Fixed-point Quantum Search Research Articles

Quantum amplitude-amplification operators

Quantum search algorithms have been shown to deliver quadratic speedups over classical ones. The authors propose a generalized form of such algorithms through the use of ``quantum amplitude-amplification operators'' (QAAOs), with a goal of implementing them on near-term quantum devices. They demonstrate three-qubit examples on cloud-based quantum computing services, and use their framework to distinguish between exact and fixed-point quantum search algorithms.

Optimal fixed-point quantum search in an interacting Ising spin system

In Grover’s search algorithm, a priori knowledge of the number of target states is needed to effectively find a solution. This is due to the inherent oscillatory nature of unitary gates in the algorithm. A fixed-point quantum search is introduced in T. J. Yoder, G. H. Low and I. L. Chuang, (Phys. Rev. Lett.)113, 210501 [9] to mitigate this oscillation even without knowing the size of the target space. This is done by modifying the phase inversion and oracle of the original Grover’s algorithm so that the phase can assume intermediate values in $$[-\pi ,\pi ]$$ . Naturally, qubits in a quantum computer are interacting among themselves and this might introduce an error in the phase assignment. In this work, we used the Ising spin chain to simulate the interactions among the qubits and demonstrate how to implement the fixed-point quantum search. By inspecting the state vectors after some finite number of iterations, we found that the error is primarily due to the phase difference of the ideal target state with that of the final state. Further investigation shows that high fidelity between the ideal and real evolution can be achieved by using a low Rabi frequency.

A query‐based quantum eigensolver

Solving eigenvalue problems is crucially important for both classical and quantum applications. Many well-known numerical eigensolvers have been developed, including the QR and the power methods for classical computers, as well as the quantum phase estimation(QPE) method and the variational quantum eigensolver for quantum computers. In this work, we present an alternative type of quantum method that uses fixed-point quantum search to solve Type II eigenvalue problems. It serves as an important complement to the QPE method, which is a Type I eigensolver. We find that the effectiveness of our method depends crucially on the appropriate choice of the initial state to guarantee a sufficiently large overlap with the unknown target eigenstate. We also show that the quantum oracle of our query-based method can be efficiently constructed for efficiently-simulated Hamiltonians, which is crucial for analyzing the total gate complexity. In addition, compared with the QPE method, our query-based method achieves a quadratic speedup in solving Type II problems.

Performance Analysis and Parameter Optimization of the Optimal Fixed-Point Quantum Search

The optimal fixed-point quantum search (OFPQS) algorithm [Phys. Rev. Lett. 113, 210501 (2014)] achieves both the fixed-point property and quadratic speedup over classical algorithms, which gives a sufficient condition on the number of iterations to ensure the success probability is no less than a given lower bound (denoted by 1 − δ 2 ). However, this condition is approximate and not exact. In this paper, we derive the sufficient and necessary condition on the number of feasible iterations, based on which the exact least number of iterations can be obtained. For example, when δ = 0 . 8 , iterations can be saved by almost 25%. Moreover, to find a target item certainly, setting directly 1 − δ 2 = 100 % , the quadratic advantage of the OFPQS algorithm will be lost, then, applying the OFPQS algorithm with 1 − δ 2 < 100 % requires multiple executions, which leads to a natural problem of choosing the optimal parameter δ . For this, we analyze the extreme and minimum properties of the success probability and further analytically derive the optimal δ which minimizes the query complexity. Our study can be a guideline for both the theoretical and application research on the fixed-point quantum search algorithms.

Implementation of a fixed-point quantum search by duality computer

The quantum algorithms where the operations are unitary can be implemented by an ordinary quantum computer, and nonunitary operations can be realized in the duality computer. The operations in the fixed-point quantum search proposed by Mizel (Phys. Rev. Lett., 102 (2009) 150501) are not unitary, and in this paper it is shown that this algorithm can effectively be implemented in duality quantum computer.

Continuous-time quantum search and time-dependent two-level quantum systems

It was recently emphasized by Byrnes, Forster and Tessler [Phys. Rev. Lett. 120 (2018) 060501] that the continuous-time formulation of Grover’s quantum search algorithm can be intuitively understood in terms of Rabi oscillations between the source and the target subspaces. In this work, motivated by this insightful remark and starting from the consideration of a time-independent generalized quantum search Hamiltonian as originally introduced by Bae and Kwon [Phys. Rev. A 66 (2002) 012314], we present a detailed investigation concerning the physical connection between quantum search Hamiltonians and exactly solvable time-dependent two-level quantum systems. Specifically, we compute in an exact analytical manner the transition probabilities from a source state to a target state in a number of physical scenarios specified by a spin-[Formula: see text] particle immersed in an external time-dependent magnetic field. In particular, we analyze both the periodic oscillatory as well as the monotonic temporal behaviors of such transition probabilities and, moreover, explore their analogy with characteristic features of Grover-like and fixed-point quantum search algorithms, respectively. Finally, we discuss from a physics standpoint the connection between the schedule of a search algorithm, in both adiabatic and nonadiabatic quantum mechanical evolutions, and the control fields in a time-dependent driving Hamiltonian.

