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Time optimal quantum control of two-qubit systems

Bin Li, Zuhuan Yu, Fei Shaoming, Xianqing Li‐Jost

Published Aug 17, 2016• 11 citations
OpenAlex

Abstract

We study the optimal quantum control of heteronuclear two-qubit systems described by a Hamiltonian containing both nonlocal internal drift and local control terms. We derive an explicit formula to compute the minimum time required to steer the system from an initial state to a specified final state. As applications the minimal time to implement Controlled-NOT gate, SWAP gate and Controlled-U gate is calculated in detail. The experimental realizations of these quantum gates are explicitly presented.

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We study the optimal quantum control of heteronuclear two-qubit systems described by a Hamiltonian containing both nonlocal internal drift and local control terms. We derive an explicit formula to compute the minimum time required to steer the system from an initial state to a specified final state. As applications the minimal time to implement Controlled-NOT gate, SWAP gate and Controlled-U gate is calculated in detail. The experimental realizations of these quantum gates are explicitly presented.
Abstract Finding minimal time and establishing the structure of the corresponding optimal controls which can transfer a given initial state of a quantum system into a given target state is a key problem of quantum control. In this work, this problem is solved for a basic component of various quantum technology processes—a qubit interacting with the environment and experiencing an arbitrary time-dependent coherent driving. We rigorously derive both upper and lower estimates for the minimal steering time. Surprisingly, we discover that the optimal controls have a very special form—they consist of two impulses, at the beginning and at the end of the control period, which can be assisted by a smooth time-dependent control in between. Moreover, an important for practical applications explicit almost optimal state transfer protocol is provided which only consists of four impulses and gives an almost optimal time of motion. The results can be directly applied to a variety of experimental situations for estimation of the ultimate limits of state control for quantum technologies.
We study the manipulation of a two level quantum mechanical system which is the basic unit of representing data in quantum information science, a qubit. This research is motivated by the design of the quantum Hadamard gate for solid state spin qubits. We consider the problem of driving the initial state of a qubit into a desired final state in a specified time, while minimizing the energies of the control fields. Optimal control theory is employed to define optimum control fields which satisfy both dynamical constraints and optimality conditions. Time evolution of the system is calculated within the Schrödinger equation and a gradient method is used for tailoring optimal control fields.
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 $({\mathrm{C}}^{n\ensuremath{-}1}\mathrm{NOT})$ in various coupling topologies of nnn 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 ${\mathrm{C}}^{n\ensuremath{-}1}\mathrm{NOT}$ gate for a completely coupled network of six qubits is nearly seven times faster.
Motivated by a bound derived in a recent work [C. Arenz, B. Russell, D. Burgarth and H. Rabitz, New J. Phys. 19 (2017) 103015], we apply optimal control theory to the dynamics of qubit systems, with the goal of investigating the lower bound of minimum gate times. In practice, we not only need to reach the desired unitary gates but we need to do so in a reasonable time (below the typical decoherence time). Therefore, we employ the recently introduced lower bound to estimate the minimum gate time and resort to numerical gate optimization in order to study the tightness of the obtained bound and our findings verify the relationship between the internal Hamiltonian and the minimum evolution time remarkably well. Finally, we discuss both challenges and ways forward for obtaining the same efficacy under realistic conditions.
Universal quantum computation requires the implementation of arbitrary control operations on the quantum register. In most cases, this is achieved by external control fields acting selectively on each qubit to drive single-qubit operations. In combination with a drift Hamiltonian containing interactions between the qubits, this allows the implementation of arbitrary gate operations. Here, we demonstrate an alternative scheme that does not require local control for all qubits: we implement one- and two-qubit gate operations on a set of target qubits indirectly, through a combination of gates on directly controlled actuator qubits with a drift Hamiltonian that couples actuator and target qubits. Experiments are performed on nuclear spins, using radio-frequency pulses as gate operations and magnetic-dipole couplings for the drift Hamiltonian.
Quantum control of systems plays an important role in modern science and technology. The ultimate goal of quantum control is to achieve high-fidelity universal control in a time-optimal way. Although high-fidelity universal control has been reported in various quantum systems, experimental implementation of time-optimal universal control remains elusive. Here, we report the experimental realization of time-optimal universal control of spin qubits in diamond. By generalizing a recent method for solving quantum brachistochrone equations [X. Wang et al., Phys. Rev. Lett. 114, 170501 (2015)], we obtained accurate minimum-time protocols for multiple qubits with fixed qubit interactions and a constrained control field. Single- and two-qubit time-optimal gates are experimentally implemented with fidelities of 99% obtained via quantum process tomography. Our work provides a time-optimal route to achieve accurate quantum control and unlocks new capabilities for the emerging field of time-optimal control in general quantum systems.