Energy-dependent barren plateau in bosonic variational quantum circuits

Abstract

Bosonic continuous-variable Variational quantum circuits (VQCs) are crucial for information processing in cavity quantum electrodynamics and optical systems, widely applicable in quantum communication, sensing and error correction. The trainability of such VQCs is less understood, hindered by the lack of theoretical tools such as $t$-design due to the infinite dimension of the physical systems involved. We overcome this difficulty to reveal an energy-dependent barren plateau in such VQCs. The variance of the gradient decays as $1/E^{Mν}$, exponential in the number of modes $M$ but polynomial in the (per-mode) circuit energy $E$. The exponent $ν=1$ for shallow circuits and $ν=2$ for deep circuits. We prove these results for state preparation of general Gaussian states and number states. We also provide numerical evidence that the results extend to general state preparation tasks. As circuit energy is a controllable parameter, we provide a strategy to mitigate the barren plateau in continuous-variable VQCs.

Figures (11)

• Figure 1: Summary of VQC trainability in DV and CV systems. The Hilbert space is finite-dimension in DV systems while for CV ones it is infinite-dimension. The universal DV VQC is built from local $2$-design unitary gates (lime green), and universal CV VQCs consist of single qubit rotations (cyan) and echoed conditional displacement (ECD) gates (pink) eickbusch2022fast. In DV VQCs, the variance of the gradient decays exponentially with the number of qubits $N$ in shallow cerezo2021cost and deep mcclean2018barren DV VQCs optimizing a global cost function. In this paper, we show the variance decays exponentially in number of modes $M$ but polynomially with circuit energy $E$ in shallow and deep region of CV VQCs.
• Figure 2: Variance of gradient ${\rm Var}[\partial_{\theta_{k}}{\cal C}]$ at $k=L/2$ in preparation of (a) displaced squeezed vacuum (DSV) state with $\gamma = 2, \zeta=\sinh^{-1}(2)$ and (b) Fock state with $E_t=8$. Orange and red dots with errorbars show numerical results of variance in shallow and deep circuits. Orange solid curve represents the analytical variance in Theorem \ref{['res:shallow']}; the dashed and solid magenta curves show the lower and upper bounds in Ineqs. \ref{['eq:LB_main']}, \ref{['eq:UB_main']}. Black dotted and dashed lines indicate the scaling of $1/E$ and $1/E^2$. Insets in (a)(b) we plot the logarithm in base ten of the upper bound in Ineq. \ref{['eq:UB_main']} versus circuit depth and energy. Green triangle (main) and line (inset) show the corresponding boundary of variance at $E_c(L)$ and $\ell_c(E)$.
• Figure 3: Variance of gradient ${\rm Var}[\partial_{\theta_{k}}{\cal C}]$ at $k=L/2$ in preparation of random CV states $\ket{\psi}_m=\sum_{n} b_n \ket{n}_m$ with $L=4$ (a) and $L=50$ (b) circuits. Curves with same color show the variance of different sample target states. Black dotted and dashed lines in (a) and (b) represent the scaling of $1/E$ and $1/E^2$. In our calculation, we have chosen cutoff $n_c\sim 2E_t$ and $\epsilon=0.1$.
• Figure 4: Correlators $C_1^{\rm Gauss}$ and $C_2^{\rm Gauss}(\bm z)$ with $\bm z=\{1/2,\cdots,1/2\}$ in Eqs. \ref{['eq:C1_gauss_modes_main']}, \ref{['eq:C2_gauss_modes_main']} versus (a) ensemble energy $E$ and (b) modes $M$. The target state $\ket{\psi}_{\bm m}$ is generated by global random passive Gaussian unitary following a single-mode squeezer with strength $r=8$.
• Figure 5: Variance of gradient ${\rm Var}[\partial_{\theta_{k}}{\cal C}]$ at $k=ML/2$ in preparation of a TMSV state $\ket{\zeta}_{\rm TMSV}$ with $\zeta = \sinh^{-1}(2)$. Orange and red dots with errorbars show numerical results of variance in shallow and deep circuits. Orange solid curve represents the $(3/4)^{2L}C_1^{\rm TMSV}/6$ for reference; the dashed and solid magenta curves show the lower and upper bounds in Ineqs. \ref{['eq:LB_main']}, \ref{['eq:UB_main']}. Black dotted and dashed lines indicate the scaling of $1/E^2$ and $1/E^4$. Inset shows the logarithm in base ten upper bound versus circuit depth and energy. Green triangle (main) and line (inset) show the corresponding boundary of variance at $E_c(L)$ and $\ell_c(E)$.

Theorems & Definitions (14)

• Lemma 1
• Definition 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6
• Corollary 7
• Proposition 8
• Proposition 9
• Proposition 10