We present a detailed analytic framework for studying multimode non-Gaussian states that are conditionally generated when few modes of a multimode Gaussian state are subject to photon-number-resolving detectors. From the output state Wigner function, we deduce that the state factorizes into a Gaussian gate applied to a finite Fock-superposition non-Gaussian state. The framework provides an approach to find the optimal strategy to generate a given target non-Gaussian state. We explore examples, such as the generation of cat states, weak cubic phase states, and bosonic code states, and achieve improvements of success probability over other schemes. Our framework also applies to the case in which the measured Gaussian state is mixed which is very important for the analysis of experimental imperfections such as photon loss. The framework has potential far-reaching implications to the generation of bosonic error-correcting codes and for the implementation of non-Gaussian gates using resource states, among other applications requiring non-Gaussianity
We present a detailed analytic framework for studying multimode non-Gaussian states that are conditionally generated when few modes of a multimode Gaussian state are subject to photon-number-resolving detectors. From the output state Wigner function, we deduce that the state factorizes into a Gaussian gate applied to a finite Fock-superposition non-Gaussian state. The framework provides an approach to find the optimal strategy to generate a given target non-Gaussian state. We explore examples, such as the generation of cat states, weak cubic phase states, and bosonic code states, and achieve improvements of success probability over other schemes. Our framework also applies to the case in which the measured Gaussian state is mixed which is very important for the analysis of experimental imperfections such as photon loss. The framework has potential far-reaching implications to the generation of bosonic error-correcting codes and for the implementation of non-Gaussian gates using resource states, among other applications requiring non-Gaussianity
Generation of high-fidelity photonic non-Gaussian states is a crucial ingredient for universal quantum computation using continuous-variable platforms, yet it remains a challenge to do this efficiently. We present a general framework for a probabilistic production of multimode non-Gaussian states by measuring a few modes of multimode Gaussian states via photon-number-resolving detectors. We use Gaussian elements consisting of squeezed displaced vacuum states and interferometers, the only non-Gaussian elements consisting of photon-number-resolving detectors. We derive analytic expressions for the output Wigner function, and the probability of generating the states in terms of the mean and the covariance matrix of the Gaussian state and the photon detection pattern. We find that the output states can be written as a Fock-basis superposition state followed by a Gaussian gate, and we derive explicit expressions for these parameters. These analytic expressions show exactly what non-Gaussian states can be generated by this probabilistic scheme. Further, it provides a method to search for the Gaussian circuit and measurement pattern that produce a target non-Gaussian state with optimal fidelity and success probability. We present specific examples such as the generation of cat states, ON states, Gottesman-Kitaev-Preskill states, NOON states, and bosonic-code states. The proposed framework has potentially far-reaching implications for the generation of bosonic error-correction codes that require non-Gaussian states and resource states for the implementation of non-Gaussian gates needed for universal quantum computation, among other applications requiring non-Gaussianity. The tools developed here could also prove useful for the quantum resource theory of non-Gaussianity.
We consider conditional photonic non-Gaussian state preparation using multimode Gaussian states and photon-number-resolving detectors in the presence of photon loss. While simulation of such state preparation is often computationally challenging, we show that obtaining the required multimode Gaussian state Fock matrix elements can be reduced to the computation of matrix functions known as loop hafnians and develop a tailored algorithm for their calculation that is faster than previously known methods. As an example of its utility, we use our algorithm to explore the loss parameter space for three specific non-Gaussian state preparation schemes: Fock state heralding, cat state heralding, and weak cubic-phase state heralding. We confirm that these schemes are fragile with respect to photon loss, yet find that there are regions in the loss parameter space that are potentially accessible in an experimental setting which correspond to heralded states with nonzero non-Gaussianity.
