Robust quantum spatial search

Quantum spatial search has been widely studied with most of the study focusing on quantum walk algorithms. We show that quantum walk algorithms are extremely sensitive to systematic errors. We present a recursive algorithm which offers significant robustness to certain systematic errors. To search N items, our recursive algorithm can tolerate errors of size O(1/\sqrt{\ln N}) which is exponentially better than quantum walk algorithms for which tolerable error size is only O(\ln N/\sqrt{N}). Also, our algorithm does not need any ancilla qubit. Thus our algorithm is much easier to implement experimentally compared to quantum walk algorithms

Quantum search algorithm tailored to clause satisfaction problems

Many important computer science problems can be reduced to clause satisfaction problem. We are given $n$ Boolean variables $x_{k}$ and $m$ clauses $c_{j}$ where each clause is a function of values of some of the variables. We want to find an assignment $i$ of variables for which all $m$ clauses are satisfied. Let $f_{j}(i)$ be a binary function which is $1$ if $j^{\rm th}$ clause is satisfied by the assignment $i$ else $f_{j}(i) = 0$. Then the solution is $r$ for which $f(i=r) = 1$, where $f(i)$ is the AND function of all $f_{j}(i)$. In quantum computing, Grover`s algorithm can be used to find $r$. A crucial component of this algorithm is the selective phase inversion $I_{r}$ of the solution state encoding $r$. $I_{r}$ is implemented by computing $f(i)$ for all $i$ in superposition which requires computing AND of all $m$ binary functions $f_{j}(i)$. Hence there must be coupling between the computation circuits for each $f_{j}(i)$. In this paper, we present an alternative quantum search algorithm which relaxes the requirement of such couplings. Hence it offers implementation advantages for clause satisfaction problems

Postprocessing can speed up general quantum search algorithms

A general quantum search algorithm aims to evolve a quantum system from a known source state $|s\rangle$ to an unknown target state $|t\rangle$. It uses a diffusion operator $D_{s}$ having source state as one of its eigenstates and $I_{t}$, where $I_{\psi}$ denotes the selective phase inversion of $|\psi\rangle$ state. It evolves $|s\rangle$ to a particular state $|w\rangle$, call it w-state, in $O(B/\alpha)$ time steps where $\alpha$ is $|\langle t|s\rangle|$ and $B$ is a characteristic of the diffusion operator. Measuring the w-state gives the target state with the success probability of $O(1/B^{2})$ and $O(B^{2})$ applications of the algorithm can boost it from $O(1/B^{2})$ to $O(1)$, making the total time complexity $O(B^{3}/\alpha)$. In the special case of Grover's algorithm, $D_{s}$ is $I_{s}$ and $B$ is very close to $1$. A more efficient way to boost the success probability is quantum amplitude amplification provided we can efficiently implement $I_{w}$. Such an efficient implementation is not known so far. In this paper, we present an efficient algorithm to approximate selective phase inversions of the unknown eigenstates of an operator using phase estimation algorithm. This algorithm is used to efficiently approximate $I_{w}$ which reduces the time complexity of general algorithm to $O(B/\alpha)$. Though $O(B/\alpha)$ algorithms are known to exist, our algorithm offers physical implementation advantages.Comment: Accepted for publication in Physical Review A. arXiv admin note: substantial text overlap with arXiv:1210.464

Adiabatic Quantum Computation with a 1D projector Hamiltonian

Adiabatic quantum computation is based on the adiabatic evolution of quantum systems. We analyse a particular class of qauntum adiabatic evolutions where either the initial or final Hamiltonian is a one-dimensional projector Hamiltonian on the corresponding ground state. The minimum energy gap which governs the time required for a successful evolution is shown to be proportional to the overlap of the ground states of the initial and final Hamiltonians. We show that such evolutions exhibit a rapid crossover as the ground state changes abruptly near the transition point where the energy gap is minimum. Furthermore, a faster evolution can be obtained by performing a partial adiabatic evolution within a narrow interval around the transition point. These results generalize and quantify earlier works.Comment: revised versio

Quantum computers can search rapidly by using almost any selective transformations

The search problem is to find a state satisfying certain properties out of a given set. Grover's algorithm drives a quantum computer from a prepared initial state to the target state and solves the problem quadratically faster than a classical computer. The algorithm uses selective transformations to distinguish the initial state and target state from other states. It does not succeed unless the selective transformations are very close to phase-inversions. Here we show a way to go beyond this limitation. An important application lies in quantum error-correction, where the errors can cause the selective transformations to deviate from phase-inversions. The algorithms presented here are robust to errors as long as the errors are reproducible and reversible. This particular class of systematic errors arise often from imperfections in apparatus setup. Hence our algorithms offer a significant flexibility in the physical implementation of quantum search.Comment: 8 pages, Accepted for publication in PR

General framework for quantum search algorithms

Grover's quantum search algorithm drives a quantum computer from a prepared initial state to a desired final state by using selective transformations of these states. Here, we analyze a framework when one of the selective trasformations is replaced by a more general unitary transformation. Our framework encapsulates several previous generalizations of the Grover's algorithm. We show that the general quantum search algorithm can be improved by controlling the transformations through an ancilla qubit. As a special case of this improvement, we get a faster quantum algorithm for the two-dimensional spatial search.Comment: revised versio