Noise tolerance via reinforcement in the quantum search problem

Abstract

We find that reinforcement exponentially reduces computation time of the quantum search problem from $\sqrt{D}$ to $\ln D$ in a $D$-dimensional system. Therefor, a reinforced quantum search is expected to exhibit an exponentially larger noise threshold compared to a standard search algorithm in a noisy environment. We use numerical simulations to characterize the level of noise tolerance via reinforcement in the presence of both coherent and incoherent noise, considering a system of $N$ qubits and a single $D$-level (qudit) system. Our results show that reinforcement significantly enhances the algorithm's success probability and improves the scaling of its computation time with system size. These findings indicate that reinforcement offers a promising strategy for error mitigation, especially when a precise noise model is unavailable.

Figures (5)

• Figure 1: An illustration of $L$ layers of quantum evolution with reinforced Hamiltonians $H_l(r)$. (a) Coherent evolution in the absence of noise. Here $\rho_l=|\psi_l\rangle\langle \psi_l|$ and $U_l(r)=e^{-\hat{i}H_l(r)}$. (b) Coherent evolution in the presence of noise $V_l$. (c) Evolution in the presence of incoherent noise, represented by channels $\mathcal{E}_l$.
• Figure 2: Quantum search in the absence of noise: Computation time $L(\delta)$ vs the system size $N$ (a) without reinforcement $r=0$, (b) with reinforcement $r=1$. $L(\delta)$ is the minimum number of evolution layers needed to have $P_{success}>1-\delta$.
• Figure 3: Quantum annealing with coherent and incoherent noise in a system of $N$ qubits using the Grover coefficients $A_l^G,B_l^G$: Success probability $P_{success}(L)$ vs the noise strength $\epsilon$ for $r=0$ ((a1),(a2)) and $r=1$ ((b1),(b2)). Here $L=50$. The points are results of averaging over at least $20$ independent realizations of the noisy Hamiltonian. Statistical errors are less than $0.05$ for coherent noise and $10^{-3}$ for incoherent noise.
• Figure 4: Quantum annealing with coherent and incoherent noise in a $D$-level (qudit) system using the Grover coefficients $A_l^G,B_l^G$: Success probability $P_{success}(L)$ vs reinforcement parameter $r$ for different noise strengths $\epsilon$. ((a1),(a2)) $D=100$, ((b1),(b2)) $D=400$, and ((c1),(c2)) $D=800$ for a fixed number of layers $L=10$.
• Figure 5: Quantum search with coherent and incoherent noise in a $D$-level (qudit) system using the locally optimal coefficients $A_l^*,B_l^*$. The computation time $L(\delta,\epsilon)$ is plotted vs dimension $D$ for different noise strengths $\epsilon$. The $L(\delta,\epsilon)$ is the minimum number of evolution layers needed to have $P_{success}>1-\delta$. Here $\delta=1/2$. Panels (a1) and (b1) show the results for coherent noise without reinforcement $r=0$ and with reinforcement $r=1$, respectively. Panels (a2) and (b2) show the results for incoherent noise without reinforcement $r=0$ and with reinforcement $r=2$, respectively.