The Rise of the Boson-Sampling Quantum Computer and The Renaissance of the Linear Optical Quantum Interferometer

在建造第一台量子计算机的长达20年的竞赛中，已经研究了用于这种计算机的许多物理硬件平台，包括半导体电路，超导电路，在电磁芯片中操纵的带电离子原子以及用激光控制的中性原子。一种落后的方法是设计一种利用光或光子的量子态的量子信息处理器。这是因为很难让光子相互作用或“交谈”，而这是制造晶体管等量子计算元件的主要要求。在2010年之前，基于光子的量子计算机电路设计在辅助量子和经典电路上有巨大的开销，甚至需要构建一个简单的双光子晶体管。在一些早期的设计中，即使是一个光子晶体管，也需要数以万计的辅助光学网络和电子开关。2010年，麻省理工学院的Aaronson Arkhipov展示了一种更简单的光学电路“线性光学干涉仪”，它只由几个光子、透镜、镜子和其他简单的光学元件构成，可以用来解决一个特别难的数学问题，其处理能力比任何经典计算机都要高得多。从那时起，这种新型光学量子计算机在世界范围内进行了大约五次实验。在这个项目中，将研究这种新型简单光学计算机的不同电路设计，并研究它可能能够解决的其他数学问题。此外，还将研究使用这种简单的光学机器制造成像设备（如显微镜）或传感器（如磁场传感器）的可能性，以获得比传统方法更高的分辨率、精度和准确度。本项目最大的智力优势在于它处于量子成像、传感和信息处理的接口，这些都属于量子计量领域。量子信息的语言提供了一种令人兴奋的工具，使得其中一个领域的问题可以用另一个领域开发的工具来看待。因此，在一个分支领域的任何进步，几乎可以立即应用到另一个分支领域，创造性和工作。这项工作在所有子领域都是协同的。所有参与该项目的研究生和本科生将接受量子力学、量子信息论、量子光学和AMO理论基础的培训。用于量子计算、成像和传感的多模、无源、线性光学干涉仪的能力将具有广泛的跨学科商业、政府和科学影响。线性光学干涉仪一直被认为不适合量子信息处理。虽然非线性干涉仪为可扩展和通用量子计算提供了一条途径，但实现这种方案所需的强光学非线性一直难以实现。即使是Knill， Laflamme和Milburn （KLM）提出的所谓线性光学量子计算（LOQC）方案也具有由检测和前馈过程产生的有效非线性。从技术角度来看，由于每个逻辑门需要大量的辅助资源，KLM的方案也被证明是令人生畏的。因此，当Aaronson和Arkhipov （AA）提出，单光子输入的无源线性光学干涉仪可以有效地解决一种计算采样问题时，量子光学界感到惊讶，这个问题在经典甚至通用量子计算机上可能是难以解决的。这一结果引发了最近一系列的实验。Dowling的团队在研究多光子fock态输入的线性光学干涉仪中的量子随机游走时，得出了与AA相似的结论。综上所述，这些新的结果表明，简单的线性光学器件包含迄今为止被忽视的计算能力，而这一能力才刚刚开始被探索。路易斯安那州立大学已经开始从量子光学的角度研究这种设备的计算复杂性，使用标准的理论工具来描述光通过线性干涉仪的量子态传播。除了为具有fock态输入的器件的复杂性提供基本的量子光学论证外，它还表明，自发参数下转换光子源是玻色子采样的可扩展资源，并且很可能与具有相干“广义cat”态叠加的线性光学干涉仪的采样数量相关的计算复杂性。本文将开展以下工作：(1)研究非高斯态输入（如光子加高斯态和高斯态减高斯态）下的数基玻色子采样的计算复杂度；(2)对开发大规模“后经典”线性光学量子信息处理器所需的实际资源进行分析；(3)研究高斯输入的非高斯（数字分辨）采样的计算复杂度；(4)对小型可编程后经典量子信息处理器进行了数值设计和测试；(5)研究用于量子计量的线性光学干涉仪的性能，包括光学传感和成像。

In the twenty year long race to build the first quantum computer a number of physical hardware platforms for such a computer have been investigated including semiconductor circuits, superconducting circuits, charged ionic atoms manipulated in electromagnetic chips, and neutral atoms controlled with lasers. One approach that has lagged behind is the design of a quantum information processor that uses quantum states of light or photons. This is because it is difficult to get photons to interact or 'talk' to each other, a primary requirement in making quantum-computing elements such as transistors. Prior to 2010, photon-based quantum computer circuit designs had a huge overhead in the ancillary quantum and classical circuitry required to build even a simple two-photon transistor. In some of the early designs, tens of thousands of ancillary optical networks and electronic switches were required to construct even a single photon transistor. In 2010 Aaronson Arkhipov at MIT showed that much simpler optical circuit 'a linear optical interferometer' constructed with just a few photons, lenses, mirrors, and other simple optical elements, could be used to solve a particularly hard mathematical problem with an exponential increase in processing power over any classical computer. Since then some five experiments on this new type of optical quantum computer have been carried out worldwide. For this project different circuit designs of this new type of simple optical computer will be investigated and a search for additional mathematical problems that it might be able to solve will be carried out. In addition the possibility of using such a simple optical machine for making imaging devices such as microscopes, or sensors such as magnetic field sensors, that operate with more resolution, precision, and accuracy than is possible classically will be investigated. The great intellectual merit of this project is that it is at the