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zePFp1(Jg>xcm`i=#hcOZRO5S)?K3&M$3$zLo!x}F--`x}T<^5$ulVFKzhX-VO}x^$ z#?$S5;E6w7W}ZYg)n$_Vl$Y~ICge`(?wbqi3|39mnNJRhu literal 0 HcmV?d00001 diff --git a/量子测控学习笔记.md b/量子测控学习笔记.md index fd1ad2e..ac12106 100644 --- a/量子测控学习笔记.md +++ b/量子测控学习笔记.md @@ -8,7 +8,7 @@ > > 版本:Fundamentals of Quantum Measurement and Control_V1.1 > -> 修订情况:20260812:初稿完成;20260813:系统化整理、补全细节、优化可读性 +> 修订情况:20260812:初稿完成;20260813:系统化整理、补全细节、优化可读性。20260820增加背景部分,优化,补充第二章2.1-2.5小节表达。 [TOC] @@ -24,9 +24,9 @@ ## 1.2 文章目的 -近年来,超导量子比特从基础研究转向规模化量子系统工程,形成了一个交叉学科——"量子工程",不同领域的人参与其中。我希望写一本体系化的书,帮助实验室的新学生完善对超导量子领域的认识。 +近年来,超导量子比特从基础研究转向规模化量子系统工程,形成了一个交叉学科——"量子工程",不同领域的人参与其中。我希望写一本体系化的教材,帮助实验室的新学生完善对超导量子领域的认识。 -**这本书叫做《超导量子测控基础》**,是我将平时的学习笔记进行整理,把我所学汇总成一本系统、有条理的书,方便以后查找相关知识,也方便我完善对应体系的知识。**我希望将自己今后的学习与心得写下来,不全依赖 AI 生成,而是以自己书写为主。** +**这本教材叫做《超导量子测控基础》**,是我将平时的学习笔记进行整理,把我所学汇总成一本系统、有条理的书,方便以后查找相关知识,也方便我完善对应体系的知识。**我希望将自己今后的学习与心得写下来,不全依赖 AI 生成,而是以自己书写为主。** ## 1.3 结构安排 @@ -37,6 +37,80 @@ - **超导量子计算机**(第四章):讲解超导量子计算机的整体认识与标定流程; - **疑问记录**(第五章):记录学习中反复思考过的问题,作为答疑与加深理解的部分。 +## 1.4 背景 + +### 1.4.1 经典计算机 + +1936年,英国数学家图灵提出图灵机的概念模型,奠定了计算机科学理论基础。1946年,世界第一台通用电子计算机在美国诞生。 + +1965年戈登-摩尔提出摩尔定律。流行说法是:集成电路上可容纳的晶体管,每隔18个月就会翻一番,性能提升一倍。现在情况是,芯片还是在按照这个规律发展,但是因为逼近物理极限了,靠着通过3D堆叠,新材料等等方式实现的,已经和最初描述的不大相同。 + +### 1.4.2 量子领域 + +1900 年,普朗克第一次提出能量量子化的概念,标志量子力学的诞生。1905年,爱因斯坦用光量子解释光电效应。1913年,波尔提出原子轨道模型。1925年,海森堡提出量子力学的矩阵形式。1926年,薛定谔提出量子力学的波动力学形式 。1982年,费曼提出可以通过 构建一个专用的量子系统来模拟物理行为,从而比经典计算机更高效地解决一些特定的问题 。 + +### 1.4.3 量子计算 + +1985年,大卫-多伊奇(牛津大学物理学家)提出量子图灵机数学模型,还涉及了第一个量子算法Deutsch算法。1994年Shor提出Shor算法,对RSA密码体系造成冲击。1997年,Grover提出Grover算法,可以降低搜索复杂度。 + +#### 1.4.3.1 量子计算的四个阶段 + +image-20260810173150088 + +| 发展阶段 | 核心特征 | 关键节点与说明 | +| :-------------------------------- | :----------------------------------------------------------- | :----------------------------------------------------------- | +| **第一阶段:理论奠基与算法发现** | 纯理论阶段,没有物理硬件。量子计算的强大潜力通过理论算法被揭示。 | 标志事件包括费曼1981年提出量子模拟构想,以及肖尔(1994年)和格罗弗(1996年)提出能显著加速的量子算法。此时的计算完全由经典计算机在“模拟”阶段。 | +| **第二阶段:经典可模拟** | 早期的物理实验,量子比特数极少(约2-10个),运行简单的量子算法。 | 1998年,首个2-qubit量子算法(Deutsch问题)被实验验证。由于规模极小,这些实验的每一步都可以用经典计算机精确模拟。 | +| **第三阶段:“NISQ”时代** | 量子比特数达到中等规模(数十到数百个),但**噪声大、无纠错**。 | 2016年IBM开放5量子比特云平台,2018年“NISQ”概念被正式提出。谷歌2019年的“量子优越性”实验是此阶段的标志性事件。此阶段的设备,其某些输出结果已**无法被经典计算机有效验证**(即难以模拟),但仍无法解决通用问题,可靠性是主要瓶颈。 | +| **第四阶段:容错量子计算 (FTQC)** | 量子比特数大幅增加(理论上可达百万级),通过**量子纠错(QEC)**有效控制噪声,实现通用、可靠的量子计算。 | 这是目前技术路线的终极目标。目前已有向此阶段过渡的迹象,例如哈佛大学团队在2023年底实现了48个逻辑量子比特的演示,证明了纠错技术的可行性。 | + +#### 1.4.3.2量子计算优越性竞赛 + +image-20260810173258064 + +**破局起步(2019–2020):** 2019 年 Google 推出 **Sycamore-53** 超导芯片,率先宣称实现量子优越性;2020 年中国科学技术大学(USTC)推出光量子计算机 **九章(Jiuzhang)**,通过高斯玻色取样实现了光量子体系的里程碑突破。 + +**多路线竞逐与迭代(2021–2023):** 中科大在双路线上加速发力,相继推出 **九章2.0/3.0** 光量子系统以及 **祖冲之2.0/2.1** 超导量子处理器;加拿大 Xanadu 公司也在 2022 年凭借 **Borealis** 在光量子领域崭露头角。 + +**规模与精度跃升(2024–2026):** 随着 Google 推出新一代超导芯片 **Willow (67&70)**,中科大升级至 **祖冲之3.0**,以及离子阱领域代表 Quantinuum 的重要突破,量子计算已从最初的“单一算力验证”逐步迈向更高保真度、更大规模与容错量子计算的关键探索阶段。 + + + +#### 1.4.3.3 量子计算物理平台 + +**科学家是怎么实现量子比特的?** + +现阶段实现量子比特的方法有很多,例如**硅中电子自旋与量子点**、**囚禁离子**、**超冷原子**、**金刚石中的氮-空位中心**以及**偏振光子**等。这些方案都将量子信息编码于天然的微观量子系统之中。相比之下,**超导量子比特**具有显著优势——其尺寸为宏观尺度,且可通过**光刻工艺**精确制备,因而在**可扩展性**和**集成化**方面展现出巨大潜力。 + +image-20260810132435086 + +所以,评价一个量子体系,可以从时间空间操控3个维度来考量,超导量子比特的空间和可操控性展现了优势。 + +在物理层面上,构建量子比特的物理系统需要满足 DiVincenzo 准则: + +**1.能造出来且能扩容(明确的比特与可扩展性)**。系统必须有清晰、稳定的两个能级来代表 $\vert{}0\rangle$ 和 $\vert{}1\rangle$;而且不能只造出 1~2 个玩玩,必须有物理途径扩展到成百上千个比特。 + +**2.能一键归零(初始化能力)**。在每次计算开始前,必须能把所有量子比特快速、准确地重置到已知的初始状态(通常是全 $\vert{}0\rangle$)。 + +**3.耐得住干扰(长相干时间)**。量子态极其脆弱,容易被环境噪声破坏(退相干)。系统的相干寿命必须远长于单次操作所需的时间,确保算法还没运行完,状态没有先崩溃。 + +**4.能完成所有运算(通用量子门集合)**。系统需要支持任意的单量子比特旋转以及双量子比特纠缠操作(如 CNOT 门)。只要具备这套“基本动作库”,就能组合出任意复杂的量子算法。 + +**5.能看清结果(特异性测量能力)**。计算完成后,系统必须能高效、无误地读出每一个特定量子比特的状态(是 0 还是 1),而不能破坏其他无关比特或读出一团乱码。 + +| **物理平台** | **核心实现方式** | **核心优势** | **主要瓶颈** | **当前前沿水平** | +| ------------------------------------ | ------------------------------------------------------------ | ------------------------------------------------------------ | ------------------------------------------------------------ | ------------------------------------------------------------ | +| **超导量子比特** *(Superconducting)* | 利用微米级超导电路中的**约瑟夫森结**构成非线性振荡器,将人工二能级系统编码为人造原子(如 Transmon)。 | 门操控极快(纳秒级)、制造工艺兼容现有半导体微纳加工。 | 相干时间相对较短(微秒级)、串扰严重、极低温(~10 mK)制冷机体积受限。 | 达到上千物理比特规模(IBM Condor/Heron、Google Willow/Sycamore);已实现多轮量子纠错与逻辑比特纠缠突破。 | +| **离子阱** *(Trapped Ion)* | 用电磁场(保罗阱)在超高真空下悬浮带电离子,以离子的**基态/超精细能级**编码比特,用激光或微波驱动库仑晶格操控。 | 比特同质性完美、门保真度极高(两比特门 > 99.9%)、全连接连通性好。 | 门速度较慢(微秒级)、激光与电极寻址系统复杂、单阱离子容量上限明显。 | 物理比特在数十至百位级(Quantinuum H2、IonQ Forte),保真度领先,已实现高保真容错逻辑比特演示。 | +| **中性原子 / 光镊** *(Neutral Atom)* | 用光学微镊阵列捕获中性原子(如铷、锶),通过激光激发至**里德堡态(Rydberg state)**利用偶极阻塞效应实现双比特门。 | 极易通过光镊实现 2D/3D 大规模阵列扩展、天然全同、比特可动态重构。 | 门操控速度中等、原子损失率与真空寿命限制连续运行能力。 | 物理比特扩展至数百到上千(QuEra、Harvard 团队),在可重构阵列中成功演示数十个纠错逻辑比特。 | +| **光量子** *(Photonic)* | 利用光子的**偏振、路径、到达时间或正交压缩态**编码量子信息,通过分束器、相移器及单光子探测器实现计算。 | 室温运行(除超导探测器外)、相干时间极长(光速传输)、天然契合量子通信与网络。 | 确定性双比特门极难(需依靠测量诱导非线性)、单光子源及波导损耗大。 | 专用机(如“九章”、“祖冲之”光量子版、Xanadu Borealis)确立“量子优越性”;PsiQuantum 等正攻坚光子纠错芯片。 | +| **半导体量子点** *(Spin Qubit)* | 在硅(Si/SiGe)或砷化镓(GaAs)异质结中用静电栅极捕获单个电子/空穴,以其**自旋状态**作为比特。 | 体积极小(纳米级)、高度兼容现有 CMOS 工业制程、工作温度容忍度稍高(~1 K)。 | 对材料界面杂质与电荷噪声极敏感、控制线引出极度密集、良率要求严苛。 | 实验室已实现 6~12 个高保真度自旋比特阵列,处于从实验室物理向先进制程代工(如 Intel、台积电合作)过渡期。 | +| **其他(拓扑量子比特等)** | 利用半导体-超导体混合纳米线两端的**马约拉纳零能模(Majorana zero modes)**进行编织操作。 | 理论上具备硬件级拓扑保护,天然抗退相干,无需庞大纠错开销。 | 物理实现极度困难,存在长期争议。 | 微软(Microsoft/Station Q)处于验证拓扑间隙与基础编织操作阶段,尚未产出通用可编程拓扑比特。 | + + + + + --- # 二、超导量子原理 @@ -45,165 +119,217 @@ ## 2.1 从经典电路到超导量子比特 -要做量子比特,首先得有一个能稳定存储量子态的电路。本节沿着"经典 LC 电路 → 量子谐振子(QHO)→ 加约瑟夫森结 → transmon"这条主线,逐步建立超导量子比特的物理图像。 - -**科学家是怎么实现量子比特的?** - -现阶段实现量子比特的方法有很多,例如**硅中电子自旋与量子点**、**囚禁离子**、**超冷原子**、**金刚石中的氮-空位中心**以及**偏振光子**等。这些方案都将量子信息编码于天然的微观量子系统之中。相比之下,**超导量子比特**具有显著优势——其尺寸为宏观尺度,且可通过**光刻工艺**精确制备,因而在**可扩展性**和**集成化**方面展现出巨大潜力。 - -image-20260810132435086 +要做量子比特,首先得有一个能稳定存储量子态的电路。本节从"经典 LC 电路 → 量子谐振子(QHO)→ 加约瑟夫森结 → transmon"这条主线,逐步讲一讲超导量子比特是怎么来的。 ### 2.1.1 从 LC 电路到量子谐振子(QHO) **怎么把量子比特电路化?** 用经典 LC 电路模拟。 -可以用**并联 LC 谐振电路**去模拟量子比特的性质:这种电路的能量会在电容的电场和电感的磁场之间来回振荡,具有和量子比特类似的性质,电路结构如下图 (a)。但经典 LC 电路不能直接当量子比特用,因为量子比特需要两个可区分的量子态($|0\rangle$ 和 $|1\rangle$),而经典 LC 电路的能量是连续的。如果把它量子化,能量不再连续,而是分成一个个能级,相邻能级的间距完全相等($\Delta E=\hbar\omega$,$\hbar$ 是约化普朗克常数,$\omega$ 是振荡频率),这样就得到了量子谐振子(QHO),能级结构对应下图 (b)。 +**1. 经典 LC 振荡与量子化(QHO)** -QHO 有个致命问题,导致它根本没法直接用:**能级等间距**(图 1b)。给它一个 "0→1" 跃迁的微波脉冲,会同时激发 "1→2" 跃迁,所以还不能直接作为量子比特用(量子比特只能在 0 和 1 之间跳动,不能跳到其他能级去)。 +经典并联 LC 电路中,能量在电容的电场(电荷 $Q$)与电感的磁场(磁通 $\Phi$)之间周期性转换。当电路在极低温下满足无耗散条件并进行量子化后,它便成为**量子谐振子(QHO)**。 + +- **分立能级:** 量子化后能量不再连续,而是形成分立能级 $\vert{}0\rangle, \vert{}1\rangle, \vert{}2\rangle \dots$ +- **能级等间距:** 相邻能级差完全相同,即 $\Delta E = \hbar\omega_r$(其中 $\omega_r = 1/\sqrt{LC}$)。 + +**2. 瓶颈:能级等间距导致的能级泄漏** + +虽然 QHO 具有量子特性,但**无法直接用作量子比特: + +- 量子计算需要将操作精准隔离在基态 $\vert{}0\rangle$ 和第一激发态 $\vert{}1\rangle$ 构成的双能级子空间内。 +- 由于所有相邻能级差相等($\hbar\omega_{01} = \hbar\omega_{12} = \dots$),当施加频率为 $\omega_{01}$ 的微波脉冲试图驱动 $\vert{}0\rangle \to \vert{}1\rangle$ 跃迁时,会不可避免地同时诱发 $\vert{}1\rangle \to \vert{}2\rangle$ 及更高阶激发(发生能级泄漏),无法实现对双能级系统的独立精准操控。 + +**3. 解决:引入非线性元件约瑟夫森结** + +为了打破能级的等间距特性,需将线性电感 $L$ 替换为**非线性、无耗散的约瑟夫森结($L_J$)**,构建如 **Transmon** 这样的超导量子比特: + +- **势阱变形:** 约瑟夫森结将原先的抛物线二次势能改造成余弦(Cosine)非谐振势能。 +- **非谐性(Anharmonicity):** 能级间距被“拉开”,产生频率差异($\hbar\omega_{01} \neq \hbar\omega_{12}$)。此时即可用特定频率 $\omega_{01}$ 准确且单独操控 $\{\vert{}0\rangle, \vert{}1\rangle\}$ 计算空间。 ### 2.1.2 约瑟夫森结 -要让电路只能在 0 到 1 之间跳动,需要引入**非线性元件——约瑟夫森结**。约瑟夫森结是用两块超导体夹一片绝缘体(SIS 结构)制成的器件,用它代替电路中的电感(如图 1c),就能产生**非简谐能级**(能级间距随能级升高而减小,图 1d),使 "0→1" 和 "1→2" 跃迁可区分。 - -这样,我们初步得到了一个电路形态的量子比特——制造了一个宏观的人造原子。 +为将电路能级严格限制在 $\vert{}0\rangle$ 和 $\vert{}1\rangle$ 的二能级子空间内,必须引入非线性元件——**约瑟夫森结(Josephson Junction)**。该器件采用超导体-绝缘体-超导体(SIS)三明治结构。当用它替代 LC 电路中的线性电感时(入图 1c),系统的势能面由简谐抛物线变为余弦势,从而打破了等间距能级分布,产生显著的**非简谐性(Anharmonicity)**(图 1d)。