Classical simulation and verification of quantum computation using matchgates and magic states

使用匹配门和魔法状态进行量子计算的经典模拟和验证

• 批准号: 2746767
• 金额: --
• 依托单位: University of Cambridge
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2746767
• 关键词: Classical; simulation; verification; quantum; computation

Quantum computers allow one to explore computational regimes which are believed to be beyond the reach of current classical computing. Therefore, it is unlikely that universal quantum computers can be efficiently simulated by classical probabilistic algorithms. This is in part because the state-of-the-art classical simulators which rely on the power of modern supercomputers struggle to simulate any quantum system beyond 50 qubits. At the same time, certain quantum information processing tasks do not require computational universality. In some scenarios, there are provable benefits, such as an exponential reduction in communication resources for some distributed computing tasks (e.g. Raz 1999) and in quantum cryptography, the ability to communicate with unconditional security against eavesdropping. To realize quantum computation in a circuit model one has to pick a universal gate set. One of the most prominent gatesets which enables universal quantum computation is made of Clifford + T gates. Clifford gates are efficiently classically simulatable, however, when you add a special single-qubit T gate you regain the full power of quantum computation. In 2016, Bravyi et al. introduced a quantity called stabilizer rank. It helps reduce this exponential scaling by significantly decreasing the scaling of resources required to classically simulate quantum systems. The ability to classically simulate generic quantum computations, while unlikely to be possible for a large number of qubits, is of great importance in the noisy intermediate-term quantum computation (NISQ). Another very natural gateset which enables universal quantum computation is made of so-called Matchgates + Magic states. Matchgates are an especially multiflorous class of two-qubit nearest neighbour quantum gates, defined by a set of algebraic constraints. They occur for example in the theory of perfect matchings of graphs, non-interacting fermions, and one-dimensional spin chains. The goal of the project is to study the analogous notion to stabilizer rank for matchgates - the so-called Gaussian rank and study the computational complexity of approximating this quantity. Currently, nearly nothing is known about Gaussian rank and unlike its stabilizer counterpart, the decompositions of n copies of magic states in terms of Gaussian states for n>3 are not known. This problem presents a unique set of challenges suitable for a strong PhD student and would require a combination of techniques: from numerical exploration for a small number of qubits to proof-based techniques which rely on the unique structural properties of Gaussian states. Computing the exact Gaussian rank for a large number of copies of magic states has a number of important applications for the emerging small-to-medium scale quantum computers. First, it would enable one to verify quantum computations for a non-trivial number of qubits (20-300), which is likely to be the milestoneSecond, it would provide unique insights into the complexity of fermionic linear optics and its abilities to achieve universal quantum computations when supplemented with magic states. Thirdly, it would allow one to design novel quantum error-correcting codes as well as efficient classical decoders.

量子计算机允许人们探索被认为超出当前经典计算能力的计算机制。因此，经典概率算法不太可能有效地模拟通用量子计算机。部分原因是依赖现代超级计算机能力的最先进的经典模拟器很难模拟任何超过 50 个量子位的量子系统。同时，某些量子信息处理任务不需要计算通用性。在某些场景中，有可证明的好处，例如某些分布式计算任务的通信资源呈指数减少（例如 Raz 1999），以及在量子密码学中，能够无条件安全地进行通信以防止窃听。 为了在电路模型中实现量子计算，我们必须选择一组通用门。实现通用量子计算的最著名的门集之一是由 Clifford + T 门组成。 Clifford 门可以有效地进行经典模拟，但是，当您添加特殊的单量子位 T 门时，您将重新获得量子计算的全部功能。 2016 年，Bravyi 等人。引入了一个称为稳定器等级的量。它通过显着减少经典模拟量子系统所需的资源扩展来帮助减少这种指数扩展。经典模拟通用量子计算的能力虽然对于大量量子位不太可能实现，但在嘈杂的中期量子计算（NISQ）中非常重要。另一种非常自然的门集可以实现通用量子计算，它由所谓的匹配门 + 魔法状态组成。匹配门是一类特别丰富的两量子位最近邻量子门，由一组代数约束定义。例如，它们出现在图、非相互作用费米子和一维自旋链的完美匹配理论中。该项目的目标是研究匹配门稳定器等级的类似概念 - 所谓的高斯等级，并研究近似该量的计算复杂性。目前，我们对高斯等级几乎一无所知，并且与稳定器对应物不同的是，n>3 时 n 个魔法状态副本在高斯状态方面的分解是未知的。这个问题提出了一系列适合优秀博士生的独特挑战，并且需要多种技术的组合：从少量量子位的数值探索到依赖高斯态独特结构特性的基于证明的技术。计算大量幻态副本的精确高斯等级对于新兴的中小型量子计算机有许多重要的应用。首先，它将使人们能够验证大量量子位（20-300）的量子计算，这可能是一个里程碑。其次，它将为费米子线性光学的复杂性及其实现通用的能力提供独特的见解。量子计算辅以魔法状态。第三，它将允许人们设计新颖的量子纠错码以及高效的经典解码器。