Measurement-based generation and preservation of cat and grid states within a continuous-variable cluster state

Miller Eaton1,2, Carlos González-Arciniegas1, Rafael N. Alexander3, Nicolas C. Menicucci3, and Olivier Pfister1

1Department of Physics, University of Virginia, Charlottesville, VA 22904, USA
2QC82, College Park, MD 20740, USA
3Centre for Quantum Computation and Communication Technology, School of Science, RMIT University, Melbourne, VIC 3000, Australia

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We present an algorithm to reliably generate various quantum states critical to quantum error correction and universal continuous-variable (CV) quantum computing, such as Schrödinger cat states and Gottesman-Kitaev-Preskill (GKP) grid states, out of Gaussian CV cluster states. Our algorithm is based on the Photon-counting-Assisted Node-Teleportation Method (PhANTM), which uses standard Gaussian information processing on the cluster state with the only addition of local photon-number-resolving measurements. We show that PhANTM can apply polynomial gates and embed cat states within the cluster. This method stabilizes cat states against Gaussian noise and perpetuates non-Gaussianity within the cluster. We show that existing protocols for breeding cat states can be embedded into cluster state processing using PhANTM.

Quantum computation with cluster states proceeds analogously to computation with qubits in a circuit model, but the cluster state model generates all prerequisite entanglement up front in the initial resource. Although computation with cluster states requires additional overhead in the number of required qubits, recent experiments have demonstrated the ability to create massively scalable cluster states with thousands or millions of modes using continuous-variable optical fields. The continuous-variable cluster states generated to date are composed of squeezed light modes, which are all Gaussian, but the addition of non-Gaussian resources will be required for universal quantum computing. This non-Gaussianity can be included through bosonic encodings, such as with GKP qubits, or through the use of gate teleportation with ancillary non-Gaussian states. Current proposals to implement the requisite non-Gaussian operations rely on the offline preparation of ancillary states, which is probabilistic in general, and then later couple these resources to the cluster state. In a sense, this defeats the purpose of a cluster state model where all required quantum resources are generated up front, but furthermore, the probabilistic nature of ancillary non-Gaussian resources poses a problem for scalability.
In this work, we devise a method to introduce the required non-Gaussianity without ancillary resources simply by performing appropriate measurements on the cluster state. These measurements take the form of photon subtraction operations followed by the normal homodyne detection to teleport the quantum information. While other methods to generate non-Gaussian states, such as the cubic phase state, can require resolution of tens of photons, we need only low photon-number resolution which is achievable with several different technologies. Although photon subtraction is probabilistic, the repeated application after teleportation from homodyne detection means that we will be nearly certain to eventually succeed and only some overhead number of modes must be consumed by measurement. When a successful photon subtraction occurs, the local state entangled to the cluster becomes non-Gaussian and is turned into a Schrӧdinger kitten state. Repeated applications of photon subtraction before teleportation increase the amplitude of the cat state to a level that depends on the squeezing present in the cluster state. Surprisingly, the process can preserve cat state amplitude even in the presence of Gaussian noise due to finite squeezing.
This process, which we call the Photon-counting-Assisted Node-Teleportation Method (PhANTM), can proceed in parallel on many separate 1-D chains on a cluster state. All but one cluster state node in each chain is consumed by measurement, but the last unmeasured node is transformed into a cat state. The local quantum information of this node can thus be used as a non-Gaussian resource, but importantly, it has remained entangled with the remainder of the cluster state resource. We then proceed to show that methods of breeding cat states to produce GKP states are compatible with the cluster state formalism, meaning that our method can both generate cat states that can then be bred into universal computational resources all by performing experimentally accessible measurements on a continuous-variable cluster state. We also motivate connections to phase estimation protocols and provide examples to indicate that our method can succeed in the presence of experimental imperfections and decoherence.

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