这使得 $\vert{}0\rangle \to \vert{}1\rangle$ 与 $\vert{}1\rangle \to \vert{}2\rangle$ 的跃迁频率彼此区分,避免了高能态的泄漏。至此,宏观电路便具备了类似天然原子的分立能级结构,构成了可独立寻址调控的人造原子——超导量子比特。 局部截取_20251004_173500 约瑟夫森结的非线性让电路的势能从抛物线变成正弦曲线,产生能级差距,进而使 "0→1" 和 "1→2" 跃迁可区分。下面进行数学推导。 -**约瑟夫森结的结构**:蓝色的线表示库珀对波函数(也叫超导序参量),可以发现它在势垒层有交叠然后衰减。库珀对波函数的相位至关重要,影响着电压电流特性。这个结的材料一般是铝-氧化铝-铝的三明治结构。 +**约瑟夫森结的结构**:蓝色的线表示库珀对波函数(也叫超导序参量),可以发现它在势垒层有交叠然后衰减。库珀对波函数的相位至关重要,影响着电压电流特性。(常见为 $\text{Al}/\text{AlO}_x/\text{Al}$ 结构) image-20260629195502141 -**四个经典约瑟夫森方程**。我简单记忆为 3 个词:相位电流、相位电压、相位磁场。波函数的相位怎么变,电学参数就会跟着变。 +**约瑟夫森结:非线性势能与等效电感推导** -$$j_s = j_c \sin\varphi$$ -$$\frac{\partial\varphi}{\partial t} = \frac{2eV}{\hbar}$$ -$$\frac{\partial\varphi}{\partial x} = \frac{2e\Lambda}{\hbar} B_y$$ -$$\frac{\partial\varphi}{\partial y} = \frac{2e\Lambda}{\hbar} B_x$$ +约瑟夫森结的核心物理量是绝缘势垒两端库珀对波函数(超导序参量)的**超导相位差 $\varphi$**。电路的电磁学行为完全由相位差 $\varphi$ 的演化决定。 -用上面的相位电流和相位电压公式,可以得到功率,对时间积分,得到**约瑟夫森结的势能**公式,简单记作"相势公式"。 +**1. 基础关系(约瑟夫森直流与交流方程)** -$$W = \int_{0}^{\varphi} IV\,dt = \int_{0}^{t} \frac{\hbar}{2e}I_c\sin\varphi\,\frac{d\varphi}{dt}\,dt = \int_{0}^{\varphi}\frac{\hbar I_c}{2e}\sin\varphi\,d\varphi = \frac{\hbar I_c}{2e}(1-\cos\varphi)$$ +超导电流与端电压分别满足: -可以把左边这一项包装成 $E_J$,当作固定份能量,这样就简单了。势能最小值为 0,最大值为 $2E_J$。这个公式取名叫约瑟夫森势能公式。 +$$I = I_c \sin\varphi \quad \text{(电流-相位关系)}$$ -经典电感公式如下,电压和电感的变化率成正比。 +$$V = \frac{\hbar}{2e} \frac{d\varphi}{dt} \quad \text{(电压-相位关系)}$$ -$$L = \frac{V}{dI/dt}$$ +其中 $I_c$ 为临界电流,$2e$ 为单个库珀对的电荷量。 -配合相位电流和相位电压公式,可以推出约瑟夫森结的电感。 +**2. 势能推导:从二次抛物势到余弦势** -$$L(\varphi) = \frac{V(\varphi)}{dI(\varphi)/dt} = \frac{\hbar}{2eI_c\cos\varphi}$$ +计算超导结吸收的瞬时功率 $P = I V$,并对时间积分即可得到势能 $U_J$: -当 $\varphi=0$ 时,剩下的记作 $L_J$,叫线性电感。可以发现线性电感是除以 $\cos\varphi$ 的形式,符号取决于 $\cos$ 值的正负。负电感可以等效于电容性。 +$$U_J = \int I V \, dt = \int \left( I_c \sin\varphi \right) \left( \frac{\hbar}{2e} \frac{d\varphi}{dt} \right) dt = \frac{\hbar I_c}{2e} \int_0^\varphi \sin\varphi' \, d\varphi'$$ -对于电容来说,流进来的电荷 $Q=2e\cdot n$,也就是 $n$ 对库珀对,储能自然和 $n$ 有关系,如下表示。把 $n^2$ 前面的系数当作 $E_C$,就可以表示电容的 $n$-能关系。 +积分得到约瑟夫森势能: -$$E = \frac{1}{2}V^2 C = \frac{1}{2}\frac{Q^2}{C} = \frac{4e^2}{2C}\,n^2$$ +$$U_J(\varphi) = E_J (1 - \cos\varphi)$$ + +其中定义 **约瑟夫森能** 为: + +$$E_J = \frac{\hbar I_c}{2e}$$ + +- 普通 LC 谐振器的势能为抛物线型($\propto \varphi^2$),导致能级严格等距; +- 约瑟夫森结的势能为余弦型($-\cos\varphi$),这种**非线性势阱**压低了高能级,使 $\omega_{01} \neq \omega_{12}$,成功实现二能级可区分。 + +**3. 非线性电感与等效响应** + +根据电感定义 $V = L \frac{dI}{dt}$,结合链式法则 $\frac{dI}{dt} = \frac{dI}{d\varphi}\frac{d\varphi}{dt}$: + +$$L(\varphi) = \frac{V}{dI/dt} = \frac{\frac{\hbar}{2e} \frac{d\varphi}{dt}}{I_c \cos\varphi \frac{d\varphi}{dt}} = \frac{\hbar}{2e I_c \cos\varphi}$$ + +令零相位处的值为**结特征电感(线性电感)** $L_J = \frac{\hbar}{2e I_c}$,则: + +$$L(\varphi) = \frac{L_J}{\cos\varphi}$$ + +电感值随相位 $\varphi$ 动态变化,体现了非线性本质;当 $\cos\varphi < 0$ 时,微分电感呈现负值,对外表现为容性阻抗。 + +**4. 结电容的充放电能(静电能)** + +约瑟夫森结本身存在极板电容 $C_J$(或并联大电容 $C$)。设极板聚集了 $n$ 对库珀对,总转移电荷量 $Q = 2e \cdot n$,其静电储能为: + +$$E = \frac{Q^2}{2C} = \frac{(2e \cdot n)^2}{2C} = \left(\frac{e^2}{2C}\right) (2n)^2 = 4 E_c n^2$$ + +其中 $E_c = \frac{e^2}{2C}$ 定义为**单电子充电能**。 + +通过将电容动能项($\sim n^2$)与非线性势能项($-E_J \cos\varphi$)结合,即可写出超导量子比特的标准量子哈密顿量: + +$$\hat{H} = 4 E_c \hat{n}^2 - E_J \cos\hat{\varphi}$$ ### 2.1.3 transmon -上面的量子比特很敏感,一点电荷噪声就会让比特能级乱晃,存的信息一下就没了(相干时间非常短)。所以需要给它做抗电荷噪声设计。科学家发现,在约瑟夫森结旁边并联一个超大电容(比结本身的电容大很多),就能让电路对电荷波动的敏感度指数级下降,相干时间大大延长。实用量子比特就这样造出来了,科学家给它起名叫 transmon。 +早期超导量子比特(如 Cooper-pair box)对环境中的电荷噪声极度敏感,背景电荷的微小波动会导致比特能级剧烈晃动,破坏量子叠加态,使相干时间短到难以实用。为解决这一痛点,物理学家在约瑟夫森结两端**并联了一个大电容 $C_s$**(远大于结自身电容 $C_J$),大幅降低了电路的单电子充电能 $E_C$。在满足 $E_J \gg E_C$(通常 $E_J/E_C \sim 50$–$100$)的深度非谐振区下,**电荷噪声对能级跃迁频率的调制被呈指数级抑制**,相干时间提升了数个数量级。这种经过电容分流改良的超导量子比特,被称为 **Transmon**(Transmission line shunted plasma oscillation qubit)。 -至于为什么并电容就能抗电荷噪声,文章里有很多公式。最后结论是:在 $E_J \gg E_C$ 的情况下($E_J/E_C$ 很大),电荷噪声对能级的影响呈指数抑制,磁通噪声的敏感性也可控,所以 transmon 才能成为主流量子比特。 +**Transmon 的三大核心优势** -**transmon 的特点** +- **抗干扰能力极强(长相干时间):** 通过指数级压制电荷噪声,相干时间跃升至百微秒($>100\ \mu\text{s}$)甚至更高量级,为底层控制电子学(如 FPGA)执行数百次门操作提供了充裕的相干时间窗口。 +- **非谐性恰到好处(抑制能级泄漏):** 尽管大电容略微弱化了非谐性(典型值在 $-100$ 至 $-300\text{ MHz}$ 之间),但该能级差已足够让微波驱动精准锁定在 $\{\vert{}0\rangle, \vert{}1\rangle\}$ 计算空间,有效避免 $\vert{}1\rangle \to \vert{}2\rangle$ 泄漏。 +- **控制频段成熟兼容:** 工作跃迁频率稳定在 **3–6 GHz**,完全落在标准射频/微波通信频段内,可直接复用成熟的经典射频元器件与微波测控技术,大幅降低了硬件工程化门槛。 -1. **频率合适**:工作频率在 3–6 GHz,刚好能用现代的微波技术控制,不用搞特殊设备; -2. **抗干扰强**:不怕电荷噪声,能稳定工作(相干时间能到 100 微秒以上,100 微秒够 FPGA 做很多事情了); -3. **非简谐够用**:虽然非简谐性不算特别大(100–300 MHz),但足够防止误跳能级了。 +### 2.1.4 电路如何表示量子 -### 2.1.4 线性谐振腔到非线性谐振腔 +#### 2.1.4.1 哈密顿量是什么? -为什么一个 LC 谐振电路能够有哈密顿量?这涉及到复杂的数学推导,想具体了解可以看李少炜博士的博士毕业论文。以下是我对他博士大论文的理解。 +哈密顿量(Hamiltonian,用 $\hat{H}$ 表示)用一句话解释:它就是量子力学里描述系统“总能量”的终极数学公式。经典力学里:总能量 = 动能 + 势能。 量子力学里:**哈密顿量 = 系统的总能量算符**。重要性体现在诸多方面。 -上面讲了约瑟夫森结与超导量子干涉器,接下来讲**线性谐振腔到非线性谐振腔**,通过这个就能明白为什么能用电路的宏观原子去模拟微观量子现象。 +**算出量子比特长什么样(求出能级)**:把电路的哈密顿量带入薛定谔方程 $\hat{H}\psi = E\psi$,解出来的能量本征值 $E_0, E_1, E_2\dots$ 就是电路允许存在的台阶能级。能级差就是比特的工作频率(比如 $\omega_{01} = (E_1 - E_0)/\hbar \approx 5\text{ GHz}$)。没有哈密顿量,芯片连用什么频率的微波去操控都不知道。 -image-20260629102705673image-20260629102737927 +**指导量子门操作(控制计算)**:薛定谔方程的核心是 $i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi$,它说明**哈密顿量直接决定了量子态随时间如何演化**。科学家打入一束微波脉冲,本质上就是在哈密顿量里加了一个“外加驱动项 $\hat{H}_{drive}(t)$”。通过精准控制哈密顿量变化多长时间,就能精确让比特完成“翻转”(X门)或“叠加”(Hadamard门)。 -谐振腔实际上就是一个 LC 谐振电路。电感的磁通储能公式和电容的磁通变化储能公式如下:把电感的当作势能,电容的当作动能。 +**设计两比特纠缠(芯片电路连接)**:把两个量子比特用电容或电感连起来时,两者的相互作用会体现为哈密顿量中的**耦合项**(如 $g(\hat{a}^\dagger \hat{b} + \hat{a}\hat{b}^\dagger)$)。只有通过哈密顿量,工程师才能精确计算出需要多强的耦合力才能在几十纳秒内做完一个纠缠门(如 CZ 门)。 -$$E = \frac{\phi^2}{2L}$$ -$$E = \frac{C}{2}\dot{\phi}^2$$ -接下来是 1.14–1.26 一共 12 个公式的"究极折磨"。我简单梳理一下:通过拉格朗日量,推到得到正则动量;然后把磁通作为正则坐标,对应的正则动量就是电感(电容)两端的电荷量,就可以写出哈密顿量了。 -$$H = P\dot{\phi} - \mathcal{L} = \frac{P^2}{2C} + \frac{\phi^2}{2L} = \frac{4e^2}{2C}\,n^2 + \frac{\Phi_0^2}{4\pi^2 L}\varphi^2$$ +### 2.1.4.2 为什么宏观 LC 电路能拥有量子哈密顿量? -因为量子力学里我们没法直接处理"电感"和"电容",只认识位置和动量。通过拉格朗日这个中间人,我们发现:**位置 $x$ = 磁通 $\Phi$,动量 $p$ = 电荷 $Q$**,$H$ 是哈密顿量(一个勒让德变换式子,不用管它)= 动能项 + 势能项,其中 $P=Q$。里面的 $\Phi_0$ 叫约化磁通量子,其实就是 $h/2e$。一个相位算符就是小 $\phi$,就是电势 $\Phi$。 +在超导量子计算中,LC 谐振腔(微波谐振器)常被当作“人造宏观原子”来研究。经典电路之所以能被量子化,本质是通过分析力学(拉格朗日与哈密顿力学)建立起经典电路变量与微观量子算符之间的**数学映射**。 -$$H = P\dot{\phi} - \mathcal{L} = \frac{P^2}{2C} + \frac{\phi^2}{2L} = \frac{4e^2}{2C}\,n^2 + \frac{\Phi_0^2}{4\pi^2 L}\varphi^2$$ +**1. 建立对应关系:把电路变量映射为坐标与动量** -用前面定义的 $E_C$ 和 $E_J$ 包装一下,简洁多了。 +在分析力学中,描述一个物理系统需要一组“广义坐标”和对应的“共轭动量”: -$$H = 4E_C n^2 + \frac{1}{2}E_L \varphi^2$$ +- **广义坐标 $x$ $\to$ 磁通量 $\Phi$(或超导相位 $\varphi$):** 电感储能只依赖磁通量,对应系统的势能项 $V = \frac{\Phi^2}{2L}$。 +- **共轭动量 $p$ $\to$ 电荷量 $Q$(或库珀对数 $n$):** 电容储能依赖电荷累积,对应系统的动能项 $T = \frac{Q^2}{2C}$。 -然后用量子算符表示公式,简单地加一个帽子就行。对易关系就是 $[A,B]=AB-BA$。 +通过拉格朗日量 $\mathcal{L} = T - V$ 进行勒让德变换,即可得到经典的哈密顿量: + +$$H = \frac{Q^2}{2C} + \frac{\Phi^2}{2L} = 4E_C n^2 + \frac{1}{2}E_L \varphi^2$$ + +其中 $E_C = \frac{e^2}{2C}$ 为单电子充电能,$E_L = \left(\frac{\Phi_0}{2\pi}\right)^2 \frac{1}{L}$ 为电感能。 + +**2. 正则量子化:从经典能量到量子算符** + +当把宏观电路放入极低温环境(毫开尔文温区)以压制热噪声后,将经典物理量提升为量子算符(“加上帽子”),它们天然满足海森堡不确定性原理下的对易关系: + +$$[\hat{\varphi}, \hat{n}] = i$$ + +此时,电路的哈密顿算符变为: $$\hat{H} = 4E_C \hat{n}^2 + \frac{1}{2}E_L \hat{\varphi}^2$$ -$\hat{n}$ 帽和 $\hat{\varphi}$ 帽能满足对易关系,从而变成量子系统。 +这与物理学中标准的一维量子谐振子(QHO)哈密顿量 $\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2 \hat{x}^2$ 形式完全一致。通过系数一一对照: -$$[\hat{n}, \hat{\varphi}] = \hat{n}\hat{\varphi} - \hat{\varphi}\hat{n} = i$$ +- 等效质量 $\frac{1}{2m} \leftrightarrow 4E_C$ +- 谐振频率 $\omega_r = \frac{\sqrt{8E_C E_L}}{\hbar} = \frac{1}{\sqrt{LC}}$ -量子力学中一维量子谐振子的哈密顿量就是动能项 + 势能项。 +**3. 算符化简与能级结构** -$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2 \hat{x}^2$$ - -还有一个重要结论,就是底下这个——一维谐振子的薛定谔方程的解析解。其实我薛定谔方程都不太了解,反正记住解是这种结构,和能级类似。 - -$$E_m = \left(m + \frac{1}{2}\right)\hbar\omega_r, \qquad m = 0,1,2,\dots$$ - -把电学的哈密顿量和上面得到的量子哈密顿量表达式进行对照。 - -$$\frac{1}{2m} = 4E_C, \qquad m\omega_r^2 = E_L$$ - -就得到一个角频率的公式。