Modular quantum-information processing by dissipation

Dissipation can be used as a resource to control and simulate quantum systems. We discuss a modular model based on fast dissipation capable of performing universal quantum computation, and simulating arbitrary Lindbladian dynamics. The model consists of a network of elementary dissipation-generated modules and it is in principle scalable. In particular, we demonstrate the ability to dissipatively prepare all single-qubit gates, and the controlled-not gate; prerequisites for universal quantum computing. We also show a way to implement a type of quantum memory in a dissipative environment, whereby we can arbitrarily control the loss in both coherence, and concurrence, over the evolution. Moreover, our dissipation-assisted modular construction exhibits a degree of inbuilt robustness to Hamiltonian and, indeed, Lindbladian errors, and as such is of potential practical relevance.

Figure 1

A four-module patch of a two-dimensional network of coupled DGMs. The $i\mathrm{th}$ module dissipates according the Liouvillian ${\mathcal{L}}_0^{\left(i\right)}$ and the bare intermodule coupling is given by the ${\mathcal{K}}_{(i,j)}=-i[K_{(i,j)},·]$. Yellow (gray) thick arrows represent strong dissipation whereas blue (black) wavy lines represent Hamiltonian interactions.

Figure 2

Two-module network. Distance between the exact evolution ${\mathcal{E}}_T$ after time $T$, and the effective one for the two Hamiltonians discussed in the main text, $K=g({\sigma }_2^{+}\otimes {\sigma }_1^{-}+{\sigma }_2^{-}\otimes {\sigma }_1^{+})$ [blue (dark gray) circles] and $K=g{\sigma }_2^z\otimes {\sigma }_1^z$ [red (light gray) circles]. We picked the dissipation time scales to be ${\tau }_1=1$ arb. units, ${\tau }_2=0.5$ arb. units. We set $gT=1$. The linear fit is obtained using the least-squares fitting on all of the data points, and the norm is the maximum singular value of the maps realized as matrices. Time is measured in arbitrary units.

Figure 3

Two modules, each containing qubits [yellow (light gray spheres), top] connected to a bosonic mode [blue (dark gray sphere), bottom], which dissipate at rate ${\tau }^{-1}$. In turn, the two modules are coherently coupled via the bosonic modes, with strength $J$.

Figure 4

Coherence of the evolved effective dynamics, Eq. (18), as a function of $J$ for a single qubit, for three values of $\tau$ (see legend). Initial qubit state $|{\psi }_0\rangle =\frac{1}{\sqrt{2}}\left(\right|0\rangle +|1\rangle )$. We set $T{g}_A^2=1$ arb. units. This result is valid only in the regime $T\gg \tau$. Time is measured in arbitrary units ($J,g_{\alpha }$ are inverse time).

Figure 5

Dynamics of entanglement, given by the concurrence $\mathcal{C}$, as a function of the boson-boson coupling strength $J$ for a two-qubit system evolving via Eq. (18), for different dissipation time scales $\tau$ (see legend). The two qubits are initialized in the maximally entangled state $|{\psi }_0\rangle =\frac{1}{\sqrt{2}}\left(\right|00\rangle +|11\rangle )$. The qubit-boson coupling parameters were chosen such that $T{g}_A^2=T{g}_B^2=1$ arb. units. This result is valid only in the regime $T\gg \tau$. Time is measured in arbitrary units ($J,g_{\alpha }$ are inverse time).

Figure 6

Robustness to Hamiltonian errors. We consider an unperturbed dissipation given by Eq. (9) plus a control Hamiltonian (11) and an error term of the form (27). The error matrix $\zeta$ has random complex entries (for two fixed, nonzero magnitudes). We plot the distance between the exact dynamics (i.e., the dynamics with error terms), to the effective dynamics as governed by Eq. (12). Blue (dark gray) data points are without error. The dissipative time scales are fixed at ${\tau }_1={\tau }_2=1$ arb. units, and we set $Tg=2$. Norms are calculated using the maximum singular value of the maps realized as matrices. Time is measured in arbitrary units.

Figure 7

Robustness to Lindbladian errors. We consider an unperturbed dissipation given by Eq. (9) plus a Lindbladian error term of the form (29) with ${\eta }_i=V$ as in Eq. (27). We show results for the unperturbed case $\zeta =0$ [blue (dark gray) points] and two nonzero perturbations. The dissipative time scales are fixed at ${\tau }_1={\tau }_2=1$ arb. units, and we set $Tg=2$. Norms are calculated using the maximum singular value of the maps realized as matrices. Time is measured in arbitrary units.

Figure 8

Distance between exact evolution and the target maximally entangled Bell state $|{\psi }^{+}\rangle$ (defined in main text). We initialize the system in the product state ${\rho }_0=|{\psi }_0\rangle \langle {\psi }_0|$, where $|{\psi }_0\rangle =\frac{|\overline{0}\rangle +|\overline{1}\rangle }{\sqrt{2}}|\overline{0}\rangle$. The exact evolution ${\mathcal{E}}_T={\mathcal{E}}_{T/7}^{\left(7\right)}{\mathcal{E}}_{T/7}^{\left(6\right)}\cdots {\mathcal{E}}_{T/7}^{\left(1\right)}$, where each ${\mathcal{E}}_{T/7}^{\left(i\right)}$ generates one of the seven gates of Eq. (A2), as described in the main text. The dissipative time scales for the two DGMs were set to ${\tau }_1={\tau }_2=1$ arb. units for this simulation. The norm is the maximum singular value of the maps realized as matrices. The linear fit is on the 10 largest $T$ values. Time is measured in arbitrary units.