Elementary quantum gates with Gaussian states

Jens Eisert, Michael M. Wolf

This article provides an elementary introduction to Gaussian channels and their capacities. We review results on the classical, quantum, and entanglement assisted capacities and discuss related entropic quantities as well as additivity issues. Some of the known results are extended. In particular, it is shown that the quantum conditional entropy is maximized by Gaussian states and that some implications for additivity problems can be extended to the Gaussian setting.

Keitaro Anai, Suguru Endo, Shuntaro Takeda +1 more

In continuous-variable quantum computing, autonomous quantum error correction (QEC) can dissipatively steer a noisy quantum state into a target state or manifold, enabling robust quantum information processing without explicit syndrome measurements and feedback. Here, we propose a nullifier-based digital autonomous QEC enabled by conditional Gaussian operations. By designing jump operators for target nullifiers and compiling the resulting Lindbladian into a Trotterized sequence of elementary conditional Gaussian operations, we demonstrate two use cases: (i) deterministic preparation of non-Gaussian resource states for universal computation, including finitely squeezed cubic phase states and approximate trisqueezed states, and (ii) autonomous suppression of dephasing error for cat and squeezed cat states. We provide explicit gate decompositions for the required conditional Gaussian operations and numerically evaluate the performance under realistic imperfections, including photon loss in the bosonic mode and ancillary-qubit decoherence. Our results clarify the resource requirements and trade-offs, such as circuit depth, time-step choices, and the required set of conditional Gaussian operations, for scalable, gate-level implementations of autonomous state preparation and error suppression.

Sergey A. Podoshvedov

Wolfgang Scherer

Kalyanapuram Rangachari Parthasarathy, Ritabrata Sengupta

The momentum and position observables in an [Formula: see text]-mode boson Fock space [Formula: see text] have the whole real line [Formula: see text] as their spectrum. But the total number operator [Formula: see text] has a discrete spectrum [Formula: see text]. An [Formula: see text]-mode Gaussian state in [Formula: see text] is completely determined by the mean values of momentum and position observables and their covariance matrix which together constitute a family of [Formula: see text] real parameters. Starting with [Formula: see text] and its unitary conjugates by the Weyl displacement operators and operators from a representation of the symplectic group [Formula: see text] in [Formula: see text], we construct [Formula: see text] observables with spectrum [Formula: see text] but whose expectation values in a Gaussian state determine all its mean and covariance parameters. Thus measurements of discrete-valued observables enable the tomography of the underlying Gaussian state and it can be done by using five one-mode and four two-mode Gaussian symplectic gates in single and pair mode wires of [Formula: see text]. Thus the tomography protocol admits a simple description in a language similar to circuits in quantum computation theory. Such a Gaussian tomography applied to outputs of a Gaussian channel with coherent input states permit a tomography of the channel parameters. However, in our procedure the number of counting measurements exceeds the number of channel parameters slightly. Presently, it is not clear whether a more efficient method exists for reducing this tomographic complexity. As a byproduct of our approach an elementary derivation of the probability generating function of [Formula: see text] in a Gaussian state is given. In many cases the distribution turns out to be infinitely divisible and its underlying Lévy measure can be obtained. However, we are unable to derive the exact distribution in all cases. Whether this property of infinite divisibility holds in general is left as an open problem.