基于非局域超导量子谐振器的贝尔态制备
Bell State Generation Based on Non-Local Superconducting Quantum Resonators
DOI: 10.12677/mp.2024.143013, PDF, HTML, XML, 下载: 58  浏览: 128 
作者: 黎 杰:天津工业大学物理科学与技术学院,天津
关键词: 腔量子电动力学量子纠缠量子计算Cavity Quantum Electrodynamics Quantum Entanglement Quantum Computation
摘要: 量子信息是量子物理和信息科学相结合的新型交叉学科,超导量子谐振器因其可被灵活设计、存储和引导微波光子以及拥有高品质因子等特点成为一个很好的量子信息处理平台。本文研究了基于非局域超导谐振器的全共振贝尔纠缠态的产生过程。使用的系统由一个长超导传输线所连接的两个非局域一维超导谐振器组成,这两个非局域谐振器又分别与两个承载信息的超导谐振器耦合。在本方案中,传输线不会被布局微波光子,从而可以抵抗长传输线的光子损失带来的影响。这也使得长传输线可以被设计的更为复杂以连接更多的非局域超导谐振器达成基于超导谐振器的大规模分布式量子计算。
Abstract: Quantum information is a new interdisciplinary field that combines quantum physics and information science. Superconducting quantum resonators have become a good platform for quantum information processing due to their flexible design, storage, and guidance of microwave photons, as well as their high quality factors. This article investigates the generation process of fully resonant Bell entangled states based on non-local superconducting resonators. The system used consists of two non-local one-dimensional superconducting resonators connected by a long superconducting transmission line, which are coupled to two superconducting resonators carrying information, respectively. In this scheme, the transmission line will not be laid out with microwave photons, so it can resist the impact of photon loss caused by long transmission lines. This also makes long transmission lines more complex to connect more non local superconducting resonators for large-scale distributed quantum computing based on superconducting resonators.
文章引用:黎杰. 基于非局域超导量子谐振器的贝尔态制备[J]. 现代物理, 2024, 14(3): 110-117. https://doi.org/10.12677/mp.2024.143013

1. 引言

量子纠缠的产生是量子信息处理(QIP) [1] 的一项基本任务,已在许多量子系统中得以实现,如在光子 [2] [3] [4] 、核磁共振 [5] [6] [7] 、金刚石氮空位中心 [8] [9] [10] [11] [12] 和腔量子电动力学(QED)中 [13] 。由于原子与光子之间一旦实现强耦合便可以对原子或光子进行快速QIP,所以人们对原子与单模光场组成的腔QED系统进行了大量研究。人工量子系统因为其具有大规模集成的良好潜力所以在模拟腔QED方面引起了广泛关注。

电路QED [14] [15] 作为一种人工量子系统,由人工原子超导量子比特(SQ)和超导谐振器(SR)组成,基于SQ或SR的通用量子计算研究也已经取得了很大进展。SQ方面,SQ一般由电容、电感和约瑟夫森结构成,其能量弛豫时间已提高到50 µs [16] 左右,近些年人们正在试图实现基于其的容错量子计算。例如,Chow等人 [17] 在2014年实现了一种可扩展的容错量子计算结构。Martinis等人 [18] 在2015年通过在一个由9个SQ组成的线性阵列的超导量子电路中,通过重复误差检测实现了量子状态的保持。SR方面,SR是基于超导量子电路的重要器件,其在存储和引导微波光子方面有重要作用。除此之外,SR还可以在超导人造原子间传递信息,充当量子数据线的作用。近些年,SR的品质因子已被提高到106 [19] ,这意味着该SR可以作为一种良好的量子信息载体。已经有大量实验和理论工作在研究基于SR的QIP,例如,Wang等人 [20] 在2011年实现了两个SR的NOON态纠缠。Taylor等人 [21] 在2013年提出了一种利用电路QED实现非线性光学量子计算的方案。与此同时,基于SR的通用量子门构造 [22] - [27] 、纠缠产生 [28] - [33] 和测量 [34] [35] [36] 等量子计算的基本任务也有很多研究工作。

基于电路量子电动力学的量子处理器的主要集成方式包括:一些SQ和一个SR [37] 的耦合,一些SR与一个SR总线 [38] 或一些SR与一个SQ [39] [40] [41] 的耦合,以及一些电路QED通过SR总线 [42] 的耦合等。在本文中,我们关注另一种集成多个SR或SQ的芯片类型——分布式量子计算(DQC) [43] ,它需要一条长传输线(TL)来连接许多本地处理器,以形成一个量子网络。在DQC中,量子器件之间的串扰可以进一步减少。为了实现远距离量子器件上的纠缠操作,人们需要在TL中使用真实的光子传播或暗光子的绝热演化。在电路QED中,为了连接多个处理器,需要对超导TL进行复杂的设计,从而降低TL的品质因子,提高了其中微波光子的泄漏率。因此,基于电路QED的分布式量子计算需要考虑超导TL中的暗光子的绝热演化过程。

