When quantum resources backfire: Non-gaussianity and symplectic coherence in noisy bosonic circuits

TL;DR

This work tackles the challenge of noise in continuous-variable bosonic circuits by introducing the displacement propagation algorithm, a CV analogue of Pauli propagation that leverages the optical equivalence theorem for characteristic functions. By defining contraction coefficients $\mathfrak c_1$, $\mathfrak c_2$, and $\mathfrak d_\epsilon$ that couple noise to CV resources (e.g., non-Gaussianity and symplectic coherence), the authors delineate regimes where circuit outputs concentrate and where classical simulation is efficient, even in the presence of cubic-phase gates. They show, somewhat counterintuitively, that strong non-Gaussianity and symplectic coherence can facilitate classical simulability under noise, while near-Gaussian regimes or high-energy growth can sustain hardness in the noiseless case but remain tractable when noise is present. The framework also provides practical unbiased and adaptive Monte Carlo strategies for estimating overlaps, local observables, and quadrature moments, offering a concrete boundary for quantum advantage in CV systems and guiding error-correction and resource-management in noisy quantum devices.

Abstract

Analyzing the impact of noise is of fundamental importance to understand the advantages provided by quantum systems. While the classical simulability of noisy discrete-variable systems is increasingly well understood, noisy bosonic circuits are more challenging to simulate and analyze. Here, we address this gap by introducing the $\textit{displacement propagation}$ algorithm, a continuous-variable analogue of Pauli propagation for simulating noisy bosonic circuits. By exploring the interplay of noise and quantum resources, we identify several computational phase transitions, revealing regimes where even modest noise levels render bosonic circuits efficiently classically simulable. In particular, our analysis reveals a surprising phenomenon: computational resources usually associated with bosonic quantum advantage, namely non-Gaussianity and symplectic coherence, can make the system easier to classically simulate in presence of noise.

Figures (6)

• Figure 1: The setting for \ref{['prop:purity_uniform']}. Computing the overlap between an arbitrary bosonic quantum state $\rho$ going through a uniform layer of noise $\Lambda_{\bar{n},\eta}^{\otimes m}$ and any other arbitrary quantum state $\sigma$ becomes trivial as the overlap rapidly decays to zero.
• Figure 2: A circuit for bosonic computations with noisy cubic phase gate. $\forall i \in \{1,\hbox{...},L\}$, $G_i$ are Gaussian unitary gates, $\gamma_i$ represents the cubic phase gate with cubicity $\gamma_i$ (acting on the first mode), and $\Lambda_{\bar{n}, \eta}$ is the thermal loss channel.
• Figure 3: Top and middle row: Contraction coefficients $\mathfrak c_1$ and $\mathfrak{c}_2$ as a function of photon loss $\eta$ and quantum resources $\sqrt{\gamma_{\min}} \sigma$ (for $\mathfrak c_1$) and $\sqrt{\gamma_{\min}} \sigma^{-1}$ for different values of average thermal photons $\bar{n}$. $\forall \eta,\bar{n}$, increasing the quantum resources leads us to the efficiently classically simulable phase (marked by the blue/green region). Bottom row: Contraction coefficient $\mathfrak d_\epsilon$ as a function of $\gamma$ for different values of $\bar{n}$, with fixed $M$ and $\epsilon$. Decreasing the value of $\gamma$ (approaching near-Gaussian circuits) leads us to the efficiently classically simulable phase of the noisy bosonic computations (marked by the violet region).
• Figure 4: Intuition behind the displacement propagation algorithm. To evaluate the characteristic function $\Tr[\mathcal{U}^* (\hat{D}(\boldsymbol{r}))\rho_0]$, we build a Markov chain by decomposing $\mathcal{U}^*$ into $L$ possibly non-physical quantum maps $\mathcal{A}_1 \circ \cdots \circ \mathcal{A}_L$ and for each of the $L$ layers, we sample a phase space point $\alpha_{i}$ according to a probability distribution that depends on both the properties of the current layer and the previously sampled phase-space point. At the end of the $L$'th layer, we obtain a phase space point $\alpha_1 \in \mathbb{C}^m$ at which we estimate the characteristic function $\Tr[\rho_0\hat{D}(\alpha_1)]$ (assuming that the characteristic function of the input state is efficiently computable). By carefully choosing the quantum channels, we ensure that the statistical average of $N$ such characteristic functions approximates $\Tr[\mathcal{U}^*(\hat{D}(\boldsymbol{r}))\rho_0]$ to arbitrary precision using a polynomial number of samples. This process is illustrated by the tree on the left and formalized in Algorithm \ref{['alg:sim-unbiased']} in the Supplementary Material. However, for certain regions of phase space, efficient sampling at every layer may not be feasible, particularly due to the non-linearity introduced by non-Gaussian cubic-phase gates. In these cases, we employ adaptive simulation algorithms, in which the layer $\mathcal{A}_j$ is replaced by an approximate map $\tilde{\mathcal{A}}_j$ (the modification is designed so that the corresponding sampled phase-space point becomes deterministic; see Section \ref{['section:biased-oracle']} of the Supplementary Material). This procedure ensures that the resulting Markov chain approximates the target characteristic function with only a negligible bias. This adaptive approach is depicted by the tree on the right and formalized in Algorithm \ref{['alg:sim-adaptive-general']} in the Supplementary Material.
• Figure 5: The circuit decomposition of the noisy bosonic circuit $\mathcal{U}$ (Eq. \ref{['appeq:noisy_bosonic']}) that helps us in proving Theorem \ref{['apptheo:global_noise_induced conc']}. The expression for $\mathcal{B}_L$ and $\mathcal{B}_j, \forall j \in \{2,\hbox{...},L-1\}$ are given by Equations \ref{['appeq:B_t']} and \ref{['appeq:B_j']} respectively. Conjugating the cubic phase gate with output operator $\hat{O}$ and upper bounding the Fourier one-norm of $\mathcal{B}_{L}\circ \hbox{...}\circ\mathcal{B}_2\circ \mathcal{C}_{1,\mathfrak{D}}\circ \Lambda_{\bar{n}, \eta}\circ \mathcal{G}_1 (\rho)$, we prove that the expectation value of output operators with at most exponential Schatten 1-norm concentrates exponentially to zero whenever $\mathfrak{c}_1 <1$ (Eq. \ref{['appeq:c_2']}) for sufficiently large circuit depth $L = \Omega(m)$.

Theorems & Definitions (64)

• Proposition 1: Overlap decay under thermal channel
• Theorem 1: Noise-induced concentration in bosonic circuits
• Theorem 2: Classically estimating overlaps under gate-based noise
• Theorem 3: Classically estimating quadratures under gate-based noise
• Definition 1: Contraction coefficients
• Lemma B.1
• Lemma B.2: Fubini-Tonelli's theorem
• Lemma B.3
• proof
• Theorem B.1: Taylor's remainder theorem