Terms Of Query Complexity Research Articles

A one-bit, comparison-based gradient estimator

We study zeroth-order optimization for convex functions where we further assume that function evaluations are unavailable. Instead, one only has access to a $\textit{comparison oracle}$, which given two points $x$ and $y$ returns a single bit of information indicating which point has larger function value, $f(x)$ or $f(y)$. By treating the gradient as an unknown signal to be recovered, we show how one can use tools from one-bit compressed sensing to construct a robust and reliable estimator of the normalized gradient. We then propose an algorithm, coined SCOBO, that uses this estimator within a gradient descent scheme. We show that when $f(x)$ has some low dimensional structure that can be exploited, SCOBO outperforms the state-of-the-art in terms of query complexity. Our theoretical claims are verified by extensive numerical experiments.

Quantum Exploration Algorithms for Multi-Armed Bandits

Identifying the best arm of a multi-armed bandit is a central problem in bandit optimization. We study a quantum computational version of this problem with coherent oracle access to states encoding the reward probabilities of each arm as quantum amplitudes. Specifically, we provide an algorithm to find the best arm with fixed confidence based on variable-time amplitude amplification and estimation. This algorithm gives a quadratic speedup compared to the best possible classical result in terms of query complexity. We also prove a matching quantum lower bound (up to poly-logarithmic factors).

Quantum Simulation Logic, Oracles, and the Quantum Advantage.

Query complexity is a common tool for comparing quantum and classical computation, and it has produced many examples of how quantum algorithms differ from classical ones. Here we investigate in detail the role that oracles play for the advantage of quantum algorithms. We do so by using a simulation framework, Quantum Simulation Logic (QSL), to construct oracles and algorithms that solve some problems with the same success probability and number of queries as the quantum algorithms. The framework can be simulated using only classical resources at a constant overhead as compared to the quantum resources used in quantum computation. Our results clarify the assumptions made and the conditions needed when using quantum oracles. Using the same assumptions on oracles within the simulation framework we show that for some specific algorithms, such as the Deutsch-Jozsa and Simon’s algorithms, there simply is no advantage in terms of query complexity. This does not detract from the fact that quantum query complexity provides examples of how a quantum computer can be expected to behave, which in turn has proved useful for finding new quantum algorithms outside of the oracle paradigm, where the most prominent example is Shor’s algorithm for integer factorization.

Near-Optimal Active Learning of Halfspaces via Query Synthesis in the Noisy Setting

In this paper, we consider the problem of actively learning a linear classifier through query synthesis where the learner can construct artificial queries in order to estimate the true decision boundaries. This problem has recently gained a lot of interest in automated science and adversarial reverse engineering for which only heuristic algorithms are known. In such applications, queries can be constructed de novo to elicit information (e.g., automated science) or to evade detection with minimal cost (e.g., adversarial reverse engineering). We develop a general framework, called dimension coupling (DC), that 1) reduces a d-dimensional learning problem to d-1 low dimensional sub-problems, 2) solves each sub-problem efficiently, 3) appropriately aggregates the results and outputs a linear classifier, and 4) provides a theoretical guarantee for all possible schemes of aggregation. The proposed method is proved resilient to noise. We show that the DC framework avoids the curse of dimensionality: its computational complexity scales linearly with the dimension. Moreover, we show that the query complexity of DC is near optimal (within a constant factor of the optimum algorithm). To further support our theoretical analysis, we compare the performance of DC with the existing work. We observe that DC consistently outperforms the prior arts in terms of query complexity while often running orders of magnitude faster.

