherent quantum state. This model was introduced by Bshouty and Jackson (1999), who showed that quantum examples are more powerful than classical examples in some fixed-distribution settings. However, Atici and Servedio (2005), improved by Zhang (2010), showed that in the PAC setting (where the learner has to succeed for every distribution), quantum examples cannot be much more powerful: the required number of quantum examples is
Ω(d1−ηϵ+d+log(1/δ)ϵ) for arbitrarily small constant η>0.
Our main result is that quantum and classical sample complexity are in fact equal up to constant factors in both the PAC and agnostic models. We give two proof approaches. The first is a fairly simple information-theoretic argument that yields the above two classical bounds and yields the same bounds for quantum sample complexity up to a log(d/ϵ) factor. We then give a second approach that avoids the log-factor loss, based on analyzing the behavior of the “Pretty Good Measurement” on the quantum state-identification problems that correspond to learning. This shows classical and quantum sample complexity are equal up to constant factors for every concept class C.