Ancilla-free continuous-variable SWAP test

TL;DR

This paper introduces an ancilla-free continuous-variable SWAP test that uses photon number measurements to estimate state overlaps, enabling efficient quantum state comparison without additional quantum registers.

Contribution

It generalizes the ancilla-free SWAP test to continuous-variable systems, providing error bounds and practical implementations for Gaussian states and multi-mode applications.

Findings

Error bounds for overlap estimation with finite Fock cutoff

Efficient overlap estimation for Gaussian states with reduced cutoff

Successful implementation on a photonic processor to estimate vacuum probability

Abstract

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.