本文是基于非局域性的超导量子谐振器的纠缠研究。在一个长超导TL连接两个SR,且这两个SR又分别与两个承载量子信息的SR耦合的系统中,提出了一种基于两个非局域超导量子SR的贝尔态产生方案。通过让所有SR和超导TL的频率相等,本方案可以使超导TL不被布局微波光子从而克服TL泄漏的影响。通过考虑可行的实验参数,贝尔态的保真度可达到99%左右。

2. 系统模型及其哈密顿量

为了构建基于两个远程超导谐振器的贝尔态纠缠产生方案。我们考虑系统如图1所示,该系统由长超导传输线(rf)连接两个远距离一维超导谐振器(Ra1和Ra2),且Ra1和Ra2又分别于两个承载量子信息的超导谐振器(b1和b2)耦合组成。

Figure 1. Diagram for preparing Bell states based on two long-range superconducting quantum resonators

图1. 基于两个远距离超导量子谐振器制备贝尔态的装置图

考虑到系统中不同原器件之间的相互作用,系统的哈密顿量在相互作用绘景下可以写为:

H ^ 1 = g 1 a 1 ( a ^ 1 + b ^ 1 e i δ 1 a 1 t + a ^ 1 b ^ 1 + e i δ 1 a 1 t ) + g 2 a 2 ( a ^ 2 + b ^ 2 e i δ 2 a 2 t + a ^ 2 b ^ 2 + e i δ 2 a 2 t ) + j = 1 g f , j I [ f ^ j ( a ^ 1 + + ( 1 ) j e i ϕ a ^ 2 + ) + H . c . ] (1)

在公式(1)中, g 1 a ^ 1 是超导量子谐振器b1与一维超导谐振器Ra1之间的耦合强度; g 2 a 2 是超导量子谐振器b2与一维超导谐振器Ra2之间的耦合强度。其中频率失谐量 δ 1 ( 2 ) a 1 ( 2 ) = ω 1 ( 2 ) ω a 1 ( a 2 ) ω 1 ( 2 ) ( ω a 1 ( a 2 ) )为谐振器b1(2) (Ra1(a2))的频率。 a ^ 1 + ( a ^ 2 + )和 b ^ 1 + ( b ^ 2 + )分别是谐振器Ra1 (Ra2)和远程谐振器b1 (b2)的产生算符。 g f , j I ( I = a b )是谐振器RaI与传输线rf在j模式下的耦合强度。 ϕ 为传输场通过长度为L的传输线rf所产生的相位,且满足公式: ϕ = 2 π ω l / c ,其中c为光速。利用精心设计的谐振器(Ra1)以及让共振操作时间远长与光子在TL中的往返时间,可以得到哈密顿量 H ^ 1 的最后一项。

在选取 2 L k f a 1 ( a 2 ) / 2 π c 1 情形下TL的短极限时,我们只需考虑rf的一个谐振模态f与Ra1和Ra2的谐振器相互作用。L是rf的长度, k f a 1 ( a 2 ) 是谐振器Ra1和Ra2的衰减率。这时,哈密顿量 H ^ 1 可以约化为:

H ^ 2 = g 1 a 1 ( a ^ 1 + b ^ 1 e i δ 1 a 1 t + a ^ 1 b ^ 1 + e i δ 1 a 1 t ) + g 2 a 2 ( a ^ 2 + b ^ 2 e i δ 2 a 2 t + a ^ 2 b ^ 2 + e i δ 2 a 2 t ) + g f a 1 ( f ^ + a ^ 1 + f ^ a ^ 1 + ) + g f a 2 ( f ^ + a ^ 2 + f ^ a ^ 2 + ) (2)

当我们可取: ω 1 / 2 π = ω 2 / 2 π = ω a 1 / 2 π = ω a 2 / 2 π = ω f / 2 π = ω ,则系统哈密顿量 H ^ 2 可以简化为:

H ^ 2 = g 1 a 1 ( a ^ 1 + b ^ 1 + a ^ 1 b ^ 1 + ) + g 2 a 2 ( a ^ 2 + b ^ 2 + a ^ 2 b ^ 2 + ) + g f a 1 ( f ^ + a ^ 1 + f ^ a ^ 1 + ) + g f a 2 ( f ^ + a ^ 2 + f ^ a ^ 2 + ) (3)

