Numerical optimizations of the microwave and radio frequency control pulses for the nitrogen vacancy electron-nuclear spin register were performed by means of a gradient ascent pulse engineering method (GRAPE). Three examples of spin control — specific quantum state preparation and the implementation of a controlled-NOT and unconditional - gate have been optimized through parallel computation. The optimized control pulses reach fidelities of F = 0:99944, F = 0:77929 and F = 0:99143 respectively, whereas simulation of Rabi oscillation based control yields corresponding results of F = 0:96325, F = 0:21884 and F = 0:42702. The fixed step size in the GRAPE algorithm imposes a tradeoff between monotone and fast convergence of optimized control pulses and the unspecific performance function may lead to erroneous optimizations. These complications require future research to be resolved. Furthermore, implementing the effect of decoherence yields no significant average fidelity improvement, but has a positive effect on reducing the spread of final density states for the case of quantum state preparation.
Numerical optimizations of the microwave and radio frequency control pulses for the nitrogen vacancy electron-nuclear spin register were performed by means of a gradient ascent pulse engineering method (GRAPE). Three examples of spin control — specific quantum state preparation and the implementation of a controlled-NOT and unconditional - gate have been optimized through parallel computation. The optimized control pulses reach fidelities of F = 0:99944, F = 0:77929 and F = 0:99143 respectively, whereas simulation of Rabi oscillation based control yields corresponding results of F = 0:96325, F = 0:21884 and F = 0:42702. The fixed step size in the GRAPE algorithm imposes a tradeoff between monotone and fast convergence of optimized control pulses and the unspecific performance function may lead to erroneous optimizations. These complications require future research to be resolved. Furthermore, implementing the effect of decoherence yields no significant average fidelity improvement, but has a positive effect on reducing the spread of final density states for the case of quantum state preparation.
Nitrogen-vacancy (NV) center ensembles have the potential to improve a wide range of applications, including nuclear magnetic resonance spectroscopy at the microscale and nanoscale, wide-field magnetometry, and hyperpolarization of nuclear spins via the transfer of optically induced NV polarization to nearby nuclear spin clusters. These NV ensembles can be coherently manipulated with microwave cavities, that deliver strong and homogeneous drivings over large volumes. However, the pulse shaping for microwave cavities presents the challenge that the external controls and intracavity field amplitudes are not identical, leading to adverse effects on the accuracy of operations on the NV ensemble. In this paper, we introduce a method based on gradient ascent pulse engineering (GRAPE) to optimize external controls, resulting in robust pulses within the cavity while minimizing the effects of cavity ringings. The effectiveness of the method is demonstrated by designing both π and π / 2 pulses. These optimized controls are then integrated into a PulsePol sequence, where numerical simulations reveal a resilience to detunings five times larger than those tolerated by the sequence constructed using standard controls.
We propose a scheme for optimizing two-qubit gate operations in a decoherence-free subspace (DFS) using the gradient ascent pulse engineering (GRAPE) algorithm, where the system consists of nitrogen-vacancy (NV) centers in diamond coupled to a whispering-gallery-mode microresonator. The logical qubits in this scheme are encoded in the DFS, which effectively avoids collective decoherence and provides a long coherence time for high-fidelity quantum operations. The GRAPE algorithm is used to design and optimize the control pulse sequences required for the target operation, thereby compensating for common errors and ensuring that the quantum gate maintains high fidelity under non-ideal conditions. As examples, we successfully implement high-fidelity and robust controlled-phase and SWAP gates. Numerical simulations demonstrate the effectiveness and superiority of the proposed scheme. The strategy of combining encoding protection with optimal control shown in this work can be extended to implement universal quantum gates in complex quantum systems.
