用於量子計算的堅固量子邏輯閘

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國立臺灣大學理學院物理學系博士論文

Department of Physics College of Science

National Taiwan University Doctoral Dissertation

用於量子計算的堅固量子邏輯閘

Robust Quantum Gates for Quantum Computation

黃嘉賢 Chia-Hsien Huang

指導教授：管希聖博士

Advisor: Hsi-Sheng Goan, Ph.D.

中華民國 106 年 7 月

July, 2017

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謝辭

首先，我要感謝在成大讀書時教導我的林水田老師、李湘楠老師、和閔振發老師幫我寫推薦信，才讓我有機會可以進入台大讀博士。

接著，我要感謝我的指導教授管希聖老師的提拔，老師給予我很大的自由進行研究工作，並指引我正確的方向，讓我有參與國際合作的機會，還讓我可以在國際會議發表演講，並且還幫助我申請工作。同時我也要感謝黃上瑜學長，在我研究起步時提供我相當多的協助，還要感謝陳建彰學長、賴彥佑學長、張晏瑞學長、簡崇欽學長和林冠廷同學，提供我許多技術上、學術上和生活上的建議在我整個博士求學過程中。還要感謝所有在 501 研究室相處過的學長、同學和學弟學妹們(陳柏文、洪常力、戴榮身、楊存毅、周宜、劉伊修、薛逸峰、楊智偉、葉宗剛、李彥賢、梁哲亮、郝鴻、廖允執、許景喻、邱博煌、方之珊、張佑誠、陳則言…)給予我的協助與腦力激盪。我要感謝澳洲的合作夥伴 Henry Yang 博士與 Andrew Dzurak 教授，提供我許多實驗上的觀點，讓我的理論成為可以在實驗上實踐的方法。我要感謝物理系辦公室季力偉先生，還有管希聖老師的秘書劉家綺女士，協助我解決處理許多非學術性的事情。我要感謝林俊達老師、蔡政達老師、李瑞光老師、和陳岳男老師，百忙中撥空來擔任我的論文口試委員。我要感謝台灣大學豐富的圖書與期刊資源，還有 Google 的搜尋引擎，讓我可以很有效率地進行研究工作。我要感謝 QQbit 伺服器不辭辛勞不舍晝夜地努力計算，讓我可以很快地蒐集數據。我要感謝 501 研究室 4 號座位，讓我可以很專心地做研究。

我要感謝 X，X 屬於我要感謝但沒有寫出來的所有一切。

最後，我要感謝我最堅強後盾，也就是我的家人，謝謝您們支持我陪伴我完成博士學位。

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摘要

要實現量子計算(quantum computation)，我們需要一組高保真(high-fidelity)而且堅固(robust)的量子邏輯閘(quantum gate)，來對抗量子位元(qubit)系統中的噪音 (noise)並容許系統參數(system parameter)的不準確性(uncertainty)。堅固控制方法 (robust control method)可以提供控制脈波(control pulse)來操控並實現高保真而且堅固的量子邏輯閘。可是大部分的堅固控制方法都假設在量子位元系統中的噪音強度並不隨時間而改變，然而這個假設並不總是對的。因此我們提供一套有系統的堅固控制方法，可以有效地處理隨機(stochastic)並且可隨時間改變(time-varying) 的噪音。我們的方法可以同時處理多個不同的噪音來源(multiple sources of

noise)，可以運用到不同的量子位元系統與不同的噪音模型，並提供連續性 (smooth)的控制脈波來操作高保真而且堅固的量子邏輯閘以實現容錯量子計算 (fault-tolerant quantum computation)。

接著，我們將此堅固控制方法運用到一個實際的量子位元系統：半導體量子點電子自旋(quantum-dot electron spin)量子位元。最近，澳洲實驗團隊將此量子位元系統建構在純化的同位素矽(isotopically purified silicon)半導體上來改善來自量子位元環境的噪音，並實現二位元(two-qubit)量子邏輯閘。然而，操控二位元量子邏輯閘會伴隨著電的噪音(electrical noise)，而這個噪音使得二位元量子邏輯閘誤差(gate error)無法達到實現容錯量子計算的門檻(threshold)。我們的堅固控制方法可提供最佳化控制脈波(optimal control pulse)，來操控可以抵抗電的噪音之二位元量子邏輯閘，使得邏輯閘誤差遠低於此門檻，並且堅固於來自於系統參數的不準確性。此外我們的最佳化控制脈波也考慮到實驗上對於控制脈波的限制，像是最強脈波強度(maximum pulse strength)限制，還有由波形產生器(waveform

generator)的有限頻寬(finite-bandwidth)所造成的濾波效應(filtering effect)。更進一步，我們在同一個控制架構下提供實驗上可實現的最佳化控制脈波，來操控高保

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真而且堅固的二位元量子邏輯閘與單一位元量子邏輯閘(single-qubit quantum gate)，為實現大尺度(large-scale)容錯量子計算提供重要的一步。

關鍵字：量子計算、最佳化控制、堅固、高保真、隨時變噪音、量子邏輯閘、

量子點量子位元

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Abstract

To realize practical quantum computation, a set of high-delity universal quantum gates robust against noise and uncertainty in the qubit system is prerequisite. Constructing control pulses to operate quantum gates which meet this requirement is an important and timely issue. In most robust control methods, noise is assumed to be quasi-static, i.e., is time-independent within the gate operation time but can vary between dierent gates.

But this quasi-static-noise assumption is not always valid. Here we develop a systematic method to nd pulses for quantum gate operations robust against both low- and high- frequency (comparable to the qubit transition frequency) stochastic time-varying noise.

Our approach, taking into account the noise properties of quantum computing systems, can output single smooth pulses in the presence of multisources of noise. Furthermore, our method can be applied to dierent system models and noise models, and will make essential steps toward constructing high-delity and robust quantum gates for fault-tolerant quantum computation (FTQC). We also discuss and compare the gate operation performance by our method with that by the lter-transfer-function method.

