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arXiv:1412.0336 · Physical Review A

Repeat-until-success cubic phase gate for universal continuous-variable quantum computation

Kevin Marshall, Raphael C. Pooser, George Siopsis, Christian Weedbrook

Published Mar 24, 2015• 78 citations
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Abstract

To achieve universal quantum computation using continuous variables, one needs to jump out of the set of Gaussian operations and have a non-Gaussian element, such as the cubic phase gate. However, such a gate is currently very difficult to implement in practice. Here we introduce an experimentally viable ``repeat-until-success'' approach to generating the cubic phase gate, which is achieved using sequential photon subtractions and Gaussian operations. We find that our scheme offers benefits in terms of the expected time until success, as well as the fact that we do not require any complex off-line resource state, although we require a primitive quantum memory.

DOI
10.1103/physreva.91.032321

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To achieve universal quantum computation using continuous variables, one needs to jump out of the set of Gaussian operations and have a non-Gaussian element, such as the cubic phase gate. However, such a gate is currently very difficult to implement in practice. Here we introduce an experimentally viable ``repeat-until-success'' approach to generating the cubic phase gate, which is achieved using sequential photon subtractions and Gaussian operations. We find that our scheme offers benefits in terms of the expected time until success, as well as the fact that we do not require any complex off-line resource state, although we require a primitive quantum memory.
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Universal quantum computing with continuous variables requires non-Gaussian resources, in addition to a Gaussian set of operations.A known resource enabling universal quantum computation is the cubic phase state, a non-Gaussian state whose experimental implementation has so far remained elusive.In this paper, we introduce two Gaussian conversion protocols that allow for the conversion of a non-Gaussian state that has been achieved experimentally, namely the trisqueezed state [Chang et al., Phys.Rev. X 10, 011011 (2020)], to a cubic phase state.The first protocol is deterministic and it involves active (inline) squeezing, achieving large fidelities that saturate the bound for deterministic Gaussian protocols.The second protocol is probabilistic and it involves an auxiliary squeezed state, thus removing the necessity of inline squeezing but still maintaining significant success probabilities and fidelities even larger than for the deterministic case.The success of these protocols provides strong evidence for using trisqueezed states as resources for universal quantum computation.
We provide an explicit construction of a universal gate set for continuous-variable quantum computation with microwave circuits. Such a universal set has been first proposed in quantum-optical setups, but its experimental implementation has remained elusive in that domain due to the difficulties in engineering strong nonlinearities. Here, we show that a realistic three-wave mixing microwave architecture based on the superconducting nonlinear asymmetric inductive element [Frattini et al., Appl. Phys. Lett. 110, 222603 (2017)APPLAB0003-695110.1063/1.4984142] allows us to overcome this difficulty. As an application, we show that this architecture allows for the generation of a cubic phase state with an experimentally feasible procedure. This work highlights a practical advantage of microwave circuits with respect to optical systems for the purpose of engineering non-Gaussian states and opens the quest for continuous-variable algorithms based on few repetitions of elementary gates from the continuous-variable universal set.
We propose a deterministic, measurement-free implementation of a cubic phase gate for continuous-variable quantum information processing. In our scheme, the applications of displacement and squeezing operations allow us to engineer the effective evolution of the quantum state propagating through an optical Kerr nonlinearity. Under appropriate conditions, we show that the input state evolves according to a cubic phase Hamiltonian, and we find that the cubic phase gate error decreases inverse quartically with the amount of quadrature squeezing, even in the presence of linear loss. We also show how our scheme can be adapted to deterministically generate a nonclassical approximate cubic phase state with high fidelity using a ratio of native nonlinearity to linear loss of only 10^{-4}, indicating that our approach may be experimentally viable in the near term even on all-optical platforms, e.g., using quantum solitons in pulsed nonlinear nanophotonics.
The cubic phase gate is the only non-Gaussian element of universal continuous-variable quantum computation. Its Fock amplitudes are known in closed form only when the input is a Fock state. For a Gaussian input the usual route is to build the cubic quadrature in a truncated Fock space and exponentiate it, which alters the generator before the exponential is taken. A closed form is obtained here for every pure single-mode Gaussian input, with the squeezing along any quadrature axis and with arbitrary displacement and boost. It is a finite combination of the Airy function and its derivative, and the argument is shared by all Fock indices, so a whole Fock profile costs two Airy evaluations. A recurrence is also derived that builds each coefficient from the four preceding ones, together with the regime in which it is stable. Evaluating the closed form at the weak nonlinearities of current experiments requires care, since its terms grow large and alternate in sign. The precision this demands is quantified, and the truncated construction is benchmarked against the exact result, whose error at cutoffs in common use is found to be of the same size as the quantity being computed.