Optimal control-based efficient synthesis of building blocks of quantum algorithms: A perspective from network complexity towards time complexity

In this paper, we demonstrate how optimal control methods can be used to speed up the implementation of modules of quantum algorithms or quantum simulations in networks of coupled qubits. The gain is most prominent in realistic cases, where the qubits are not all mutually coupled. Thus the shortest times obtained depend on the coupling topology as well as on the characteristic ratio of the time scales for local controls vs nonlocal (i.e., coupling) evolutions in the specific experimental setting. Relating these minimal times to the number of qubits gives the tightest known upper bounds to the actual time complexity of the quantum modules. As will be shown, time complexity is a more realistic measure of the experimental cost than the usual gate complexity. In the limit of fast local controls (as, e.g., in NMR), time-optimized realizations are shown for the quantum Fourier transform (QFT) and the multiply controlled NOT gate $\left({\mathrm{C}}^{n-1}\mathrm{NOT}\right)$ in various coupling topologies of $n$ qubits. The speed-ups are substantial: in a chain of six qubits the quantum Fourier transform so far obtained by optimal control is more than eight times faster than the standard decomposition into controlled phase, Hadamard and SWAP gates, while the ${\mathrm{C}}^{n-1}\mathrm{NOT}$ gate for a completely coupled network of six qubits is nearly seven times faster.

Figure 1

(Color online) Global phase dependence of the times needed to implement the three-qubit QFT on a linear chain $\left(L_3\right)$ of nearest-neighbor interactions with uniform weak scalar $J$ couplings. The right curves (●) show the special unitary implementation of a QFT with the smallest global phase ${\varphi }_0=1\pi ∕16$, where it takes $2.53J^{-1}$ to reach a fidelity $F_{\mathrm{tr}}≔(1∕N)\mathrm{Re}\mathrm{tr}\left\{e^{-i{\varphi }_p}{U}_G^{†}U\left(T\right)\right\}⩾0.99999$. The left curves (엯) display the fastest QFT implementations obtained. They are attained with the global phase ${\varphi }_1=5\pi ∕16$. Fidelities $⩾0.99999$ are reached after $2.05J^{-1}$. Times for local controls are assumed to be negligible in this limit matching the typical NMR scenario, where the time cost is determined by coupling evolutions. (a) Gives the trace fidelities against time, while (b) shows the devitiations from full fidelity in a semilogarithmic way.

Figure 2

Four ordered connected graphs with four vertices representing the topology of pairwise couplings (edges) between four qubits (vertices). Times rounded to $0.01J^{-1}$ are given for the shortest QFT-realizations obtained thus far by numerical time-optimal control with trace fidelities $(1∕N)\mid \mathrm{tr}\left\{U_G^{†}U\left(T\right)\right\}\mid ⩾0.99999$.

Figure 3

(Color online) (a) Gate complexity of the QFT in linear coupling topologies $L_n$. Standard-gate decomposition (●) [58] and optimized scalable gate decomposition (▴) [54]. (b) Time complexity of the QFT in linear coupling topologies. Upper traces give analytical times associated with the decompositions of part (a): standard-gate decompositions (●) [58] and optimized scalable gate decompositions (▴) [54]; (▵) gives a special (i.e., nonscalable) five-qubit decomposition into standard gates obtained by simulated annealing [54]. Lowest trace: speed-up by time-optimal control with shortest numerical realizations obtained (엯) rounded to $0.01J^{-1}$. Further details in Table .

Figure 4

(Color online) (a) Network complexity of the ${\mathrm{C}}^{n-1}\mathrm{\text{NOT}}$ gate on complete coupling topologies $K_n$ [57]. (b) Time complexity of the ${\mathrm{C}}^{n-1}\mathrm{\text{NOT}}$ gate on complete coupling topologies. Upper trace: analytical times for decomposition into standard gates (●) [57]. Lower trace: speed-up by time-optimal control with shortest times (엯) currently needed for realizing ${\mathrm{C}}^{n-1}\mathrm{\text{NOT}}$ by numerical control rounded to $0.01J^{-1}$.

Figure 5

(Color online) Time course of controls for the shortest realizations obtained for the following quantum modules: (a) the QFT on a linear coupled four-qubit system $\left(L_4\right)$; (b) the ${\mathrm{C}}^3\mathrm{NOT}$ gate on a fully coupled four-qubit system $\left(K_4\right)$. Traces in blue: amplitudes for ${\sigma }_{\ell x}$ controls ($x$ pulses); red: amplitudes for ${\sigma }_{\ell y}$ controls ($y$ pulses) on the spins $\ell =1$, 2, 3, 4.