哈密顿量 H ^ 2 在薛定谔绘景下可以表示成:

H ^ 3 = ω a ^ 1 + a ^ 1 + ω a ^ 2 + a ^ 2 + ω f + f + ω b ^ 1 + b ^ 1 + ω b ^ 2 + b ^ 2 + g 1 a 1 ( a ^ 1 + b ^ 1 + a ^ 1 b ^ 1 + ) + g 2 a 2 ( a ^ 2 + b ^ 2 + a ^ 2 b ^ 2 + ) + g f a 1 ( f ^ + a ^ 1 + f ^ a ^ 1 + ) + g f a 2 ( f ^ + a ^ 2 + f ^ a ^ 2 + ) (4)

通过取 g f a 1 = g f a 2 = g 和规范变换 C ^ ± = 1 2 ( a ^ 1 + a ^ 2 ± 2 f ^ ) C ^ = 2 2 ( a ^ 1 a ^ 2 ) ,哈密顿量 H 3 可表示为:

H ^ 4 = ω b ^ 1 + b ^ 1 + ω b ^ 2 + b ^ 2 + ω C ^ + C ^ + ( ω + 2 g ) C ^ + C ^ + + + ( ω 2 g ) C ^ C ^ + + 1 2 [ g 1 a 1 ( C ^ + + C ^ + 2 C ^ ) b ^ 1 + + g 1 a 1 ( C ^ + + + C ^ + + 2 C ^ + ) b ^ 1 + g 2 a 2 ( C ^ + + C ^ 2 C ^ ) b ^ 2 + + g 2 a 2 ( C ^ + + + C ^ + 2 C ^ + ) b ^ 2 ] (5)

公式(5)中, C ^ C ^ ± 是三种相互不耦合的玻色子模式。当Ra1、Ra2和rf的频率相等时, C ^ ± C ^ 的频率被分成三个不同的部分,频率分别为 ω C + = ω + 2 g ω C = ω 2 g ω C = ω 。当取 g { g 1 a 1 , g 2 a 2 } 时,当 ω ± 2 g 与模式C、b1和b2的频率大失谐时,模式 C ± 可以被忽略。因此,系统的哈密顿量可以简化为(相互作用绘景中):

H ^ e f f = 1 2 [ g 1 a 1 ( C ^ b ^ 1 + + C ^ + b ^ 1 ) g 2 a 2 ( C ^ b ^ 2 + + C ^ + b ^ 2 ) ] (6)

这里只剩下模式 C ^ = 2 2 ( a ^ 1 a ^ 2 ) ,这意味着在超导传输线rf不参与整个演化过程,不会被布局微波光子。

假设哈密顿量 H e f f 所描述的有效系统的初始状态为 | ϕ 1 = | 1 1 | 0 2 | 0 c ( | 0 c | 0 a | 0 b | 0 f ),系统的演化可以表示为:

| ϕ ( t ) = e i H ^ e f f t | ϕ 1 = 1 G [ ( g 2 a 2 ) 2 + ( g 1 a 1 ) 2 cos ( G 2 t ) ] | ϕ 1 g 1 a 1 g 2 a 2 G [ cos ( G 2 t ) 1 ] | ϕ 2 i g 1 a 1 G sin ( G 2 t ) | ϕ c (7)

其中, G = ( g 1 a 1 ) 2 + ( g 2 a 2 ) 2 | ϕ 2 = | 0 1 | 1 2 | 0 c | ϕ c = | 0 1 | 0 2 | 1 c 。显然,如果取 g 1 a 1 / g 2 a 2 = 2 + 1 ,有效系统的状态可以演化为:

| ϕ = 1 2 ( | 0 1 | 1 2 | 1 1 | 0 2 ) | 0 c (8)

其操作时间为 G 2 t = k π ( k = 1 , 2 , 3 , )。此外,如果取 g 1 a 1 / g 2 a 2 = 2 1 ,就可以得到另一个贝尔纠缠态:

| ϕ + = 1 2 ( | 0 1 | 1 2 + | 1 1 | 0 2 ) | 0 c (9)

其操作时间为 G 2 t = k π ( k = 1 , 2 , 3 , )。

3. 数值模拟

为了研究系统在演化过程中每个器件的能量布局的变化,我们数值模拟了rf中的能量布局随时间变化(如图2所示)、Ra1和Ra2中能量布局随时间变化(分别如图3(a)和图3(b)所示)、b1和b2中的能量布局随时间变化(分别如图3(c)和图3(d)所示)。从图2中可以看出,随着 g / g 1 a 1 越大,传输线中能量布局概率越小,即传输线的泄露率对整个演化过程的影响越小。

