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臺灣大學物理研究所學位論文

量子邏輯閘 (CNOT) 之時間最佳化研究

蔡東邦

Published Jan 1, 2008• 0 citations
Publisher

Abstract

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.

DOI
10.6342/ntu.2008.00828

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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.
In this paper, we demonstrate how gradient ascent pulse engineering (GRAPE) optimal control methods can be implemented on donor electron spin qubits in semiconductors with an architecture complementary to the original Kane’s proposal. We focus on the high fidelity controlled‐NOT (CNOT) gate and we explicitly find the digitized control sequences for a controlled‐NOT gate by optimizing its fidelity using the effective, reduced donor electron spin Hamiltonian with external controls over the hyperfine A and exchange J interactions. We then simulate the CNOT‐gate sequence with the full spin Hamiltonian and find that it has an error of 10−6 that is below the error threshold of 10−4 required for fault‐tolerant quantum computation. Also the CNOT gate operation time of 100 ns is 3 times faster than 297 ns of the proposed global control scheme.
We demonstrate how gradient ascent pulse engineering optimal control methods can be implemented on donor-electron-spin qubits in Si semiconductors with an architecture complementary to the original Kane's proposal. We focus on the high-fidelity-controlled-NOT (CNOT) gate and explicitly find its digitized control sequences by optimizing its fidelity over the external controls of the hyperfine AAA and exchange JJJ interactions. This high-fidelity-CNOT gate has an error of about ${10}^{\ensuremath{-}6}$, below the error threshold required for fault-tolerant quantum computation, and its operation time of 100 ns is about three times faster than 297 ns of the proposed global control scheme. It also relaxes significantly the stringent distance constraint of two neighboring donor atoms of 10--20 nm as reported in the original Kane's proposal to about 30 nm in which surface AAA and JJJ gates may be built with current fabrication technology. The effects of the control voltage fluctuations, the dipole-dipole interaction, and the electron-spin decoherence on the CNOT gate fidelity are also discussed.
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.
We propose a scheme for quantum information processing based on donor electron spins in semiconductors, with an architecture complementary to the original Kane proposal. We show that a na\"{\i}ve implementation of electron spin qubits provides only modest improvement over the Kane scheme, however through the introduction of global gate control we are able to take full advantage of the fast electron evolution timescales. We estimate that the latent clock speed is 100--1000 times that of the nuclear spin quantum computer with the ratio T2∕Tops{T}_{2}∕{T}_{\mathrm{ops}}T2​∕Tops​ approaching the 106{10}^{6}106 level.
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.