Fixed-Point Quantum Search with an Optimal Number of Queries

Abstract

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. Grover's quantum search algorithm [1] provides a quadratic speedup over classical algorithms for solving a broad class of problems. Included are the many important, yet computationally prohibitive nondeterministic polynomial time (NP) problems [2], which can always be solved, albeit inefficiently, by searching the space of possible solutions. Because the problem Grover's algorithm solves is so simple to understand-given an oracle function that recognizes marked items, locate one of M such marked items among N unsorted items-its classical time complexity OðN=MÞ is obvious, making the quantum speedup that much more conclusive.Conceptually also, Grover's algorithm is compellingthe iterative application of the oracle and initial state preparation rotates from a superposition of mostly unmarked states to a superposition of mostly marked states in just Oð ffiffiffiffiffiffiffiffiffiffi ffi N=M p Þ steps [3]. This interpretation of Grover's algorithm as a rotation is very natural because the Grover iterate is a unitary operator. However, this same unitarity is also a weakness. Without knowing exactly how many marked items there are, there is no knowing when to stop the iteration. This leads to the soufflé problem [4], in which iterating too little "undercooks" the state, leaving mostly unmarked states, and iterating too much "overcooks" the state, passing by the marked states and leaving us again with mostly unmarked states.The most direct solution of the soufflé problem is to estimate M by either using full-blown quantum counting [5,6] or a trial-and-error scheme where iterates are applied an exponentially increasing number of times [5,7]. Although scaling quantumly, these strategies are unappealing for search as they work best not by monotonically amplifying marked states, but rather by getting "close enough" before resorting to classical random sampling.An alternative approach, in line with what we advocate here, is to construct, either recursively or dissipatively, operators that avoid overcooking by always amplifying marked states. Such algorithms are known as fixed-point searches. For example, running Grover's π=3 algorithm [8] or the comparable ancilla algorithm [9] longer can only ever improve its success probability....

Finding spin glass ground states using quantum walks

“…Hence, a quantum walk search with a random measurement time should on average only need to be repeated twice to locate the marked state; knowing the exact time to measure for the optimal success probability is not necessary for the success of the algorithm. Fixed point quantum search algorithms (Yoder et al 2014, Dalzell et al 2017 are another approach that avoids the need to know how long to run the algorithm for.…”

Quantum search for unknown number of target items by hybridizing fixed-point method with trail-and-error method*

“…[32], the final state gradually converges to the target states along with the iterations, avoiding "overcooking" the state. However, the algorithm loses the valuable quantum speedup of quantum search [33], which has also been proven asymptotically optimal [34,35].…”

“…Fortunately, Yoder et al [33] creatively reduces the "fixed point" to a bounded region of the target states, and claimed that the quadratic speedup over classical search is maintained consequently, which has got many recognitions [36][37][38]. For example, in Ref.…”

Measurement-enhanced quantum search

“…Otherwise, the operation must be applied indefinitely and might give a low probability of success due to the inherent oscillations in the unitary operators. Several approaches were reported in the literature to suppress the oscillations [3,4,5], one of which was the application of a measurement right after the iteration. This effectively damps the search and yet preserves the quadratic speed up of the original method [6].…”