进一步,我们绘制了谐振器Ra1、Ra2、b1和b2的能量布局随时间的变化图3

图3可以发现谐振器b1、b2的能量布局图在100~200 ns区间内有一处b1与b2的峰值同时处于50%的位置,也就是说:在此时刻,谐振器b1中50%的概率有一个光子,同时在b2腔内也有50%的概率有一个光子。联合来看,当微波光子在b1和b2中均有一半概率存在时,在Ra1、Ra2和rf中没有微波光子,那么此时谐振器b1与b2处于贝尔纠缠态 | ϕ 。这里使用的能量在b1、b2、Ra1、Ra2和rf中的布局 P m ( m = 1 , 2 , a 1 , a 2 , f )的定义为:

P m = | ϕ m | e i H ^ 1 t | ϕ 1 | 2 (10)

其中 ϕ 1 = | 1 1 | 0 2 | 0 a 1 | 0 a 2 | 0 f ϕ 2 = | 0 1 | 1 2 | 0 a 1 | 0 a 2 | 0 f ϕ a 1 = | 0 1 | 0 2 | 1 a 1 | 0 a 2 | 0 f ϕ a 2 = | 0 1 | 0 2 | 0 a 1 | 1 a 2 | 0 f ϕ f = | 0 1 | 0 2 | 0 a 1 | 0 a 2 | 1 f 。在这里,我们取 g f a 1 ( a 2 ) = 20 g 1 a 1 ( a 2 ) 可以完美的抑制 C ± (如图3(c)所示)。

Figure 2. (a), (b), and (c) show the energy layout in the transmission line at = 5, 10, and 20, respectively

图2. (a)、(b)和(c)分别为 g / g 1 a 1 = 5 、10和20时的传输线rf中的能量布局图

Figure 3. (a), (b), (c), and (d) show the energy layout of resonators Ra1, Ra2, b1, and b2 with time, respectively

图3. (a)、(b)、(c)和(d)分别为谐振器Ra1、Ra2、b1和b2的能量布局随时间变化图

最后我们需要计算我们的贝尔态的保真度,利用计算贝尔态的保真度的定义式:

F = | ϕ m | e i H ^ 1 t | ϕ 1 | 2 (11)

基于(11)式我们给出了态 | ϕ | ϕ + 的保真度随时间变化,分别如图4(a)和图4(b)所示。其中, | ϕ 的保真度在131.7 ns可以达到99.8%, | ϕ + 的保真度在55.4 ns达到99.4%。

Figure 4. (a) and (b) show the changes in fidelity over time for | ϕ and | ϕ + , respectively

图4. (a)和(b)分别为 | ϕ | ϕ + 的保真度随时间变化图

纠缠态是量子计算中的基础资源,无论是构造量子比特还是实现量子门操作都需要以其为基础。而在超导量子计算中,大量超导量子比特的集成会带来严重的串扰问题。为了规避串扰问题,我们可以考虑将分布式量子计算。而分布式计算中我们需要传输线连接各个超导谐振器,并且希望传输线中的能量布局概率越低越好,为此我们进行了一些数值模拟以保证我们的设计的贝尔态生成方案的可行性。

此外,无论是比特还是量子比特都依赖于超导量子电路或者半导体晶体管这些真实存在的元件。所以我们必须确保其正常运行的概率足够高,即量子态的保真度要足够高。所以,我们根据态保真度的定义计算我们方案中所制备的贝尔态的保真度,结果表明, | ϕ 的保真度在131.7 ns可以达到99.8%, | ϕ + 的保真度在55.4 ns达到99.4%,结果较好,证明了我们的贝尔态生成方案的可行性。

4. 总结

我们基于非局域超导量子谐振器的设计处理贝尔态纠缠产生方案。我们首先构建了适当的系统,并根据系统中各器件及其相互作用关系确定了系统的原始哈密顿量,再引入特定参数关系,从原始哈密顿量获得有效哈密顿量。紧接着我们研究了系统在特定初态下的演化过程,并成功生成了基于两个非局域超导量子谐振器的贝尔纠缠态。而为了为验证我们方案的实验可行性,我们利用Python和QuTiP程序包进行了数值模拟。在采用真实实验参数的前提下,数值模拟的结果表明,通过我们的方案制备的贝尔态的保真度均在99%以上。鉴于超导谐振器的泄露时间约为50微秒,而贝尔态制备时间在50~100纳秒范围内,所以即使考虑谐振器泄露率的影响,我们的贝尔态保真度仍将高于99%,结果较好,理论上可以基于此方案进一步构建量子比特和量子门。我们的工作为分布式超导量子计算提供了一定理论支持。

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