Non-Gaussian quantum states of light are important resources for implementing quantum information tasks, ranging from fault-tolerant optical quantum information processing to long distance quantum communication. These states can be generated by applying non-Gaussian operations, such as photon addition and subtraction, to an input Gaussian quantum state. Also, the use of Gaussian operations (like homodyne measurement) allows the transformation of non-Gaussian input quantum states while maintaining their non-Gaussianity. It has been shown that in such context, iterative tailoring of arbitrary quantum states is possible, by cascading elementary operation blocks. This invitably comes at the cost of increasing the protocol complexity. Spectral modes, on the other hand, can be used to handily carry multiple quantum states in a single spatial mode. Moreover, multimode states populating such modes are readily available in commonly exploited processes such as spontaneous parametric down conversion.In this work, we propose a protocol to generate non-Gaussian quantum states using spectral multiplexing of input states and Gaussian operations. We give the analytical expression of the output non-Gaussian state as a function of the number of input state, the Gaussian operation and the properties of spectral modes carrying the input quantum states. Taking as an example the cat breeding operation, we show that with a single mixing and measurement operation, we can obtain an amplified output cat state comparable to the one obtained using iterative breeding scheme with multiple mixing operations and measurements with the same number of input states. We show also that the generation of approximated GKP states is possible with this protocol. We believe that the proposed protocol will have a major impact on the field of non-Gaussian quantum states generation, and can be used to extend the study beyond the cases considered in our work.
We consider a complete description of a multi-mode bosonic quantum state in the coherent-state basis (which in this paper is denoted as the ``K'' function), which---up to a phase---is the square root of the well-known Husimi Q representation. We express the K function of any N-mode Gaussian state as a function of its covariance matrix and displacement vector, and also that of a general continuous-variable cluster state in terms of the modal squeezing and graph topology of the cluster. This formalism lets us characterize the non-Gaussian state left over when one measures a subset of modes of a Gaussian state using photon number resolving detection, the fidelity of the obtained non-Gaussian state with any target state, and the associated heralding probability, all analytically. We show that this probability can be expressed as a Hafnian, reinterpreting the output state of a circuit claimed to demonstrate quantum supremacy termed Gaussian boson sampling. As an example application of our formalism, we propose a method to prepare a two-mode coherent-cat-basis Bell state with fidelity close to unity and success probability that is fundamentally higher than that of a well-known scheme that splits an approximate single-mode cat state---obtained by photon number subtraction on a squeezed vacuum mode---on a balanced beam splitter. This formalism could enable exploration of efficient generation of cat-basis entangled states, which are known to be useful for quantum error correction against photon loss.
Photonic quantum computing has gained significant interest in recent years due to its potential for scaling to large numbers of qubits. A critical requirement for fault-tolerant quantum computation is the reliable generation of non-Gaussian quantum states, typically achieved using Gaussian operations and photon-number-resolving detectors. However, the probabilistic nature of quantum measurement typically results in low success rates for state preparation. Conventionally, these circuits are optimized to herald a single specific target outcome, thereby disregarding the potential utility of alternative measurement patterns generated by the same physical setup. In this work, we propose and demonstrate a multi-outcome optimization strategy that increases the overall acceptance probability by allowing a single circuit to produce useful quantum states across several measurement patterns. To evaluate this approach, we apply the framework to the generation of Gottesman-Kitaev-Preskill core states, Schrodinger cat states, binomial codes, and cubic phase states using both two-mode and three-mode Gaussian circuits. We demonstrate that the success probability can be enhanced through two distinct mechanisms: first, by simultaneously targeting a diverse set of useful resource states, and second, by aggregating degenerate outcomes to maximize the production rate of a single target state.
We investigate the generation of squeezed Fock states (SFSs) via particle-number measurements in the modes of multimode Gaussian states. We identify a universal class of N-mode Gaussian states for which measuring N−1 modes results in the generation of SFSs. The key feature of these states is that the generated SFSs depend only on the total number of detected particles and are independent of their distribution among the detectors. Based on the general form of the wave functions of multimode Gaussian states, we propose a universal scheme for SFS generation. For this scheme, we evaluate the probability of SFS generation and analyze the robustness of the process against imperfections in particle-number-resolving detectors. In addition, we compare the universal scheme with a nonuniversal scheme, in which the generation of SFSs depends on a specific distribution of particle numbers across the detectors. We demonstrate that the universal scheme provides a higher probability of SFS generation, at the cost of increased experimental resources.