Protocol Achieves Efficient Estimation of Multivariate Traces with Two Photons

Scientists have developed a novel method for efficiently estimating complex quantum states using photons, addressing a key challenge in quantum information processing. Leonardo Novo, Marco Robbio, and Ernesto F. Galvão, from the International Iberian Nanotechnology Laboratory and the Universidade Federal Fluminense, alongside Nicolas J. Cerf et al from the Ecole polytechnique de Bruxelles, present a linear-optical protocol that estimates multivariate traces of ‘Bargmann invariants’ , states crucial to various areas of mechanics. This photon-native approach, akin to a quantum version of the circuit model’s cycle test, significantly improves upon existing techniques by offering sample efficiency and opening doors to advancements in areas like kernel estimation, eigenspectrum analysis, and the precise characterisation of multiphoton indistinguishability.

Cerf et al from the Ecole polytechnique de Bruxelles, present a linear-optical protocol that estimates multivariate traces of ‘Bargmann invariants’, states crucial to various areas of mechanics. This photon-native approach, akin to a quantum version of the circuit model’s cycle test, significantly improves upon existing techniques by offering sample efficiency and opening doors to advancements in areas like kernel estimation, eigenspectrum analysis, and the precise characterisation of multiphoton indistinguishability.

Photon-native estimation of Bargmann invariants via interferometry offers

This breakthrough addresses a significant gap in the field by providing a photon-native method applicable to many-photon multimode quantum states, unlike existing techniques limited to single photons or exponentially large Hilbert spaces. Galvão, and Nicolas J. Cerf, achieved this by drawing inspiration from the well-known Hong-Ou-Mandel test and the quantum SWAP test, effectively creating a photon-native analogue of the circuit model’s cycle test. Their protocol leverages Fourier interferometry and photo-counting measurements to determine these traces without requiring postselection or auxiliary photons, representing a substantial advancement in quantum computation.
Experiments demonstrate that the protocol is sample-efficient and generalises suppression laws previously known for input Fock states to encompass arbitrary many-photon multimode states of light. This work opens avenues for impactful applications, including efficient Quantum Kernel reconstruction for Quantum Machine Learning, precise eigenspectrum estimation, and improved characterisation of multiphoton indistinguishability, critical for advancing quantum technologies. The team’s innovation provides a deterministic, photon-native quantum algorithm, circumventing the need for costly two-qubit gates or probabilistic schemes often required in photonic quantum computing. This level of detail underscores the practicality and potential for implementation of the proposed method. The study’s findings not only bridge a computational gap between the circuit model and linear optics for overlap estimation but also establish a foundation for exploring new possibilities in photonic quantum information processing and quantum machine learning applications.,.

Bargmann Invariant Estimation via Fourier Interferometry is a

This work pioneers a photon-native approach analogous to the cycle test, enabling the estimation of Tr[ρ1ρ2. ρM] for M quantum states, a quantity known as Bargmann invariants. The research addresses a limitation in existing linear optics methods, which previously struggled with multi-photon, multi-mode states, and delivers a sample-efficient technique applicable to arbitrary states of light. The study employed Fourier interferometry and photo-counting measurements to estimate these multivariate traces without requiring postselection or auxiliary photons. Researchers engineered a system where M bosonic quantum states {ρ1, ., ρM}, each belonging to identical Fock spaces, were subjected to a linear interference process.

This process utilized a M × M unitary matrix U, acting on creation operators, defined by Ua†j,αU† = Σk=1M Uk,ja†k,α for all j and α, effectively mixing the states while preserving their internal degrees of freedom. The team generalized previously known suppression laws for Fourier interferometers, originally defined for Fock states, to encompass arbitrary many-photon multimode states, demonstrating that violations of these laws are key to accurately estimating the multivariate traces. This innovative methodology enables several impactful applications, including efficient Quantum Kernel reconstruction for Quantum Machine Learning, estimation of state spectra, and precise characterization of photonic indistinguishability. By harnessing the power of linear optics in a new way, the research provides a powerful tool for exploring and manipulating complex quantum states, offering a significant advancement in quantum information science and technology.,.

Multimode State Characterisation via Cycle Testing is crucial

Experiments revealed a generalized suppression law for Fourier interferometers involving diverse multimode many-photon states, going beyond previous studies limited to Fock-state inputs. The team measured probabilities, denoted as Pj, defined as the probability of obtaining a specific outcome S such that f(S) = j, where j ranges from 0 to M-1. These probabilities are intricately linked to expectation values, Xk, via a discrete Fourier transform: Pj = 1/M Σk=0 to M-1 ω−jkXk, with ω representing the complex exponential exp(2πi/M). Results demonstrate that by estimating these probabilities from experimental samples, the values of Xk can be determined through an inverse discrete Fourier transform.

Notably, X1 corresponds to the multivariate trace of the M input states, providing a direct measurement of state overlap. Measurements confirm that X2 equals Tr [ρ1ρ3] Tr [ρ2ρ4] for a four-state system, while X3 is the complex conjugate of X1, revealing relational information about the input states. The corollary proves that if the input state Ω is invariant under cyclic permutations, then P0 equals 1, meaning any outcome S where f(S) = 0 is forbidden, a clear indication of state indistinguishability. This work assumes only that Ω is an M-party bosonic state within a tensor product of M identical Fock spaces, allowing for the study of both entangled and separable states.

A generalized Hong-Ou-Mandel test, depicted with M = 2, was implemented to estimate overlaps between states ρ1 and ρ2, potentially involving superpositions of photon numbers in various internal degrees of freedom. In this scenario, P0 represents the probability of measuring even photon numbers in mode 1, with X0 always equalling 1 and X1 corresponding to Tr [ρ1ρ2]. This breakthrough delivers a pathway for quantum kernel methods, potentially accelerating machine learning tasks by efficiently learning relationships in exponentially large Hilbert spaces.

Bargmann Invariants via Linear Optical Fourier Interferometry

This protocol, akin to a photon-native SWAP test, leverages Fourier interferometry, photon counting, and classical data processing to directly measure these invariants from general bosonic states. The findings establish a strong equivalence between the cycle test, a computational primitive in quantum computing, and Fourier interferometry using linear optics, potentially circumventing the need for complex photonic entangling gates. This advancement generalizes the established connection between the SWAP and Hong-Ou-Mandel tests, opening avenues for exploration in near-term photonic quantum computing applications such as generalized suppression laws, kernel estimation, eigenspectrum estimation, and the characterization of multiphoton indistinguishability. The authors acknowledge limitations related to the complexity of implementing high-order invariant measurements and suggest future work could focus on refining the protocol for specific quantum algorithms and exploring its scalability.