Oracle Operator Research Articles

Scalable quantum search using trapped ions

We propose a scalable implementation of Grover's quantum search algorithm in a trapped-ion quantum information processor. The system is initialized in an entangled Dicke state by using simple adiabatic techniques. The inversion-about-average and the oracle operators take the form of single off-resonant laser pulses, addressing, respectively, all and half of the ions in the trap. This is made possible by utilizing the physical symmetrie of the trapped-ion linear crystal. The physical realization of the algorithm represents a dramatic simplification: each logical iteration (oracle and inversion about average) requires only two physical interaction steps, in contrast to the large number of concatenated gates required by previous approaches. This does not only facilitate the implementation, but also increases the overall fidelity of the algorithm.

Solving a class of variational inequalities with inexact oracle operators

Consider a class of variational inequality problems of finding $${x^*\in S}$$ , such that $$f(x^*)^\top (z-x^*)\geq 0,\quad \forall z\in S,$$ where the underlying mapping f is hard to evaluate (sometimes its explicit form is unknown), and S has the following structure $$S=\{x\in R^n\; | \; Ax\le b, x\in K\}.$$ For any given Lagrangian multiplier y associated with the linear inequality constraint in S, a solution of the relaxed variational inequality problem of finding $${\hat x\in K}$$ , such that $$(x^\prime-\hat x)^\top (f(\hat x)+A^\top y)\geq 0 \quad\forall x^\prime \in K \quad\quad\quad\quad (1)$$ can be given by an oracle. This class of problems arises frequently in economics and engineering. In this paper, we focus on considering the above problems where the underlying mapping f, though is unknown, is strongly monotone. We propose an iterative method for solving this class of variational inequality problems. At each iteration, the method consists of two steps: predictor and corrector. At the predictor step, a trial multiplier is given and the oracle is called for a solution of the relaxed variational inequality problem (1); then at the corrector step, the multiplier y is updated, using the information from the predictor step. We allow the oracle to give just an inexact solution of the relaxed variational inequality problem at the predictor step, which makes the method very efficient and practical. Under some suitable conditions, the global convergence of the method is proved. Some numerical examples are presented to illustrate the method.

Quantum search in a nonclassical database of trapped ions

We propose an implementation of the Grover's search algorithm in a nonclassical database using a linear chain of trapped ions. The database comprises all collective states with two ionic excitations; hence the number of database entries scales quadratically with the number of ions $N$. The system is initialized in an even superposition of all register states, i.e., in the symmetric Dicke state with $N$ ions sharing two excitations. The reflection-about-the-mean operator is produced merely by global addressing of the ion string by an off-resonant pulse with a suitable area, whereas the oracle operator is a control-phase gate. This simplification should allow a demonstration of quantum search in a database of hundreds of elements without needing to synthesize multiqubit quantum gates. The technique does not rely on coherent addition or subtraction of phonons, and hence the ion chain can be sympathetically cooled throughout the process.

Distributed implementation of standard oracle operators

The standard oracle operator corresponding to a function $f$ is a unitary operator that computes this function coherently, i.e., it maintains superpositions. This operator acts on a bipartite system, where the subsystems are the input and output registers. In distributed quantum computation, these subsystems may be spatially separated, in which case we will be interested in its classical and entangling capacities. For an arbitrary function $f$, we show that the unidirectional classical and entangling capacities of this operator are ${\mathrm{log}}_{2}({n}_{f})$ bits (ebits) where ${n}_{f}$ is the number of different values this function can take. An optimal procedure for bidirectional classical communication with a standard oracle operator corresponding to a permutation on ${\mathbb{Z}}_{M}$ is given. The bidirectional classical capacity of such an operator is found to be $2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}(M)$ bits. The proofs of these capacities are facilitated by an optimal distributed protocol for the implementation of an arbitrary standard oracle operator.

Unambiguous discrimination among oracle operators

We address the problem of unambiguous discrimination among oracle operators. The general theory of unambiguous discrimination among unitary operators is extended with this application in mind. We prove that entanglement with an ancilla cannot assist any discrimination strategy for commuting unitary operators. We also obtain a simple, practical test for the unambiguous distinguishability of an arbitrary set of unitary operators on a given system. Using this result, we prove that the unambiguous distinguishability criterion is the same for both standard and minimal oracle operators. We then show that, except in certain trivial cases, unambiguous discrimination among all standard oracle operators corresponding to integer functions with fixed domain and range is impossible. However, we find that it is possible to unambiguously discriminate among the Grover oracle operators corresponding to an arbitrarily large unsorted database. The unambiguous distinguishability of standard oracle operators corresponding to totally indistinguishable functions, which possess a strong form of classical indistinguishability, is analysed. We prove that these operators are not unambiguously distinguishable for any finite set of totally indistinguishable functions on a Boolean domain and with arbitrary fixed range. Sets of such functions on a larger domain can have unambiguously distinguishable standard oracle operators, and we provide a complete analysis of the simplest case, that of four functions. We also examine the possibility of unambiguous oracle operator discrimination with multiple parallel calls and investigate an intriguing unitary superoperator transformation between standard and entanglement-assisted minimal oracle operators.