Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1310.0297 · Physical Review A

Evidence for the conjecture that sampling generalized cat states with linear optics is hard

Peter P. Rohde, Keith R. Motes, Paul Knott, Joseph F. Fitzsimons, William J. Munro, Jonathan P. Dowling

Published Jan 30, 2015• 28 citations
arXivHTMLPDF

Abstract

Boson sampling has been presented as a simplified model for linear optical quantum computing. In the boson-sampling model, Fock states are passed through a linear optics network and sampled via number-resolved photodetection. It has been shown that this sampling problem likely cannot be efficiently classically simulated. This raises the question as to whether there are other quantum states of light for which the equivalent sampling problem is also computationally hard. We present evidence, without using a full complexity proof, that a very broad class of quantum states of light---arbitrary superpositions of two or more coherent states---when evolved via passive linear optics and sampled with number-resolved photodetection, likely implements a classically hard sampling problem.

DOI
10.1103/physreva.91.012342

Related papers

Boson sampling with displaced single-photon Fock states versus single-photon-added coherent states: The quantum-classical divide and computational-complexity transitions in linear optics

Kaushik Parasuram Seshadreesan, Jonathan P. Olson, Keith R. Motes +2 more

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.

arXiv:1402.0531•Physical Review A•Feb 27, 2015•40 citations

Sampling arbitrary photon-added or photon-subtracted squeezed states is in the same complexity class as boson sampling

Jonathan P. Olson, Kaushik Parasuram Seshadreesan, Keith R. Motes +2 more

Boson sampling is a simple model for nonuniversal linear optics quantum computing using far fewer physical resources than universal schemes. An input state comprising vacuum and single-photon states is fed through a Haar-random linear optics network and sampled at the output by using coincidence photodetection. This problem is strongly believed to be classically hard to simulate. We show that an analogous procedure implements the same problem, using photon-added or -subtracted squeezed vacuum states (with arbitrary squeezing), where sampling at the output is performed via parity measurements. The equivalence is exact and independent of the squeezing parameter, and hence provides an entire class of quantum states of light in the same complexity class as boson sampling.

arXiv:1406.7821•Physical Review A•Feb 17, 2015•48 citations

Non-linear Boson Sampling

Nicolò Spagnolo, Daniel J. Brod, Ernesto F. Galvão +1 more

Abstract Boson Sampling is a task that is conjectured to be computationally hard for a classical computer, but which can be efficiently solved by linear-optical interferometers with Fock state inputs. Significant advances have been reported in the last few years, with demonstrations of small- and medium-scale devices, as well as implementations of variants such as Gaussian Boson Sampling. Besides the relevance of this class of computational models in the quest for unambiguous experimental demonstrations of quantum advantage, recent results have also proposed the first applications for hybrid quantum computing. Here, we introduce the adoption of non-linear photon–photon interactions in the Boson Sampling framework, and analyze the enhancement in complexity via an explicit linear-optical simulation scheme. By extending the computational expressivity of Boson Sampling, the introduction of non-linearities promises to disclose novel functionalities for this class of quantum devices. Hence, our results are expected to lead to new applications of near-term, restricted photonic quantum computers.

arXiv:2110.13788•npj Quantum Information•Jan 9, 2023•24 citations

Boson-Sampling in the light of sample complexity

Christian Gogolin, Martin Kliesch, Leandro Aolita +1 more

Boson-Sampling is a classically computationally hard problem that can - in principle - be efficiently solved with quantum linear optical networks. Very recently, a rush of experimental activity has ignited with the aim of developing such devices as feasible instances of quantum simulators. Even approximate Boson-Sampling is believed to be hard with high probability if the unitary describing the optical network is drawn from the Haar measure. In this work we show that in this setup, with probability exponentially close to one in the number of bosons, no symmetric algorithm can distinguish the Boson-Sampling distribution from the uniform one from fewer than exponentially many samples. This means that the two distributions are operationally indistinguishable without detailed a priori knowledge. We carefully discuss the prospects of efficiently using knowledge about the implemented unitary for devising non-symmetric algorithms that could potentially improve upon this. We conclude that due to the very fact that Boson-Sampling is believed to be hard, efficient classical certification of Boson-Sampling devices seems to be out of reach.

arXiv:1306.3995•Jun 17, 2013•40 citations

The Computational Complexity of Linear Optics

Scott Aaronson, Arkhipov, Alex

We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.

arXiv:1011.3245•Electronic colloquium on computational complexity•Jan 25, 2010•20 citations

Classical Algorithms for Function Computation in Gaussian Boson Sampling

Ruoting Dou, Hao Zhan, Shengjun Wu +2 more

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.

quant-pharXiv:2609.38882•Oct 1, 2026•0 citations
Boson sampling has been presented as a simplified model for linear optical quantum computing. In the boson-sampling model, Fock states are passed through a linear optics network and sampled via number-resolved photodetection. It has been shown that this sampling problem likely cannot be efficiently classically simulated. This raises the question as to whether there are other quantum states of light for which the equivalent sampling problem is also computationally hard. We present evidence, without using a full complexity proof, that a very broad class of quantum states of light---arbitrary superpositions of two or more coherent states---when evolved via passive linear optics and sampled with number-resolved photodetection, likely implements a classically hard sampling problem.
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.
Boson sampling is a simple model for nonuniversal linear optics quantum computing using far fewer physical resources than universal schemes. An input state comprising vacuum and single-photon states is fed through a Haar-random linear optics network and sampled at the output by using coincidence photodetection. This problem is strongly believed to be classically hard to simulate. We show that an analogous procedure implements the same problem, using photon-added or -subtracted squeezed vacuum states (with arbitrary squeezing), where sampling at the output is performed via parity measurements. The equivalence is exact and independent of the squeezing parameter, and hence provides an entire class of quantum states of light in the same complexity class as boson sampling.
Abstract Boson Sampling is a task that is conjectured to be computationally hard for a classical computer, but which can be efficiently solved by linear-optical interferometers with Fock state inputs. Significant advances have been reported in the last few years, with demonstrations of small- and medium-scale devices, as well as implementations of variants such as Gaussian Boson Sampling. Besides the relevance of this class of computational models in the quest for unambiguous experimental demonstrations of quantum advantage, recent results have also proposed the first applications for hybrid quantum computing. Here, we introduce the adoption of non-linear photon–photon interactions in the Boson Sampling framework, and analyze the enhancement in complexity via an explicit linear-optical simulation scheme. By extending the computational expressivity of Boson Sampling, the introduction of non-linearities promises to disclose novel functionalities for this class of quantum devices. Hence, our results are expected to lead to new applications of near-term, restricted photonic quantum computers.
Boson-Sampling is a classically computationally hard problem that can - in principle - be efficiently solved with quantum linear optical networks. Very recently, a rush of experimental activity has ignited with the aim of developing such devices as feasible instances of quantum simulators. Even approximate Boson-Sampling is believed to be hard with high probability if the unitary describing the optical network is drawn from the Haar measure. In this work we show that in this setup, with probability exponentially close to one in the number of bosons, no symmetric algorithm can distinguish the Boson-Sampling distribution from the uniform one from fewer than exponentially many samples. This means that the two distributions are operationally indistinguishable without detailed a priori knowledge. We carefully discuss the prospects of efficiently using knowledge about the implemented unitary for devising non-symmetric algorithms that could potentially improve upon this. We conclude that due to the very fact that Boson-Sampling is believed to be hard, efficient classical certification of Boson-Sampling devices seems to be out of reach.
We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.
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.