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arXiv:2305.11215

Simulating Gaussian Boson Sampling with Tensor Networks in the Heisenberg picture

Dario Cilluffo, Nicola Lorenzoni, Martin Bodo Plenio

Published May 18, 2023• 2 citations
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Abstract

Although the Schr{ö}dinger and Heisenberg pictures are equivalent formulations of quantum mechanics, simulations performed choosing one over the other can greatly impact the computational resources required to solve a problem. Here we demonstrate that in Gaussian boson sampling, a central problem in quantum computing, a good choice of representation can shift the boundary between feasible and infeasible numerical simulability. To achieve this, we introduce a novel method for computing the probability distribution of boson sampling based on the time evolution of tensor networks in the Heisenberg picture. In addition, we overcome limitations of existing methods enabling simulations of realistic setups affected by non-uniform photon losses. Our results demonstrate the effectiveness of the method and its potential to advance quantum computing research.

DOI
10.48550/arxiv.2305.11215

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Although the Schr{ö}dinger and Heisenberg pictures are equivalent formulations of quantum mechanics, simulations performed choosing one over the other can greatly impact the computational resources required to solve a problem. Here we demonstrate that in Gaussian boson sampling, a central problem in quantum computing, a good choice of representation can shift the boundary between feasible and infeasible numerical simulability. To achieve this, we introduce a novel method for computing the probability distribution of boson sampling based on the time evolution of tensor networks in the Heisenberg picture. In addition, we overcome limitations of existing methods enabling simulations of realistic setups affected by non-uniform photon losses. Our results demonstrate the effectiveness of the method and its potential to advance quantum computing research.
Tensor network formalisms have emerged as powerful tools for simulating quantum state evolution. While widely applied in the study of optical quantum circuits, such as boson sampling, existing tensor network approaches fail to address the complexity mismatch between tensor contractions and the calculation of photon-counting probability amplitudes. Here, we present an alternative tensor network framework that exploits the input-output relations of quantum optical circuits encoded in the unitary interferometer matrix. Our approach bridges the complexity gap by enabling the computation of the permanent—central to boson sampling—with the same computational complexity as the best known classical algorithm based on a graphical representation of the operator-basis matrix product states that we introduce. Furthermore, we exploit the flexibility of tensor networks to extend our formalism to incorporate partial distinguishability and photon loss, two key imperfections in practical interferometry experiments. This work offers a significant step forward in the simulation of large-scale quantum optical systems and the understanding of their computational complexity.
Gaussian boson sampling, a computational model that is widely believed to admit quantum supremacy, has already been experimentally demonstrated and is claimed to surpass the classical simulation capabilities of even the most powerful supercomputers today. However, whether the current approach limited by photon loss and noise in such experiments prescribes a scalable path to quantum advantage is an open question. To understand the effect of photon loss on the scalability of Gaussian boson sampling, we analytically derive the asymptotic operator entanglement entropy scaling, which relates to the simulation complexity. As a result, we observe that efficient tensor network simulations are likely possible under the ${N}_{\text{out}}\ensuremath{\propto}\sqrt{N}$ scaling of the number of surviving photons Nout{N}_{\text{out}}Nout​ in the number of input photons NNN. We numerically verify this result using a tensor network algorithm with U(1)\mathrm{U}(1)U(1) symmetry, and we overcome previous challenges due to the large local Hilbert-space dimensions in Gaussian boson sampling with hardware acceleration. Additionally, we observe that increasing the photon number through larger squeezing does not increase the entanglement entropy significantly. Finally, we numerically find the bond dimension necessary for fixed accuracy simulations, providing more direct evidence for the complexity of tensor networks.
The continuous variable quantum computing platform constitutes a promising candidate for realizing quantum advantage, as exemplified in Gaussian Boson Sampling. While noise in the experiments makes the computation attainable for classical simulations, it has been suggested that the addition of non-linear elements to the experiment will help retain the quantum advantage. We propose a classical simulation tool for general continuous variable sampling problems, including Gaussian Boson Sampling and beyond. We reformulate the sampling problem as that of finding the ground state of a simple few-body Hamiltonian. This allows us to employ powerful variational methods based on tensor networks and to read off the simulation error directly from the expectation value of the Hamiltonian. We validate our method by simulating Gaussian Boson Sampling, where we achieve results comparable to the state of the art. We also consider a non-Gaussian sampling problem, for which we develop novel local basis optimization techniques based on a non-linear parameterization of the implicit basis, resulting in high effective cutoffs with diminished computational overhead.
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.
If classical algorithms have been successful in reproducing the estimation of expectation values of observables of some quantum circuits using off-the-shelf computing resources, matching the performance of the most advanced quantum devices on sampling problems usually requires extreme cost in terms of memory and computing operations, making them accessible to only a handful of supercomputers around the world. In this work, we demonstrate for the first time a classical simulation outperforming Gaussian boson sampling experiments of one hundred modes on established benchmark tests using a single CPU or GPU. Being embarrassingly parallelizable, a small number of CPUs or GPUs allows us to match previous sampling rates that required more than one hundred GPUs. We believe algorithmic and implementation improvements will generalize our tools to photo-counting, single-photon inputs, and pseudo-photon-number-resolving scenarios beyond one thousand modes. Finally, most of the innovations in our tools remain valid for generic probability distributions over binary variables, rendering it potentially applicable to the simulation of qubit-based sampling problems and creating classical surrogates for classical-quantum algorithms.
Abstract Boson sampling is the problem of sampling from the same distribution as indistinguishable single photons at the output of a linear optical interferometer. It is an example of a non-universal quantum computation which is believed to be feasible in the near term and cannot be simulated on a classical machine. Like all purported demonstrations of ‘quantum computational supremacy’, this motivates optimising classical simulation schemes for a realistic model of the problem, in this case boson sampling when the implementations experience lost or distinguishable photons. Although current simulation schemes for sufficiently imperfect boson sampling are classically efficient, in principle the polynomial runtime can be infeasibly large. In this work, we develop a scheme for classical simulation of boson sampling under uniform distinguishability and loss, based on the idea of sampling from distributions where at most k photons are indistinguishable. We show that asymptotically this scheme can provide a polynomial improvement in the runtime compared to classically simulating idealised boson sampling. More significantly, we show that in the regime considered experimentally relevant, our approach gives an substantial improvement in runtime over other classical simulation approaches.