Then we apply our robust control method for a realistic system of electron spin qubits in semiconductor (silicon) quantum dots, a promising solid-state system compatible with existing manufacturing technologies for practical quantum computation. A two-qubit controlled-NOT (CNOT) gate, realized by a controlled-phase (C-phase) gate together with some single-qubit gates, has been experimentally implemented recently for quantum-dot electron spin qubits in isotopically puried silicon. But the indelity of the two-qubit C- phase gate is, primarily due to the electrical control noise, still higher than the required error threshold for FTQC. Here we apply our robust control method to construct high-

delity CNOT gates with single smooth control pulses robust against the electrical noise and the system parameter uncertainty. The experimental constraints on the maximum pulse strength due to the power limitation of the on-chip electron spin resonance (ESR)

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line and the ltering eects on the pulses due to the nite bandwidth of waveform generators are also accounted for. The robust and high-delity single-qubit gates, together with the two-qubit CNOT gates, can be performed within the same control framework in our scheme, paving the way for large-scale FTQC.

Keywords: quantum computation, optimal control, robust, high-delity, time- varying noise, quantum gate, quantum dot qubit

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List of Figures

2.1. J1+ hJ₂i versus (a) γZZ for Z-noise (σZZ = 10⁻³, σXX = 0) and (b) γXX

for X-noise (σXX = 10⁻³, σZZ = 0). The J1 + hJ₂i values are obtained using the optimal control parameter sets of the Hadamard gate from the IDG strategy (blue triangles), QSN strategy (red circles), and TVN strategy (yellow squares). Ten realizations of OU noise βOU(t)with σOU = 10⁻³ for γOU/ω0 equal to (c) 10⁻⁷, (d) 10⁻³, and (e) 10⁻¹. . . 12 2.2. J1+ hJ2i values versus γZZ for Z-noise (σZZ = 10⁻³, σXX = 0) and versus

γ_XX for X-noise (σXX = 10⁻³, σZZ = 0) obtained from the IDG strategy (blue triangles), QSN strategy (red circles), and TVN strategy (yellow squares) for the phase gate shown in (a) and (b), respectively, and for the π/8 gate in (c) and (d), respectively. . . 12 2.3. Robust performance of the Hadamard gate of the IDG strategy (blue tri-

angles), QSN strategy (red circles), and TVN strategy (yellow squares) for low-frequency (γZZ = γXX = 10⁻⁷ω0) (a) Z-noise, (b) X-noise, and (c) Z-&-X-noise. The corresponding optimal control pulses of the TVN strategy for Z-noise, X-noise, and Z-&-X-noise are shown in (d), (e), and (f), respectively. The number of control parameters kmax=10 for ΩX(t)in (d)-(f). 14

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2.4. Robust performance of the Hadamard gate of the IDG strategy (blue triangles), QSN strategy (red circles), and TVN strategy (yellow squares) for high-frequency (γZZ = γ_XX = 10⁻¹ω₀) (a) Z-noise, (b) X-noise, and (c) Z-&-X-noise. For TVN strategy with an additional Y control (green pentagrams) in (b), γY Y = γ_XX = 10⁻¹ω₀ and σY Y = σ_XX, and in (c), γ_{Y Y} = γ_ZZ = γ_XX = 10⁻¹ω₀ and σY Y = σ_XX = σ_ZZ. Optimal control pulses of the TVN strategy (d) for Z-noise and of the TVN strategy with an additional Y control and accompanying Y -noise (e) for X-noise and (f) for Z-&-X-noise. The number of control parameters kmax=10 for ΩX(t) in (d) and kmax=20 for both ΩX(t)and ΩY(t)in (e) and (f). . . 15 2.5. Robust performance of CNOT gates of the IDG strategy (ω0t_f = 100, blue

triangles), QSN strategy (ω0t_f = 100, red circles), TVN strategy (ω0t_f = 100, yellow squares; and ω0t_f = 20, purple pentagrams) for high-frequency (γZZ1 = γ_ZZ2 = γ_XX1 = γ_XX2 = γ_{J J} = 10⁻¹ω₀) (a) Z-noise, (b) X-

&-J-noise, and (c) Z-&-X-&-J-noise. The optimal control pulses of the TVN strategy (ω0t_f = 100) for the Z-noise, X-&-J-noise, and Z-&-X-&- J-noise are shown in (d), (e), and (f), respectively. The numbers of control parameters kmax=16, 16, and 8 for ΩX1(t), ΩX2(t), and J(t), respectively, in (d) and (f); kmax=12, 12, and 6 for ΩX1(t), ΩX2(t), and J(t), respectively, in (e). . . 18 2.6. The behavior of Fz(ω)/(2πω²)

obtained using the optimal control parameter sets from the IDG strategy (thick dotted blue line), FTF strategy (thin dash-dotted red line), and TVN strategy (thick solid yellow line) for the noise PSD S(ω) with (a) γ = 0.1ω0, (b) γ = 0.3ω0, and (c) γ = 0.5ω0 is shown in (d), (e), and (f), respectively. (g) The corresponding hJ2i values. 19 3.1. Loss-and-DiVincenzo's model for quantum dot spin qubit (courtesy of Daniel

Loss and David P. DiVincenzo, 1998). . . 29 3.2. The architecture of single-spin qubit (left panel) and singlet-triplet qubit

(right panel) in GaAs semiconductor quantum dots. . . 30 3.3. Quantum dots in silicon (courtesy of W. H. Lim et al., 2009). . . 31

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3.4. Architecture of the quantum-dot electron spin qubits in isotopically puried silicon (courtesy of M. Veldhorst et al., 2015). . . 32 3.5. The eective detuning frequencies ν↑↓ and ν↓↑ versus U − for δEZ =

