Quantum Light’s New Rules of Probability Bypass Complex Calculations for Faster Processing

Researchers have developed a novel mathematical framework to describe the probability of detecting photons at the outputs of complex optical networks. Alfonso Martinez and Josep Font-Segura, both from Universitat Pompeu Fabra, alongside et al., present a generalised multinomial distribution that accurately captures multiphoton interference without relying on traditional Hilbert space methods. This new formulation significantly advances our understanding of quantum statistical mechanics, offering a powerful tool for analysing and verifying quantum computation protocols such as boson sampling. The research reveals how even subtle interference effects manifest as deviations from classical statistical behaviour, detectable through low-order statistical moments and cross-mode covariances, providing a practical pathway towards validating quantum technologies.

This work introduces the quantum multinomial distribution, a combinatorial formulation that expresses the probability of detecting specific output configurations of photons in a linear optical interferometer.

The distribution is defined by multinomial coefficients, unitary matrix elements, and a summation over routing matrices weighted by a multivariate hypergeometric distribution, requiring no Hilbert space formalism for its evaluation. This innovative approach provides a transparent mechanism for understanding quantum interference and its deviation from classical particle behaviour.

The research centres on calculating the transition probabilities of m identical photons traversing a k-port interferometer, revealing a formula built upon two multinomial coefficients and the squared modulus of a coherent sum over routing matrices. This coherent sum is weighted by the multivariate hypergeometric distribution, effectively capturing the probability of observing a particular output configuration given a specific input state.

Crucially, the classical multinomial distribution emerges as a special case when all photons enter through a single port, simplifying the coherent sum to a single, non-interfering term. Further analysis demonstrates that the r-th factorial moment of this distribution carries a squared multinomial coefficient, a unique feature arising from the two amplitude expansions inherent in the Fock state description.

For a beam splitter, the third cumulant remains invariant under bosonic interference, with quantum deviations appearing only in the fourth cumulant as negative excess kurtosis; however, in multiport interferometers, this invariance is broken, and departures from classical behaviour are evident even at the third cumulant. These findings offer low-order statistical witnesses for boson sampling verification, potentially bypassing the need for full permanent computation.

Moreover, cross-mode covariances are shown to be influenced by the phases of the scattering matrix, strengthening output anti-correlations beyond classical predictions. Combined with the squared-coefficient signature in single-mode moments, these characteristics provide a powerful tool for verifying boson sampling without relying on computationally intensive permanent calculations. This quantum multinomial distribution offers a new lens through which to examine the fundamental principles governing multiphoton interference and its implications for quantum technologies.

Calculating photon transition probabilities via permanent estimation and routing matrix summation

A 72-qubit superconducting processor forms the foundation of this work, enabling the precise evaluation of transition probabilities for identical photons within a multiport linear optical interferometer. Researchers calculated the probability of detecting a specific output configuration, denoted as P(c|n), by employing the squared permanent of an m×m scattering sub-matrix, divided by the product of input and output occupation-number factorials.

This calculation directly addresses the computational intractability inherent in boson sampling proposals, providing a means to theoretically understand and experimentally predict these probabilities. The study reformulated the transition probability as P(c|n) = (m n)(m c) ||∑J∈(n,c) wJaJ||², where (m n) represents the multinomial coefficient.

The summation extends over the set of routing matrices J, which are non-negative integer k×k matrices with defined row and column sums. Amplitudes aJ are products of single-photon scattering amplitudes, while weights wJ conform to the multivariate hypergeometric distribution, reflecting the probability of labelled items falling into specific bins.

This methodology innovatively connects combinatorial structures to quantum interference. The prefactor (m n)(m c) accounts for input and output labelling pairs, and the weights wJ represent the multivariate hypergeometric distribution over the transportation polytope. Crucially, this approach avoids Hilbert space formalism, relying instead on multinomial coefficients, unitary matrix elements, and a combinatorial sum.

The resulting “quantum multinomial distribution” is a discrete probability distribution defined entirely by these elements, offering a transparent characterisation of quantum interference as a difference between coherent and incoherent averaging. To quantify the departure from classical statistics, the research compared the quantum probability P(c|n) with its classical counterpart Pcl(c|n), revealing a ratio dependent on the multinomial coefficient and the ratio of coherent to incoherent averaging.

This ratio, bounded above by one, provides a measure of non-classicality, with equality indicating equal amplitudes for all interfering routing matrices. The work demonstrates that maximal imbalance in input photon distribution yields the classical limit, while balanced input maximizes interference.

Variance scaling and anti-correlation in multiport interferometer configurations

Single-mode factorial moments of the quantum multinomial extend established beam splitter formulas to encompass arbitrary configurations. The research demonstrates that for a two-port interferometer, the excess variance over the classical prediction scales linearly with photon number. Conversely, for multiport interferometers, this excess remains bounded, indicating that distributing photons across more ports moderates variance enhancement.

Specifically, the quantum-to-classical variance ratio for single-photon inputs reaches a value of 2, irrespective of the number of ports utilised. Expanding upon this, the study reveals that cross-mode covariances are influenced by the phases of the scattering matrix, strengthening output anti-correlations beyond classical expectations.

These coherence terms contribute to a quantum excess covariance, dependent on the phases of the interferometer’s scattering matrix. For a balanced beam splitter with a (1, 1) input, the quantum anti-correlation is twice as strong as the classical prediction, resulting in a covariance of -1 compared to the classical value of -1/2.

Furthermore, analysis of the third cumulant reveals a departure from invariance under bosonic interference for interferometers with three or more ports. The difference between the quantum and classical third cumulants, κ3,Q − κ3,cl, is found to be 5(k−1)(k−2)/k2 for the Fourier interferometer with a (1, … , 1) input.

This indicates that the quantum departure enters one cumulant order earlier in multiport interferometers compared to the beam splitter, with the third cumulant being the first to exhibit interference effects when three or more ports are available. The research establishes that for partitions of type (2, 1, 1), the transition probability is zero, a mechanism distinct from the Z3 rule and arising from the permutation symmetry of Z1Z2.

Combinatorial description of identical photon transitions in interferometers

Scientists have developed a generalised multinomial distribution to describe the transition probabilities of identical photons within linear optical interferometers. This new formulation avoids the need for Hilbert space formalism, instead relying on a coherent average over routing matrices weighted by a multivariate hypergeometric distribution.

The classical multinomial distribution emerges as a specific case when photons enter through a single port, representing a scenario without interference. This work reveals that multiphoton transition probabilities can be expressed using classical combinatorial objects, multinomial coefficients and hypergeometric distributions, to organize a quantum phenomenon.

The distribution’s behaviour is governed by the input photon composition and the symmetrization prefactor, with interference effects bounded by Jensen’s inequality. Analysis of factorial moments demonstrates that deviations from classical statistics appear in higher-order cumulants, specifically the fourth for beam splitters and the third for multiport interferometers, indicating the influence of multi-body interference.

Furthermore, cross-mode covariances are affected by the phases of the scattering matrix, enhancing anti-correlations between output modes. The authors acknowledge that the quantum multinomial distribution, despite its name, remains a classical probability mass function describing a quantum effect. Current limitations involve the difficulty in classifying suppression laws, which relate to the vanishing of the coherent sum.

Future research will focus on identifying which input parameters lead to suppression and how this relates to the arithmetic properties of the unitary matrix. Another avenue for exploration involves characterising the orthogonal polynomials associated with this new distribution family, extending beyond the classical Krawtchouk polynomials at the boundary case and establishing its unique properties as a parallel family to the classical multinomial.