Quantum Sampling Problems, BosonSampling and Quantum Supremacy

There is a large body of evidence for the potential of greater computational power using information carriers that are quantum mechanical over those governed by the laws of classical mechanics. But the question of the exact nature of the power contributed by quantum mechanics remains only partially answered. Furthermore, there exists doubt over the practicality of achieving a large enough quantum computation that definitively demonstrates quantum supremacy. Recently the study of computational problems that produce samples from probability distributions has added to both our understanding of the power of quantum algorithms and lowered the requirements for demonstration of fast quantum algorithms. The proposed quantum sampling problems do not require a quantum computer capable of universal operations and also permit physically realistic errors in their operation. This is an encouraging step towards an experimental demonstration of quantum algorithmic supremacy. In this paper, we will review sampling problems and the arguments that have been used to deduce when sampling problems are hard for classical computers to simulate. Two classes of quantum sampling problems that demonstrate the supremacy of quantum algorithms are BosonSampling and IQP Sampling. We will present the details of these classes and recent experimental progress towards demonstrating quantum supremacy in BosonSampling.Comment: Survey paper first submitted for publication in October 2016. 10 pages, 4 figures, 1 tabl

Adaptive Phase Measurements in Linear Optical Quantum Computation

Photon counting induces an effective nonlinear optical phase shift on certain states derived by linear optics from single photons. Although this no nlinearity is nondeterministic, it is sufficient in principle to allow scalable linear optics quantum computation (LOQC). The most obvious way to encode a qubit optically is as a superposition of the vacuum and a single photon in one mode -- so-called "single-rail" logic. Until now this approach was thought to be prohibitively expensive (in resources) compared to "dual-rail" logic where a qubit is stored by a photon across two modes. Here we attack this problem with real-time feedback control, which can realize a quantum-limited phase measurement on a single mode, as has been recently demonstrated experimentally. We show that with this added measurement resource, the resource requirements for single-rail LOQC are not substantially different from those of dual-rail LOQC. In particular, with adaptive phase measurements an arbitrary qubit state $\alpha \ket{0} + \beta\ket{1}$ can be prepared deterministically

Exact Boson Sampling using Gaussian continuous variable measurements

BosonSampling is a quantum mechanical task involving Fock basis state preparation and detection and evolution using only linear interactions. A classical algorithm for producing samples from this quantum task cannot be efficient unless the polynomial hierarchy of complexity classes collapses, a situation believe to be highly implausible. We present method for constructing a device which uses Fock state preparations, linear interactions and Gaussian continuous-variable measurements for which one can show exact sampling would be hard for a classical algorithm in the same way as Boson Sampling. The detection events used from this arrangement does not allow a similar conclusion for the classical hardness of approximate sampling to be drawn. We discuss the details of this result outlining some specific properties that approximate sampling hardness requires

Classes of two-photon states defined by linear interactions and destructive two-photon quantum interference in a single mode

We describe a two-photon quantum interference effect which differs from the Hong-Ou-Mandel effect in that the destructive quantum inference occurs on a component of the state where two photons are in a single output mode while maintaining the two-photon events in the alternative mode. This effect is manifestly nonclassical but requires more sophisticated technology to observe than the Hong-Ou-Mandel effect. The theory outlined in this paper can also be used to classify two-photon states into classes which are related by the ability to transform the states within the class by using only linear optical interactions. This theory shows that there is an infinite number of these classes of two photon states when there are two or more modes which can support the photons

Conditional Production of Superpositions of Coherent States with Inefficient Photon Detection

It is shown that a linear superposition of two macroscopically distinguishable optical coherent states can be generated using a single photon source and simple all-optical operations. Weak squeezing on a single photon, beam mixing with an auxiliary coherent state, and photon detecting with imperfect threshold detectors are enough to generate a coherent state superposition in a free propagating optical field with a large coherent amplitude ($\alpha>2$) and high fidelity ($F>0.99$). In contrast to all previous schemes to generate such a state, our scheme does not need photon number resolving measurements nor Kerr-type nonlinear interactions. Furthermore, it is robust to detection inefficiency and exhibits some resilience to photon production inefficiency.Comment: Some important new results added, to appear in Phys.Rev.A (Rapid Communication

Coherent state LOQC gates using simplified diagonal superposition resource states

In this paper we explore the possibility of fundamental tests for coherent state optical quantum computing gates [T. C. Ralph, et. al, Phys. Rev. A \textbf{68}, 042319 (2003)] using sophisticated but not unrealistic quantum states. The major resource required in these gates are state diagonal to the basis states. We use the recent observation that a squeezed single photon state ($\hat{S}(r) \ket{1}$) approximates well an odd superposition of coherent states ($\ket{\alpha} - \ket{-\alpha}$) to address the diagonal resource problem. The approximation only holds for relatively small $\alpha$ and hence these gates cannot be used in a scaleable scheme. We explore the effects on fidelities and probabilities in teleportation and a rotated Hadamard gate.Comment: 21 pages, 12 figure

Fault-tolerant linear optical quantum computing with small-amplitude coherent states

Quantum computing using two optical coherent states as qubit basis states has been suggested as an interesting alternative to single photon optical quantum computing with lower physical resource overheads. These proposals have been questioned as a practical way of performing quantum computing in the short term due to the requirement of generating fragile diagonal states with large coherent amplitudes. Here we show that by using a fault-tolerant error correction scheme, one need only use relatively small coherent state amplitudes ($\alpha > 1.2$) to achieve universal quantum computing. We study the effects of small coherent state amplitude and photon loss on fault tolerance within the error correction scheme using a Monte Carlo simulation and show the quantity of resources used for the first level of encoding is orders of magnitude lower than the best known single photon scheme. %We study this reigem using a Monte Carlo simulation and incorporate %the effects of photon loss in this simulation

Boson Sampling from Gaussian States

We pose a generalized Boson Sampling problem. Strong evidence exists that such a problem becomes intractable on a classical computer as a function of the number of Bosons. We describe a quantum optical processor that can solve this problem efficiently based on Gaussian input states, a linear optical network and non-adaptive photon counting measurements. All the elements required to build such a processor currently exist. The demonstration of such a device would provide the first empirical evidence that quantum computers can indeed outperform classical computers and could lead to applications