Squeezing-Enhanced Photon-Number Measurements for GKP State Generation

TL;DR

The paper tackles scalable, fault-tolerant CV quantum computing with GKP qubits by developing a passive, time-multiplexed cluster approach that tightly integrates squeezing, PhANTM, and adaptive breeding to produce high-quality GKP states. By introducing a teleportation-based squeezing gate, a reset mechanism, and optimized PhANTM sequences, the authors demonstrate an end-to-end path from Gaussian resources to GKP states that supports RHG surface-code error correction with a threshold of $11.53$ dB cluster squeezing. This represents a ~1.4 dB improvement over prior work and emphasizes a realistic, loss-tolerant, switch-free architecture compatible with current detector capabilities. The results highlight the viability of achieving fault-tolerant photonic quantum computing with Gaussian resources and probabilistic non-Gaussian operations, while outlining concrete directions for loss modeling and further optimization.

Abstract

We present an architecture for the generation of GKP states in which quadrature squeezing operations are used to control the average photon number statistics of probabilistic photon number measurements on Gaussian resource states. Specifically, we present an architecture employing a teleportation-based squeezing protocol and polynomial-gate applications integrated into a time-multiplexed multi-mode cluster state to generate cat states with high amplitudes, which are consequently used to generate GKP states with high quadrature effective squeezing. Compared to our previous work, in addition to using squeezing as a resource, the present architecture reduces damping and noise by minimizing the number of homodyne measurements required in GKP state generation. We demonstrate the effectiveness of these improvements - including dynamic input-state resetting and an improved breeding algorithm - by achieving a fault-tolerance threshold of 11.5 dB cluster squeezing using the RHG surface code for error correction, without requiring active switching or photon-number resource states.

Figures (12)

• Figure 1: (a) Circuit for PhANTM on a dual-rail quantum wire. $O$ stands for multiple photon subtractions, i.e., $O$ represents multiple $\mathcal{O}$ in Eq. \ref{['eq:subtraction']} combined. (b) Circuit for squeezing gate using dual-rail quantum wire which resembles the PhANTM circuit, with key differences being the removal of photon subtraction and the change of the homodyne detection angles depending on the desired squeezing. In both circuits, arrow represents a beam splitter and line with circular ends represents a $C_Z$ interaction.
• Figure 2: Effect of noise on cat states. a) ratio of cat state size, between after and before noise channel $\mathcal{N}$, as a function of the cluster squeezing $r$ of the channel. b) ratio of cat state squeezing as a function of the cluster squeezing $r$. c) Wigner function of cat states. An input state is taken with a size of $\alpha_i=2$ and a squeezing of $r^\prime_{i}=0.5$. The output state of the noise channel are shown with $r=11.5$ dB and $r=7$ dB. d) $\tilde{n}$ (left axis) and $\mathcal{P}_0$ (right axis) as a function the anti-squeezing parameter $r_a$ where the anti-squeezing gate is applied before PhANTM (solid line). We assume cat state input with $\alpha_i= 3$, $r^\prime=0.5$, and squeezing in the cluster state as well as the anti-squeezing gate to be $r=11.5$ dB. For these simulations, we consider one photon subtraction in a PhANTM step. For reference, we show $\mathcal{P}_0$ and $\tilde{n}$ in the case where PhANTM is applied without an anti-squeezing gate (dashed line). Subscript $i$ denotes input or initial parameter.
• Figure 3: Cat state generation protocol and results.(a) PhANTM and squeezing are applied successively in a dual rail frequency-time cluster state. In PhANTM, homodyne bases are $p$ and $q$ while for squeezing, the angle $\theta_{i,a}$ and $\theta_{i,b}$ are chosen dynamically (see Alexander2016a) depending on how much the cat state needs to be squeezed (see eq.\ref{['eq:Sgate']}). (b) Result from PhANTM simulation: $\alpha_c$ (corrected amplitude) as a function of cluster squeezing $r$. Each dot is the mean of a Monte Carlo simulation with 1000 trials, while error bars show the standard deviation of the mean. For baseline comparison without anti-squeezing gate, PhANTM is applied both on mode $\nu_1$ and $\nu_2$.
• Figure 4: GKP effective squeezing as a function of $r$ from adaptive breeding simulations. The circle shows the average over 1000 Monte Carlo runs, while the bars show the standard deviation. Simulations are performed with a Fock dimension of 65.
• Figure 5: Logical error rate as a function of the cluster squeezing $r$ for different code distances. The dashed line indicate the position of the quantum error correction threshold. Each dot is determined from $10^5$ points and by using MWPM decoding method Dennis2001.