Digital quantum simulator in the presence of a bath

For a digital quantum simulator (DQS) imitating a target system, we ask the following question: Under what conditions is the simulator dynamics similar to that of the target in the presence of coupling to a bath? In this paper, we derive conditions for close simulation for three different physical regimes, replacing previous heuristic arguments on the subject with rigorous statements. In fact, we find that the conventional wisdom that the simulation cycle time should always be short for good simulation need not always hold up. Numerical simulations of two specific examples strengthen the evidence for our analysis and we go beyond to explore broader regimes.

Figure 1

A periodic pulse sequence, with cycle time $T$. Each pulse sequence comprises a set of $M$ pulses that altogether take time ${\tau }_g$ to complete. The $i\mathrm{th}$ pulse, of strength $H_{g_i}-H_{\mathrm{S}}$, starts at time $t_i$ and lasts for duration ${\tau }_i$. Together with the natural Hamiltonian $H_{\mathrm{S}}$, the pulse implements the gate $g_i\equiv e^{-i{H}_{g_i}{\tau }_i}$.

Figure 2

The simulator and the target system interact with the bath in the same manner.

Figure 3

A small piece of the square lattice in the toric-code model of Ref. [12]. Qubits (circles) sit on the edges of the lattice; qubits around each plaquette of the lattice (in cyan) are acted upon by the local four-body $ZZZZ$ terms of the toric-code model; the qubits around each lattice vertex (in yellow) are acted upon by the $XXXX$ terms. The qubits labeled 1 to 4 around the vertex are the four qubits in our $H_{\mathrm{tar}}$ model of Eq. (62).

Figure 4

Dynamics of the toric-code vertex target and simulator, in the regime where $x_c\ll \epsilon$, or, equivalently, where $\omega {\tau }_{\mathrm{B}}\ll 1$. The left column [plots marked (A)] gives the situation where all parameters but $R={\tau }_g/T$ are fixed (to be regarded as varying ${\tau }_g$ for fixed $T$; see main text). The right column [plots marked (B)] gives the case where all parameters but the simulation cycle time $T$ are fixed. Plots marked (i) give the ground-space population; those marked (ii) give the trace distances between the target and simulator states; those marked (iii) are the trace distances between the time $t$ state $\rho \left(t\right)$ and the initial state, for the target and the simulator. Parameters for plots A: $\epsilon =0.1,x_c=\epsilon /5=0.02,\tilde{\eta }=0.02$, and $\tilde{\beta }=40$. Parameters for plots B: $\omega =20\mathrm{kHz},{\nu }_c=4\mathrm{kHz},\beta =0.2\mathrm{ms},\eta =0.02$, and ${\tau }_g=50\mathrm{ns}$. The blue dashed line is for the target. In plot A(ii), the $R=0.01$ line essentially lie on the horizontal axis, for the plotted vertical scale; in plot B(ii), the $T=5\mu \mathrm{s}$ line is also nearly on the horizontal axis.

Figure 5

The dynamics of the target and the DQS, with varying $x_c={\nu }_{c}T=T/{\tau }_{\mathrm{B}}\sim 1$ values. The other parameters are kept fixed: $\epsilon =0.005,\tilde{\eta }=5\times {10}^{-4},\tilde{\beta }=2000$, and $R=0.01$. The plots are labeled by $x_c/\epsilon =100,200$, and 400, corresponding to $x_c=0.5,1$, and 2, respectively.

Figure 6

Dynamics of the toric-code vertex target and the simulator, in the regime where $x_c\gg \epsilon$, or, equivalently, where $\omega {\tau }_{\mathrm{B}}\gg 1$. The left column [plots marked (A)] gives the situation where all parameters but $R={\tau }_g/T$ are fixed. The right column [plots marked (B)] gives the case where all parameters but the simulation cycle time $T$ and ${\tau }_g$ (with $R={\tau }_g/T$ held constant) are fixed. Plots marked (i) give the ground-space population; those marked (ii) are the trace distance between the target and simulator states; those marked (iii) are the trace distance between the time $t$ state $\rho \left(t\right)$ and the initial state, for the target and the simulator. Parameters for plots A: $\epsilon =4\times {10}^{-4},x_c=8\gg \epsilon ,\tilde{\eta }=5\times {10}^{-4}$, and $\tilde{\beta }=2500$. Parameters for plots B: $R$ is fixed to be 0.01, $\omega =1\mathrm{kHz},{\nu }_c=400\mathrm{kHz},\beta =1\mathrm{ms},\eta =5\times {10}^{-4}$, and $\delta t=2\mu \mathrm{s}$.

Figure 7

Dynamics of the five-qubit code target and the simulator, in the regime where $x_c\ll \epsilon$, or, equivalently, where $\omega {\tau }_{\mathrm{B}}\ll 1$. (This is similar to Fig. 4, but for the five-qubit code, rather than the toric-code vertex.) The left column [plots marked (A)] gives the situation where all parameters but $R={\tau }_g/T$ are fixed; the right column [plots marked (B)] gives the case where all parameters but the simulation cycle time $T$ are fixed. Plots marked (i) give the ground-space population; those marked (ii) are the trace distance between the target and simulator states; those marked (iii) are the trace distance between the time $t$ state $\rho \left(t\right)$ and the initial state, for the target and the simulator. Parameters for plots A: $\epsilon =0.1,x_c=0.2,\tilde{\eta }=0.02$, and $\tilde{\beta }=40$; $R={\tau }_g/T$ is varied over the values $0.01,0.1,0.4$, and 0.7 (to be regarded as varying ${\tau }_g$ for fixed $T)$. Parameters for plots B: $\omega =20\mathrm{kHz},{\nu }_c=4\mathrm{kHz},\beta =0.2\mathrm{ms},\eta =0.02,{\tau }_g=50\mathrm{ns}$, and $\delta t=5\mu \mathrm{s}$.