Near-optimal quantum circuit for Grover's unstructured search using a transverse field

Abstract

Inspired by a class of algorithms proposed by Farhi et al. (arXiv:1411.4028), namely the quantum approximate optimization algorithm (QAOA), we present a circuit-based quantum algorithm to search for a needle in a haystack, obtaining the same quadratic speedup achieved by Grover's original algorithm. In our algorithm, the problem Hamiltonian (oracle) and a transverse field are applied alternately to the system in a periodic manner. We introduce a technique, based on spin-coherent states, to analyze the composite unitary in a single period. This composite unitary drives a closed transition between two states that have high degrees of overlap with the initial state and the target state, respectively. The transition rate in our algorithm is of order Θ(1/ √ N ), and the overlaps are of order Θ(1), yielding a nearly optimal query complexity of T √ N (π/2 √ 2 ). Our algorithm is a QAOA circuit that demonstrates a quantum advantage with a large number of iterations that is not derived from Trotterization of an adiabatic quantum optimization (AQO) algorithm. It also suggests that the analysis required to understand QAOA circuits involves a very different process from estimating the energy gap of a Hamiltonian in AQO.

Training the quantum approximate optimization algorithm without access to a quantum processing unit

“…Eq. (11). Tensor networks of quantum circuits can be very memory efficient ways to save quantum states.…”

How quantum is the speedup in adiabatic unstructured search?

“…Explicitly, the quantum adiabatic gap can be computed to be g(s) = 1 − 4s(1 − s)(1 − 1/N ) with a minimum gap on the order of 1/ √ N that is centered around s = 1/2 to arXiv:1811.08302v2 [quant-ph] 14 Apr 2019 within a region of width 1/ √ N [33,35]-implying that in order to maintain the quadratic speedup as the problem scales up, an exponentially precise annealing schedule s(t) is required. To wit, the 'digitization' of the algorithm, as prescribed by the polynomial equivalence between adiabatic quantum computing and the quantum circuit model [27,36,37], does not in fact preserve the quadratic speedup, but instead yields a classical O(N ) scaling.…”

Variational learning of Grover's quantum search algorithm

“…This result looks promising and points out that there is a set of variational quantum problems with parameters that can be efficiently found on a classical computer for arbitrary number of qubits.Grover's algorithm [2] is one of the most celebrated quantum algorithms, enabling quantum computers to quadratically outperform classical computers at database search provided database access is restricted to a 'black box' -called the oracle model. In addition to the wide application scope of database search, Grover's algorithm has further applications as a subroutine used in a variety of other quantum algorithms.Variational hybrid quantum/classical algorithms have recently become an area of significant interest [3][4][5][6][7][8][9][10]. These algorithms have shown several advantages such as robustness to quantum errors and low coherence time requirements [11], which make them ideal for implementations in current quantum computer architectures.…”

“…Variational hybrid quantum/classical algorithms have recently become an area of significant interest [3][4][5][6][7][8][9][10]. These algorithms have shown several advantages such as robustness to quantum errors and low coherence time requirements [11], which make them ideal for implementations in current quantum computer architectures.…”

“…We sometimes call (2) the driver Hamiltonian or diffusion operator [6]. The state is varied to minimize this orthogonal component (4).…”