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Physical Review A

Control of a coupled two-spin system without hard pulses

Viswanath Ramakrishna, Raimund J. Ober, Kathryn L. Flores, HERSCHEL A. RABITZ

Published Jun 6, 2002• 31 citations
Publisher

Abstract

Constructive techniques for preparing a specific unitary generator from a coupled two-spin system, via bounded amplitude radio frequency selective single spin pulses, are presented. The frequency of any pulse in the sequence is one of the two Larmor frequencies, while the phase takes one of two values. The fact that the amplitude is bounded implies that the Larmor frequency separation between the two spins need not be large to maintain selectivity. In addition, the contribution of the J coupling term to single spin evolution is not neglected, but instead plays a crucial role. The procedure is based on a certain decomposition of SU(4) available in the literature. A method for determining the parameters entering this factorization, in terms of the entries of the target unitary generator, is also provided.

DOI
10.1103/physreva.65.063405

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Constructive techniques for preparing a specific unitary generator from a coupled two-spin system, via bounded amplitude radio frequency selective single spin pulses, are presented. The frequency of any pulse in the sequence is one of the two Larmor frequencies, while the phase takes one of two values. The fact that the amplitude is bounded implies that the Larmor frequency separation between the two spins need not be large to maintain selectivity. In addition, the contribution of the J coupling term to single spin evolution is not neglected, but instead plays a crucial role. The procedure is based on a certain decomposition of SU(4) available in the literature. A method for determining the parameters entering this factorization, in terms of the entries of the target unitary generator, is also provided.
Constructive procedures for the control of a class of coupled spin systems with bounded amplitude sinusoidal pulses are described. Full details are first illustrated for a single spin 1/2 particle. The key steps for a pair of such particles are then sketched. The fact that the amplitudes can be arbitrarily bounded implies that the separation between the Larmor frequencies of the individual spin 1/2 particles can be quite small and yet the system can be controlled by addressing each spin individually.
Constructive procedures for the control of a class of coupled spin systems with bounded amplitude sinusoidal pulses are described. Full details are first illustrated for a single spin 1/2 particle. The key steps for a pair of such particles are then sketched. The fact that the amplitudes can be arbitrarily bounded implies that the separation between the Larmor frequencies of the individual spin 1/2 particles can be quite small and yet the system can be controlled by addressing each spin individually.
Most quantum processors requires pulse sequences for controlling quantum states. Here we present an alternative algorithm for computing an optimal pulse sequence so as to perform a specific task, being an implementation of a quantum gate or a quantum-state preparation. In our method, we reduce greatly the number of parameters to be fitted by using a limited number of functions as the modulations for the amplitude and phase of the radio-frequency pulses. Unlike other algorithms, ours uses some approximations to make it fast and scalable, and uses some efficient methods to obtain pulses that are robust to common types of experimental errors. We demonstrate the success of the proposed algorithm by performing several real experiments for 4-, 7-, and 12-quantum-bit systems using NMR. In addition, we show the efficiency of the algorithm, finding pulses for controlling with good fidelity the quantum states of spins in fictional square two-dimensional lattices containing 16, 36, and 100 qubits.
Controlling ensembles of weakly coupled spins typically requires computationally expensive multispin optimisations. We present a compact framework that enables control of weakly coupled spin systems (of any spin), but using RF pulses optimised for a single spin-1/2. We do this by explicitly creating a GRAPE pulse with fixed 'active' evolution times using single spin-1/2 methods, and pulsing on one spin at a time. By enforcing this form uniformly across offsets ('band-schematic' pulses),chemical shift and scalar coupling evolution of the entire system can be precisely controlled. We demonstrate the approach by constructing band-schematic pulses and a continuously irradiated joint INEPT (JINEPT) that achieves band-selective transfer Iz→2IzSzI_z \rightarrow 2I_zS_zIz​→2Iz​Sz​. The framework is implemented in the software Seedless, which both rapidly generates such pulses and analyses the schematic form of arbitrary pulses, enabling robust multi-spin control, without multi-spin optimisation.
We introduce a scheme for realizing arbitrary controlled-unitary operations in a two-qubit system. If the $2\ifmmode\times\else\texttimes\fi{}2$ unitary matrix is special unitary (has unit determinant), the controlled-unitary gate operation can be realized in a single pulse operation. The pulse in our scheme will constitute varying one of the parameters of the system between an arbitrarily maximum and a ``calculated'' minimum value. This parameter will constitute the variable parameter of the system while the other parameters, which include the coupling between the two qubits, will be treated as fixed parameters. The values of the parameters are what we solve for using our approach in order to realize an arbitrary controlled-unitary operation. We further show that the computational complexity of the operation is no greater than that required for a controlled-NOT (CNOT) gate. Since conventional schemes realize a controlled-unitary operation by breaking it into a sequence of single-qubit and CNOT gate operations, our method is an improvement because we not only require less time duration, but also fewer control lines to implement the same operation. To demonstrate improvement over other schemes, we show, as examples, how two controlled-unitary operations, one being the controlled-Hadamard gate, can be realized in a single pulse operation using our scheme. Furthermore, our method can be applied to a wide range of coupling schemes and can be used to realize gate operations between two qubits coupled via Ising, Heisenberg, and anisotropic interactions.