−40MHzand t0 = 900MHz. . . 34 3.6. Ensemble average probability hP (|↑↓i)i simulation for stochastic and static

electrical noise βU −with mean value µU −= 0and with standard deviation σ_{U −}= 0, 3, 10GHz. . . 40 3.7. The experimentally measured hP (|↑↓i)i (courtesy of Veldhorst et al., 2015). 40 3.8. J2,t0 versus hJ2,U −i with σU − = 3GHzfor αt0 = 9MHz and 4.5MHz. . . . 41 3.9. ΩX(ω) and ΩY(ω) of the optimal CNOT gate with tf = 500ns. . . 44 3.10. The ltered pulse Ω^filt_X (t)and Ω^filt_Y (t)of the optimal CNOT gate by the trans-

fer function with the response function of the lter F (ω) = exp(−ω²/ω²₀). . 45 3.11. J1 and hJ2,U −idegradation from the ltering eects. . . 45 3.12. The optimized J1 values versus optimization iterations after the J1 opti-

mization for the CNOT gate. . . 49 3.13. The distribution of the maximum pulse strength Ω^Max_X and Ω^Max_Y after the

J1 optimization for the CNOT gate. . . 49 3.14. The hJ2,U −i topography of the CNOT gate with σU − = 3GHz versus

maximum pulse strength Ω^Max_X and Ω^Max_Y for the J1 optimized ensemble (1100 samples). . . 50 3.15. hIi of the CNOT gate versus αt0 after the two-step optimization and after

the ne-tuning optimization. . . 51 3.16. Pulse shift after the ne-tuning optimization for the CNOT gate. . . 51 3.17. Robust performance against uncertainty αt0 in t0 for the optimal CNOT

gates of Ω<1mT (red diamond-line) and Ω<1.5mT (yellow square-line), and the C-phase gate (blue circle-line). . . 52 3.18. The optimal control pulses of the CNOT gate with Ω<1mT and Ω<1.5mT. . . 53 3.19. hJ2,U −i topography of I2⊗ X₁ gate with σU − = 3GHz versus maximum

pulse strength Ω^Max_X and Ω^Max_Y for the J1 optimized ensemble (2389 samples). 54 3.20. Robust performance against uncertainty αt0 in t0 for the optimal single-

qubit I2⊗ X₁ gate and H2⊗ I₁ gate. . . 55 3.21. The optimal control pulses of the single-qubit I2⊗ X₁ gate and H2⊗ I₁ gate. 55

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1. Introduction

A bit is the fundamental storage and computing unit in classical computers. This fundamental unit in quantum computers is called a quantum bit (qubit), which is dened through a realistic two-level system. The two-level states, |0i and |1i, of a qubit corre- spond to 0 and 1 of a classical bit, respectively. However, a qubit can according to quantum mechanics be in the superposition state of |0i and |1i, α |0i + β |1i, and a classical bit can only be 0 or 1. Combining with other properties of quantum mechanics such as entangle- ment, quantum computers via quantum algorithms can tackle certain problems, which can not be solved by the existing classical supercomputers. These quantum algorithms such as Shor's algorithm and Grover's algorithm command the quantum computers to execute a specic task via a composition of quantum gates. A quantum gate is operated on the qubits by our controls in the system, and is just the propagator for the qubit system. The dynamics of a quantum gate is governed by the Schr¨odinger equation. However, there exist noise and uncertainty in a realistic system, causing the gate error or indelity for each gate. In this case, a quantum algorithm, composed of many quantum gates, may easily fail. Fortunately, fault-tolerant quantum computation (FTQC) via a set of universal quantum gates, in terms of which any unitary operation can be expressed to arbitrary accuracy, can correct these errors if gate error of each universal quantum gate is below some threshold, for example, 10⁻² for surface codes [1]. Therefore, our goal is to make all universal quantum gates robust against the strength of noise and uncertainty to meet the FTQC threshold requirement and to realize practical quantum computation.

Quantum gates in open quantum systems have been investigated by various methods such as dynamical decoupling methods [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and optimal control methods [12, 13, 14, 15, 16, 17, 18, 19]. For classical noise, there are many robust control methods such as composite pulses [20, 21, 22, 23, 24, 25, 26, 27, 28, 29], soft uniaxial positive control for orthogonal drift error (SUPCODE) [30, 31, 32, 33, 34, 35], sampling-based

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learning control method [36, 37, 38], inhomogeneous control methods [39, 40], analytical method [41], single-shot pulse method [42], optimal control methods [43, 44, 45], invariant- based inverse engineering method [46, 47], and lter-transfer-function (FTF) methods [48, 49, 50, 51]. However, in most of these methods [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], noise is assumed to be quasi-static. We call these robust control strategies the quasi-static-noise (QSN) methods. But this QSN assumption is not always valid [52]. The robust performance of control pulses obtained by the QSN methods under time-dependent noise (e.g., 1/f^α noise) [32, 33, 35, 53] have been investigated, and it was found that they can still work well for relatively low-frequency non-Markovian noise (e.g., 1/f^α noise with α ? 1) .

Stochastic time-dependent noise is treated in the FTF method [48, 49, 50] in which the area of the lter-transfer function in the frequency region, where the noise power spectral density (PSD) is non-negligible, is minimized. However, in this approach only the lter-transfer function overlapping with the noise PSD in the preset frequency region is considered, but the detailed information of the distribution of the noise PSD is not included in the optimization cost function. Here we develop an optimal control method in time domain by choosing the ensemble average gate indelity (error) as our cost function for optimization. As a result, the noise correlation function (CF) or equivalently the detailed noise PSD distribution appears naturally in our chosen optimization cost function.

Therefore our method can have better robust performance against noise in a general case.

The idea of our method is simple, and our method is not limited to particular system models, noise models, and noise CFs. We demonstrate our robust control method for classical noise, but our method can be easily generalized to the case of quantum noise by replacing the ensemble average for classical noise with the trace over the degrees of freedom of the quantum noise (environment) [51]. In other words, our method can be applied to systems with both classical noise and quantum noise present simultaneously.

