Efficiency of classical simulations of a noisy Grover algorithm

Abstract

We analyze the modification of entanglement dynamics in the Grover algorithm when the qubits are subjected to single-qubit amplitude-damping or phase-flip noise. We compare quantum trajectories with full density-matrix simulations, analyzing the dynamics of averaged trajectory entanglement (TE) and operator entanglement (OE), in the respective state representation. Although not a genuine entanglement measure, both TE and OE are connected to the efficiency of matrix product state simulations and thus of fundamental interest. As in many quantum algorithms, at the end of the Grover circuit entanglement decreases as the system converges towards the target product state. While we find that this is well captured in the OE dynamics, quantum trajectories rarely follow paths of reducing entanglement. Optimized unraveling schemes can lower TE slightly, however we show that deep in the circuit OE is generally smaller than TE. This implies that matrix product density operator (MPDO) simulations of quantum circuits can in general be more efficient than quantum trajectories. In addition, we investigate the noise-rate scaling of success probabilities for both amplitude-damping and phase-flip noise in Grover's algorithm.

Figures (4)

• Figure 1: Grover search algorithm - (a) Representation of the circuit implementing the Grover search algorithm. In the following we will assume that after each Grover iteration $\hat{U}_\omega\hat{U}_s$ a single-qubit noise channel is applied on each qubit. (b) Evolution of the success probability $P_\omega$ and the von Neumann entropy $S_{vN}$ for an equal bipartition of the qubit string as a function of the number of iterations of the Grover search. Calculations are performed for $n=24$ qubits.
• Figure 2: TE and OE evolution in the Grover search algorithm subjected to (a) phase flip, and (b) amplitude noise channels ($n=10$ qubits). In each panel we plot the von Neumann entropy of $n_T=2000$ individual trajectories unraveled via a a naive (first row) or NUMU (second row) procedure. Different columns show results for different noise rates $p_{\rm pf/ad} = 0.5 \times 10^{-2}$, $1.0 \times 10^{-2}$, $2.0 \times 10^{-2}$, and $4.0 \times 10^{-2}$. We compare the averaged TE, $S_\mathrm{T}$, to the OE, $S_\mathrm{OE}$. The entropies are relevant to the efficiency of matrix product decompositions of the trajectory states, or the full density matrix, respectively.
• Figure 3: (a) Final success probability $P_\omega^f$ after $M = (\pi/4) \sqrt{N}$ iterations as a function of the phase-flip rate $p_{\rm pf}$ for various number of qubits $8 \leq n \leq 16$ (log-log scale). The dashed black line stands for an algebraically decaying function of the form $p_{\rm pf}^{-\alpha}$ where $\alpha = 1.735$. Results are obtained for an exact MPDO representation (converged with $\chi = 4$). (b)-(c) Scaling analysis of the final success probability with respect to $p_{\rm pf}$ (upper panel) and number of qubits $n$ (lower panel). Dashed lines indicate the algebraic ($P^f_\omega \propto p_{\rm pf}^{-\alpha}$) and exponential ($P^f_\omega \propto e^{-\beta n}$) dependence, respectively.
• Figure 4: (a) Final success probability $P_\omega^f$ shifted by $2^{-n}$ after $M = (\pi/4) \sqrt{N}$ iterations as a function of the amplitude-damping rate $p_{\rm ad}$ for various numbers of qubits $8 \leq n \leq 16$ (log-log scale). The dashed black line stands for an algebraically decaying function of the form $p_{\rm pf}^{-\alpha}$ where $\alpha = 1.7$. Results are obtained for an exact MPDO representation (converged with $\chi = 4$). (b)-(c) Scaling analysis of the shifted final success probability with respect to $p_{\rm ad}$ (upper panel) and number of qubits $n$ (lower panel). Dashed lines indicate the algebraic ($P^f_\omega - 2^{-n} \propto p_{\rm ad}^{-\gamma}$) and exponential ($P^f_\omega - 2^{-n} \propto e^{-\delta n}$) dependence, respectively.