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Gaussian Boson Sampling Complexity

Andrew Tanggara

Published Jan 1, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.25911/j9r5-e059

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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.
Quantum computers solve intractable problems which classically require an exponentially long time to compute. With the development of large-scale experiments that claim quantum advantage, a vital issue has now emerged. What are the errors, and how do they affect the complexity of the problem solved? Large-scale Gaussian boson sampling (GBS) experiments give an example in which random numbers are generated. Despite classical hardness, these have computable benchmarks for checking data validity. While there are other quantum computing architectures, Gaussian boson sampling is uniquely testable at all scales. Several large, pioneering quantum computing (QC) experiments have been carried out to investigate quantum advantage. Here, we introduce a highly scalable but classical algorithm that can solve GBS approximately. Our numerical simulation of the output count data is closer to the exact solution than current experiments up to 1152 modes. This algorithm outperforms all previous classical, approximate algorithms and scales efficiently to larger experiments. Our results show that effects beyond losses can cause the errors that allow classical simulability. This work will lead to more precise algorithms and is a step towards understanding how QC quantum advantage is affected by the underlying physics.
Gaussian boson sampling is a model of photonic quantum computing that has attracted attention as a platform for building quantum devices capable of performing tasks that are out of reach for classical devices. There is therefore significant interest, from the perspective of computational complexity theory, in solidifying the mathematical foundation for the hardness of simulating these devices. We show that, under the standard Anti-Concentration and Permanent-of-Gaussians conjectures, there is no efficient classical algorithm to sample from ideal Gaussian boson sampling distributions (even approximately) unless the polynomial hierarchy collapses. The hardness proof holds in the regime where the number of modes scales quadratically with the number of photons, a setting in which hardness was widely believed to hold but that nevertheless had no definitive proof. Crucial to the proof is a new method for programming a Gaussian boson sampling device so that the output probabilities are proportional to the permanents of submatrices of an arbitrary matrix. This technique is a generalization of Scattershot BosonSampling that we call BipartiteGBS. We also make progress towards the goal of proving hardness in the regime where there are fewer than quadratically more modes than photons (i.e., the high-collision regime) by showing that the ability to approximate permanents of matrices with repeated rows/columns confers the ability to approximate permanents of matrices with no repetitions. The reduction suffices to prove that GBS is hard in the constant-collision regime.
Quantum computing holds the promise of enabling algorithms that are beyond the reach of classical computation, hence demonstrating quantum computational advantage. However, current quantum devices face significant limitations, primarily due to the lack of error correction. In this context, the following question arises: can quantum computational advantage be demonstrated with today's noisy hardware? Recently, Gaussian boson sampling (GBS) has emerged as a leading candidate to achieve this milestone, due to its relative experimental feasibility. The first part of this thesis focuses on two central challenges associated with GBS. The first one is the impact of noise on the computational complexity of the task. Using a phase-space approach, we establish a necessary non-classicality condition that any experimental proof of quantum advantage must satisfy, and demonstrate that there exists a threshold temperature at which quantum sampling experiments become classically simulable. We then investigate the possibility of introducing higher-order non-linearities as a mean to enhance the protocol’s robustness against noise. Then, we turn our attention to the problem of validating a GBS experiment, and propose the use binned-detector probability distributions as a suitable quantity to perform efficient statistical validation. In the second part of the thesis, we present a novel approach to protect a quantum observable from a given noise model.We start by introducing the concept of quantum observables over time, an operator that jointly describes two observables at two distinct time points, and use the latter to establish a notion of time-reversal for non-unitary quantum channels with respect to a reference observable, enabling the systematic construction of recovery maps that preserve the latter. These recovery maps, although generally non-physical, can still be used in noiseless expectation value estimation tasks and can achieve optimal performance in terms of sampling overhead, outperforming known error mitigation protocols.
A universal quantum computer of moderate scale is not available yet, however intermediate models of quantum computation would still permit demonstrations of a quantum computational advantage over classical computing and could challenge the Extended Church-Turing Thesis. One of these models based on single photons interacting via linear optics is called Boson Sampling. Although Boson Sampling was demonstrated and the threshold to claim quantum computational advantage was achieved, the question on how to scale up Boson Sampling experiments remains. To make progress with this problem, here we present a practically achievable pathway to scale Boson Sampling experiments with current technologies by combining continuous-variables quantum information and temporal encoding. We propose the use of switchable dual-homodyne and single-photon detections, the temporal loop technique and scattershot based Boson Sampling. This proposal gives details as to what the required assumptions are and a pathway for scaling Boson Sampling experiments. Furthermore, this particular combination of techniques permits a single efficient implementation of Boson Sampling and efficient certification in a single experimental setup.
A universal quantum computer of large scale is not available yet, however, intermediate models of quantum computation would still permit demonstrations of a quantum computational advantage over classical computing and could challenge the extended Church-Turing thesis. One of these models based on single photons interacting via linear optics is called boson sampling. Although boson sampling was demonstrated and the threshold to claim quantum computational advantage was achieved, the question of how to scale up boson sampling experiments remains. To make progress with this problem, here we present a practically achievable pathway to scale boson sampling experiments by combining continuous-variable quantum information and temporal encoding. We propose the combination of switchable dual-homodyne and single-photon detections, the temporal loop technique, and scattershot-based boson sampling. We detail the required assumptions for concluding computational hardness for this configuration. Furthermore, this particular combination of techniques moves towards an efficient scaling and certification of boson sampling, all in a single experimental setup.
A universal quantum computer of moderate scale is not available yet, however intermediate models of quantum computation would still permit demonstrations of a quantum computational advantage over classical computing and could challenge the Extended Church-Turing Thesis. One of these models based on single photons interacting via linear optics is called Boson Sampling. Proof-of-principle Boson Sampling has been demonstrated, but the number of photons used for these demonstrations is below the level required to claim quantum computational advantage. To make progress with this problem, here we conclude that the most practically achievable pathway to scale Boson Sampling experiments with current technologies is by combining continuous-variables quantum information and temporal encoding. We propose the use of switchable dual-homodyne and single-photon detections, the temporal loop technique and scattershot based Boson Sampling. This proposal gives details as to what the required assumptions are and a pathway for a quantum optical demonstration of quantum computational advantage. Furthermore, this particular combination of techniques permits a single efficient implementation of Boson Sampling and efficient verification in a single experimental setup.