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arXiv:0908.1476 · Physical Review A

Resources for universal quantum-state manipulation and engineering

Petr Marek, Jaromı́r Fiurášek

Published Jun 23, 2009• 12 citations
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Abstract

We investigate which non-Gaussian resources are needed, in addition to Gaussian operations and measurements, for implementation of arbitrary quantum gates on multimode quantum states of light. We show that an arbitrary set of pure non-Gaussian states with finite expansion in Fock basis is sufficient for this task. As an illustration we present an explicit scheme for probabilistic implementation of the nonlinear sign gate using resource non-Gaussian states and Gaussian operations.

DOI
10.1103/physreva.79.062321

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We investigate which non-Gaussian resources are needed, in addition to Gaussian operations and measurements, for implementation of arbitrary quantum gates on multimode quantum states of light. We show that an arbitrary set of pure non-Gaussian states with finite expansion in Fock basis is sufficient for this task. As an illustration we present an explicit scheme for probabilistic implementation of the nonlinear sign gate using resource non-Gaussian states and Gaussian operations.
Non-Gaussian quantum gates are essential components for optical quantum information processing. However, the efficient implementation of practically important multimode higher-order non-Gaussian gates has not been comprehensively studied. We propose a measurement-based method to directly implement general, multimode, and higher-order non-Gaussian gates using only fixed non-Gaussian ancillary states and adaptive linear optics. Compared to existing methods, our method allows for a more resource-efficient and experimentally feasible implementation of multimode gates that are important for various applications in optical quantum technology, such as the two-mode cubic quantum nondemolition gate or the three-mode continuous-variable Toffoli gate, and their higher-order extensions. Our results will expedite the progress toward fault-tolerant universal quantum computing with light. Published by the American Physical Society 2024
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.
This thesis presents an investigation into the generation and characterisation of non-Gaussian states in continuous variable quantum optics. Beginning by placing the study of continuous variables within the context of quantum information processing more generally, we then motivate the need for non-Gaussianity within quantum computing protocols. The focus then narrows to the consideration of two particular sets of non-Gaussian states: orthogonal superposition states and the cubic resource state. The superposition of two orthogonal states has been shown to enhance certain continuous variable quantum information processing protocols that rely on entanglement, and within this section two distinct methods for generating such states are considered. While one of these methods provides a specific example, the other introduces a general orthogonaliser that relies solely on knowledge of the expectation value for the chosen orthogonalising operator in order to produce the superposition. The second case for non-Gaussian state generation employs current results demonstrating the production of a weak approximation of the cubic resource state in order to illustrate the use of such single-mode states to generate multimode states with similar features. These states are implemented as ancillas for a deterministic non-Gaussian gate operation on a system of choice. We show that generating multimode states through this distribution scheme allows an enhancement of the output state to better approximate the nonlinear features. Finally, we consider methods of characterising non-Gaussian states. In particular, we introduce a witness for pure states existing outside of the Gaussian convex hull. Such states exclude Gaussian pure states as well as non-Gaussian states generated from mixtures of squeezed and coherent states, and therefore consist of non-Gaussian states generated from non-Gaussian operations. We present a detection-independent bound for such states based on the generalised quasiprobability distribution.
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.
In this work, we study the generation of non-Gaussian quantum states of light using multimode quantum resources. We first propose a theoretical framework to describe photon subtraction operation - a well known non-Gaussian operation - applied to a multimode Gaussian state. We apply this framework to the case of Schrödinger kitten state generation using a multimode squeezed vacuum as input state. We show that a monomode non-Gaussian state with high purity can be generated by a better choice of the different experimental parameters. Then, we propose a new protocol to generate high amplitude non-Gaussian states using spectral multiplexing of input states and Gaussian operations. We show that this protocol can implement the cat breeding operation with less complexity. We show also that this protocol can be used for the generation of approximated GKP state.
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.