Electron spin qubits in semiconductor quantum dots [54] are promising solid-state systems to realize quantum computation. Signicant progresses of quantum-dot spin qubits for quantum information processing have been made with III-V semiconductors such as GaAs [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67], but the coherence time of the qubits is limited by the strong dephasing from the environment nuclear spins [68]. On the other hand, the coherence time is substantially improved by using the Si-based host substrate

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[69, 70, 71, 72, 73, 74]. Recently, important quantum gate operations for quantum-dot spin qubits in isotopically puried silicon have been demonstrated experimentally [71, 72].

There, the single-qubit gates have been demonstrated with fault-tolerant control-delity [71], but the two-qubit gate delity [72] has not yet reached the criterion for surface codes, primarily due to the noise of the electrical voltage control used to realize the two-qubit gate.

Our goal is to construct robust quantum gates for quantum-dot spin qubits in puried silicon with delity enabling large-scale FTQC by our robust control method. In the experiment of this system [72], a single-qubit gate is realized by tuning down the detuning energy to a small constant value to decouple the two-qubit coupling, and the qubit working in this detuning energy region is not sensitive to the electrical noise. Inversely, a two-qubit gate is realized by tuning up the detuning energy to a large constant value to increase the coupling between two qubits, and the qubit working in this detuning energy region is very sensitive to the electrical noise. Therefore, for two-qubit gates, the electrical noise is the dominant source for delity degradation. Besides, when operating a sequence of single- qubit gates and two-qubit gates, the rise and fall times of the detuning energy between two-qubit gate and single-qubit gates would cause gate errors. And changing detuning energy accompanies stark shifts on the quantum-dot qubits, which may result in additional gate errors if the calibration is not precise. Therefore, we propose to keep the detuning energy as a constant value when operating a sequence of single-qubit and two-qubit gates to prevent the delity degradation from tuning the detuning energy up and down. After

nishing a sequence of gate operations, the detuning energy can be pulled to a small value for the idle time.

Therefore, we keep the detuning energy as a constant value, and only control two AC magnetic elds to operate single-qubit gates and two-qubit gates against the electrical noise with realistic system parameters from the experiment [72]. In addition to the electrical noise, we also consider other factors degrading the gate delity in our realistic model for simulation such as the uncertainty in the system parameter and the ltering eects due to the nite bandwidth of waveform generators. In experiment, the interdot tunnel coupling is obtained by tting the experimental data, and thus there may exist some uncertainty in the interdot tunnel coupling, and the uncertainty will degrade the gate

delity. When we apply our optimal control pulses in an experiment, their shape will be

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altered due to the ltering eects, and the pulse distortion will also contribute extra gate errors. Then we apply our robust control method to minimize the gate error contributions from the noise, the uncertainty, and the ltering eects by our optimal control pulses, which satisfy the constraint of the maximum magnetic eld strength smaller than 1mT due to the power limitation through the on-chip ESR line. Instead of decomposing a controlled-NOT (CNOT) gate into a C-phase gate and several single-qubit gates in series as in the experiment [72], we can construct single smooth pulses for CNOT gates directly to reduce the gate operation time and the accumulated gate errors from the decomposed gates. Finally, we demonstrate that our optimal CNOT gate with maximum magnetic eld strength smaller than 1mT can suppress the indelity contribution from the electrical noise to ∼ 10⁻⁵ (around two orders of magnitude improvement compared with the simulation of the realized ideal C-phase gate in experiment) and can be robust against the uncertainty

∼ 10% of the interdot tunnel coupling for the threshold of surface codes (10⁻²). For our optimal single-qubit gates with maximum magnetic eld strength smaller than 1mT, the indelity contribution from the electrical noise is also suppressed to ∼ 10⁻⁵ and the robustness against the uncertainty error of the interdot tunnel coupling is over 15% for the threshold of surface codes. The gate operation time of our optimal single-qubit gates is also improved to 200 ∼ 250ns from 1.5µs (π pulse in the experiment [72]). The gate indelities mentioned above have been recovered from the delity degradation due to the

ltering eects for both single-qubit gates and CNOT gates. To conclude, our robust control strategy can provide high-delity and robust single-qubit gates and CNOT gates for quantum-dot spin qubits in isotopically puried silicon, paving an essential step toward large-scale FTQC.

The thesis is organized as follows. In Chapter 2, we introduce our robust control method and demonstrate its performance. In Chapter 3, we apply our robust control method introduced in Chapter 2 to construct high-delity and robust single-qubit gates and CNOT gates for quantum-dot spin qubits in isotopically puried silicon. In Chapter 4, we conclude what we did and show the future development directions for robust quantum gates.

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2. Robust quantum gates for stochastic time-varying noise

In this chapter, we rst introduce the concept of ensemble average indelity and our robust control method. We then demonstrate the performance of our method through comparing with the quasi-static-noise (QSN) method and the lter-transfer-function (FTF) method, and nally generalize our method to open quantum system.

2.1. Ensemble average indelity

We describe the dynamics of the n−qubit system by its propagator

U (t) = T₊exp[−i ˆ _t

0

H(t⁰)dt⁰], (2.1)

where H(t) is the Hamiltonian of the system (in this chapter we set ~ = 1), and T+ is the time-ordering operator. We can control H(t) from t = 0 to t = tf to obtain U(tf) by Eq.

(2.1), and U(tf) is just a quantum gate for the n−qubit system with operation time tf. Assume that UT is our target gate, we can dene the gate error (gate indelity) as

I ≡ 1 − 1 4ⁿ

Tr

h

U_T^†U (t_f) i

2

, (2.2)

where Tr denotes a trace over the n-qubit system state space. In a realistic system, there may exist noise, and thus the Hamiltonian of the system H(t) should include two parts

H(t) = H_I(t) + HN(t), (2.3)

where HI(t)is the ideal system Hamiltonian and HN(t)is the noise Hamiltonian. If there is no noise in the system (HN(t) = 0), the Hamiltonian H(t) will recover to the ideal

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system Hamiltonian HI(t), and then the system propagator U(t) = UI(t), where

U_I(t) = T₊exp[−i ˆ _t

0

H_I(t⁰)dt⁰] (2.4)

is the ideal system propagator. For there may exist many sources of noise, the general form of the noise Hamiltonian is

H_N(t) =X

j

βj(t)H_N_j(t), (2.5)

where βj(t) is the strength of the j-th noise and HNj(t) is the corresponding system coupling operator term. In general, βj(t) is time-varying and stochastic, but if βj(t)is a constant and non-stochastic, βj can be regarded as a systematic error or uncertainty.

