Polynomial approximation of non-Gaussian unitaries by counting one photon at a time

In quantum computation with continous-variable systems, quantum advantage can only be achieved if some non-Gaussian resource is available. Yet, non-Gaussian unitary evolutions and measurements suited for computation are challenging to realize in the lab. We propose and analyze two methods to apply a polynomial approximation of any unitary operator diagonal in the amplitude quadrature representation, including non-Gaussian operators, to an unknown input state. Our protocols use as a primary non-Gaussian resource a single-photon counter. We use the fidelity of the transformation with the target one on Fock and coherent states to assess the quality of the approximate gate.Comment: 11 pages, 7 figure

Effect of one-, two-, and three-body atom loss processes on superpositions of phase states in Bose-Josephson junctions

In a two-mode Bose-Josephson junction formed by a binary mixture of ultracold atoms, macroscopic superpositions of phase states are produced during the time evolution after a sudden quench to zero of the coupling amplitude. Using quantum trajectories and an exact diagonalization of the master equation, we study the effect of one-, two-, and three-body atom losses on the superpositions by analyzing separately the amount of quantum correlations in each subspace with fixed atom number. The quantum correlations useful for atom interferometry are estimated using the quantum Fisher information. We identify the choice of parameters leading to the largest Fisher information, thereby showing that, for all kinds of loss processes, quantum correlations can be partially protected from decoherence when the losses are strongly asymmetric in the two modes.Comment: 23 pages, 8 figures, to be published in Eur. Phys. J.

Macroscopic superpositions in Bose-Josephson junctions: Controlling decoherence due to atom losses

We study how macroscopic superpositions of coherent states produced by the nondissipative dynamics of binary mixtures of ultracold atoms are affected by atom losses. We identify different decoherence scenarios for symmetric or asymmetric loss rates and interaction energies in the two modes. In the symmetric case the quantum coherence in the superposition is lost after a single loss event. By tuning appropriately the energies we show that the superposition can be protected, leading to quantum correlations useful for atom interferometry even after many loss events.Comment: 6 pages, 3 figure

Random coding for sharing bosonic quantum secrets

We consider a protocol for sharing quantum states using continuous variable systems. Specifically we introduce an encoding procedure where bosonic modes in arbitrary secret states are mixed with several ancillary squeezed modes through a passive interferometer. We derive simple conditions on the interferometer for this encoding to define a secret sharing protocol and we prove that they are satisfied by almost any interferometer. This implies that, if the interferometer is chosen uniformly at random, the probability that it may not be used to implement a quantum secret sharing protocol is zero. Furthermore, we show that the decoding operation can be obtained and implemented efficiently with a Gaussian unitary using a number of single-mode squeezers that is at most twice the number of modes of the secret, regardless of the number of players. We benchmark the quality of the reconstructed state by computing the fidelity with the secret state as a function of the input squeezing.Comment: Updated figure 1, added figure 2, closer to published versio

Compact Gaussian quantum computation by multi-pixel homodyne detection

We study the possibility of producing and detecting continuous variable cluster states in an optical set-up in an extremely compact fashion. This method is based on a multi-pixel homodyne detection system recently demonstrated experimentally, which includes classical data post-processing. It allows to incorporate the linear optics network, usually employed in standard experiments for the production of cluster states, in the stage of the measurement. After giving an example of cluster state generation by this method, we further study how this procedure can be generalized to perform gaussian quantum computation.Comment: Eqs.(20)-(21) correcte

Study of noise in virtual distillation circuits for quantum error mitigation

Virtual distillation has been proposed as an error mitigation protocol for estimating the expectation values of observables in quantum algorithms. It proceeds by creating a cyclic permutation of $M$ noisy copies of a quantum state using a sequence of controlled-swap gates. If the noise does not shift the dominant eigenvector of the density operator away from the ideal state, then the error in expectation-value estimation can be exponentially reduced with $M$. In practice, subsequent error-mitigation techniques are required to suppress the effect of noise in the cyclic permutation circuit itself, leading to increased experimental complexity. Here, we perform a careful analysis of noise in the cyclic permutation circuit and find that the estimation of expectation value of observables diagonal in the computational basis is robust against dephasing noise. We support the analytical result with numerical simulations and find that $67\%$ of errors are reduced for $M=2$, with physical dephasing error probabilities as high as $10\%$. Our results imply that a broad class of quantum algorithms can be implemented with higher accuracy in the near-term with qubit platforms where non-dephasing errors are suppressed, such as superconducting bosonic qubits and Rydberg atoms.Comment: 12 pages, 5 figure

Efficient simulation of Gottesman-Kitaev-Preskill states with Gaussian circuits

We study the classical simulatability of Gottesman-Kitaev-Preskill (GKP) states in combination with arbitrary displacements, a large set of symplectic operations and homodyne measurements. For these types of circuits, neither continuous-variable theorems based on the non-negativity of quasi-probability distributions nor discrete-variable theorems such as the Gottesman-Knill theorem can be employed to assess the simulatability. We first develop a method to evaluate the probability density function corresponding to measuring a single GKP state in the position basis following arbitrary squeezing and a large set of rotations. This method involves evaluating a transformed Jacobi theta function using techniques from analytic number theory. We then use this result to identify two large classes of multimode circuits which are classically efficiently simulatable and are not contained by the GKP encoded Clifford group. Our results extend the set of circuits previously known to be classically efficiently simulatable

Sufficient condition for universal quantum computation using bosonic circuits

We present a new method for quantifying the resourcefulness of continuous-variable states in the context of promoting otherwise simulatable circuits to universality. The simulatable, albeit non-Gaussian, circuits that we consider are composed of Gottesman-Kitaev-Preskill states, Gaussian operations, and homodyne measurements. We first introduce a general framework for mapping a continuous-variable state into a qubit state. We then express existing maps in this framework, including the modular subsystem decomposition and stabilizer subsystem decomposition. Combining these results with existing results in discrete-variable quantum computation provides a sufficient condition for achieving universal quantum computation. These results also allow us to demonstrate that for states symmetric in the position representation, the modular subsystem decomposition can be interpreted in terms of resourceless (simulatable) operations - i.e., included in the class of Gaussian circuits with input stabilizer Gottesman-Kitaev-Preskill states. Therefore, the modular subsystem decomposition is an operationally relevant mapping to analyze the logical content of realistic Gottesman-Kitaev-Preskill states, among other states.Comment: 30 pages, 13 figure

Classical simulation of Gaussian quantum circuits with non-Gaussian input states

We consider Gaussian quantum circuits supplemented with non-Gaussian input states and derive sufficient conditions for efficient classical strong simulation of these circuits. In particular, we generalise the stellar representation of continuous-variable quantum states to the multimode setting and relate the stellar rank of the input non-Gaussian states, a recently introduced measure of non-Gaussianity, to the cost of evaluating classically the output probability densities of these circuits. Our results have consequences for the strong simulability of a large class of near-term continuous-variable quantum circuits.Comment: 8+6 pages, 3 figures. Comments welcome