[Paper Review] Marginal probabilities in boson samplers with arbitrary input states

Jelmer J. Renema|arXiv (Cornell University)|Dec 29, 2020

Quantum Information and Cryptography12 references4 citations

TL;DR

This paper presents a general method to compute and sample from marginal detection probabilities in boson samplers with arbitrary input quantum states and arbitrary photon distinguishabilities. Building on Clifford and Clifford's framework, it enables efficient generation of samples with correct k-th order marginals for any k, resolving a key challenge in verifying large-scale quantum interference in photonic quantum advantage experiments.

ABSTRACT

With the recent claim of a quantum advantage demonstration in photonics by Zhong et al, the question of the computation of lower-order approximations of boson sampling with arbitrary quantum states at arbitrary distinguishability has come to the fore. In this work, we present results in this direction, building on the results of Clifford and Clifford. In particular, we show: 1) How to compute marginal detection probabilities (i.e. probabilities of the detection of some but not all photons) for arbitrary quantum states. 2) Using the first result, how to generalize the sampling algorithm of Clifford and Clifford to arbitrary photon distinguishabilities and arbitrary input quantum states. 3) How to incorporate truncations of the quantum interference into a sampling algorithm. 4) A remark considering maximum likelihood verification of the recent photonic quantum advantage experiment.

Motivation & Objective

• To address the open problem of generating samples with correct higher-order marginal distributions in boson sampling under arbitrary input states and photon distinguishabilities.
• To extend the Clifford-Clifford sampling algorithm beyond Fock-state inputs to general quantum states and arbitrary distinguishability.
• To enable diagnostic verification of large-scale quantum interference using marginal probabilities, particularly for recent photonic quantum advantage experiments.
• To clarify the limitations of lower-order marginals in adversarial settings and establish a hierarchy for spoofing and verification.

Proposed method

• The method projects the arbitrary input quantum state onto the n-photon subspace, using an orthonormal basis of Fock states with exactly n photons.
• It computes marginal detection probabilities via a trace formula involving the unitary transformation matrix U, the input state's amplitude coefficients, and permutations of photon paths.
• The approach generalizes the permanent-based formalism of boson sampling to include interference terms from superpositions and entangled states, including phase-dependent contributions from differing photon emission histories.
• It incorporates truncations of quantum interference by restricting the number of contributing terms in the marginal probability sum, enabling efficient sampling.
• The algorithm uses the structure of the input state (e.g., Gaussian, symmetric) to avoid full enumeration of all basis states, improving computational efficiency.
• It enables maximum likelihood estimation of distinguishability parameters by leveraging the information contained in k-th order marginals, even in the presence of loss or imperfections.

Experimental results

Research questions

• RQ1 How can marginal detection probabilities be computed for arbitrary quantum input states and arbitrary photon distinguishabilities in boson sampling?
• RQ2 Can the Clifford-Clifford sampling algorithm be generalized to work with arbitrary input states and non-identical photon overlaps?
• RQ3 To what extent do k-th order marginals contain information about large-scale quantum interference, and can they be used to verify quantum advantage in an adversarial setting?
• RQ4 How do losses and truncations affect the reliability of marginal distributions as diagnostics for quantum interference?
• RQ5 Can quantum states be constructed that are resistant to classical simulation attempts based on marginal distributions?

Key findings

• The paper derives a general formula for computing k-th order marginal detection probabilities in boson sampling with arbitrary input states and arbitrary photon distinguishabilities, extending prior results limited to Fock states.
• It demonstrates that second-order marginals contain phase-dependent interference terms not present in standard boson sampling, arising from different photon emission histories.
• The method enables efficient sampling with correct k-th order marginals for any k, resolving a key bottleneck in verifying higher-order quantum interference in photonic quantum advantage experiments.
• The results show that higher-order marginals carry information about k-photon interference processes, forming a hierarchy where k-order marginals can verify up to k-photon interference but not higher.
• The framework reveals that certain entangled input states, such as superpositions of disjoint Fock states, can lead to exponentially enhanced n-photon interference, making them resistant to classical approximation methods.
• It clarifies that marginal distributions cannot reliably measure photon loss effects when losses are uniform, as they commute with the interferometer and do not alter the marginal structure.