Functional matrix product state simulation of continuous variable quantum circuits

Abstract

We introduce a functional matrix product state (FMPS) based method for simulating the real-space representation of continuous-variable (CV) quantum computation. This approach efficiently simulates non-Gaussian CV systems by leveraging their functional form. By addressing scaling bottlenecks, FMPS enables more efficient simulation of shallow, multi-mode CV quantum circuits with non-Gaussian input states. The method is validated by simulating random shallow and cascaded circuits with highly non-Gaussian input states, showing superior performance compared to existing techniques, also in the presence of loss.

Figures (8)

• Figure 1: Functional matrix product state decomposition. Undulating lines represent continuous variables, while straight lines represent discrete variables.
• Figure 2: Simulating a 10-step cascaded circuit of cat states with a gate fidelity of $0.999$ and interpolation with cubic splines, we see that the maximum bond dimension increases linearly with the number of grid points $N$ for a while, before it falls back down and converges. Once the bond dimension no longer changes with $N$, we have a faithful representation of the state. The dashed line is included as a guide to the eye.
• Figure 3: When rotating the domain of our state, it is important that we keep track of the original domain such that the bounding box does not grow unnecessarily large upon repeated rotations. Here this is illustrated with a two-dimensional domain undergoing two $\pi/4$ rotations. At step b, the dashed bounding box defines the new domain, but further rotating that domain to step c results in an unnecessarily large domain (dotted box), mostly without information of the state. Instead, keeping track of the original domain lets us shrink the domain in step c to the solid line.
• Figure 4: Noisy simulations. The loss noise is commuted through to the end of the computation. The trace can naturally be applied to evaluate expectation values in the tensor network framework, by contracting the indices with their conjugate copy.
• Figure 5: (a) To investigate the behavior of the FMPS representation, we consider a two-mode state with squeezing parameters $r_1 = 1, r_2 = -1$. We choose the two-dimensional wave function to be real-valued, and apply a beam splitting gate, effectively rotating the wave function in the $(q_1, q_2)$ space, as plotted in for $\theta = 0, \pi/4$ and $\pi/2$. (b) Rotating the squeezed two-mode state on $N=200$ points in each dimension, we see that the bond dimension required to reach $0.999$ gate fidelity peaks at $\theta = \pi/4$. At either extreme the state reduces to a product state, corresponding to bond dimension 1. (c) The singular values at $\theta = \pi/4$ fall off exponentially fast, meaning that we can achieve a high fidelity approximation with low bond dimension.