Abstract Gaussian Boson Sampling (GBS) is the problem of sampling from the output of photon-number-resolving measurements of squeezed states input to a linear optical interferometer. For purposes of demonstrating quantum computational advantage as well as practical applications, a large photon number is often desirable. However, producing squeezed states with high photon numbers is experimentally challenging. In this work, we examine the computational complexity implications of increasing the photon number by introducing coherent states. This displaces the state in phase space and as such we call this modified problem Displaced GBS. By utilising a connection to the matching polynomial in graph theory, we first describe an efficient classical algorithm for Displaced GBS when displacement is high or when the output state is represented by a non-negative graph. Then we provide complexity theoretic arguments for the quantum advantage of the problem in the low-displacement regime and numerically quantify where the complexity transition occurs.
Abstract Gaussian Boson Sampling (GBS) is the problem of sampling from the output of photon-number-resolving measurements of squeezed states input to a linear optical interferometer. For purposes of demonstrating quantum computational advantage as well as practical applications, a large photon number is often desirable. However, producing squeezed states with high photon numbers is experimentally challenging. In this work, we examine the computational complexity implications of increasing the photon number by introducing coherent states. This displaces the state in phase space and as such we call this modified problem Displaced GBS. By utilising a connection to the matching polynomial in graph theory, we first describe an efficient classical algorithm for Displaced GBS when displacement is high or when the output state is represented by a non-negative graph. Then we provide complexity theoretic arguments for the quantum advantage of the problem in the low-displacement regime and numerically quantify where the complexity transition occurs.
Gaussian boson sampling (GBS) is a quantum sampling task in which one has to draw samples from the photon-number distribution of a large-dimensional nonclassical squeezed state of light.In an effort to make this task intractable for a classical computer, experiments building GBS machines have mainly focused on increasing the dimensionality and squeezing strength of the nonclassical light.However, no experiment has yet demonstrated the ability to displace the squeezed state in phase space, which is generally required for practical applications of GBS.In this work, we build a GBS machine that achieves the displacement by injecting a laser beam alongside a two-mode squeezed vacuum state into a 15-mode interferometer.We focus on two new capabilities.Firstly, we use the displacement to reconstruct the multimode Gaussian state at the output of the interferometer.Our reconstruction technique is in situ and requires only three measurement settings regardless of the state dimension.Secondly, we study how the addition of classical laser light in our GBS machine affects the complexity of sampling its output photon statistics.We introduce and validate approximate semiclassical models that reduce the computational cost when a significant fraction of the detected light is classical.
Gaussian Boson Sampling (GBS) is capable of solving certain classes of graph problems owing to the samples produced by such a device having a connection to the hafnian matrix function.
In particular, a GBS device has been shown to provide an enhancement in the search of cliques -- or complete subgraphs -- in undirected weighted graphs over classical algorithms.
A graph can be mapped to a GBS experiment by configuring the squeezing parameters of the input states and programming the linear optical network.
In practice, limited squeezing and photon loss degrade the performance of the GBS device for max-clique search.
In comparison, coherent states -- often considered a classical resource due to their Poissonian statistics -- can be readily prepared across many modes using an attenuated laser.
In this paper, we report an enhancement of the success rate of GBS in finding maximum weighted cliques by adding displacements under lossy conditions or when a limited amount of squeezing is available.
Moreover, we report that this enhancement can be scaled up to large graphs with limited resource overheads.
Gaussian boson sampling (GBS) seeks quantum advantage by sampling photon-number patterns generated with squeezed inputs and passive linear optics. Many proposed GBS applications instead target function computation by applying functions to mode-resolved photon-number outcomes---a natural form of experimental postprocessing that produces classical outputs. Sampling hardness alone, however, does not determine the complexity of these tasks. By analyzing the irreducible decomposition of fixed-photon-number operator spaces, we prove that the expectation value of every such function in the average case over passive linear-optical networks can be classically evaluated for inputs with finite squeezing strength. We also provide a classical algorithm that estimates this value to inverse-polynomial additive error. The result provides new theoretical tools for analyzing linear-optical quantum systems, helps clarify the origin of current GBS hardness evidence, and inspires new applications of GBS with genuine quantum advantages.
Gaussian boson sampling (GBS) is not only a feasible protocol for demonstrating quantum computational advantage, but also mathematically associated with certain graph-related and quantum chemistry problems. In particular, it is proposed that the generated samples from the GBS could be harnessed to enhance the classical stochastic algorithms in searching some graph features. Here, we use Jiǔzhāng, a noisy intermediate-scale quantum computer, to solve graph problems. The samples are generated from a 144-mode fully connected photonic processor, with photon click up to 80 in the quantum computational advantage regime. We investigate the open question of whether the GBS enhancement over the classical stochastic algorithms persists-and how it scales-with an increasing system size on noisy quantum devices in the computationally interesting regime. We experimentally observe the presence of GBS enhancement with a large photon-click number and a robustness of the enhancement under certain noise. Our work is a step toward testing real-world problems using the existing noisy intermediate-scale quantum computers and hopes to stimulate the development of more efficient classical and quantum-inspired algorithms.
