Time-optimal control of collisional $\sqrt{\mathrm{SWAP}}$ gates in ultracold atomic systems

We use quantum optimal control to identify fast collision-based two-qubit $\sqrt{\mathrm{SWAP}}$ gates in ultracold atoms. We show that a significant speedup can be achieved by optimizing the full gate instead of separately optimizing the merge-wait-separate sequence of the trapping potentials. Our optimal strategy does not rely on the atoms populating the lowest eigenstates of the merged potential, and it crucially includes the accumulation of quantum phases before the potentials are fully merged. Our analyses transcend the particular trapping geometry, but for comparison with previous works, we present systematic results for an optical lattice and find greatly improved gate durations and fidelities.

Figure 1

Schematic of the merge-wait-separate sequence implementing the $\sqrt{\mathrm{SWAP}}$ gate. Completing the sequence transfers a state of initially opposite spins into a spin-entangled state.

Figure 2

Merging stage ($\beta =0.52\pi \times t/T, \theta =-0.474\pi , V_0/h=122$) kHz of the lattice unit cell in the independent-particle picture. (a) $t=0$: Atoms are initially prepared in the separated double-well configuration. (b) $t=0.7T$: Intermediate snapshot of the merging process. (c) $t=T$: Atoms occupy orthogonal states in the merged single-well configuration. In each snapshot the value of the energy difference $U_{a,b}$ between corresponding $|{\mathrm{\Psi }}_{a,b}^{\pm }\rangle$ is shown in units of $\mathrm{kHz}·h$ [see Eqs. (6, 7, 8, 9)].

Figure 3

Illustration of how the spins are distributed due to the relative phase in the first two quadrants. Here $\alpha =\pi /10$.

Figure 4

Adiabatic lattice merging of the atoms into a single-well configuration as in Fig. 2. Top: Density plots ${\left|a(x,t)\right|}^2$ (top, blue shade) and ${\left|b(x,t)\right|}^2$ (bottom, red shade) of single-particle states $|a\left(0\right)\rangle |b\left(0\right)\rangle =|Lg\rangle |Rg\rangle \to |e\rangle |g\rangle$ [51]. The merge duration $T^{\mathrm{m}}$ is marked by the dashed line and at $t>T^{\mathrm{m}}$ the potential is static. Bottom: Interaction energy $U_{a\left(t\right),b\left(t\right)}$, Eq. (9) (dash-dotted green line), and total accumulated relative phase $\alpha \left(t\right)={\hslash }^{-1}{\int }_0^t{U}_{a\left(t^{\prime }\right),b\left(t^{\prime }\right)}d{t}^{\prime }$ (solid purple line). The phase acquired during merging $\alpha \left(T^{\mathrm{m}}\right)$ is nonzero.

Figure 5

Schematic of different idealized optimal trajectories in Hilbert space, both leading to $\sqrt{\mathrm{SWAP}} |{\uparrow }_{Lg}{,\downarrow }_{Rg}\rangle$. The three upper trajectories (solid blue curves) corresponds to implementing the simple merge-wait-separate sequence. In this case, each trajectory must pass through $|{\mathrm{\Phi }}_{\text{t}}^{\text{m}}\rangle$, whence it can be connected to $|{\mathrm{\Phi }}_{\text{t}}^{\text{f}}\rangle$ as explained in the text. The three lower trajectories (dash-dotted red curves) correspond to implementing the full gate without distinct stages. Here there is no requirement to pass through a particular intermediate state.

Figure 6

Optimization results (lower is better) for the merging subproblem. The lower (upper) dashed horizontal line marks the 0.99 (0.97) fidelity threshold. $1-{\mathcal{F}}^{\prime \mathrm{m}}$ is shown for each solution in a batch optimized for ${\alpha }_{\text{t}}=0.33$. The distribution density is indicated by the translucency. The best solutions for each $T$ are shown by filled red circles. Of 2544 seeds, only 3 optimized to ${\mathcal{F}}^{\prime \mathrm{m}}=0.99$. The quantum speed limit bound in this optimization batch is $T_{\text{QSL}}^{\mathrm{m}}\le 0.0888$ ms.

