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arXiv:1909.03498 · Physical Review Research

Impossibility to produce arbitrary non-Gaussian states using zero-mean Gaussian states and partial photon number resolving detection

Christos N. Gagatsos, Saikat Guha

Published Dec 15, 2021• 7 citations
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Abstract

Gaussian states and measurements collectively are not powerful-enough resources for quantum computing, as any Gaussian dynamics can be simulated efficiently, classically. However, it is known that any one non-Gaussian resource---a state, a unitary operation, or a measurement---together with Gaussian unitaries, makes for universal quantum resources. Photon number resolving (PNR) detection, a readily realizable non-Gaussian measurement, has been a popular tool to try and engineer non-Gaussian states for universal quantum processing. In this paper, we consider PNR detection of a subset of the modes of a zero-mean pure multimode Gaussian state as a means to herald a target non-Gaussian state on the undetected modes. This is motivated from the ease of scalable preparation of Gaussian states that have zero mean, using squeezed vacuum and passive linear optics. We calculate upper bounds on the fidelity between the actual heralded state and the target state. We find that this fidelity upper bound is $1/2$ when the target state is a multimode coherent cat-basis cluster state, a resource sufficient for universal quantum computing. This proves that there exist non-Gaussian states that are not producible by this method. Our fidelity upper bound is a simple expression that depends only on the target state represented in the photon-number basis, which could be applied to other non-Gaussian states of interest.

DOI
10.1103/physrevresearch.3.043182

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Gaussian states and measurements collectively are not powerful-enough resources for quantum computing, as any Gaussian dynamics can be simulated efficiently, classically. However, it is known that any one non-Gaussian resource---a state, a unitary operation, or a measurement---together with Gaussian unitaries, makes for universal quantum resources. Photon number resolving (PNR) detection, a readily realizable non-Gaussian measurement, has been a popular tool to try and engineer non-Gaussian states for universal quantum processing. In this paper, we consider PNR detection of a subset of the modes of a zero-mean pure multimode Gaussian state as a means to herald a target non-Gaussian state on the undetected modes. This is motivated from the ease of scalable preparation of Gaussian states that have zero mean, using squeezed vacuum and passive linear optics. We calculate upper bounds on the fidelity between the actual heralded state and the target state. We find that this fidelity upper bound is 1/21/21/2 when the target state is a multimode coherent cat-basis cluster state, a resource sufficient for universal quantum computing. This proves that there exist non-Gaussian states that are not producible by this method. Our fidelity upper bound is a simple expression that depends only on the target state represented in the photon-number basis, which could be applied to other non-Gaussian states of interest.
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 a complete description of a multi-mode bosonic quantum state in the coherent-state basis (which in this paper is denoted as the ``KKK'' function), which---up to a phase---is the square root of the well-known Husimi QQQ representation. We express the KKK function of any NNN-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.
In this work, we investigate the generation of squeezed Schrödinger cat states in schemes based on photon-number-resolving measurements on multimode Gaussian states. We derive analytical expressions for the states generated in two- and three-mode schemes, as well as formulas for their fidelity with squeezed Schrödinger cat states. We analyze how the amplitude of the generated states scales with the number of detected particles. Furthermore, we derive an upper bound on the achievable generation fidelity and identify the conditions under which multimode schemes can enhance the quality of the generated states.
It is important to find feasible measurement bounds for quantum information protocols. We present analytic bounds for quantum illumination with Gaussian states when using an on-off detection or a photon number resolving (PNR) detection, where its performance is evaluated with signal-to-noise ratio. First, for coincidence counting measurement, the best performance is given by the two-mode squeezed vacuum (TMSV) state which outperforms the coherent state and the classically correlated thermal (CCT) state. However, the coherent state can beat the TMSV state with increasing signal mean photon number in the case of the on-off detection. Second, the performance is enhanced by taking Fisher information approach with all counting probabilities including non-detection events. In the Fisher information approach, the TMSV state still presents the best performance but the CCT state can beat the TMSV state with increasing signal mean photon number in the case of the on-off detection. Furthermore, we show that it is useful to take the PNR detection on the signal mode and the on-off detection on the idler mode, which reaches similar performance of using PNR detection on both modes.
Non-Gaussian quantum states of light are critical resources for optical quantum information processing, but methods to generate them efficiently remain challenging to implement. Here we introduce a generic approach for non-Gaussian state production from input states populating discrete frequency bins. Based on controllable unitary operations with a quantum frequency processor, followed by photon-number-resolved detection of ancilla modes, our method combines recent developments in both frequency-based quantum information and non-Gaussian state preparation. Leveraging and refining the KKK-function representation of quantum states in the coherent basis, we develop a theoretical model amenable to numerical optimization and, as specific examples, design quantum frequency processor circuits for the production of Schr\"odinger cat states, exploring the performance tradeoffs for several combinations of ancilla modes and circuit depth. Our scheme provides a valuable framework for producing complex quantum states in frequency bins, paving the way for single-spatial-mode, fiber-optic-compatible non-Gaussian resources.
Gaussian states are ubiquitous in quantum optics and information processing, and it is essential to have effective tools for their characterization. One such tool is a photon-number-resolving detector, and the simplest configuration involves counting the total number of photons in the state to be characterized. This motivates the following question: What properties of a multimode Gaussian state are determined by the signal from one detector that measures the total number of photons up to some bound? We find that if the Gaussian state occupies SSS modes and the probabilities of nnn photons for all $n\ensuremath{\le}8S$ are known, then we can determine the spectrum of the Gaussian covariance matrix and the magnitude of the displacements in each eigenspace of the covariance matrix. Nothing more can be learned, even if all photon-number probabilities are known. When the state is pure, the covariance matrix spectrum determines the squeezing parameters of the state.