Query Complexity Lower Bounds for Reconstruction of Codes

We investigate the problem of local reconstruction, as deflned by Saks and Seshadhri (2008), in the context of error correcting codes. The flrst problem we address is that of message reconstruction, where given oracle access to a corrupted encoding w 2 f0;1g n of some message x 2 f0;1g k our goal is to probabilistically recover x (or some portion of it). This should be done by a procedure (reconstructor) that given an index i as input, probes w at few locations and outputs the value of xi. The reconstructor can (and indeed must) be randomized, with all its randomness specifled in advance by a single random seed, and the main requirement is that for most random seeds, all values are reconstructed correctly (notice that swapping the order of \for most random seeds and \for all x1;:::;xk makes the deflnition equivalent to standard Local Decoding). A message reconstructor can serve as a \fllter that allows evaluating certain classes of algorithms on x safely and e-ciently. For instance, to run a parallel algorithm, one can initialize several copies of the reconstructor with the same random seed, and then they can autonomously handle decoding requests while producing outputs that are consistent with the original message x. Another motivation for studying message reconstruction arises from the theory of Locally Decodable Codes. The second problem that we address is codeword reconstruction, which is similarly deflned, but instead of reconstructing the message the goal is to reconstruct the codeword itself, given an oracle access to its corrupted version. Error correcting codes that admit message and codeword reconstruction can be obtained from Locally Decodable Codes (LDC) and Self Correctible Codes (SCC) respectively. The main contribution of this paper is a proof that in terms of query complexity, these are close to be the best possible constructions, even when we disregard the length of the encoding.

Claw finding algorithms using quantum walk

The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. Given two functions, f and g , with domain sizes N and M ( N ≤ M ) , respectively, and the same range, the goal of the problem is to find x and y such that f ( x ) = g ( y ) . This problem has been considered in both quantum and classical settings in terms of query complexity. This paper describes an optimal algorithm that uses quantum walk to solve this problem. Our algorithm can be slightly modified to solve the more general problem of finding a tuple consisting of elements in the two function domains that has a prespecified property. It can also be generalized to find a claw of k functions for any constant integer k > 1 , where the domain sizes of the functions may be different.

Entanglement tensor for a general pure multipartite quantum state

In the black-box model, problems constrained by a `promise' are the only ones that admit a quantum exponential speedup over the best classical algorithm in terms of query complexity. The most prominent example of this is the Deutsch-Jozsa algorithm. More recently, Wim van Dam put forward an algorithm for unstructured problems (i.e., those without a promise). We consider the Deutsch-Jozsa algorithm with a less restrictive (or `broken') promise and study the transition to an unstructured problem. We compare this to the success of van Dam's algorithm. These are both compared with a standard classical sampling algorithm. The Deutsch-Jozsa algorithm remains good as the problem initially becomes less structured, but the van Dam algorithm can be adapted so as to become superior to the Deutsch-Jozsa algorithm as the promise is weakened.

Broken promises and quantum algorithms

In the black-box model, problems constrained by a "promise" are the only ones that admit a quantum exponential speedup over the best classical algorithm in terms of query complexity. The most prominent example of this is the Deutsch-Jozsa algorithm. More recently, Wim van Dam put forward an algorithm for unstrucred problems (i.e., those without a promise). We consider the Deutsch-Jozsa algorithm with a less restrictive (or "broken") promise and study the transition to an unstructured problem. We compare this to the success of van Dam's algorithm. These are both compared with a standard classical sampling algorithm. The Deutsch-Jozsa algorithm remains good as the problem initially becomes less structured, but the van Dam algorithm can be adapted so as to become superior to the Deutsch-Jozsa algorithm as the promise is weakened.

Broken promises and quantum algorithms

In the black-box model, problems constrained by a 'promise' are the only ones that admit a quantum exponential speedup over the best classical algorithm in terms of query complexity. The most prominent example of this is the Deutsch-Jozsa algorithm. More recently, Wim van Dam put forward an algorithm for unstructured problems (i.e., those without a promise). We consider the Deutsch-Jozsa algorithm with a less restrictive (or 'broken') promise and study the transition to an unstructured problem. We compare this to the success of van Dam's algorithm. These are both compared with a standard classical sampling algorithm. The Deutsch-Jozsa algorithm remains good as the problem initially becomes less structured, but the van Dam algorithm can be adapted so as to become superior to the Deutsch-Jozsa algorithm as the promise is weakened.