Process characterization with Monte Carlo wave functions

We present an efficient method to simulate a quantum process subject to dissipation and noise. To describe the effect on any input state we evolve Monte Carlo wave functions for a principal and ancilla system, prepared initially in an entangled state. In analogy to experimental process tomography, the simulated propagator for the system density matrix is conveniently described by a process $\chi$ matrix - directly determined from the stochastic state vectors. Our method significantly reduces the computational complexity compared with standard theoretical characterization methods. It also delivers an upper bound on the trace distance between the ideal and simulated process based on the evolution of only a single wave function of the entangled system.

Figure 1

Implementation of AAPC with Monte Carlo wave functions: In each trajectory $i$, the evolution of the entangled system and ancilla is simulated, subject to the process $\mathcal{E}\otimes I$. From the expansion coefficients ${\lambda }_j$, we determine the vector of components ${\zeta }_m$. Averaging over many simulation outcomes ${\zeta }_m^i{\zeta }_n^{i*}$ the $\chi$ matrix is obtained.

Figure 2

Rydberg-mediated controlled phase gate. (a) Illustration of (1) a resonant transfer between $|1\rangle$ and $|r\rangle$ of the control atom $c$, (2) subsequent coherent excitation and de-excitation between $|1\rangle$ and $|r\rangle$ of the target atom $t$, yielding the controlled phase shift on the $|1\rangle$ component, unless the control atom is excited and providing the Blockade shift $\mathcal{B}$, and (3) de-excitation of the control atom. (b) Illustration of the pulse sequence and listing the Hamiltonian and driving terms. The identity operator $I$ signifies individual addressing of atoms, ${\mathcal{L}}^{\otimes 2}=-\frac{i\hslash }{2}({\sum }_m{L}_m^{†}{L}_m\otimes I+I\otimes {\sum }_m{L}_m^{†}{L}_m)$ describes decoherence in the system, and the two-atom interaction $\mathcal{B}|rr\rangle \langle rr|$ prevents both atoms from occupying the Rydberg state. The single-qubit Hamiltonian $\widehat{H}$ is discussed in detail in the text.

Figure 3

For numerical efficiency the short-lived, intermediate state $|p\rangle$ is adiabatically eliminated and yields an effective description with couplings (${\Omega }_{\mathrm{eff}}$) and dissipation terms (${\widehat{L}}_{{\gamma }_d}$,${\widehat{L}}_{{\gamma }_p,j}$) shown in the left part of the figure. For convenience, $|p\rangle$ is formally reintroduced to represent decay events from $|r\rangle$ (see text).

Figure 4

Convergence of the process fidelity and trace distance determined by the ancilla-assisted wave function characterization method. For selected values of $n$ the mean value and standard deviation of the fidelity (top) and trace-distance (bottom) is obtained using 50 samples. Each sample involves a process matrix simulation with $n$ trajectories. The solid lines represent calculations using the parameters in Table with ${\gamma }_d/2\pi =$ 1.0 kHz. The dashed lines are obtained with the higher dephasing rate ${\gamma }_d/2\pi =$ 2.0 kHz.

Figure 5

Simulating the $\chi$ matrix with AAWF. (a) Real and (b) imaginary part of the difference between the ideal and simulated process matrix for a C-PHASE gate [N.B. Sub-tick labels to the right of each $\mathcal{W}$I read $\mathcal{W}$X, $\mathcal{W}$Y, $\mathcal{W}$Z, where $\mathcal{W}\in \{I,X,Y,Z\}$]. (c) Trace distance as a function of the blue laser Rabi frequency for three experimental realizations of blockade strength: full AAWF treatment (solid line) and “no-jump wave function” upper bound (dotted line). The (red) dot on the solid $\mathcal{B}=20$ MHz line indicates the parameters used in parts (a) and (b) of the figure. We use the parameters listed in Table with ${\gamma }_d/2\pi =$ 1.0 kHz.