To see the noise contribution in the gate indelity I, we transform the system to the interaction picture by UI(t), and then the system Hamiltonian in the interaction picture is

H˜_N(t) = Σjβj(t)Rj(t), (2.6)

where

R_j(t) ≡ U_I^†(t)H_N_j(t)U_I(t). (2.7) Then the system propagator in the interaction picture is

U (t˜ _f) = T₊exp[−i ˆ _t_f

0

H˜_N(t⁰)dt⁰] (2.8)

If noise strength is not too strong, we can expand ˜U (t_f) by Dyson series [75] as the form U (t˜ f) = I + Ψ1+ Ψ2+ · · · , where the rst two terms of Ψj are

Ψ₁= −i ˆ _t_f

0

H˜_N(t⁰)dt⁰, (2.9)

Ψ2= − ˆ _t_f

0

dt1

ˆ _t₁

0

dt2H˜_N(t1) ˜H_N(t2). (2.10)

Now we transform the propagator in the interaction picture ˜U (t_f) back to the original frame to obtain

U (t_f) = U_I(t_f) · [I + Ψ₁+ Ψ₂+ · · · ], (2.11)

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and substitute it into the gate indelity denition in Eq. (2.2). The expanded indelity I (see Appendix A) takes the form

I = J₁+ J₂+ + O( ˜H^m_N, m ≥ 3), (2.12) J₁ ≡ 1 − 1

4ⁿ Trh

U_T^†U_I(t_f)i

2

, (2.13)

J2 ≡ − 1

2ⁿ⁻¹Re [Tr (Ψ2)] − 1

4ⁿ|Tr (Ψ₁)|². (2.14) Here J1 is the denition of gate indelity for the ideal system HI(t), J2 is the lowest-order contribution of the noise to the gate indelity, (detailed form shown in Appendix A) denotes an extra contribution that is correlated to J1 and the Dyson expansion terms Ψj, and O( ˜H^m_N, m ≥ 3)represents other higher-order terms excluding . If noise strength is not too strong such that |Ψj+1| |Ψ_j|, the extra contribution will become negligible when J₁ is getting small (see discussion in Appendix A). The symbol Re in Eq. (2.14) denotes taking the real part of the quantity it acts on. Because noise βj(t)is stochastic in general, we denote the ensemble average of the indelity over the dierent noise realizations as

hIi = J₁+ hJ2i + hi +D

O( ˜H^m_N, m ≥ 3) E

. (2.15)

Here

hJ₂i = X

j,k

ˆ _t_f

0

dt1

ˆ _t₁

0

dt2Cjk(t1, t2)Tr [Rj(t1)Rk(t2)]

2ⁿ⁻¹

−X

j,k

ˆ _t_f

0

dt1

ˆ _t_f

0

dt2Cjk(t1, t2)Tr [R_j(t₁)] Tr [R_k(t₂)]

4ⁿ , (2.16)

where Cjk(t1, t2) = hβj(t1)β_k(t2)iis the CF for noise βj(t1)and βk(t2). The rst-order noise term proportional to Re[Tr(Ψ1)]vanishes due to the fact that Tr(Ψ1) is purely imaginary rather than the assumption of hβj(t)i = 0 (see Appendix A). If dierent sources of noise are independent, Cjk(t1, t2) = 0 for j 6= k, and if noise Hamiltonian HN(t)is traceless, the second term in Eq. (2.16) vanishes. If βj is a systematic error or uncertainty, Cjj(t₁, t₂) = β_j².

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2.2. Optimization method and noise suppression

The ideal Hamiltonian HI(t) is a function of the control eld Ω(t), that is HI(t) = H_I(Ω(t)), and the control eld Ω(t) is chosen to be a function of a set of control parameters [a1, a₂, · · · ]. Then UI(t) and each term of the ensemble average indelity hIi in Eq. (2.15) are also a function of the control parameter set [a1, a₂, · · · ]. Our goal is to search the optimal parameter set [a1, a₂, · · · ]that minimizes the ensemble average indelity hIi.

If the noise strength or uctuation is not large, then the dominant noise contribution to hIi is from hJ2i as the higher order terms hO( ˜H^m_N, m ≥ 3)i can be neglected (see Ap- pendix B). J1 can generally be made suciently small so that the extra term hi in hIi of Eq. (2.15) can be safely ignored. So we concentrate on the minimization of hIi ∼= J1+ hJ2i for obtaining the optimal control parameter set.

We use the two-step optimization to achieve this goal. The rst step is called the J1

optimization in which J1 is the cost function. The gate indelities J1 in an ideal unitary system with gate-operation-controllability and a sucient number of control parameters can be made as low as one wishes, limited only by the machine precision of the computation.

