Universal quantum computing models: a perspective of resource theory

Wang Dong-Sheng

doi:10.7498/aps.73.20240893

Abstract

Quantum computing has been proven to be powerful, however, there are still great challenges for building real quantum computers due to the requirements of both fault-tolerance and universality. There is still no systematic method to design fast quantum algorithms and identify the key quantum resources. In this work, we develop a resource-theoretic approach to characterize universal quantum computing models and the universal resources for quantum computing. Our theory combines the framework of universal quantum computing model (UQCM) and the quantum resource theory (QRT). The former has played major roles in quantum computing, while the later was developed mainly for quantum information theory. Putting them together proves to be ‘win-win’: on one hand, using QRT can provide a resource-theoretic characterization of a UQCM, the relation among models and inspire new ones, and on the other hand, using UQCM offers a framework to apply resources, study relation among resources and classify them. In quantum theory, we mainly study states, evolution, observable, and probability from measurements, and this motivates the introduction of different families of UQCMs. A family also includes generations depending on a hierarchical structure of resource theories. We introduce a table of UQCMs by first classifying two categories of models: one referring to the format of information, and one referring to the logical evolution of information requiring quantum error-correction codes. Each category contains a few families of models, leading to more than one hundred of them in total. Such a rich spectrum of models include some well-known ones that people use, such as the circuit model, the adiabatic model, but many of them are relatively new and worthy of more study in the future. Among them are the models of quantum von Neumann architectures established recently. This type of architecture or model circumvents the no-go theorems on both the quantum program storage and quantum control unit, enabling the construction of more complete quantum computer systems and high-level programming. Correspondingly, each model is captured by a unique quantum resource. For instance, in the state family, the universal resource for the circuit model is coherence, for the local quantum Turing machine is bipartite entanglement, and for the cluster-state based, also known as measurement-based model is a specific type of entanglement relevant to symmetry-protected topological order. As program-storage is a central feature of the quantum von Neumann architecture, we find the quantum resources for it are quantum memories, which are dynamical resources closely related to entanglement. In other words, our classification of UQCMs also serves as a computational classification of quantum resources. This can be used to resolve the dispute over the computing power of resources, such as interference, entanglement, or contextuality. In all, we believe our theory lays down a solid framework to study computing models, resources, and design algorithms.

Keywords:

Authors and contacts

Wang Dong-Sheng^1,2

1.
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
2.
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Corresponding author: Wang Dong-Sheng, wds@itp.ac.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12047503, 12105343).

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Cited By

图 1 经典与量子信息领域的一些发展阶段. 经典(上部): 在世纪之交, 希尔伯特提出了著名的23个问题, 其中一个启发了图灵对于计算的研究, 直接奠定了计算机科学的理论基础. 香农证明了通信的三大定理, 为纠错码理论奠定基础. 同时, 冯·诺依曼提出了通用计算机的架构理论. 之后, PN结和三极管的发明奠定了电子计算机的硬件基础, 然后发展到大规模可编程集成电路(IC). 量子(下部): 早期有EPR和Bell关于量子纠缠和非定域性的探讨. 之后, 经Holevo, Kraus等人将量子信道演化、退相干、测量等数学形式发展出来. BB84是首个利用量子不确定性的保密通信方案, 整个领域从此开始起步. 在理论方面, 量子资源理论(QRT)作为描述量子信息的完备理论逐渐发展成熟

Figure 1. Development of classical and quantum information science. Classical (up): From the 23 problems of Hilbert, Turing laid the foundation of computation science. Shannon established the theory of communication, and von Neumann established the architecture of computers. The next breakthrough include PN junction and transistor, forming the building blocks of modern integrated circuits. Quantum (down): With the early study of EPR and Bell, the mathematical formalism of quantum channel, decoherence, and measurement were developed by Holevo, Kraus, etc. The BB84 secure protocol boosted the field. The theoretical achievement is the recent development of quantum resource theory as the theory of quantum information.