所以角频率是**需要算出电学的哈密顿量,再把磁通当坐标、电荷当动量,最后和量子力学的哈密顿量对照**,比出来的。本来一个电路没有角频率之类的东西,但是科学家直接把东西映射到量子模型了,才有了角频率、能级一说。$\omega_r$ 有了,能级也就能画了。 - -$$\omega_r = \frac{\sqrt{8E_C E_L}}{\hbar}$$ - -为了比较好算,还引入了一个湮灭算符。但是我感觉没有好算多少,而且多了一个产生算符(有角标 $+$)和湮灭算符。 - -$$\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}\left( \hat{a} + \hat{a}^\dagger \right)$$ -$$\hat{p} = -i\sqrt{\frac{\hbar m\omega}{2}}\left( \hat{a} - \hat{a}^\dagger \right)$$ - -能级可以写成下面这样,相位和电子对数也可以写成这样。 +为了直观地数出电路里有多少个“微波光子”,引入**湮灭算符 $\hat{a}$** 与**产生算符 $\hat{a}^\dagger$**: $$\hat{H} = \hbar\omega_r \left( \hat{a}^\dagger \hat{a} + \frac{1}{2} \right)$$ -$$\hat{\varphi} = \left( \frac{8E_C}{E_J} \right)^{1/4} \frac{\hat{a}^\dagger + \hat{a}}{\sqrt{2}}$$ -$$\hat{n} = \left( \frac{E_J}{8E_C} \right)^{1/4} \frac{\hat{a}^\dagger - \hat{a}}{\sqrt{2}\,i}$$ -对 LC 谐振电路加一个外加驱动,电子对数会发生变化。 +求解对应的薛定谔方程,得到的本征能级为: -$$\hat{H} = 4E_C\left( \hat{n} - n_d(t) \right)^2 + \frac{1}{2}E_L \hat{\varphi}^2$$ +$$E_m = \left(m + \frac{1}{2}\right)\hbar\omega_r, \quad m = 0, 1, 2, \dots$$ -把平方项展开,可以变成原先的能量加上一个驱动项能量。这个感觉说明不了什么,只能说明电容耦合驱动后总哈密顿量会怎么变化。 +- **外加微波驱动:** 当向电路施加交变电压驱动 $V_d(t)$ 时,会在哈密顿量中引入微扰项 $\hat{H}_d(t) \propto \hat{a} + \hat{a}^\dagger$,用于在谐振腔内激发或抽取微波光子。 -$$\hat{H} = 4E_C \hat{n}^2 + \frac{1}{2}E_L \varphi^2 + \hat{H}_d(t)$$ -$$\hat{H}_d = 8E_C n_d \hat{n} = 8E_C \frac{V_d(t)C_d}{2e}\,\hat{n}$$ +**核心意义与应用** -关键在于能级是等间距的,不是一个二能级系统,不能当比特来用,但比特读取的时候可以用到这个结构。 +- **为何不能做量子比特:** LC 电路的能级是**严格等间距**的($\Delta E = \hbar\omega_r$),微波脉冲会引发连续跃迁而造成能级泄漏,因此必须改用约瑟夫森结(Transmon)来引入非线性。 +- **谐振腔的真正用途:** 虽然无法作为比特,但该线性 LC 谐振腔在电路量子电动力学(cQED)中是不可或缺的**量子总线(Bus)\**与\**色散读取腔(Readout Cavity)**,专门用于辅助量子比特的状态测量与耦合互联。 -**transmon 的能级是怎么来的呢?** +image-20260820111347809image-20260630104101113 -对于下面这个 transmon 结构,通过前面的推导,可以知道哈密顿量等于电容储能加电感储能,电容是动能项,电感是势能项。 +#### 2.4.1.3 Transmon 的能级来源? -$$\hat{H}_q = 4E_C \hat{n}^2 + E_J\left( 1 - \cos\hat{\varphi} \right)$$ +Transmon 的物理结构由一个大电容并联一个约瑟夫森架构成(如图 a 所示)。其哈密顿量同样遵循“动能(电容储能)+ 势能(约瑟夫森结储能)”的形式: -image-20260630104101113 +$$\hat{H}_q = 4E_C \hat{n}^2 - E_J \cos\hat{\varphi}$$ -解能级时并没有精确的解析解。在 $E_J \gg E_C$ 时,把 $\cos$ 展开到第四项做近似。 +其中 $E_C$ 为充电能,$E_J$ 为约瑟夫森耦合能。 -得到能级的近似表达式。 +**1. 势能展开与微扰近似** -$$E_m = -E_J + \sqrt{8E_J E_C}\left( m + \frac{1}{2} \right) - \frac{E_C}{2}\left( 6m^2 + 6m + 3 \right)$$ -$$h\omega_{m,m+1} = \sqrt{8E_J E_C} - (m+1)E_C$$ +该系统的薛定谔方程严格数学解对应马丢函数(Mathieu Functions),没有初等的精确解析解。但由于 Transmon 工作在 **$E_J \gg E_C$** 的深度非谐振区(相位涨落较小),可将余弦势能在底部($\hat{\varphi} \approx 0$)展开到四阶项: -这个公式有误差,但是有帮助。可以看到能级有个 $E_C$ 的二次函数项,导致能级不等间距、能级差不同。 +$$\cos\hat{\varphi} \approx 1 - \frac{\hat{\varphi}^2}{2!} + \frac{\hat{\varphi}^4}{4!}$$ -$$\hbar\omega_{01} = \sqrt{8E_J E_C} - 1\cdot E_C$$ -$$\hbar\omega_{12} = \sqrt{8E_J E_C} - 2\cdot E_C$$ +代入哈密顿量可拆分为两部分: + +- **谐振子项(二次项):** $4E_C \hat{n}^2 + \frac{1}{2}E_J \hat{\varphi}^2$,对应基准振荡频率 $\omega_p \approx \frac{\sqrt{8E_J E_C}}{\hbar}$。 +- **非谐性微扰项(四阶项):** $-\frac{E_J}{24}\hat{\varphi}^4$,该项直接打破了抛物线势阱的理想谐振性。 + +**2. 能级公式与不等间距特性** + +利用一阶微扰论对上述哈密顿量求解,得到本征能级 $E_m$ 的近似解析解: + +$$E_m \approx -E_J + \sqrt{8E_J E_C}\left(m + \frac{1}{2}\right) - \frac{E_C}{12}\left(6m^2 + 6m + 3\right)$$ + +由上式可直接求出相邻能级之间的跃迁能量差: + +$$\hbar\omega_{m, m+1} = E_{m+1} - E_m \approx \sqrt{8E_J E_C} - (m + 1)E_C$$ + +具体到低能级跃迁: + +- **计算区间跃迁($\vert{}0\rangle \to \vert{}1\rangle$):** $\hbar\omega_{01} \approx \sqrt{8E_J E_C} - E_C$ +- **泄漏能级跃迁($\vert{}1\rangle \to \vert{}2\rangle$):** $\hbar\omega_{12} \approx \sqrt{8E_J E_C} - 2E_C$ + +**3. 非谐性(Anharmonicity)的物理意义** + +两个相邻跃迁频率的差值定义为系统的**绝对非谐性 $\alpha$**: + +$$\alpha = \omega_{12} - \omega_{01} \approx -\frac{E_C}{\hbar}$$ + +- **能级不等距:** 由于负非谐性 $\alpha \approx -E_C$ 的存在,$\omega_{12}$ 比 $\omega_{01}$ 低了约 $100$–$300\text{ MHz}$。 +- **隔离双能级:** 当使用频率为 $\omega_{01}$ 的微波脉冲驱动系统时,微波频率与 $\omega_{12}$ 产生失谐(Detuned),无法诱发 $\vert{}1\rangle \to \vert{}2\rangle$ 跃迁,从而成功将动力学限制在 $\{\vert{}0\rangle, \vert{}1\rangle\}$ 二维计算子空间内。 ## 2.2 量子比特的优化 @@ -211,45 +337,70 @@ transmon 比特虽然好用,但不同场景需要不同特性的比特:有 ### 2.2.1 transmon 的发展与优化 -image-20251004185137650 +image-20251004185137650 -上排是优化后的电路结构(图 2a、2c、2e、2g),分别是 Split Transmon、不对称 Split Transmon、Flux Qubit 和 Fluxonium 的部分电路图。每个图对应的下排图的简单解读如下: +图 2 上排(a、c、e、g)展示了四种主流电容分流型量子比特的电路拓扑,下排(b、d、f、h)给出了对应的能级跃迁频率随外加磁通 $\Phi_{\text{ext}}$ 的响应曲线。 -> 横轴是**外部磁通偏置**($\Phi_{\text{ext}}$,以磁通量子 $\Phi_0$ 为单位,范围 $-\pi$ 到 $\pi$),纵轴是**跃迁频率(GHz)**。蓝色线为 $\omega_{01}$(基态 $|0\rangle$ → 第一激发态 $|1\rangle$ 的频率),红色虚线为 $\omega_{12}$($|1\rangle$ → $|2\rangle$ 的频率),两者的差值就是**非简谐性**($\alpha = \omega_{12} - \omega_{01}$),决定比特是否容易"误跳到高能级"。曲线平缓段越多,代表对磁通噪声越不敏感,相干时间越长。 +- **横轴**:外加磁通偏置 $\Phi_{\text{ext}}$。 +- **纵轴**:跃迁频率(GHz)。蓝实线为基态跃迁频率 $\omega_{01}$($\vert{}0\rangle \to \vert{}1\rangle$),红虚线为第一激发态跃迁频率 $\omega_{12}$($\vert{}1\rangle \to \vert{}2\rangle$)。 +- **核心指标**:两者差值定义为**非简谐性($\alpha = \omega_{12} - \omega_{01}$)**,决定系统能否将跃迁严格限制在二能级子空间;曲线的极值点(平缓处)对应**磁通稳健点(Sweet Spot)**,此处斜率 $\frac{\partial\omega_{01}}{\partial\Phi_{\text{ext}}} \approx 0$,对一阶磁通噪声免疫,相干时间最长。 -**Split Transmon**。图 2a 中 Transmon 的频率是固定的,做量子计算时经常需要两个比特"对话"——比如让比特 A 和比特 B 的频率对齐(实现两比特门)。能调频率的 Split Transmon 就派上用场了:原来的 Transmon 用一个约瑟夫森结,Split Transmon 把这个结换成了"两个结组成的环路"(叫 DC-SQUID,如图 2a)。加外部磁场(磁通),就能改变两个结的有效约瑟夫森能量,进而改变比特的频率。加不同的磁通,就能让比特工作在不同频率(调谐范围通常有 1 GHz)。 +#### 1. 对称可调 Transmon (Symmetric Split Transmon) -这个超导量子干涉仪(SQUID)听名字很唬人,但实际上就是两个约瑟夫森结穿个环。这样做的目的,是为了实现可调的等效 $E_J$:如果 $E_J$ 可调,量子比特的频率才能变,要不然就是固定死的。 +- **电路原理(图 2a)**: -image-20260629202715133 +- image-20260629202715133 -公式推导不难,用到了和差化积公式。但是有个地方不好理解:加一个磁通后,上面和下面波函数的相位差就不相等了。前面说 $E_J$ 里面包含着 $I_c$,是等价的,可以看到新的等价 $E_J$ 后面多了个磁通项。这也是为什么 Z 芯片要发波形控制磁通、最终实现调量子频率的原因。 + 将单结 Transmon 的固定结替换为由两个相同约瑟夫森结($I_c$ 相同)构成的闭合环路——**直流超导量子干涉仪(DC-SQUID)**。通过环路的外加磁通 $\Phi_{\text{ext}}$ 会在两结间诱导相位差,实现等效约瑟夫森能量 $E_J(\Phi_{\text{ext}})$ 的连续调控: -$$I = I_{c1}\sin\varphi_1 + I_{c2}\sin\varphi_2$$ -$$\varphi_2 = \varphi_1 + \frac{2\pi\phi_a}{\Phi_0}, \qquad \Phi_0 = \frac{h}{2e}$$ -$$I = 2I_c \cos\left( \frac{\pi\phi_a}{\Phi_0} \right) \sin\left( \varphi_1 + \frac{\pi\phi_a}{\Phi_0} \right)$$ -$$I'_c = 2I_c\left| \cos\left( \frac{\pi\phi_a}{\Phi_0} \right) \right|, \qquad E_J' = 2E_J\left| \cos\left( \frac{\pi\phi_a}{\Phi_0} \right) \right|$$ + $$I = I_{c1}\sin\varphi_1 + I_{c2}\sin\varphi_2$$ -**图 2b 对称 Transmon 的频率曲线**。曲线呈对称的碗状:当 $\Phi_{\text{ext}}=0$ 时频率最高,偏离 0 时频率下降。非简谐性为负($\omega_{12} < \omega_{01}$),且 $\Phi_{\text{ext}}=0$ 时非简谐性的绝对值较小——这意味着 "0→1" 和 "1→2" 跃迁容易混淆,需额外技术(如 DRAG 脉冲)抑制高能级泄漏。 + 由于磁通量子化条件,环路满足 $\varphi_2 - \varphi_1 = \frac{2\pi \Phi_{\text{ext}}}{\Phi_0}$。对称情况下($I_{c1}=I_{c2}=I_c$),利用和差化积化简总超导电流: -**不对称 Split Transmon**。图 2c,早期的 Split Transmon 有个问题:调频率时特别怕磁通噪声。解决办法是把两个结做得不一样大(不对称),这样磁通噪声对频率的影响会大大降低。 + $$I = 2I_c \cos\left( \frac{\pi \Phi_{\text{ext}}}{\Phi_0} \right) \sin\left( \varphi_1 + \frac{\pi \Phi_{\text{ext}}}{\Phi_0} \right)$$ -**图 2d 不对称 Split Transmon 的频率曲线**。因结不对称($\gamma=2.5$),曲线不再严格对称,且"碗状"更平缓。此时磁通偏置对频率的影响变弱,说明抗磁通噪声能力提升(即使磁通有波动,频率也不易乱晃),同时仍保留一定调谐范围。 + 对应的等效临界电流与有效约瑟夫森能为: -**Flux Qubit 和 Fluxonium**。Transmon 虽然好用,但非简谐性不算特别大,有时候还是会不小心跳到 $|2\rangle$ 态。如果想让比特更可靠,就需要非简谐性更大的设计——Flux Qubit(磁通比特)和 Fluxonium。 + $$I'_c(\Phi_{\text{ext}}) = 2I_c \left\vert{}\cos\left( \frac{\pi \Phi_{\text{ext}}}{\Phi_0} \right)\right\vert{}, \qquad E'_J(\Phi_{\text{ext}}) = 2E_J \left\vert{}\cos\left( \frac{\pi \Phi_{\text{ext}}}{\Phi_0} \right)\right\vert{}$$ -**Flux Qubit**。图 2e,Flux Qubit 是非简谐性为正的比特。普通 Transmon 的非简谐性是"负的"($|1\rangle \to |2\rangle$ 比 $|0\rangle \to |1\rangle$ 容易),相当于二楼到三楼的台阶比一楼到二楼的低,在一、二楼之间跳可能会不小心跳到三楼。但 Flux Qubit 的非简谐性是"正的"($|1\rangle \to |2\rangle$ 比 $|0\rangle \to |1\rangle$ 难),相当于二楼到三楼的台阶比一楼到二楼高很多,所以只能在一楼和二楼之间跳。它的电路设计方法是:用 3 个或 4 个约瑟夫森结组成一个小环路,让势能曲线变成双阱,比特的态能在两个阱之间稳定切换,不容易跑到更高的能级。 + 外加磁通可使有效 $E'_J$ 在 $0 \sim 2E_J$ 之间调谐,从而大范围改变比特工作频率(图 2b)。 -**图 2f Flux Qubit 的频率曲线**。左侧放大图显示:在特定磁通偏置下(如 $-\pi/2$ 附近),$\omega_{01}$ 有很深的"谷",而 $\omega_{12}$ 的"谷"更浅,非简谐性为正($\omega_{12} > \omega_{01}$)——这意味着 "1→2" 比 "0→1" 更难发生,能更严格限制在两能级,适合对"非简谐性要求极高"的场景。 +- **能谱特征(图 2b)**: -**Fluxonium**。图 2g,Fluxonium 是 Flux Qubit 的升级版,是记得久的比特(相干时间长)。方法:用很多个约瑟夫森结(比如 100 个)组成"结阵列"。它有两个优势: + 能谱呈深周期性凹陷,非简谐性为负($\omega_{12} < \omega_{01}$,$\alpha \approx -E_c$)。由于 $\alpha$ 的绝对值相对有限(约 $-200 \sim -300\text{ MHz}$),高速控制时通常需结合 DRAG(Derivative Removal by Adiabatic Gate)微波脉冲技术以抑制 $\vert{}1\rangle \to \vert{}2\rangle$ 的泄漏。在半磁通偏置点处频率下潜至零,调谐范围过宽导致其对磁通抖动极为敏感。 -1. **相干时间长**:能稳定保存信息到"毫秒级",比 Transmon 长得多; -2. **非简谐性超大**:几乎不会跳到 $|2\rangle$ 态。 +#### 2. 不对称可调 Transmon (Asymmetric Split Transmon) + +- **电路设计(图 2c)**: + + 为缓解对称 Transmon 在调频时对磁通噪声高度敏感的缺陷,将 SQUID 环路的两个结设计为不同尺寸(结不对称比 $\gamma = I_{c2}/I_{c1} = 2.5$)。 + +- **能谱特征(图 2d)**: + + 等效约瑟夫森能在极小值点不再下潜归零,而是保留了底限 $\sim (\gamma - 1)E_{J1}$。能谱曲线起伏显著变平缓(频率调谐范围收窄至数 GHz 内),大幅降低了 $\frac{\partial\omega_{01}}{\partial\Phi_{\text{ext}}}$ 的斜率,有效抑制了磁通漂移引起的退相位,兼顾了频率可调性与抗噪能力。 + +#### 3. 电容分流磁通比特 (C-shunted Flux Qubit) + +- **电路设计(图 2e)**: + + 采用一个主约瑟夫森结与两只较大面积的结($\gamma C_J$)构成超导环路,并额外并联大电容进行分流。其势能面呈现双阱特征,基态与激发态由环路中的顺时针与逆时针持续电流态线性叠加而成。 + +- **能谱特征(图 2f)**: + + 在半磁通偏置点($\Phi_{\text{ext}} = \pm \pi/2$ 附近,即 $\pm 0.5\,\Phi_0$),$\omega_{01}$ 出现深窄的能隙极小点,且**非简谐性转为显著的正值($\omega_{12} > \omega_{01}$,$\alpha > 0$)**。正非简谐性使得高能级跃迁门槛变高,天然杜绝了向 $\vert{}2\rangle$ 态泄漏的通道。 + +#### 4. 电容分流磁通量子 (C-shunted Fluxonium) + +- **电路设计(图 2g)**: + + 由一个小面积约瑟夫森结和一个由数十至上百个大结组成的超电感阵列(Superinductance Array)闭环并联,外加电容分流。超电感提供了极大的线性电感且没有电阻损耗,彻底消除了传统磁通比特中的相位滑移与电荷噪声。 + +- **能谱特征(图 2h)**: + + - **超大正非简谐性**:在半磁通偏置点($\pm \pi/2$)处,$\omega_{01}$ 处于深能谷(通常只有几百 MHz 到 1~2 GHz),而 $\omega_{12}$ 维持在数 GHz 之上($\omega_{12} \gg \omega_{01}$),呈现极强的正非简谐性,彻底消除能级泄漏。 + - **超长相干时间**:计算态波函数在半磁通点具有强烈的空间正交偶极矩保护,加上极低的跃迁频率,使得能级跃迁对介电损耗和磁通噪声具备天然双重屏蔽(毫秒级相干时间),已成为构建高精度容错量子计算的重要前沿路线。 -缺点是设计复杂、成本高,目前主要用在对相干时间要求高的场景(比如量子存储器)。 -**图 2h Fluxonium 的频率曲线**。曲线呈现宽而深的"碗状",且 $\omega_{12}$ 与 $\omega_{01}$ 差距很大(非简谐性为负且绝对值大)。这说明:① 频率可调范围大;② 高能级泄漏极难发生;③ 对磁通噪声不敏感(曲线平缓段多),因此相干时间长(信息能保存更久)。 ## 2.3 量子比特相互作用 @@ -326,10 +477,10 @@ $T_2$ 代表总退相干速度,等于 $T_1$ 导致的能量丢失加上 $T_\va 局部截取_20251002_162928 -- 图 5 (a) 测 $T_1$:给 qubit 发 Xπ 脉冲(让它从 $|0\rangle\to|1\rangle$),不断测量量子比特状态,结果是指数下降,拟合出 $T_1=85\mu$s($1/e$ 状态处)。 -- 图 5 (b) 测 $T_2^*$(没抵消低频噪声的 $T_2$):用 Ramsey 干涉,发两个 Xπ/2 脉冲,中间等时间 $\tau$,结果是振荡衰减,衰减快,拟合出 $T_2^*=95\mu$s。 -- 图 5 (c) 测 $T_2^E$(抵消了部分低频噪声的 $T_2$):用 Hahn 回波,在两个 Xπ/2 中间加一个 Xπ 脉冲(相当于重置相位),衰减变慢,拟合出 $T_2^E=120\mu$s(比 $T_2^*$ 长,说明抵消了部分 1/f 噪声)。 -- 图 5 (d) 分解衰减原因:把图 5 (b) 的数据拆成指数部分($T_1$ 导致)和高斯部分(1/f 噪声导致),两个加起来刚好拟合实验数据,说明 1/f 噪声确实让衰减变了样。 +- 图 (a) 测 $T_1$:给 qubit 发 Xπ 脉冲(让它从 $|0\rangle\to|1\rangle$),不断测量量子比特状态,结果是指数下降,拟合出 $T_1=85\mu$s($1/e$ 状态处)。 +- 图 (b) 测 $T_2^*$(没抵消低频噪声的 $T_2$):用 Ramsey 干涉,发两个 Xπ/2 脉冲,中间等时间 $\tau$,结果是振荡衰减,衰减快,拟合出 $T_2^*=95\mu$s。 +- 图 (c) 测 $T_2^E$(抵消了部分低频噪声的 $T_2$):用 Hahn 回波,在两个 Xπ/2 中间加一个 Xπ 脉冲(相当于重置相位),衰减变慢,拟合出 $T_2^E=120\mu$s(比 $T_2^*$ 长,说明抵消了部分 1/f 噪声)。 +- 图 (d) 分解衰减原因:把图 5 (b) 的数据拆成指数部分($T_1$ 导致)和高斯部分(1/f 噪声导致),两个加起来刚好拟合实验数据,说明 1/f 噪声确实让衰减变了样。 ### 2.4.4 常见的噪声例子 @@ -418,15 +569,33 @@ CNOT 门你会发现矩阵更大,因为是双比特门,有四个态 00、01 ![局部截取_20251003_150945](./量子测控学习笔记.assets/局部截取_20251003_150945.png) -经典 NOT 门(图 11a):输入 0→输出 1,输入 1→输出 0,就是比特值的翻转,只有两种可能。 +**经典 NOT 门与量子 X 门的对比及本质区别** -量子 X 门(图 11b):输入 $|0\rangle\to|1\rangle$,输入 $|1\rangle\to|0\rangle$;但如果输入是叠加态(比如 $(|0\rangle-i|1\rangle)/\sqrt{2}$),X 门会翻转叠加态的振幅(变成 $(|1\rangle-i|0\rangle)/\sqrt{2}$),不仅翻转了值,还会保留量子态的相位信息。 +1. **经典 NOT 门(非门,图 11a)**: -**最关键的 3 个区别** + 作用于离散经典比特 $b \in \{0, 1\}$,输出为反相值 $\bar{b}$。仅有 $0 \to 1$ 与 $1 \to 0$ 两种确定性的翻转状态,不涉及任何概率幅或相位。 -1. **可逆性**:所有量子门都是可逆的(幺正操作的逆操作仍为幺正操作),数学上可逆;但经典门大多不可逆——AND 门输出 0,没法知道输入是 00、01 还是 10。 -2. **叠加态操控**:量子门能直接操控叠加态,比如 H 门把 $|0\rangle$ 变成叠加态;经典门只能操控 0 和 1。 -3. **门的组合**:量子通用门需要任意单量子比特门 + 一个纠缠双量子比特门,才能实现所有量子计算;经典通用门只要有一组(如单独的 NAND 门)就能实现所有经典计算。 +2. **量子 X 门(比特翻转门,图 11b)**: + + 作用于二维希尔伯特空间中的态矢 $\vert{}\psi_{\text{in}}\rangle = a\vert{}0\rangle + b\vert{}1\rangle$(矩阵形式为 $\begin{bmatrix} a \\ b \end{bmatrix}$)。 + + 其矩阵表示为泡利矩阵 $\hat{X} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$,作用后输出为: + + $$\vert{}\psi_{\text{out}}\rangle = \hat{X}\vert{}\psi_{\text{in}}\rangle = b\vert{}0\rangle + a\vert{}1\rangle \quad \left(\text{即 } \begin{bmatrix} b \\ a \end{bmatrix}\right)$$ + + 不仅能将计算基态 $\vert{}0\rangle \leftrightarrow \vert{}1\rangle$ 互换,更能**线性地交换任意叠加态中的两项概率幅**(例如 $\frac{\vert{}0\rangle - i\vert{}1\rangle}{\sqrt{2}} \xrightarrow{\hat{X}} \frac{-i\vert{}0\rangle + \vert{}1\rangle}{\sqrt{2}}$),并在变换过程中完整保留全局与相对相位信息。 + +**量子逻辑门与经典逻辑门的三大核心区别** + +1. **操作的可逆性(Reversibility)**: + - **量子门**:所有闭合量子系统的门操作均由幺正算符($U^\dagger U = I$)描述,天然无信息损失,数学上严格可逆(即每个量子门都可以通过作用其共轭转置门 $U^\dagger$ 完美还原输入)。 + - **经典门**:大多数多输入经典逻辑门(如 AND、OR、NAND)具有多对一的映射关系,无法仅凭输出反推输入,属于不可逆操作,伴随信息擦除与朗道尔能量耗散。 +2. **状态空间的连续性与叠加态调控(Superposition Handling)**: + - **量子门**:定义在连续的复向量空间(布洛赫球表面),支持生成、旋转和干涉连续变化的叠加态(如 Hadamard 门可将确定态制备为等权叠加态 $\frac{\vert{}0\rangle + \vert{}1\rangle}{\sqrt{2}}$)。 + - **经典门**:仅能在离散的二值布尔集合 $\{0, 1\}$ 间进行离散逻辑映射,无法处理中间态。 +3. **通用门集合的构建(Universality of Gate Sets)**: + - **量子计算**:量子状态空间为连续流形,通用量子计算需要包含任意单比特连续旋转门(SU(2))加上至少一个双比特纠缠门(如 CNOT、CZ)构成的通用门集合(如 Clifford + T 门集),以覆盖整个多比特复合纠缠空间。 + - **经典计算**:经典逻辑只需离散的通用逻辑门(如单独的 NAND 门或 NOR 门),即可通过简单级联组合出任意复杂的布尔函数。 ### 2.5.4 单量子比特门操控 @@ -434,11 +603,49 @@ CNOT 门你会发现矩阵更大,因为是双比特门,有四个态 00、01 #### 2.5.4.1 微波控制 -如何用微波脉冲控制超导量子比特的旋转?超导量子比特(如 transmon)主要靠微波脉冲控制,核心是微波驱动线和量子比特的电容耦合。 - ![局部截取_20251003_151702](./量子测控学习笔记.assets/局部截取_20251003_151702.png) -左边是室温下的微波线路,右边是低温下的芯片上的超导量子比特,两者通过电容($C_d$)耦合。微波驱动线输出随时间变化的电压 $V_d(t)$,这个电压会通过电容传递到量子比特,驱动量子比特的状态旋转。 +如何用微波脉冲控制超导量子比特的旋转?超导量子比特(如 transmon)主要靠微波脉冲控制,核心是微波驱动线和量子比特的电容耦合。 + +图 12 展示了典型的微波驱动线与 Transmon 量子比特之间的电容耦合控制电路。整个微波操控链路分为三个核心部分: + +1. **室温控制端(Room Temperature)**: + + 由任意波形发生器(AWG)与微波源合成时变微波电压信号 $V_d(t) = V_0(t)\cos(\omega_d t + \phi_d)$。其中,载波频率 $\omega_d$、包络幅度 $V_0(t)$ 与初始相位 $\phi_d$ 均可独立精确编程。 + +2. **低温传输布线(Wiring)**: + + 微波信号通过制冷机内部的同轴线缆下行传递至毫开尔文(mK)低温区,电路中的 $R_w$ 代表同轴传输线的特征阻抗与衰减衰减网络,起到阻抗匹配与滤除室温热噪声的作用。 + +3. **芯片端量子比特(On-Chip Qubit)**: + + 位于低温芯片上的 Transmon 量子比特(由约瑟夫森结与分流电容 $C$ 并联构成),通过极小的弱驱动耦合电容 $C_d$(通常 $C_d \ll C$)接入微波控制线。 + +**微波脉冲如何驱动量子比特在布洛赫球上旋转** + +1. **驱动哈密顿量的引入(能量注入)**: + + 微波信号通过耦合电容 $C_d$ 在量子比特极板上诱导时变感应电荷 $q_d(t) \approx C_d V_d(t)$,在系统哈密顿量中引入电荷驱动微扰项: + + $$\hat{H}_d(t) \approx 2e \frac{C_d}{C_{\Sigma}} V_d(t) \hat{n}$$ + + 其中 $C_{\Sigma} = C + C_d$ 为总对地电容,$\hat{n}$ 为库珀对数算符。在二能级子空间近似下,该项可映射为泡利算符 $\hat{\sigma}_x$。 + +2. **旋转轴的确定(由微波相位 $\phi_d$ 决定)**: + + 当微波频率与比特跃迁频率共振($\omega_d = \omega_{01}$)时,转换到随微波频率旋转的坐标系(旋转波近似 RWA)下,驱动哈密顿量简化为: + + $$\hat{H}_{\text{RWA}} = \frac{\hbar \Omega_R(t)}{2} \left( \cos\phi_d \, \hat{\sigma}_x + \sin\phi_d \, \hat{\sigma}_y \right)$$ + + 改变微波的相位 $\phi_d$,即可自由指定布洛赫球赤道面上的旋转轴(如 $\phi_d = 0$ 对应绕 $X$ 轴旋转,$\phi_d = \pi/2$ 对应绕 $Y$ 轴旋转)。 + +3. **旋转角度的确定(由脉冲包络积分决定)**: + + 驱动强度(拉比频率)$\Omega_R(t) \propto V_0(t)$ 与微波脉冲幅度成正比。比特在布洛赫球上的旋转角度 $\theta$ 由脉冲包络的积分面积严格决定: + + $$\theta = \int \Omega_R(t) \, dt$$ + + 通过精确调制微波脉冲的宽度与幅度,即可实现任意角度的单比特量子逻辑门(例如 $\theta = \pi$ 对应将 $\vert{}0\rangle$ 完全翻转为 $\vert{}1\rangle$ 的 X 门,$\theta = \pi/2$ 则制备等权叠加态)。 #### 2.5.4.2 旋转的实现 @@ -508,7 +715,7 @@ $$|\psi\rangle_I = \alpha|0\rangle + \beta|1\rangle$$ $$R(\gamma, \varphi) P(\theta) = P(\theta) R(\gamma, \varphi + \theta)$$ -这个公式含义是:先绕 Z 旋转 $\theta$,再绕赤道 $\varphi$ 转 $\gamma$ 度,等价于先让它绕赤道面夹角为 $\varphi + \theta$ 的轴旋转 $\gamma$,**后**补一个绕 Z 轴旋转 $\theta$。这个比较难想,要配图来想。我想了比较久,但是没想明白。**反正知道虚拟相位门就是不需要物理 Z 门了,用 XY 波形相位的改变来等效。** 通过选取 XY 平面的轴,我们可以把量子态转到任意一个我们想要的相位。我们在意的是把量子态投到南北极,所以直接把这个 $P(\theta)$ 省略掉,就不需要加了。 +这个公式含义是:先绕 Z 旋转 $\theta$,再绕赤道 $\varphi$ 转 $\gamma$ 度,等价于先让它绕赤道面夹角为 $\varphi + \theta$ 的轴旋转 $\gamma$,后补一个绕 Z 轴旋转 $\theta$。这个比较难想,要配图来想。我想了比较久,但是没想明白。**反正知道虚拟相位门就是不需要物理 Z 门了,用 XY 波形相位的改变来等效。** 通过选取 XY 平面的轴,我们可以把量子态转到任意一个我们想要的相位。我们在意的是把量子态投到南北极,所以直接把这个 $P(\theta)$ 省略掉,就不需要加了。 image-20260630140929855 @@ -1099,11 +1306,11 @@ JPA 是约瑟夫森参量放大器,作用是放大量子比特的微弱读出 超导量子计算机发展不是为了取代量子计算机,而是配合使用,解决一些经典计算机解决不了的问题。 -image-20260810173150088 + image-20260810173229271 -image-20260810173258064 + ## 4.2 超导量子计算机的标定