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arXiv:2504.05079 · npj Quantum Information

Experimental verification of threshold quantum state tomography on a fully-reconfigurable photonic integrated circuit

Eugenio Caruccio, Diego Maragnano, Giovanni Rodari, Davide Picus, Giovanni Garberoglio, Daniele Binosi, Riccardo Albiero, Niki Di Giano, Francesco Ceccarelli, Giacomo Corrielli, Nicolò Spagnolo, Roberto Osellame, Maurizio Dapor, Marco Liscidini, Fabio Sciarrino

Published Nov 7, 2025• 3 citations
arXivHTMLPDF

Abstract

Abstract Reconstructing the state of a quantum system represents a pivotal task for quantum information applications. The standard approach based on quantum state tomography requires a number of measurements that scales exponentially with the number of qubits. Other methods have been proposed and tested to reduce the number of measurements, or to focus on specific properties of the output state rather than on its complete reconstruction. Here, we show experimentally the application of an approach, called threshold quantum state tomography, in an advanced hybrid photonic platform with states up to n = 4 qubits. This method does not require prior knowledge and selects only the informative projectors starting from the measurement of the density matrix diagonal. We demonstrate its effectiveness by showing that a consistent reduction in the number of measurements is obtained for relevant states, with only very limited loss of information. These results open perspective for its application in larger systems.

DOI
10.1038/s41534-025-01111-z

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Abstract Reconstructing the state of a quantum system represents a pivotal task for quantum information applications. The standard approach based on quantum state tomography requires a number of measurements that scales exponentially with the number of qubits. Other methods have been proposed and tested to reduce the number of measurements, or to focus on specific properties of the output state rather than on its complete reconstruction. Here, we show experimentally the application of an approach, called threshold quantum state tomography, in an advanced hybrid photonic platform with states up to n = 4 qubits. This method does not require prior knowledge and selects only the informative projectors starting from the measurement of the density matrix diagonal. We demonstrate its effectiveness by showing that a consistent reduction in the number of measurements is obtained for relevant states, with only very limited loss of information. These results open perspective for its application in larger systems.
Abstract Quantum information systems are on a path to vastly exceed the complexity of any classical device. The number of entangled qubits in quantum devices is rapidly increasing, and the information required to fully describe these systems scales exponentially with qubit number. This scaling is the key benefit of quantum systems, however it also presents a severe challenge. To characterize such systems typically requires an exponentially long sequence of different measurements, becoming highly resource demanding for large numbers of qubits. Here we propose and demonstrate a novel and scalable method for characterizing quantum systems based on expanding a multi-photon state to larger dimensionality. We establish that the complexity of this new measurement technique only scales linearly with the number of qubits, while providing a tomographically complete set of data without a need for reconfigurability. We experimentally demonstrate an integrated photonic chip capable of measuring two- and three-photon quantum states with statistical reconstruction fidelity of 99.71%.
Quantum state tomography aims at reconstructing the state of a quantum system. The standard approach is based on a tomographically complete set of observables and a maximum likelihood estimation of the density matrix. However, the number of measurements scales exponentially with the number of qubits or qudits of the system. By leveraging some properties of any physical density matrix, we propose a method in which the number of necessary measurements depends on the state under investigation but without the need for any a priori information. In some cases, this can lead to a significant reduction of the resources necessary for state characterization.
Summary form only given. Full characterization of quantum states is increasingly important as ever more complex quantum systems are realized. Quantum state tomography of these systems is a difficult task because individual measurements only give partial information about the underlying density matrix of the system. Thus, conventionally the measurement apparatus must be reconfigured to take multiple measurements in order to fully determine the density matrix. It is accepted that for an W-qubit system an exponentially increasing number of measurements are needed [1]. As a consequence, the complexity of typical quantum state tomography grows exponentially with increasing photon number, which makes characterizing large quantum systems highly challenging. We formulate a concept of on-chip quantum tomography where the complexity scales linearly with the number of photons. This is achieved by using a single specially designed non-reconfigurable linear optical circuit as in Fig. 1(a). Crucially, the number of waveguides in this circuit scales linearly with the number of photons in the state to be measured. Furthermore, our approach is simple and robust, and we demonstrate it experimentally with a specially designed photonic chip.
Quantum state tomography (QST) aims at reconstructing the state of a quantum system. However, in conventional QST, the number of measurements scales exponentially with the number of qubits. Here, we propose a QST protocol, in which the introduction of a threshold allows one to drastically reduce the number of measurements required for the reconstruction of the state density matrix without compromising the result accuracy. In addition, one can also use the same approach to reconstruct an approximated density matrix tailoring the number of measurements on the available resources. We experimentally demonstrate this protocol by performing the tomography of states up to 7 qubits. We show that our approach can lead to results in agreement with those obtained by QST even when the number of measurements is reduced by more than two orders of magnitude.
This work proposes to carry out a study on integrated photonic circuits. Our goal is to develop a methodology for obtaining the parameters of a photonic circuit that allows us to perform a set of quantum operations necessary for the realization of a given task or quantum processing. More specifically, the quantum tasks involve the implementation of a POVM (Positive Operator Value Measure) through integrated photonic circuits. Thus, we seek to obtain optimized configurations for these devices so that the reduction of their structural complexity simplifies their manufacturing process. In addition, it is also our goal that such objects are robust against errors and losses and that the experimental applications that involve them present a minimization of the time necessary for their realization. We studied the symmetries of the proposed POVMs to, from them, configure the circuit in an optimized way. Initially, we studied the case of quantum state tomography for qubits and qutrits. We obtained planar circuits that had a smaller number of optical elements and an optical depth than those obtained by methods already known in the literature. In addition, these circuits implement all POVM elements at once, making it possible to obtain all their associated probabilities at once, minimizing the time spent on the experiment compared to experiments on an optical table or with reconfigurable circuits, where it is necessary to make changes to the configuration of the experimental apparatus from one POVM element to another. Looking for a way to generalize our protocol to larger systems, we developed a method to obtain three-dimensional photonic circuits that can be used in the experimental realization of quantum state tomography in qudits of any finite dimension. Our protocol generates circuits whose number of beam splitters and optical depth scale with functions less than in other protocols already known, showing a drastic reduction in their complexity. In addition, these circuits were highly robust in relation to the occurrence of losses. The same protocol used to obtain three-dimensional circuits for performing quantum tomography can be used to obtain circuits to implement quantum coherence measurements. The circuits obtained here also have the same advantages as the tomographic case.
ventional full state tomography reaches its limit already for a few qubits and hence novel methods for the verification and benchmarking of quantum devices are called for. We show how the complete reconstruction of density matrices is possible even if one relies only on local information about the state. This results in an experimental effort that is linear in the number of qubits and efficient post-processing – in stark contrast to the exponential scaling of standard tomography. Whenever full tomography is not needed but instead less information required, one would expect that even fewer measurements suffice. Taking entanglement content of solid state samples and bosons in lattices as an example, we show how it may be quantified unconditionally using already routinely performed measurements only.