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范围的计算机制。因此，不太可能用经典的概率算法来有效地模拟通用量子计算机。这在一定程度上是因为最先进的经典模拟器依赖于现代超级计算机的能力，难以模拟任何超过50量子比特的量子系统。同时，某些量子信息处理任务不需要计算普适性。在一些情况下，存在可证明的好处，例如用于某些分布式计算任务的通信资源的指数减少(例如，RAZ 1999)，以及在量子密码学中，以无条件安全的方式进行通信以防止窃听的能力。要在电路模型中实现量子计算，必须选择一个通用的门集。实现普遍量子计算的最突出的门之一是由Clifford+T门组成。Clifford门是有效的经典模拟，然而，当你添加一个特殊的单量子比特T门时，你就重新获得了量子计算的全部力量。2016年，Bravyi等人。引入了一个称为稳定阶数的量。它通过显著减少经典模拟量子系统所需的资源比例来帮助减少这种指数级的比例。经典模拟一般量子计算的能力，虽然不太可能在大量的量子比特上实现，但在嘈杂的中期量子计算(NISQ)中是非常重要的。另一个能够实现普遍量子计算的非常自然的门是由所谓的匹配门+魔术态组成的。匹配门是一类特别多花的两量子比特最近邻量子门，由一组代数约束定义。例如，它们出现在图的完美匹配理论、非相互作用费米子和一维自旋链中。该项目的目标是研究与匹配门稳定器等级类似的概念-所谓的高斯等级，并研究近似这个量的计算复杂性。目前，关于高斯阶几乎一无所知，而且与它的稳定器不同的是，n&gt；3的n个魔态副本关于高斯态的分解是未知的。这个问题提出了一组独特的挑战，适合于一个优秀的博士生，并将需要技术的组合：从对少量量子比特的数值探索到依赖于高斯态的独特结构属性的基于证明的技术。计算大量魔态副本的精确高斯阶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非平凡数量的量子比特(20-300)的量子计算，这很可能是里程碑。其次，它将提供对费米子线性光学的复杂性的独特见解，以及当补充魔术态时它实现通用量子计算的能力。第三，它将使人们能够设计出新的量子纠错码和高效的经典解码器。