# Ancilla-free continuous-variable SWAP test

T. J. Volkoff1 and Yiğit Subaşı2

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
2Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545 USA.

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### 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.

The SWAP test is a basic quantum algorithm for estimation of the fidelity, or overlap, between two quantum states. Originally introduced in the context of quantum fingerprinting, it is used in numerous applications including quantum machine learning. The original SWAP test required an ancilla register and Fredkin (controlled-SWAP) gates. Subsequently, an ancilla-free SWAP test was developed which also has a much simpler quantum circuit.

In this paper we propose a continuous-variable (CV) SWAP test that requires no ancilla register, thereby generalizing the ancilla-free SWAP test for qubits. The quantum circuit for the algorithm consists of a single layer of 50-50 beam splitter unitaries followed by photon number detection measurements. The overlap can be estimated by running this quantum circuit repeatedly and processing the measurement outcomes classically. In an ideal setting the estimator is unbiased. We upper-bound the systematic error that results when the photon detectors have a threshold beyond which they saturate in practice. We also upper-bound the number of times the circuit needs to be ran to achieve a desired overall error.

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. We also observe that the ancilla-free CV SWAP test along with the analogous algorithm for qubits can be applied in hybrid quantum systems, such as cavity quantum electrodynamics. Finally, we use the ancilla-free CV SWAP test to calculate the vacuum probability of a two-mode squeezed state on Xanadu’s 8-mode photonic processor.

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### Cited by

[1] Tyler J. Volkoff, "Strategies for Variational Quantum Compiling of a Zero-Phase Beam Splitter on the Xanadu X8 Processor", Journal of Russian Laser Research 43 4, 448 (2022).

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