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AIP conference proceedings

Timeoptimal Quantum Computing via Optimal Control in Finite Dimensions: Spin Systems and beyond

Thomas Schulte‐Herbrüggen

Published Jan 1, 2004• 0 citations
Publisher

Abstract

Timeoptimal quantum computing provides a dramatic speed‐up in realistic coupling networks of qubits, wherever the time‐limiting steps are coupling evolutions, while local controls are fast. Already in a chain of six spin qubits the timeoptimal quantum Fourier transform obtained by optimal control is some six times faster than the standard decomposition into controlled phase and Hadamard gates. Here, new results of quantum control are presented within their Lie‐theoretical framework sketched from the preconditions of controllability and reachability up to recent methods of gradient flows.

DOI
10.1063/1.1834416

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Timeoptimal quantum computing provides a dramatic speed‐up in realistic coupling networks of qubits, wherever the time‐limiting steps are coupling evolutions, while local controls are fast. Already in a chain of six spin qubits the timeoptimal quantum Fourier transform obtained by optimal control is some six times faster than the standard decomposition into controlled phase and Hadamard gates. Here, new results of quantum control are presented within their Lie‐theoretical framework sketched from the preconditions of controllability and reachability up to recent methods of gradient flows.
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.
In this paper, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. Time optimal control helps to minimize relaxation losses. The ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems. The generators of evolution, $\g$, are decomposed into fast generators $\k$ (local Hamiltonians) and slow generators $\p$ (couplings) as a Cartan decomposition $\g = \p \oplus \k$. Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems. In this paper, we carry out a second order analysis of time optimality.
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.
Quantum Fourier transform (QFT) is a quantum analogue of the classical discrete Fourier transform. It is a fundamental quantum gate in quantum algorithms which has an exponential advantage over the classical computation and has been excessively studied. Normally, an n-qubit quantum Fourier transform could be resolved into the tensor product of n single-qubit operations, and each operation could be implemented by a Hadamard gate and a controlled phase gate. Then the complexity of an n-qubit QFT is of order O(n2). To reduce the complexity of quantum operations, optimal control (OC) method has recently been used successfully to find the minimum time for implementing a quantum operation. Up to now, two types of quantum optimal control methods have been presented, i.e. analytical and numerical methods. The analytical approach is to change the problem of efficient synthesis of unitary transformations into the geometrical one of finding the shortest paths. Numerical optimal control procedures are based on the gradient methods (GRAPE, Gradient Ascent Pulse Engineering) and Krotov methods. Notable application mainly focus on nuclear magnetic resonance fields, including imaging, liquid-state NMR, solid-state NMR, and NMR quantum computation. One obvious advantage of optimal control NMR quantum computation is that the OC unitary evolution transformation pulse sequences are normally shorter than the conventional corresponding ones. Here we use the optimal control method to find the minimum duration for implementing QFT quantum gate. A linear spin chain with nearest-neighbor Ising interaction is used to find the optimization. And the optimized pulse sequence is experimentally demonstrated on an NMR quantum information processor. By using optimal control method with numerical calculation, a three-qubit QFT in an indirect-linear-coupling chain system is optimized. The duration of the OC QFT is obviously shorter than that of conventional approaches. The OC pulse sequence has been experimentally implemented on a liquid-state NMR spectrometer. To verify the optimally controlled pulse sequence for the three-qubit QFT, different initial states are assumed. The accuracy of the OC pulse sequence could be demonstrated by the consistency of theoretical simulation spectra with the experimental results. The good consistency between the simulation and the experimental spectra demonstrates that the OC QFT is of high fidelity.
Abstract In quantum control, quantum speed limits provide fundamental lower bounds on the time that is needed to implement certain unitary transformations. Using Lie algebraic methods, we link these speed limits to symmetries of the control Hamiltonians and provide quantitative bounds that can be calculated without solving the controlled system dynamics. In particular we focus on two scenarios: On one hand, we provide bounds on the time that is needed for a control system to implement a given target unitary U and on the other hand we bound the time to implement the most difficult to reach evolution e − i θ H s of a target Hamiltonian H s . We apply our abstract bounds on physically relevant systems like coupled qubits, spin chains, globally controlled Rydberg atoms and NMR molecules and compare our results to the existing literature. We hope that our bounds can aid experimentalists to identify bottlenecks and design faster quantum control systems.
In this work, we develop a protocol to find the optimal control pulse for implementing the Quantum Fourier Transform (QFT) on qudit-based hardware. We focus on two types of controls: (1) qubit-qudit dispersive coupling with qubit and qudit drives, and (2) qubit-qudit system with tunable coupling and qubit drive. We observe that for both systems, the minimum gate time grows linearly as a function of the size of the logical space. We find that the latter system supports faster pulses, requiring 45% shorter pulse time.