So using an ensemble of random control parameter sets as initial guesses, we obtain after the J1 optimization an ensemble of optimized control parameters sets all with very low values of J1. The second step is called the J1 + hJ2i optimization. We take J1 + hJ2i as a cost function and randomly choose some optimized control parameter sets in the

rst optimization step as initial guesses to run the optimal control algorithm. After the J1+ hJ2i optimization, we obtain an ensemble of control parameter sets with low values of J1 + hJ₂i, and then choose the lowest one as the optimal control parameter set. The purpose of using the two-step optimization is to improve optimization eciency. If we run the J1+ hJ₂i optimization directly from an ensemble of random control parameter sets, we need more optimization iterations to achieve the goal, and the success rate is relatively low compared with the two-step optimization. Besides, the J1+ hJ2ioptimization enables us to know separately the optimized values of J1 and hJ2i. When hJ2i can be minimized to a very small value as in the case of static or low-frequency noise, one has to use a small time step for simulation to make J1 smaller than hJ2i. However, for high-frequency noise, hJ2i is hard to be minimized to a very small value, and one can instead choose a suitable larger time step to make J1just one or two orders of magnitude smaller than hJ2i,

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saving substantially the optimization time especially for multiqubits and multiple sources of noise. We use the gradient-free and model-free Nelder-Mead (NM) algorithm [76] in both the J1 and J1+ hJ₂i optimization steps. However, the NM algorithm may be stuck in local traps in the J1+ hJ₂iparameter space topography. To overcome this problem, we use the repeating-NM algorithm in the J1+ hJ₂ioptimization step. The control parameter set from the rst J1+ hJ₂ioptimization may lie in a local trap. Therefore, we add random

uctuations to this control parameter set and try to pull it out of the trap. Then we use this shifted control parameter set as an initial guess to run the second J1 + hJ2i optimization. We repeat the same procedure many times until the values of J1+ hJ2i can not be improved (reduced) anymore, and then output the corresponding control parameter set. Our optimization method employing the gradient-free and model-free NM algorithm is quite general, capable of dealing with dierent forms or structures of the ideal system Hamiltonian HI(t), control eld Ω(t), noise Hamiltonian HN(t), and noise CF Cjk(t₁, t₂) for a few qubit systems.

The robustness of our method can be understood as follows. After the two-step optimization, one can obtain small J1+ hJ2i. Generally, J1 can be even a few orders of magnitude smaller than hJ2i, and then hIi ∼= hJ2i. For simplicity, let us assume that there is only one source of traceless noise present in the system with correlation function given by C(t1, t2) =

¯

σ²C(t˜ 1, t2), where ¯σ is the standard deviation of the noise strength uctuation. Then from Eq. (2.16), we have hIi ∼= hJ2i = ¯σ²{´_t_f

0 dt₁´_t₁

0 dt₂C(t˜ ₁, t₂)Tr[R(t₁)R(t₂)]/2ⁿ⁻¹} . If the value of {´_t_f

0 dt₁´_t₁

0 dt₂C(t˜ ₁, t₂) Tr[R(t₁)R(t₂)]/2ⁿ⁻¹} can be reduced more, then larger noise ¯σ² can be tolerated under the same error (indelity) threshold, that is, the quantum gate can be more robust to noise uctuation. The indelity hIi to the lowest noise order is proportional to ¯σ²; but if ¯σ is too large, then the higher order terms hO( ˜H^m_N, m ≥ 3)i should be considered. Therefore, robust performance can be demonstrated by showing the relation of full-order hIi versus ¯σ. The full-order hIi we use to show the robust performance is calculated using the full evolution of the total system-noise Hamiltonian without any approximation. By inputting the optimal control parameter set obtained by the optimization strategy into the total system-noise Hamiltonian H(t) = HI(t) + H_N(t)to obtain numerically the full propagator for a single noise realization, we can calculate the gate indelity I using Eq. (2.2) for the noise realization. The procedure is repeated for many dierent noise realizations. Then we take an ensemble average of the indelities over the

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dierent noise realizations to obtain hIi.

2.3. Demonstration of our optimal control method

In principle, we could deal with any given form of the noise correlation function (or equivalently the noise PSD) to insert into Eq. (2.16) for the J1+ hJ2i optimization. But as a particular example, we choose the Ornstein-Uhlenbeck (OU) process βOU(t) to simulate stochastic time-varying noise [77]. Studying the inuence of and developing robust strategies against time-dependent noise is an important subject of research in quantum control problems both theoretically and experimentally [48, 49, 50, 52, 53]. If the initial noise β_OU(t = 0) is a normal distribution with zero mean and with standard deviation σOU, then the noise CF of the OU process βOU(t)is

COU(t1, t2) = σ_OU² exp (−γOU|t₁− t₂|) (2.17)

with the noise correlation time τ ∼ (1/γOU), and the corresponding noise PSD is Lorentzian

SOU(ω) = 2σ_OU² γ_OU

(γ_OU² + ω²). (2.18)

Lorentzian PSDs of spin noise resulting in a uctuating magnetic eld at the location of the qubits in InGaAs semiconductor quantum dots have been measured experimentally [78, 79]. Generally, a small γOU corresponds to low-frequency or quasi-static noise; a large γOU corresponds to high-frequency noise. The noise βOU(t)can be simulated through the formula βOU(t + dt) = (1 − γOUdt) βOU(t) + σOU

√2γOUdW (t), where W (t) is a Wiener process [77]. Figures 2.1(c), (d), and (e) show the dierent realizations of the noise βOU(t) with σOU = 10⁻³ for dierent values of γOU/ω₀ = 10⁻⁷, 10⁻³, and 10⁻¹, respectively, where ω0 is the typical system frequency. We note here that the particular choice of the OU noise should by no means diminish the value of our work or the power of our method.

Any given or experimentally measured well-behaved noise PSD or noise CF can be dealt with. We will demonstrate later that our method can also work eectively for another form of noise PSD dierent from that of the OU noise when we compare the performance of our method with that of the FTF method. The reason for using the OU noise in the system-noise Hamiltonian here is that it is relatively easy to simulate its stochastic noise

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realizations in the time domain. Therefore, we can calculate the full-order ensemble average indelity hIi to show that our J1+ hJ2i optimization, which minimizes the second-order noise contribution to the average indelity hIi, can indeed work rather well for not too strong a noise uctuation.