Figure 7

Full $\sqrt{\mathrm{SWAP}}$ gate operation based on the time-optimal merging control. The merge-wait-separate stages are indicated by the dashed vertical lines. Top: Instantaneous fidelities with various states. Bottom: Corresponding independent single-particle densities [51]. The target state $|{\mathrm{\Phi }}_{\mathrm{t}}^{\mathrm{m}}\rangle$ is obtained at $T^{\mathrm{m}}$ with ${\mathcal{F}}^{\prime \text{m}}=0.99$. The system then exhibits two-level dynamics before the separation onset and enters $|{\mathrm{\Phi }}_{\mathrm{t}}^{\mathrm{f}}\rangle$ with suboptimal final fidelity ${\mathcal{F}}^{\text{f}}=0.983$ at $T^{\mathrm{f}}$.

Figure 8

Optimization results (lower is better) for the full gate problem when using the optimized extended solutions from Fig. 6 as seeds. The lower (upper) horizontal dashed line marks the 0.99 (0.97) fidelity threshold. $1-{\mathcal{F}}^{\mathrm{f}}$ is shown for each solution. The monotonically best and optimal solutions are shown by filled red circles. Of 2323 seeds, a total of 277 optimized to ${\mathcal{F}}^{\mathrm{f}}=0.99$. The quantum speed limit bound in this optimization batch is $T_{\text{QSL}}^{\mathrm{f}}\le 0.1377\mathrm{ms}$.

Figure 9

Full $\sqrt{\mathrm{SWAP}}$ gate operation from the time-optimal full gate control. The initial merge-wait-separate stages for the seed are indicated by the dashed vertical lines. Top: Instantaneous fidelities with various states. Bottom: Corresponding independent single-particle densities [51]. The merging target state $|{\mathrm{\Phi }}_{\mathrm{t}}^{\mathrm{m}}\rangle$ is only partially populated, with at most ${\mathcal{F}}^{\prime \text{m}}\approx 0.52$. The atoms exhibit in-phase oscillations before the separation onset and enters $|{\mathrm{\Phi }}_{\mathrm{t}}^{\mathrm{f}}\rangle$ with ${\mathcal{F}}^{\text{f}}=0.99$ at $T^{\mathrm{f}}$.

Figure 10

Comparison of instantaneous fidelities with the initial state in the 1D and 2D case. Their difference roughly corresponds to the leakage out of the ground state in the $\widehat{y}$ direction.

Figure 11

Numerical two-particle states in the initial configuration (top row) and the merged configuration (bottom row). Lengths are in units of $a=408\mathrm{nm}$ and energies are in units of $\mathrm{kHz}·\mathrm{h}$. (a)–(c) Lowest-lying eigenstates. The (ground) states $|{\tilde{\mathrm{\Psi }}}_{Lg,Lg}^{\pm }\rangle$ corresponding to singly occupied wells are degenerate since both wave functions are vanishing for $x_1=x_2$. The excited state $|{\tilde{\mathrm{\Psi }}}_{Lg+Rg,Lg-Rg}^{\pm }\rangle$ corresponding to doubly occupied wells has an increased energy due to the interaction. (d) Numerical initial state $|{\mathrm{\Psi }}_0\rangle$. (e)–(g) Lowest-lying eigenstates. The symmetric excited state $|{\tilde{\mathrm{\Psi }}}_{e,g}^{+}\rangle$ has an increased energy compared to the antisymmetric one $|{\tilde{\mathrm{\Psi }}}_{e,g}^{-}\rangle$. (h) Numerical target state $|{\mathrm{\Psi }}_{\text{t}}^{\text{m}}\rangle$ (here ${\alpha }_{\text{t}}=0$).

Figure 12

The set of (scaled) optimal controls and their seed at the quantum speed limit bound $T_{\text{QSL}}^{\mathrm{m}}=0.0888\mathrm{ms}$. Thin dotted blue line: Initial control. Thick red line: Optimized control.

Figure 13

The set of (scaled) optimal controls and their seed at the quantum speed limit bound $T_{\text{QSL}}^{\mathrm{f}}=0.1377\mathrm{ms}$. Thin dotted blue line: Initial control. Thick red line: Optimized control.