A universal adiabatic quantum query algorithm

Quantum query complexity is known to be characterized by the so-called quantum adversary bound. While this result has been proved in the standard discrete-time model of quantum computation, it also holds for continuous-time (or Hamiltonian-based) quantum computation, due to a known equivalence between these two query complexity models. In this work, we revisit this result by providing a direct proof in the continuous-time model. One originality of our proof is that it draws new connections between the adversary bound, a modern technique of theoretical computer science, and early theorems of quantum mechanics. Indeed, the proof of the lower bound is based on Ehrenfest's theorem, while the upper bound relies on the adiabatic theorem, as it goes by constructing a universal adiabatic quantum query algorithm. Another originality is that we use for the first time in the context of quantum computation a version of the adiabatic theorem that does not require a spectral gap.Comment: 22 pages, compared to v1, includes a rigorous proof of the correctness of the algorithm based on a version of the adiabatic theorem that does not require a spectral ga

Adiabatic quantum search algorithm for structured problems

The study of quantum computation has been motivated by the hope of finding efficient quantum algorithms for solving classically hard problems. In this context, quantum algorithms by local adiabatic evolution have been shown to solve an unstructured search problem with a quadratic speed-up over a classical search, just as Grover's algorithm. In this paper, we study how the structure of the search problem may be exploited to further improve the efficiency of these quantum adiabatic algorithms. We show that by nesting a partial search over a reduced set of variables into a global search, it is possible to devise quantum adiabatic algorithms with a complexity that, although still exponential, grows with a reduced order in the problem size.Comment: 7 pages, 0 figur

Information-theoretic lower bounds for quantum sorting

We analyze the quantum query complexity of sorting under partial information. In this problem, we are given a partially ordered set $P$ and are asked to identify a linear extension of $P$ using pairwise comparisons. For the standard sorting problem, in which $P$ is empty, it is known that the quantum query complexity is not asymptotically smaller than the classical information-theoretic lower bound. We prove that this holds for a wide class of partially ordered sets, thereby improving on a result from Yao (STOC'04)

Finding a marked node on any graph by continuous-time quantum walk

Spatial search by discrete-time quantum walk can find a marked node on any ergodic, reversible Markov chain $P$ quadratically faster than its classical counterpart, i.e.\ in a time that is in the square root of the hitting time of $P$. However, in the framework of continuous-time quantum walks, it was previously unknown whether such general speed-up is possible. In fact, in this framework, the widely used quantum algorithm by Childs and Goldstone fails to achieve such a speedup. Furthermore, it is not clear how to apply this algorithm for searching any Markov chain $P$. In this article, we aim to reconcile the apparent differences between the running times of spatial search algorithms in these two frameworks. We first present a modified version of the Childs and Goldstone algorithm which can search for a marked element for any ergodic, reversible $P$ by performing a quantum walk on its edges. Although this approach improves the algorithmic running time for several instances, it cannot provide a generic quadratic speedup for any $P$. Secondly, using the framework of interpolated Markov chains, we provide a new spatial search algorithm by continuous-time quantum walk which can find a marked node on any $P$ in the square root of the classical hitting time. In the scenario where multiple nodes are marked, the algorithmic running time scales as the square root of a quantity known as the extended hitting time. Our results establish a novel connection between discrete-time and continuous-time quantum walks and can be used to develop a number of Markov chain-based quantum algorithms.Comment: This version deals only with new algorithms for spatial search by continuous-time quantum walk (CTQW) on ergodic, reversible Markov chains. Please see arXiv:2004.12686 for results on the necessary and sufficient conditions for the optimality of the Childs and Goldstone algorithm for spatial search by CTQ

Quantum circuit implementation of the Hamiltonian versions of Grover's algorithm

We analyze three different quantum search algorithms, the traditional Grover's algorithm, its continuous-time analogue by Hamiltonian evolution, and finally the quantum search by local adiabatic evolution. We show that they are closely related algorithms in the sense that they all perform a rotation, at a constant angular velocity, from a uniform superposition of all states to the solution state. This make it possible to implement the last two algorithms by Hamiltonian evolution on a conventional quantum circuit, while keeping the quadratic speedup of Grover's original algorithm.Comment: 5 pages, 3 figure

Search via Quantum Walk

We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.Comment: 21 pages. Various modifications and improvements, especially in Section

Symmetry-assisted adversaries for quantum state generation

We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum state generation might be a route to tackle the {\sc Graph Isomorphism} problem. We show that for the related problem of {\sc Index Erasure} our method leads to a lower bound of $\Omega(\sqrt N)$ which matches an upper bound obtained via reduction to quantum search on $N$ elements. This closes an open problem first raised by Shi [FOCS'02]. Our approach is based on two ideas: (i) on the one hand we generalize the known additive and multiplicative adversary methods to the case of quantum state generation, (ii) on the other hand we show how the symmetries of the underlying problem can be leveraged for the design of optimal adversary matrices and dramatically simplify the computation of adversary bounds. Taken together, these two ideas give the new result for {\sc Index Erasure} by using the representation theory of the symmetric group. Also, the method can lead to lower bounds even for small success probability, contrary to the standard adversary method. Furthermore, we answer an open question due to \v{S}palek [CCC'08] by showing that the multiplicative version of the adversary method is stronger than the additive one for any problem. Finally, we prove that the multiplicative bound satisfies a strong direct product theorem, extending a result by \v{S}palek to quantum state generation problems.Comment: 35 pages, 5 figure

Information transmission via entangled quantum states in Gaussian channels with memory

Gaussian quantum channels have recently attracted a growing interest, since they may lead to a tractable approach to the generally hard problem of evaluating quantum channel capacities. However, the analysis performed so far has always been restricted to memoryless channels. Here, we consider the case of a bosonic Gaussian channel with memory, and show that the classical capacity can be significantly enhanced by employing entangled input symbols instead of product symbols.Comment: 13 pages, 5 figures, Workshop on Quantum entanglement in physical and information sciences, Pisa, December 14-18, 200

Simulating quantum correlations as a distributed sampling problem

It is known that quantum correlations exhibited by a maximally entangled qubit pair can be simulated with the help of shared randomness, supplemented with additional resources, such as communication, post-selection or non-local boxes. For instance, in the case of projective measurements, it is possible to solve this problem with protocols using one bit of communication or making one use of a non-local box. We show that this problem reduces to a distributed sampling problem. We give a new method to obtain samples from a biased distribution, starting with shared random variables following a uniform distribution, and use it to build distributed sampling protocols. This approach allows us to derive, in a simpler and unified way, many existing protocols for projective measurements, and extend them to positive operator value measurements. Moreover, this approach naturally leads to a local hidden variable model for Werner states.Comment: 13 pages, 2 figure