2.3.1. Comparison with the quasi-static-noise method

2.3.1.1. Single-qubit gates

We demonstrate as an example the implementation of single-qubit gates in the presence of time-varying noise using our method. The ideal system Hamiltonian for the qubit is

H_I(t) = ω₀Z

2 + Ω_X(t)X

2, (2.19)

where X and Z stand for the Pauli matrices, ω0 is the qubit transition frequency, and ΩX(t)is the control eld in the X term. The noise Hamiltonian is written as

H_N(t) = β_Z(t)ω₀Z

2 + β_X(t)Ω_X(t)X

2 . (2.20)

We call βZ(t)the Z-noise and βX(t)the X-noise, and assume that they are independent OU

noises with CFs CZZ(t1, t2) = σ²_ZZexp (−γZZ|t₁− t₂|)and CXX(t1, t2) =σ²_XXexp (−γXX|t₁− t₂|) as the form of Eq. (2.17). We choose the control pulse as a composite sine pulse expressed

as

Ω_X(t) =

kmax

X

k=1

a_ksin

m_kπ t

t_f

, (2.21)

where the set of the strengths of the single sine pulses is the control parameter set [ak] = [a₁, a₂, · · · , a_k_max] and {mk} is a set of integers, chosen depending on the nature of the system Hamiltonians and the target gates as well as the properties of the noise models.

We dene below three optimization strategies, namely, the ideal-gate (IDG) strategy, quasi-static-noise (QSN) strategy, and time-varying-noise (TVN) strategy. The IDG strategy is to perform the rst-step optimization (J1optimization) only and to show the performance of an ideal gate pulse in the presence of noise. The TVN strategy is our proposed method described earlier above, in which the actual γZZ and γXX values are used in the noise CFs of the cost function hJ2i for the second-step optimization. The QSN strategy uses the same optimization procedure as the TVN strategy, but with γZZ = γ_XX = 0 for

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γ_ZZ/ω0

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 J1+hJ2i

10^-12 10^-11 10^-10 10^-9 10^-8 10^-7 10^-6 10^-5 10^-4 10^-3

(a)Hadamard gate, Z-noise

γ_XX/ω0

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10^-12

10^-11 10^-10 10^-9 10^-8 10^-7 10^-6 10^-5 10^-4 10^-3

(b)Hadamard gate, X-noise

IDG QSN TVN

ω0t

0 10 20

103βOU(t) -4

0

4 (c)γOU = 10⁻⁷ω0

ω0t

0 10 20

-4 0

4 (d)γOU= 10⁻³ω0

ω0t

0 10 20

-4 0

4 (e)γOU = 10⁻¹ω0

σ_ZZ = 10⁻³, σXX= 0 σ_XX= 10⁻³, σZZ= 0

Figure 2.1.: J1 + hJ2i versus (a) γZZ for Z-noise (σZZ = 10⁻³, σXX = 0) and (b) γXX

for X-noise (σXX = 10⁻³, σZZ = 0). The J1 + hJ₂i values are obtained using the optimal control parameter sets of the Hadamard gate from the IDG strategy (blue triangles), QSN strategy (red circles), and TVN strategy (yellow squares). Ten realizations of OU noise βOU(t) with σOU = 10⁻³ for γOU/ω₀ equal to (c) 10⁻⁷, (d) 10⁻³, and (e) 10⁻¹.

10^-710^-610^-510^-410^-310^-2 10^-1 J1+hJ2i

10^-12 10^-11 10^-10 10^-9 10^-8 10^-7 10^-6 10^-5 10^-4

10^-3 (a)Phase gate, Z-noise

10^-7 10^-610^-510^-410^-310^-2 10^-1 10^-12

10^-11 10^-10 10^-9 10^-8 10^-7 10^-6 10^-5 10^-4

10^-3 (b)Phase gate, X-noise

γ_ZZ/ω0

10^-710^-610^-510^-410^-310^-2 10^-1 J1+hJ2i

10^-12 10^-11 10^-10 10^-9 10^-8 10^-7 10^-6 10^-5 10^-4

10^-3 (c)π/8 gate, Z-noise

γ_XX/ω0

10^-7 10^-610^-510^-410^-310^-2 10^-1 10^-12

10^-11 10^-10 10^-9 10^-8 10^-7 10^-6 10^-5 10^-4

10^-3 (d)π/8 gate, X-noise

IDG QSN TVN

σ_ZZ = 10⁻³, σXX= 0

σ_XX= 10⁻³, σZZ = 0 σ_ZZ = 10⁻³, σXX= 0

σ_XX= 10⁻³, σZZ = 0

Figure 2.2.: J1+hJ₂ivalues versus γZZfor Z-noise (σZZ = 10⁻³, σXX = 0) and versus γXX

for X-noise (σXX = 10⁻³, σZZ = 0) obtained from the IDG strategy (blue triangles), QSN strategy (red circles), and TVN strategy (yellow squares) for the phase gate shown in (a) and (b), respectively, and for the π/8 gate in (c) and (d), respectively.

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the noise CFs in the cost function hJ2i. Thus it is regarded to represent the QSN methods.

We choose the gate operation time tf = 20/ω0. After the optimizations of Hadamard gate, we plot the corresponding J1 + hJ₂i values obtained from these three strategies versus γ_ZZ in Figure 2.1(a) for the Z-noise and versus γXX in Figure 2.1(b) for the X-noise.

For low-frequency (quasi-static) noise (γZZ = γ_XX = 10⁻⁷ω₀), the performance of the TVN strategy and the QSN strategy are about the same but they are several orders of magnitude better in indelity J1+ hJ2i value than the IDG strategy which does not take the noise into account at all. As the noise goes from the low frequency to high frequency (γZZ = γXX = 10⁻¹ω0), the TVN strategy taking account of the TVN information in the cost function gets better and better (from a factor-level to an order-of-magnitude-level) improvement in J1 + hJ₂i values than the QSN strategy in which noise is assumed to be quasi-static. In addition to the Hadamard gate, we perform calculations for other quantum gates, namely the phase gate, π/8 gate and controlled-NOT (CNOT) gate, in the fault-tolerant universal set in terms of which any unitary operation can be expressed to arbitrary accuracy. The J1+ hJ₂i values versus γZZ and versus γXX obtained from the three strategies are shown in Figures 2.2(a) and (b), respectively, for the phase gate and in Figures 2.2(c) and (d), respectively, for the π/8 gate. Their performances are similar to those in Figure 2.1(a) and (b) of the Hadamard gate. The optimization results for the two-qubit CNOT gate are presented in Sec. 2.3.1.2.

