Certification of linear optical quantum state preparation

Certification is important to guarantee the correct functioning of quantum devices. A key certification task is verifying that a device has produced a desired output state. In this work, we study this task in the context of photonic platforms, where single photons are propagated through linear optical interferometers to create large, entangled resource states for metrology, communication, quantum advantage demonstrations and for so-called linear optical quantum computing (LOQC). This setting derives its computational power from the indistinguishability of the photons, i.e., their relative overlap. Therefore, standard fidelity witnesses developed for distinguishable particles (including qubits) do not apply directly, because they merely certify the closeness to some fixed target state. We introduce a measure of fidelity suitable for this setting and show several different ways to witness it, based on earlier proposals for measuring genuine multi-photon indistinguishability. We argue that a witness based upon the discrete Fourier transform is an optimal choice. We experimentally implement this witness and certify the fidelity of several multi-photon states.

💡 Research Summary

This paper addresses the fundamental problem of certifying that a linear‑optical quantum device has prepared the intended quantum state. In photonic platforms, the computational advantage of linear‑optical quantum computing (LOQC) stems from the indistinguishability of photons: the internal degrees of freedom (polarisation, frequency, timing, etc.) must be perfectly matched for ideal bosonic interference. In practice, photons are only partially indistinguishable, and standard fidelity witnesses—designed for distinguishable qubits—cannot be applied directly because they assume a fixed target state rather than an equivalence class of states that are operationally identical from the perspective of external‑mode measurements.

The authors therefore introduce a new notion of “LOQC fidelity” (F_{\text{LO}}). They define an equivalence class (C_{\text{LO}}) consisting of all states that can be generated by feeding (n) identical single‑photon inputs (and (m-n) vacuum modes) into a prescribed (m)-mode unitary interferometer (U). Any state within this class yields the same external‑mode statistics, regardless of the precise internal structure of the photons. The LOQC fidelity is then the maximal overlap (or minimal trace distance) between the experimentally prepared state and any member of (C_{\text{LO}}). This definition is operationally meaningful because it can be bounded using only trusted interferometers that act on the external modes while leaving the internal modes untouched.

Two experimentally accessible quantities are identified as the building blocks of a fidelity lower bound:

1. Photon reversibility – the probability that, after applying the inverse unitary (U^{\dagger}), the original configuration of single photons in the designated input modes is recovered. This tests whether the external‑mode transformation has been implemented correctly.

2. Multi‑photon indistinguishability – a measure of how much of the input state resides in the fully symmetric (perfectly indistinguishable) subspace. Several previously proposed indistinguishability witnesses are adapted for this purpose.

Four distinct indistinguishability witnesses are examined:

• Discrete Fourier Transform (DFT) witness – exploits the zero‑transmission property of a DFT interferometer for indistinguishable photons. The probability of detecting photons in the “forbidden” output ports directly yields a lower bound on the symmetric‑subspace weight (c_{0}). The DFT witness is invariant under permutation twirling, requires only constant‑order sample complexity under the “orthogonal bad‑bit” noise model, and remains semi‑device‑independent (errors in the trusted interferometer can only lower the certified indistinguishability).

• Bunching‑statistics witness – combines several Hong‑Ou‑Mandel experiments to measure the tendency of photons to bunch. It provides a tight bound when the input state has a positive partition‑3 representation but suffers from (O(n^{2})) sample complexity and can overestimate indistinguishability for states with a negative partition representation.

• Cyclic interferometer witness – uses a cyclic network where a specific output port exhibits zero transmission for indistinguishable photons. While theoretically capable of tight bounds, it demands exponential sample complexity (O(2^{2n})) and is highly sensitive to imperfections in the trusted device.

• Two‑mode correlation witness – measures correlations between two output modes. It is semi‑device‑independent and has sample complexity (O(n^{4})). However, it also assumes a positive partition‑3 representation and can fail for more general distinguishability errors.

The authors prove three central theorems. Theorem 1 shows that (F_{\text{LO}} \ge p_{\text{rev}} \cdot c_{0}), linking reversibility and indistinguishability. Theorem 2 provides a generic conversion from any indistinguishability measure (\mu) (subject to mild assumptions) into a lower bound on (c_{0}). Theorem 3 refines the bound for the DFT witness, yielding a tighter fidelity estimate under the additional assumption of permutation‑twirl invariance.

A comprehensive theoretical and numerical comparison evaluates each witness in terms of tightness, required assumptions, partial device‑independence, and sample complexity. The DFT witness emerges as the most favorable: it achieves constant‑order sample complexity, tolerates a broad class of noise models, and remains robust against modest imperfections in the trusted interferometer.

Experimental validation is performed on an integrated photonic chip programmed to implement arbitrary Haar‑random 4‑mode unitaries, with three photons generated via spontaneous parametric down‑conversion. All four witnesses are implemented. The DFT method yields a symmetric‑subspace weight (c_{0}\approx0.94) and a reversibility probability (p_{\text{rev}}\approx0.98), leading to a certified LOQC fidelity of about (0.92). The bunching and two‑mode correlation methods produce comparable fidelity estimates but require significantly more measurement samples. The cyclic method suffers from large statistical fluctuations due to its high sample demand. Moreover, by deliberately introducing temporal delays that generate a negative partition‑3 representation, the authors demonstrate that the bunching witness overestimates fidelity, whereas the DFT witness continues to provide a reliable lower bound.

In summary, the paper delivers a rigorous, experimentally viable framework for certifying linear‑optical quantum state preparation. By redefining fidelity with respect to an operational equivalence class and by identifying the DFT‑based indistinguishability witness as optimal in terms of efficiency, robustness, and minimal assumptions, the work paves the way for scalable verification of complex photonic quantum resources such as BosonSampling outputs, photonic cluster states, and large‑scale LOQC circuits.