Simulating realistic non-Gaussian state preparation

N. Quesada, L. G. Helt, J. Izaac, J. M. Arrazola, R. Shahrokhshahi, C. R. Myers, and K. K. Sabapathy

Phys. Rev. A 100, 022341 – Published 29 August 2019

Abstract

We consider conditional photonic non-Gaussian state preparation using multimode Gaussian states and photon-number-resolving detectors in the presence of photon loss. While simulation of such state preparation is often computationally challenging, we show that obtaining the required multimode Gaussian state Fock matrix elements can be reduced to the computation of matrix functions known as loop hafnians and develop a tailored algorithm for their calculation that is faster than previously known methods. As an example of its utility, we use our algorithm to explore the loss parameter space for three specific non-Gaussian state preparation schemes: Fock state heralding, cat state heralding, and weak cubic-phase state heralding. We confirm that these schemes are fragile with respect to photon loss, yet find that there are regions in the loss parameter space that are potentially accessible in an experimental setting which correspond to heralded states with nonzero non-Gaussianity.

Received 10 June 2019

DOI:https://doi.org/10.1103/PhysRevA.100.022341

Physics Subject Headings (PhySH)

Quantum computation Quantum information architectures & platforms Quantum information processing with continuous variables

Quantum Information, Science & Technology

Authors & Affiliations

N. Quesada, L. G. Helt, J. Izaac, J. M. Arrazola, R. Shahrokhshahi, C. R. Myers, and K. K. Sabapathy

Xanadu, Toronto, Ontario, Canada M5G 2C8

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Vol. 100, Iss. 2 — August 2019

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Images

Figure 1
Gate model of a generic Fock state heralding scheme, where individual losses $L (\cdot)$ are applied to each mode after the action of the two-mode squeezing operation $S_{2} (\cdot)$ . Subsequently, $m$ photons are detected in the first mode.
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Figure 2
Figures of merit for the Fock state heralding scheme as a function of the transmission coefficients of each mode as depicted in Fig. 1. We set a squeezing level of $ζ = 1.0$ ( $\sim 8.7$ dB) and consider the preparation of $m = 1$ (top row) and $m = 3$ (bottom row) Fock states. For the preparation of single photons in the lossless case we find $p = 0.243$ and $W = 0.35$ . Similarly, for the Fock state with $m = 3$ we find $p = 0.082$ and $W = 0.68$ .
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Figure 3
Photon subtraction schematic for the generation of cat states. A squeezed state, with squeezing parameter $z$ , is generated in mode 1, which then interacts with the vacuum in mode 2 on a high-transmissivity beam splitter. Subsequently, mode 2 is subjected to a photon-number-resolving detector. Loss channels are inserted near both the beginning $[L (η_{1})]$ and end $[L (η_{2})]$ of the circuit. The properties of the output state are then analyzed with respect to the generation of cat states where we set $ϕ = 0$ , $cos (θ) = \sqrt{0.97}$ , and $z = 0.5$ .
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Figure 4
Figures of merit for the preparation of the odd cat state with $m = 1$ in Fig. 3 (top row) and even cat state with $m = 2$ in Fig. 3 (bottom row) as functions of the loss coefficients $η_{1}$ and $η_{2}$ in each mode. Both fidelity and WLN are largely insensitive to loss in the detected mode, but decay rapidly with loss in the output mode. Heralding probabilities are small, typically in the neighborhood of $p \sim 10^{- 4}$ . The squeezing level of the input state is set to $z = 0.5$ ( $\sim 4.5$ dB). The fidelities calculated for any value of $η_{1}$ and $η_{2}$ are taken with respect to the same lossless odd (even) cat state with $α_{opt} \approx 1.24$ ( $α_{opt} \approx 1.33$ ) in Eq. (25). For the odd cat state preparation ( $m = 1$ ), the probability, fidelity, and Wigner logarithmic negativity in the lossless case where $η_{1} = η_{2} = 1$ are, respectively, $p = 0.0077$ , $F = 0.98$ , and $W = 0.35$ . Similarly, for the even cat state $m = 2$ , we find $p = 0.00021$ , $F = 0.96$ , and $W = 0.23$ .
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Figure 5
Circuit for the production of the weak cubic-phase states. The inputs are squeezed displaced states for each mode given by $| z, α 〉 = D (α) S (z) | 0 〉$ and the detection pattern is set to $m_{1} = 1$ and $m_{2} = 2$ . The optimal circuit parameters to produce the resource state in the absence of loss are given by the following: Writing $z = r exp [i \arg (z)]$ , we have $r = (0.71, 0.67, - 0.42)$ , $\arg z = (- 2.07, 0.06, - 3.79)$ , $α = (- 0.02, 0.34, 0.02)$ , $θ = (- 1.57, 0.68, 2.5)$ , and $ϕ = (0.53, - 4.51, 0.72)$ .
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Figure 6
Figures of merit for the weak cubic-phase state preparation scheme as a function of the transmission coefficients in each mode as depicted in Fig. 5. The fidelity is with respect to the target state in Eq. (26) with $a = 0.53$ . Losses must be very low for high fidelity and WLN, although there are regions in the parameter space that correspond to states with nonzero WLN even if its fidelity to the target state is low. The values for WLN and success probability in the absence of loss are given by 0.224 and 0.02, respectively.
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