We propose a continuous-variable (CV) SWAP test that requires no ancilla register, thereby generalizing the ancilla-free SWAP test for qubits. In this ancilla-free CV SWAP test, the computational basis measurement is replaced by photon number-resolving measurement, and we calculate an upper bound on the error of the overlap estimate obtained from a finite Fock cutoff in the detector. As an example, we show that estimation of the overlap of pure, centered, single-mode Gaussian states of energy $E$ and squeezed in opposite quadratures can be obtained to error $\epsilon$ using photon statistics below a Fock basis cutoff $O(E\ln \epsilon^{-1})$. This cutoff is greatly reduced to $E + O(\sqrt{E}\ln \epsilon^{-1})$ when the states have rapidly decaying Fock tails, such as coherent states. We show how the ancilla-free CV SWAP test can be extended to many modes and applied to quantum algorithms such as variational compiling and entanglement spectroscopy in the CV setting. For the latter we also provide a new algorithm which does not have an analog in qubit systems. The ancilla-free CV SWAP test is implemented on Xanadu's 8-mode photonic processor in order to estimate the vacuum probability of a two-mode squeezed state.

中文翻译：

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Ancilla-free 连续变量 SWAP 测试

我们提出了一种不需要辅助寄存器的连续变量 (CV) SWAP 测试，从而推广了量子比特的无辅助 SWAP 测试。在这个无辅助的 CV SWAP 测试中，计算基础测量被光子数解析测量所取代，我们计算了从探测器中的有限 Fock 截止获得的重叠估计误差的上限。作为一个例子，我们展示了使用低于 Fock 基截止值 $O 的光子统计量来估计纯的、中心的、单模高斯能量状态 $E$ 和在相反正交中压缩的重叠的估计误差 $\epsilon$ (E\ln \epsilon^{-1})$。当状态具有快速衰减的 Fock tail（例如相干状态）时，此截止值大大减少到 $E + O(\sqrt{E}\ln \epsilon^{-1})$。我们展示了如何将无辅助 CV SWAP 测试扩展到多种模式并应用于量子算法，例如 CV 设置中的变分编译和纠缠光谱。对于后者，我们还提供了一种在量子比特系统中没有模拟的新算法。无辅助 CV SWAP 测试在 Xanadu 的 8 模式光子处理器上实施，以估计双模式压缩状态的真空概率。