Fixed-point adiabatic quantum search

Fixed-point quantum search algorithms succeed at finding one of $M$ target items among $N$ total items even when the run time of the algorithm is longer than necessary. While the famous Grover's algorithm can search quadratically faster than a classical computer, it lacks the fixed-point property --- the fraction of target items must be known precisely to know when to terminate the algorithm. Recently, Yoder, Low, and Chuang gave an optimal gate-model search algorithm with the fixed-point property. Meanwhile, it is known that an adiabatic quantum algorithm, operating by continuously varying a Hamiltonian, can reproduce the quadratic speedup of gate-model Grover search. We ask, can an adiabatic algorithm also reproduce the fixed-point property? We show that the answer depends on what interpolation schedule is used, so as in the gate model, there are both fixed-point and non-fixed-point versions of adiabatic search, only some of which attain the quadratic quantum speedup. Guided by geometric intuition on the Bloch sphere, we rigorously justify our claims with an explicit upper bound on the error in the adiabatic approximation. We also show that the fixed-point adiabatic search algorithm can be simulated in the gate model with neither loss of the quadratic Grover speedup nor of the fixed-point property. Finally, we discuss natural uses of fixed-point algorithms such as preparation of a relatively prime state and oblivious amplitude amplification.

Steering quantum dynamics via bang-bang control: Implementing optimal fixed-point quantum search algorithm

A robust control over quantum dynamics is of paramount importance for quantum technologies. Many of the existing control techniques are based on smooth Hamiltonian modulations involving repeated calculations of basic unitaries resulting in time complexities scaling rapidly with the length of the control sequence. On the other hand, the bang-bang controls need one-time calculation of basic unitaries and hence scale much more efficiently. By employing a global optimization routine such as the genetic algorithm, it is possible to synthesize not only highly intricate unitaries, but also certain nonunitary operations. Here we demonstrate the unitary control through the first implementation of the optimal fixed-point quantum search algorithm in a three-qubit NMR system. More over, by combining the bang-bang pulses with the twirling process, we also demonstrate a nonunitary transformation of the thermal equilibrium state into an effective pure state in a five-qubit NMR system.

Fixed-point quantum search with an optimal number of queries.

Grover's quantum search and its generalization, quantum amplitude amplification, provide a quadratic advantage over classical algorithms for a diverse set of tasks but are tricky to use without knowing beforehand what fraction λ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction but, as a consequence, lose the very quadratic advantage that makes Grover's algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup. Our result incorporates an adjustable bound on the failure probability and, for a given number of oracle queries, guarantees that this bound is satisfied over the broadest possible range of λ.

Scale invariance of entanglement dynamics in Grover's quantum search algorithm

We calculate the amount of entanglement of the multiqubit quantum states employed in the Grover algorithm, by following its dynamics at each step of the computation. We show that genuine multipartite entanglement is always present. Remarkably, the dynamics of any type of entanglement as well as of genuine multipartite entanglement is independent of the number $n$ of qubits for large $n$, thus exhibiting a scale invariance property. We compare this result with the entanglement dynamics induced by a fixed-point quantum search algorithm. We also investigate criteria for efficient simulatability in the context of Grover's algorithm.

Experimental implementation of a fixed-point duality quantum search algorithm in the nuclear magnetic resonance quantum system

In this work, we demonstrated a fixed-point quantum search algorithm in the nuclear magnetic resonance (NMR) system. We constructed the pulse sequences for the pivotal operations in the quantum search protocol. The experimental results agree well with the theoretical predictions. The generalization of the scheme to the arbitrary number of qubits has also been given.

PHASE MATCHING IN FIXED-POINT QUANTUM SEARCH ALGORITHM

We investigate the phase matching problem in the fixed-point quantum search algorithm proposed by Grover [Phys. Rev. Lett.95 (2005) 150501]. We show that the optimal phase shift is π/3.61, which replaces the π/3 phase shift in fixed-point quantum search algorithm. The π/3.61-phase algorithm can be achieved in [Formula: see text] with the success probability of at least 94.11%, which offsets disadvantage that the success probability of getting correct results usually decreases with the increase of marked items when original Grover quantum search algorithm is applied to search an unordered database. In the meantime, this work also indicates that Grover quantum search algorithm is considerably robust to certain kinds of perturbations and is robust against modest noise in the initialization procedure.

Critically Damped Quantum Search

Although measurement and unitary processes can accomplish any quantum evolution in principle, thinking in terms of dissipation and damping can be powerful. We propose a modification of Grover's algorithm in which the idea of damping plays a natural role. Remarkably, we find that there is a critical damping value that divides between the quantum O(sqrt[N]) and classical O(N) search regimes. In addition, by allowing the damping to vary in a fashion we describe, one obtains a fixed-point quantum search algorithm in which ignorance of the number of targets increases the number of oracle queries only by a factor of 1.5.

Fixed-point quantum search for different phase shifts

Grover recently presented the fixed-point search algorithm. In this letter, we study the fixed-point search algorithm obtained by replacing equal phase shifts of $\pi /3$ by different phase shifts.

Error tolerance in an NMR implementation of Grover’s fixed-point quantum search algorithm

We describe an implementation of Grover's fixed-point quantum search algorithm on a nuclear magnetic resonance quantum computer, searching for either one or two matching items in an unsorted database of four items. In this algorithm the target state (an equally weighted superposition of the matching states) is a fixed point of the recursive search operator, so that the algorithm always moves towards the desired state. The effects of systematic errors in the implementation are briefly explored.