Exponential convergence dynamics in Grover's search algorithm

TL;DR

The paper tackles the soufflé problem in Grover's search by introducing a dissipative variant that converts oscillatory dynamics into exponential decay toward the solution subspace using ancilla reservoir engineering. It develops both a continuous-time and a discrete, Trotterized circuit formulation and shows, via mapping to the Bixon-Jortner model, that the decay rate and revival times can be engineered to preserve Grover-like scaling (O(sqrt(N/M))). The approach yields robust convergence without requiring prior knowledge of M, at the cost of additional qubits for the reservoir and more complex circuitry, and demonstrates resilience to control errors relative to fixed-point alternatives. This work suggests practical, hardware-friendly routes for robust quantum search primitives and highlights connections between quantum algorithms and open-system dynamics.

Abstract

Grover's search algorithm is the cornerstone of many applications of quantum computing, providing a quadratic speed-up over classical methods. One limitation of the algorithm is that it requires knowledge of the number of solutions to obtain an optimal success probability, due to the oscillatory dynamics between the initial and solutions states (the ``soufflé problem''). While various methods have been proposed to solve this problem, each has their drawbacks in terms of inefficiency or sensitivity to control errors. Here, we modify Grover's algorithm to eliminate the oscillatory dynamics, such that the search proceeds as an exponential decay into the solution states. The basic idea is to convert the solution states into a reservoir by using ancilla qubits such that the initial state is nonreflectively absorbed. Trotterizing the continuous algorithm yields a quantum circuit that gives equivalent performance, which has the same quadratic quantum speedup as the original algorithm.

Figures (7)

• Figure 1: Grover's algorithm in continuous Hamiltonian formulation. (a) Standard Grover's algorithm and (b) dissipative Grover's algorithm proposed in this work. Examples shown are for $M = 2$ solutions for both cases.
• Figure 2: Success probability (\ref{['fidelity']}) of obtaining a solution state under various versions of Grover's algorithm. Standard Grover's algorithm corresponds to $r = 0$, and any of the values with $r > 0$ correspond to dissipative Grover's algorithm. In (a)(b), the $r = 0$ line is obtained by time evolving the Hamiltonian (\ref{['standardgroverham']}) from the initial state $| \psi_0 \rangle$, while the $r = 3,4$ line is obtained using Hamiltonian (\ref{['mainham']}). In (c)(d), the $r = 0$ line is obtained by standard Grover's algorithm, while the $r = 3,4$ line is obtained using (\ref{['trottered']}) with $\delta t = \pi$. Dashed lines are the fidelity curves (\ref{['fidbj2']}) for the BJ model. All figures were obtained with $n=3$, $M=1$, $\Delta = 0.1$.
• Figure 3: Quantum circuit for the discrete version of the dissipative Grover algorithm (\ref{['trottered']}). Explicit circuits corresponding to the operators $U_S$ and $U_+$ defined in (\ref{['uplusdef']}) and (\ref{['usdef']}) respectively are shown. The oracle is defined as $O_S = \sum_{m=0}^{M-1} |S_m \rangle \langle S_m | \otimes X + \sum_{m=0}^{N-M-1} | \cancel{S}_m \rangle \langle \cancel{S}_m | \otimes I$, where $| \cancel{S}_m \rangle$ are the non-solution states, and the Pauli operator $X$ acts on the ancilla qubit nielsen2010quantumyoder2014fixed. The conditional reservoir operator is defined as $U_R = |1 \rangle \langle 1 | \otimes \exp( -i \delta t \sum_{k=0}^{R-1}E_k\ketbra{k}{k} ) + |0 \rangle \langle 0 \otimes | I$, where the $|0 \rangle, | 1 \rangle$ act on the ancilla qubit. $H$ is the Hadamard operator.
• Figure 4: Effect of gate control errors on dissipative Grover's algorithm. We perform (\ref{['trottered']}) where the time step $\delta t$ in the gates (\ref{['uplusdef']}) and (\ref{['usdef']}) are randomly adjusted by a relative fraction $\epsilon$. (a) Three random instances with $\epsilon = 0.05$ (solid lines) as well as the ideal evolution $\epsilon = 0$ (dotted line). The evolution according to the BJ model is also shown (dashed line). (b) The mean deviation $\delta F = E[ | F(\epsilon) - F(\epsilon=0) | ]$ evaluated over 100 independent runs. We use parameters $M =1$, $n = 6$, $r = 3$, $\Delta =3 \sqrt{M(N-M)}/NR$ for all plots.
• Figure 5: Energy levels of the Bixon-Jortner model. The state $\ket{a}$ is coupled to an infinite ladder of states $\ket{b_m}$ via uniform coupling $\beta$. The reservoir levels are regularly spaced by the energy $\Delta$.