超導體量子位元的回饋與最佳化控制的理論研究

Abstract

量子資訊處理 (quantum information processing) 的基本前提是要能夠對量子系統或者量子位元 (quantum bit or qubit) 進行精準的控制。因此本論文致力於對具擴展性 (scalable) 的固態量子計算系統--超導體量子位元 (superconducting qubit) 的控制與操作進行研究。我們的研究內容包含了兩種不同類型的超導體量子位元與結構：電路型空腔量子電動力學 (circuit cavity quantum electrodynamics) 和通量量子位元的耦合系統 (coupled superconducting flux qubits)。 在第一部分，我們對超導體電路型空腔量子電動力學系統提出了一個簡單且有希望實現的回饋方案來產生並穩定三個量子位元的 W 糾纏態 (three-qubit entangled W state) 的回饋方案。我們利用量子軌跡法 (quantum trajectory method) 與極化子變換法 (polaron-type transformation method)所得到的等效隨機主方程式 (stochastic master equation) 來模擬量子回饋控制的方案。我們的結果顯示在受到一般不太強的環境去相干(environmental decoherence) 的影響下，W 糾纏態的保真度不僅 可以高於 0.9，而且還可以被維持比單一量子位元的去相干時間 (dcoherence time)還要長的時間上。除此之外，這個回饋控制方案也被證實在測量效率低下，或是個別量子衰變率存在差異的時候還是相當地穩固。最後，本論文也對使用極化子變換法以及常用的絕熱消去法 (adiabatic elimination method) 來消去空腔態做了比較。 在第二部分中，我們應用了以克羅托夫法 (Krotov method) 為基礎的最佳化控制理論來實現在電感耦合超導體通量量子位元系統上的單一量子位元的 X 和 Z 邏輯閘，以及雙量子位元的 CNOT 邏輯閘，此超導量子通量子系統具有固定的量子躍遷頻率以及固定的量子位元耦合強度。在我們的方案中，量子位元處在最佳相干點運作，而且邏輯閘的運行時間（單量子位元邏輯閘 <1 奈秒； CNOT邏輯閘 2 奈秒）比相對應的量子位元的去相干時間要短得多。CNOT 邏輯閘或其它普遍的量子邏輯閘可以藉由最佳化的方式找出單一脈衝序列的來運行, 而不須將其分解成若干個單個 量子位元操控並且串聯一些糾纏的雙量子位元操控的複合脈衝序列方式來運作。我們利用最佳化控制的方式所構造出來的量子邏輯閘都具有很高的保真度（非常低的誤差），因為我們的方法把量子位元固定的頻率失調 (detuning)，固定的量子位元之間的作用力，以及其由於時變的磁通量所引起的耦合的各種因素都考慮進來。 此外，我們探討在測量效率因素以及去相干影響下的連續測量過程中以非絕熱消除的方式 (nonadiabatic elimination method)來消除與感興趣的系統耦合的輔助探測場。當被消除的輔助場演化的時間尺度在大於或者接近系統演化以及衰減的時間尺度的時候，相較於絕熱消去法，我們所發展出來的非絕熱消除法就顯得特別重要，因為在這種情況下，輔助場不僅存在有限的記憶效應，而且系統本身的動力學也變為非馬爾可夫過程。我們以與輔助空腔模具線性相互作用的光學機械系統的精確可解模型來進行研究，並且利用這個模型來闡明我們的方法。

An essential prerequisite for quantum information processing (QIP) is precise coherent control of the dynamics of quantum systems or quantum bits (qubits). This thesis is devoted to the study of quantum control and manipulation of superconducting qubits that are promising candidates for scalable solid-state quantum computing. We study two different types of superconducting qubits and architectures: Circuit cavity quantum electrodynamics (QED) and coupled flux qubit systems. In the first part, we present a simple and promising quantum feedback control scheme for deterministic generation and stabilization of a three-qubit entangled W state in the superconducting circuit QED system. We simulate the dynamics of the proposed quantum feedback control scheme using the quantum trajectory approach with an effective stochastic maser equation obtained by a polaron-type transformation method and demonstrate that in the presence of moderate environmental decoherence, the average state fidelity higher than 0.9 can be achieved and maintained for a considerably long time (much longer than the single-qubit decoherence time). This control scheme is also shown to be robust against measurement inefficiency and individual qubit decay rate differences. Finally, the comparison of the polaron-type transformation method to the commonly used adiabatic elimination method to eliminate the cavity mode is presented. In the second part, we apply the quantum optimal control theory based on the Krotov method to implement single-qubit X and Z gates and two-qubit CNOT gates for inductively coupled superconducting flux qubits with fixed qubit transition frequencies and fixed off-diagonal qubit-qubit coupling. The qubits in our scheme are operated at the optimal coherence points and the gate operation times (single-qubit gates <1 ns; CNOT gates 2 ns) are much shorter than the corresponding qubit decoherence time. A CNOT gate or other general quantum gates can be implemented in a ingle run of pulse sequence rather than being decomposed into several single-qubit and some entangled two-qubit operations in series by composite pulse sequences. Quantum gates constructed via our scheme are all with very high delity (very low error) as our optimal control scheme takes into account the fixed qubit detuning and fixed two- qubit interaction as well as all other time-dependent magnetic-eld-induced single- qubit interactions and two-qubit ouplings. In addition, we also investigate the effects of inefficient measurement and additional decoherence on the problems of nonadiabatic elimination of an auxiliary mode coupled to the system of interest in continuous quantum measurements. In contrast to the adiabatic elimination method, the eveloped nonadiabatic elimination approach is particularly important when the eliminated auxiliary mode evolves at a time scale larger than or comparable to the typical system evolution or decay time scale as, in this case, the auxiliary mode has finite memory, and the resultant dynamics of the system alone becomes non-Markovian. We investigate an exactly solvable model of an optomechanical system with a linear interaction with an auxiliary cavity mode to illustrate our approach.