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arXiv:quant-ph/0207103 · Physical Review A

Error rate of the Kane quantum computer controlled-NOT gate in the presence of dephasing

Austin G. Fowler, Cameron J. Wellard, Lloyd C. L. Hollenberg

Published Jan 3, 2003• 17 citations
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Abstract

We study the error rate of controlled-NOT (CNOT) operations in the Kane solid-state quantum computer architecture [B. Kane, Nature 393, 133 (1998)]. A spin Hamiltonian is used to describe the system. Dephasing is included as exponential decay of the off-diagonal elements of the system's density matrix. Using available spin-echo decay data, the CNOT error rate is estimated at $\ensuremath{\approx}{10}^{\ensuremath{-}3}.$

DOI
10.1103/physreva.67.012301

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We study the error rate of controlled-NOT (CNOT) operations in the Kane solid-state quantum computer architecture [B. Kane, Nature 393, 133 (1998)]. A spin Hamiltonian is used to describe the system. Dephasing is included as exponential decay of the off-diagonal elements of the system's density matrix. Using available spin-echo decay data, the CNOT error rate is estimated at \ensuremath≈10\ensuremath−3.\ensuremath{\approx}{10}^{\ensuremath{-}3}.\ensuremath≈10\ensuremath−3.
In this paper we investigate the effect of dephasing on proposed quantum gates for the solid-state Kane quantum computing architecture. Using a simple model of the decoherence, we find that the typical error in a controlled-NOT gate is $8.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$. We also compute the fidelities of ZZZ, XXX, swap, and controlled ZZZ operations under a variety of dephasing rates. We show that these numerical results are comparable with the error threshold required for fault tolerant quantum computation.
We propose and find pulse sequences for non‐adiabatic two‐qubit gates (CNOT, swap and controlled Z gates) for the Kane silicon‐based electron mediated nuclear spin quantum computer architecture. These quantum gates are simpler, with higher fidelity and faster than existing proposals. Any two qubit gate may be implemented using similar pulse sequences. In this paper, we investigate the effect of decoherence on the operation of these quantum gates and some simple algorithms for a variety of dephasing rates. Our model shows that conservative estimates of the dephasing rates in the Kane architecture give gate fidelities comparable to that required for fault tolerant quantum computing.
The method of iterated resolvents is used to obtain an effective Hamiltonian for neighboring qubits in the Kane solid-state quantum computer. In contrast to the adiabatic gate processes inherent in the Kane proposal we show that free evolution of the qubit-qubit system, as generated by this effective Hamiltonian, combined with single-qubit operations, is sufficient to produce a controlled-NOT gate. Thus the usual set of universal gates can be obtained on the Kane quantum computer without the need for adiabatic switching of the controllable parameters as prescribed by Kane [Nature (London) 393, 133 (1998)]. Both the fidelity and the gate time of this nonadiabatic controlled-NOT gate are determined by numerical simulation.
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.
The Kane proposal for the construction of a solid state quantum computer calls for the placement of a metallic A-gate above each nuclear qubit. Voltage biases applied to these A-gates are to be used to control the evolution of the qubits. In this paper, we consider the decohering effect of thermal fluctuations, the so-called Johnson noise, in the A-gate biases, on the average evolution of qubits undergoing single qubit rotations, as well as those that are not undergoing any operation at all. We find that although the errors introduced are of a different nature for each case, phase errors for the static qubits and both phase and bit-flip errors for qubits undergoing operations, the net error probability is similar for both, and low enough when compared to the timescale of single qubit operations for them to be corrected efficiently.
An off-resonant error for a driven quantum system refers to interactions due to the input drives having nonzero spectral overlap with unwanted system transitions. For the cross-resonance gate, this includes leakage as well as off-diagonal computational interactions that lead to bit-flip error on the control qubit. In this paper, we quantify an off-resonant error, with more focus on the less studied off-diagonal control interactions, for a direct controlled-not (cnot) gate implementation. Our results are based on a numerical simulation of the dynamics while we demonstrate the connection to time-dependent Schrieffer-Wolff and Magnus perturbation theories. We present two methods for suppressing such error terms. First, pulse parameters need to be optimized so off-resonant transition frequencies coincide with the local minima due to the pulse spectrum sidebands. Second, we show the advantage of a YYY-DRAG pulse on the control qubit in mitigating the off-resonant error. Depending on qubit-qubit detuning, the proposed methods can improve the average off-resonant error from approximately ${10}^{\ensuremath{-}3}$ closer to the ${10}^{\ensuremath{-}4}$ level for a direct cnot calibration.