It remains questionable whether all physically realizable computational model can be simulated by currently available digital computer in an efficient manner. The theory of Quantum Mechanics offers an answer to this question through one of its branches that intersects with Computer Science, known as Quantum Computing. It claims that current computers, which is operating based on classical mechanics cannot efficiently simulate quantum mechanical process. Under this claim, a computer that is operating based on quantum mechanics holds more computational power than classical computers, promising a possibility to computationally solve problems that are classically intractable. To show that this claim holds, numerous quantum computational models have been proposed and analyzed through the lens of Computational Complexity Theory, providing a rigorous notion of efficiency. Models that are more feasible experimentally, although not necessarily universal, have also been of a particular interests given the numerous complicated challenges surrounding the experimental implementation of a universal quantum computer. One of this non-universal models is called Gaussian Boson Sampling (GBS), which embraces the methods of Quantum Optics by sampling the number of photon outputs from a linear optical network as a means to process quantum information. Compared to its prototype model, Boson Sampling, GBS is more experimentally feasible due to its method of generating photon that is based on the Continuous Variable (CV) paradigm of Quantum Information. Nevertheless as with Boson Sampling, it was also shown that classical simulation of GBS cannot be performed efficiently under some highly plausible conjectures. In the work presented in this thesis, we probe the complexity of classically constructing the probability distribution induced by GBS using the method of Quantum Homodyne Tomography. Instead of photon number measurement, this method further embraces the CV paradigm in the measurement of GBS by using Homodyne measurements that outputs real numbers. Sample data obtained from the measurement outcomes is then classically processed using Pattern Functions, which we use to approximate the probability distribution of GBS. A series of computer simulations that approximates some probability values of GBS are presented here, which shows that the number of samples needed to decrease the error of the approximation increases rapidly for more accurate approximation. We have derived a bound for the approximation error which can be achieved given that the sampled data satisfies a certain statistical property. This statistical property can eventually be satisfied as the number of Homodyne samples increases. Having shown another evidence for the complexity of approximating GBS using Homodyne Tomography, we further investigate the nature of the complexity of GBS by analyzing its classical-quantum computational boundary in the spectrum of amount of noise present in a GBS setup. In doing this, we built our discussions on the measure of non-classicality using the presence of negative values in the Quasi-probability distribution (QPD) representation of states of a quantum system. We then discuss existing results that shows the possibility of efficient classical simulation of a quantum computational model given a sufficient amount of noise. These results impose a necessary criteria of the minimum amount of noise that an experiment has to surpass in order for it to be impossible to simulate classically in an efficient manner. We extend this idea for GBS, where we propose a modified GBS where we consider noise in its measurements that opens a way for a possible formulation of a noise condition that a GBS experiment has to fulfill to rule out possible efficient classical simulation. To further probe the classical-quantum computational boundary, we also propose a different quantum computational model called the Photon-added Thermal State CV-Sampling (PATS CVS), which can be formulated similarly to our noisy GBS model. We then discuss existing results that analyzes the negativity of the QPD representation of a noisy Photon-added Thermal state (PATS), which constitutes the inputs for PATS CVS. Using the result that shows the quantum-classical transition in terms of the QPD negativity for a PATS state when the amount of noise changes, we discuss the potential of PATS CVS model as a possible avenue to further probe the transition between efficiently simulable and non-efficiently simulable quantum mechanical experiment.
Boson sampling is a specific quantum computation, which is likely hard to implement efficiently on a classical computer. The task is to sample the output photon-number distribution of a linear-optical interferometric network, which is fed with single-photon Fock-state inputs. A question that has been asked is if the sampling problems associated with any other input quantum states of light (other than the Fock states) to a linear-optical network and suitable output detection strategies are also of similar computational complexity as boson sampling. We consider the states that differ from the Fock states by a displacement operation, namely the displaced Fock states and the photon-added coherent states. It is easy to show that the sampling problem associated with displaced single-photon Fock states and a displaced photon-number detection scheme is in the same complexity class as boson sampling for all values of displacement. On the other hand, we show that the sampling problem associated with single-photon-added coherent states and the same displaced photon-number detection scheme demonstrates a computational-complexity transition. It transitions from being just as hard as boson sampling when the input coherent amplitudes are sufficiently small to a classically simulatable problem in the limit of large coherent amplitudes.