interface of quantum imaging, sensing, and information processing all within the field of quantum metrology. The language of quantum information provides an exciting tool such that problems in one of these fields can be viewed using tools developed in another. Hence any advance in one subfield can almost immediately be applied, with creativity and work, to another subfield. The work is synergistic across all the subfields. All the graduate and undergraduate students involved in this project will be trained in the foundations of quantum mechanics, quantum information theory, quantum optics, and AMO theory. The power of multimode, passive, linear optical interferometers for quantum computation, imaging, and sensing will have broad cross-disciplinary commercial, governmental, and scientific impact.Linear optical interferometers have been thought to be unsuitable for quantum information processing. While nonlinear interferometers provide a route to scalable and universal quantum computation, the strong optical nonlinearities required to implement such schemes have been difficult to attain. Even the so-called linear optical quantum computing (LOQC) scheme proposed by Knill, Laflamme, and Milburn (KLM) has effective nonlinearities that are generated by the detection and feed-forward processes. The KLM scheme has also proved daunting from a technological standpoint due to the immense number of ancilla resources required per logical gate. It thus came as a surprise to the quantum optics community when Aaronson and Arkhipov (AA) proposed that passive linear optical interferometers with single photon inputs could efficiently solve a type of computational sampling problem, a problem that is likely intractable on a classical or even a universal quantum computer. This result has let to a flurry of recent experiments. Dowling's group was led to a similar conclusion as that of AA in the study of quantum random walks in linear optical interferometers with multiphoton Fock-state inputs. Taken together, these new results indicate that simple linear optical devices contain a hitherto overlooked computational capability that has only yet begun to be explored. LSU has begun an investigation of the computational complexity of such devices from a quantum optics point of view using the standard theoretical tools for describing the propagation of quantum states of lights through linear interferometers. In addition to providing an elementary quantum optical argument for the complexity of the devices with Fock-state inputs, it has been shown that spontaneous parametric down conversion photon sources are a scalable resource for boson sampling and that there is very likely a computational complexity associated with the number sampling of linear optical interferometers with superpositions of coherent 'generalized cat' states. The following tasks will be carried out: (1) investigate the computational complexity of boson sampling in the number basis with non-Gaussian state inputs such as photon added and subtracted Gaussian states; (2) carry out a realistic resource analysis of what is required in practice to develop a large-scale 'post-classical' linear optical quantum information processor; (3) investigate the computational complexity non-Gaussian (number-resolved) sampling with Gaussian inputs; (4) numerically design and test a small-scale programmable post-classical quantum information processor; (5) investigate the performance of linear optical interferometers for the purposes of quantum metrology including optical sensing and imaging.