Oracles Are Subtle But Not Malicious

Theoretical computer scientists have been debating the role of oracles since the 1970's. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an oracle relative to which PP has linear-sized circuits, by proving a new lower bound for perceptrons and low- degree threshold polynomials. This oracle settles a longstanding open question, and generalizes earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More importantly, it implies the first nonrelativizing separation of "traditional" complexity classes, as opposed to interactive proof classes such as MIP and MA-EXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does not have circuits of size n^k for any fixed k. We present an alternative proof of this fact, which shows that PP does not even have quantum circuits of size n^k with quantum advice. To our knowledge, this is the first nontrivial lower bound on quantum circuit size. Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean circuits in ZPP^NP. We show that the NP queries in this algorithm cannot be parallelized by any relativizing technique, by giving an oracle relative to which ZPP^||NP and even BPP^||NP have linear-size circuits. On the other hand, we also show that the NP queries could be parallelized if P=NP. Thus, classes such as ZPP^||NP inhabit a "twilight zone," where we need to distinguish between relativizing and black-box techniques. Our results on this subject have implications for computational learning theory as well as for the circuit minimization problem.Comment: 20 pages, 1 figur

Improved Bounds on Quantum Learning Algorithms

In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples. Hunziker et al. conjectured that for any class C of Boolean functions, the number of quantum black-box queries which are required to exactly identify an unknown function from C is $O(\frac{\log |C|}{\sqrt{{\hat{\gamma}}^{C}}})$, where $\hat{\gamma}^{C}$ is a combinatorial parameter of the class C. We essentially resolve this conjecture in the affirmative by giving a quantum algorithm that, for any class C, identifies any unknown function from C using $O(\frac{\log |C| \log \log |C|}{\sqrt{{\hat{\gamma}}^{C}}})$ quantum black-box queries. We consider a range of natural problems intermediate between the exact learning problem (in which the learner must obtain all bits of information about the black-box function) and the usual problem of computing a predicate (in which the learner must obtain only one bit of information about the black-box function). We give positive and negative results on when the quantum and classical query complexities of these intermediate problems are polynomially related to each other. Finally, we improve the known lower bounds on the number of quantum examples (as opposed to quantum black-box queries) required for $(\epsilon,\delta)$-PAC learning any concept class of Vapnik-Chervonenkis dimension d over the domain $\{0,1\}^n$ from $\Omega(\frac{d}{n})$ to $\Omega(\frac{1}{\epsilon}\log \frac{1}{\delta}+d+\frac{\sqrt{d}}{\epsilon})$. This new lower bound comes closer to matching known upper bounds for classical PAC learning.Comment: Minor corrections. 18 pages. To appear in Quantum Information Processing. Requires: algorithm.sty, algorithmic.sty to buil

Average-Case Quantum Query Complexity

We compare classical and quantum query complexities of total Boolean functions. It is known that for worst-case complexity, the gap between quantum and classical can be at most polynomial. We show that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms. Under non-uniform distributions the gap can even be super-exponential. We also prove some general bounds for average-case complexity and show that the average-case quantum complexity of MAJORITY under the uniform distribution is nearly quadratically better than the classical complexity.Comment: 14 pages, LaTeX. Some parts rewritten. This version to appear in the Journal of Physics

Quantum Algorithms for Learning and Testing Juntas

In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: - whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; - with no access to any classical or quantum membership ("black-box") queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; - which require only a few quantum examples but possibly many classical random examples (which are considered quite "cheap" relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: - We give an algorithm for testing k-juntas to accuracy $\epsilon$ that uses $O(k/\epsilon)$ quantum examples. This improves on the number of examples used by the best known classical algorithm. - We establish the following lower bound: any FS-based k-junta testing algorithm requires $\Omega(\sqrt{k})$ queries. - We give an algorithm for learning $k$-juntas to accuracy $\epsilon$ that uses $O(\epsilon^{-1} k\log k)$ quantum examples and $O(2^k \log(1/\epsilon))$ random examples. We show that this learning algorithms is close to optimal by giving a related lower bound.Comment: 15 pages, 1 figure. Uses synttree package. To appear in Quantum Information Processin