Quantum optimal control, a powerful toolbox for engineering an optimal control field modulation that most precisely implements a desired quantum operation in the best way possible, has evolved into one of the cornerstones for enabling quantum technologies. The gradient ascent pulse engineering (GRAPE) algorithm is a widely used method in quantum optimal control, which has achieved great success in different physical platforms. However, its computational complexity increases exponentially with the number of qubits, making it challenging to be implemented for large-scale quantum systems. To mitigate this issue, we present the iterative GRAPE algorithm (iGRAPE), which reduces the optimization problem into a series of lower-dimensional subproblems by incorporating disentanglement operations. Our numerical simulations on physical platforms such as nuclear magnetic resonance and superconducting quantum systems demonstrate that iGRAPE significantly enhances state preparation speed. Specifically, compared to GRAPE, iGRAPE achieves up to a five-fold acceleration in preparing Greenberger--Horne--Zeilinger states using a 12-qubit implementation, and up to a 13-fold acceleration for arbitrary state preparation with eight qubits. To further validate our findings, we conduct experimental validation of iGRAPE on a four-qubit nuclear magnetic resonance system. Overall, iGRAPE offers an efficient solution for implementing optimal control in large-scale quantum systems, holding great potential for advancing quantum technologies during the noisy intermediate-scale quantum era.
We investigate how pulse-sequences and operation times of elementary quantum gates can be optimized for silicon-based donor electron spin quantum computer architecture, complementary to the original Kane's nuclear spin proposal. This gate-sequence-optimal or time-optimal quantum gate control in a quantum circuit is in addition to the more conventional concept of optimality in terms of the number of elementary gates needed in a quantum transformation. The optimal control method we use is the so-called gradient ascent pulse engineering (GRAPE) scheme. We focus on the high fidelity controlled-NOT (CNOT) gate and explicitly find the digitized control sequences by optimizing the effective, reduced donor electron spin Hamiltonian, with external controls over the hyperfine A and exchange J interactions. We first try different piecewise constant control steps and numerically calculate the fidelity (error) against the time needed to implement a CNOT gate with stopping criteria of error in the optimizer set to 〖10〗^(-9) in order to economize the simulation time. Here, the error is defined as 1-F, where F is fidelity. The error is less than 〖10〗^(-8) for times longer than 100ns, and it is found that 30 piecewise constant control steps for the CNOT gate operation will be sufficient to meet the required fidelity (error), and the performance would not be improved further with more steps. With operation time t=100ns and stopping criteria of error set to 〖10〗^(-16), we can find that the near time-optimal, high-fidelity CNOT gate control sequence has an error of 〖1.11×10〗^(-16). We then simulate the control sequences of the CNOT gate, obtained from reduced Hamiltonian simulations, with the full spin Hamiltonian. We find the error of about 〖10〗^(-6) which is below the error threshold required for fault-tolerant (〖10〗^(-4)) quantum computation. The CNOT gate operation time of 100ns is 3 times faster than the globally controlled electron spin scheme of 297ns. One of the great advantages of this near optimal-time high fidelity CNOT gate is that the exchange interaction is not required to be strong (the maximum value is J/h=20MHz compared to the typical value of 10.2GHz. This relaxes significantly the stringent distance constraint of two neighboring donor atoms of about 10nm as reported in the original Kane's proposal to be about 30nm which is within the reach of the current fabrication technology.
Quadratic convergence throughout the active space is achieved for the gradient ascent pulse engineering (GRAPE) family of quantum optimal control algorithms. We demonstrate in this communication that the Hessian of the GRAPE fidelity functional is unusually cheap, having the same asymptotic complexity scaling as the functional itself. This leads to the possibility of using very efficient numerical optimization techniques. In particular, the Newton-Raphson method with a rational function optimization (RFO) regularized Hessian is shown in this work to require fewer system trajectory evaluations than any other algorithm in the GRAPE family. This communication describes algebraic and numerical implementation aspects (matrix exponential recycling, Hessian regularization, etc.) for the RFO Newton-Raphson version of GRAPE and reports benchmarks for common spin state control problems in magnetic resonance spectroscopy.
Gradient ascent pulse engineering algorithm (GRAPE) is a typical method to solve quantum optimal control problems. However, it suffers from an exponential resource in computing the time evolution of quantum systems with the increasing number of qubits, which is a barrier for its application in large-qubit systems. To mitigate this issue, we propose an iterative GRAPE algorithm (iGRAPE) for preparing a desired quantum state, where the large-scale, resource-consuming optimization problem is decomposed into a set of lower-dimensional optimization subproblems by disentanglement operations. Consequently these subproblems can be solved in parallel with less computing resources. For physical platforms such as nuclear magnetic resonance (NMR) and superconducting quantum systems, we show that iGRAPE can provide up to 13-fold speedup over GRAPE when preparing desired quantum states in systems within 12 qubits. Using a four-qubit NMR system, we also experimentally verify the feasibility of the iGRAPE algorithm.