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图 2 计算机系统的层次化设计原理. 从整体上看, 可以分为硬件层次和软件层次, 也可以区分出软硬件之间的架构层次, 即微体系结构的设计

Figure 2. Hierarchy of computer system. There are layers of hardware and software, and also the layers of system architecture.

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图 3 量子线路模型示意及算法设计结构. 基本结构(左上)包括某经典算法A和它设计的量子线路U以及测量方式(三角符号). 也可以扩展为经典-量子混合的迭代结构(右上), 或等价地表示为线性方式(下)

Figure 3. Structures of quantum circuit model and quantum algorithms. Basic structure (top-left) has a classical algorithm A that designs the quantum circuit U and measurement. It extends to the iterative classical-quantum algorithms (top-right), which can be “stretched” into a linear flow (bottom).

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图 4 量子纠错过程示意, 即用量子超信道$ \hat{{\cal{S}}} $将$ \varPhi^{\otimes n} $近似地转化为$ {1 1}^{\otimes k} $

Figure 4. Structure of quantum error correction that converts $ \varPhi^{\otimes n} $ into $ {1 1}^{\otimes k} $ approximately by a superchannel $ \hat{{\cal{S}}} $.

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图 5 量子资源理论框架下的计算模型. 第I类模型主要是针对输入端, 即信息的不同表示形式; 第II类模型主要是针对编码后的逻辑操作的形式

Figure 5. Structure of quantum computing model via quantum resource theory. The Category-I (-II) models are defined for different types of input (logical operations).

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图 6 通用量子计算模型分类表. 第I类模型即形式类有12个模型, 第II类模型即演化类有9个模型, 因而一共108个完备的模型(灰色方格). 其中研究最多的是基于线路模型的各种方案. 信道家族的模型统称为量子冯·诺依曼模型或架构. 模型之间也可以进行混合搭配

Figure 6. The classification table of universal quantum computing models. There are 12 (9) Category-I (-II) models, hence in total 108 complete models (grey boxes). The most well-studied are those based on circuit model. The channel-family models are all von Neumann architecture or models. Hybridization among models are also allowed.

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图 7 矩阵乘积态的等价表示方式. 张量形式(上): 横线是纠缠空间, 竖线是不同的物理空间, 方框代表张量(或矩阵). VBS或AKLT形式^[69](中): 张量由圈代表的算子构造, 横向线段代表Bell态, 对应(11)式. 量子线路形式(下): 每个张量可以由幺正过程(大框)实现

Figure 7. Representations of matrix-product states. Tensor form (Top): the top register is the entanglement space, the vertical wires are physical sites, the boxes are the tensors or matrices. VBS or AKLT form^[69] (Middle): tensors are defined by local operators (circles) acting on Bell states Eq.(11). Circuit form (Bottom): each tensor is realized by a unitary circuit (big boxes).

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图 8 量子冯·诺依曼架构示意图(左)与量子线路模型(右). 对比来看, 在通常的线路模型中不考虑量子的控制单元和量子的程序存储

Figure 8. Schematics of the quantum von Neumann architecture (left) and the circuit model (right). For the later, there is no explicit quantum control unit and storage of quantum programs.

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图 9 量子超算法结构示意. 其母算法(阴影部分)将输入的数据(方框)转化为所需的算法即子算法, 完成上端数据系统的输入输出过程(自左向右). 经典-量子混合架构是其特例(图3), 且MPS结构(图7)也可以看作其特例. 输入(方框)之间也可以存在量子关联(未表示)

Figure 9. Schemetics of quantum super-algorithm. The “mother” algorithm (shaded) maps the input data (boxes) into the desired “child” algorithm, which acts on the data system (top register). The classical-quantum hybrid algorithm (Fig. 3) is a special case, and the MPS formula (Fig. 7) is also a special case of it. There can also be quantum correlation or memory (unshown) between the input (boxes).

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图 10 在现有的经典计算和未来通用量子计算之间, 还存在其他的研究范式, 比如专用光芯片、忆阻器、适用于人工智能的GPU以及量子模拟等

Figure 10. In between the current classical computers and the universal quantum computers in the future, there are other research paradigm, such as optical chips, memristors, GPU for AI, and quantum simulators etc.

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