Next, we take the optimal control parameter sets of the Hadamard gate from these three strategies to show their robust performance against Z-noise, X-noise, and Z-&-X-noise at a low frequency (γZZ = γ_XX = 10⁻⁷ω₀) in Figures 2.3(a), (b), and (c) and at a high frequency (γZZ = γ_XX = 10⁻¹ω₀) in Figures 2.4(a), (b), and (c). For low-frequency noise and for low noise strength (σXX < 10⁻¹, σZZ < 10⁻¹), one can see in Figure 2.3 that the full-order ensemble average indelity hIi scales for the IDG strategy as the second power of the noise standard deviation (σZZ, σXX) but scales for the TVN and QSN strategies as the fourth power. This implies that hIi ∼= hJ2i for the IDG strategy, but the TVN and QSN strategies can nullify the contribution from hJ2i for the low-frequency (quasi-static) noise and the dominant contribution in hIi comes from the next higher-order term, i.e., hIi ∼= hO( ˜H⁴_N)i. In this case, our method, the TVN strategy, still performs slightly better than the QSN strategy. For gate error (indelity) less than the error threshold of 10⁻² of surface codes required for FTQC, the Hadamard gate of TVN strategy can be robust

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σZZ

10^-3 10^-2 10^-1 10⁰

hIi

10^-12 10^-11 10^-10 10^-9 10^-8 10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1

10⁰ (a)γZZ= 10⁻⁷ω0

σXX

10^-3 10^-2 10^-1 10⁰ 10^-12

10^-11 10^-10 10^-9 10^-8 10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1

10⁰ (b)γXX= 10⁻⁷ω0

σZZ= σXX

10^-3 10^-2 10^-1 10⁰ 10^-12

10^-11 10^-10 10^-9 10^-8 10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1

10(c)γ⁰ ZZ= γXX= 10⁻⁷ω0

IDG QSN TVN

ω0t

0 10 20

ΩX(t)/ω0

-4 0 4 (d)

ω0t

0 10 20

-4 0 4 (e)

ω0t

0 10 20

-4 0 4 (f)

Figure 2.3.: Robust performance of the Hadamard gate of the IDG strategy (blue triangles), QSN strategy (red circles), and TVN strategy (yellow squares) for low-frequency (γZZ = γXX = 10⁻⁷ω0) (a) Z-noise, (b) X-noise, and (c) Z-&- X-noise. The corresponding optimal control pulses of the TVN strategy for Z-noise, X-noise, and Z-&-X-noise are shown in (d), (e), and (f), respectively.

The number of control parameters kmax=10 for ΩX(t)in (d)-(f).

to σZZ ∼ 30% for low-frequency Z-noise (i.e., against noise uctuation with a standard deviation up to about 30% of ω0/2), robust to σXX ∼ 20% for the X-noise [i.e., against noise uctuation with a standard deviation up to about 20% of ΩX(t)/2], and robust to σ_ZZ = σ_XX ∼ 10%for Z-&-X-noise as shown in Figures 2.3(a), (b), and (c), respectively.

The corresponding optimal control pulses of the TVN strategy are shown in Figures 2.3(d), (e), and (f), respectively.

For high-frequency noise shown in Figure 2.4, the full-order ensemble average indelity hIi scales as the second power of the noise standard deviation (σZZ, σXX) for all three strategies and noises. This indicates that for high-frequency noise hJ2i is not nullied completely, and is only minimized. Even in this case, the TVN strategy still has over two orders of magnitude improvement in hIi compared with the IDG strategy, and over one order of magnitude improvement compared with the QSN strategy for the Z-noise at small noise strengths as shown in Figure 2.4(a). For hIi . 10⁻² less than the FTQC error threshold of the surface codes, the Hadamard gate implemented by our optimal control pulse shown in Figure 2.4(d) can be robust to σZZ ∼ 20% for the Z-noise. On the other hand, for the high-frequency X-noise, hIi obtained by the QSN strategy has

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σ_ZZ

10^-3 10^-2 10^-1 10⁰

hIi

10^-8 10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1

10⁰ (a)γZZ= 10⁻¹ω0

σ_XX

10^-3 10^-2 10^-1 10⁰ 10^-8

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1

10⁰ (b)γXX = 10⁻¹ω0

σ_ZZ= σXX

10^-3 10^-2 10^-1 10⁰ 10^-8

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1

10(c)γ⁰ ZZ= γXX= 10⁻¹ω0

IDG QSN TVN TVN,Y

-10 0 10 (d)

-5 0 5 (e)

ω0t

0 2 4 6 8 10 12 14 16 18 20

-10 0 10 (f)

ΩX(t)/ω0 ΩY(t)/ω0

Figure 2.4.: Robust performance of the Hadamard gate of the IDG strategy (blue triangles), QSN strategy (red circles), and TVN strategy (yellow squares) for high-frequency (γZZ = γ_XX = 10⁻¹ω₀) (a) Z-noise, (b) X-noise, and (c) Z-&-X-noise. For TVN strategy with an additional Y control (green pentagrams) in (b), γY Y = γXX = 10⁻¹ω0 and σY Y = σXX, and in (c), γ_{Y Y} = γ_ZZ = γ_XX = 10⁻¹ω₀ and σY Y = σ_XX = σ_ZZ. Optimal control pulses of the TVN strategy (d) for Z-noise and of the TVN strategy with an additional Y control and accompanying Y -noise (e) for X-noise and (f) for Z-&-X-noise. The number of control parameters kmax=10 for ΩX(t) in (d) and kmax=20 for both ΩX(t)and ΩY(t)in (e) and (f).