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arXiv:2103.10219

Experimental SWAP test of infinite dimensional quantum states

Chi Huan Nguyen, Ko-Wei Tseng, Gleb Maslennikov, Hongfeng Gan, Dzmitry Matsukevich

Published Mar 18, 2021• 7 citations
arXivHTMLPDF

Abstract

Efficient overlap estimation of high-dimensional quantum states is an important task in quantum information and a core element in computational speedups of quantum machine learning. Here we experimentally demonstrate the SWAP test that measures the overlap of two motional states in a system of trapped $^{171}\mathrm{Yb}^+$ ions. To illustrate the versatility of our implementation, we report the overlap measurement of a variety of quantum states: Fock states, coherent states, squeezed vacuum states, and cat states. We highlight applications of the SWAP test by measuring the purity of mixed states. Our results enable quantum information processing with high dimensional quantum states.

DOI
10.48550/arxiv.2103.10219

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Efficient overlap estimation of high-dimensional quantum states is an important task in quantum information and a core element in computational speedups of quantum machine learning. Here we experimentally demonstrate the SWAP test that measures the overlap of two motional states in a system of trapped 171Yb+^{171}\mathrm{Yb}^+171Yb+ ions. To illustrate the versatility of our implementation, we report the overlap measurement of a variety of quantum states: Fock states, coherent states, squeezed vacuum states, and cat states. We highlight applications of the SWAP test by measuring the purity of mixed states. Our results enable quantum information processing with high dimensional quantum states.
We study the estimation of the overlap between two unknown pure quantum states of a finite-dimensional system, given M and N copies of each type. This is a fundamental primitive in quantum information processing that is commonly accomplished from the outcomes of N swap tests, a joint measurement on one copy of each type whose outcome probability is a linear function of the squared overlap. We show that a more precise estimate can be obtained by allowing for general collective measurements on all copies. We derive the statistics of the optimal measurement and compute the optimal mean square error in the asymptotic pointwise and finite Bayesian estimation settings. Besides, we consider two strategies relying on the estimation of one or both states and show that, although they are suboptimal, they outperform the swap test. In particular, the swap test is extremely inefficient for small values of the overlap, which become exponentially more likely as the dimension increases. Finally, we show that the optimal measurement is less invasive than the swap test and study the robustness to depolarizing noise for qubit states.
Abstract Estimating the overlap between two states is an important task with several applications in quantum information. However, the typical swap test circuit can only measure a sole pair of quantum states at a time. In this study, a recursive quantum circuit is designed to measure overlaps of n quantum states $\left | {\phi _{1} } \right \rangle ,\left | {\phi _{2} } \right \rangle ,\ldots\left | {\phi _{n} }\right \rangle $ | ϕ 1 〉 , | ϕ 2 〉 , … | ϕ n 〉 concurrently with O(k2k)O(k2^{k})O(k2k) O ( k 2 k ) controlled-swap(CSWAP) gates and O(k)O(k)O(k) O ( k ) ancillary qubits, where $k=\left \lceil {\log n} \right \rceil $ k = ⌈ log n ⌉ . All pairwise overlaps among input quantum states ∣⟨ϕi∣ϕj⟩∣2|\langle \phi _{i}|\phi _{j}\rangle |^{2}∣⟨ϕi​∣ϕj​⟩∣2 | 〈 ϕ i | ϕ j 〉 | 2 can be obtained in this circuit. Compared with existing scheme for measuring the overlap of multiple quantum states, the circuit provides higher precision and less consumption of ancillary qubits. In addition, some simulation experiments are performed on IBM quantum cloud platform to verify the superiority of this algorithm.
Estimating the overlap between two states is an important task with several applications in quantum information. However, the typical swap test circuit can only measure a sole pair of quantum states at a time. In this study we designed a recursive quantum circuit to measure overlaps of multiple quantum states ∣ϕ1...ϕn⟩|ϕ_1...ϕ_n\rangle∣ϕ1​...ϕn​⟩ concurrently with O(nlog⁡n)O(n\log n)O(nlogn) controlled-swap (CSWAP) gates and O(log⁡n)O(\log n)O(logn) ancillary qubits. This circuit enables us to get all pairwise overlaps among input quantum states ∣⟨ϕi∣ϕj⟩∣2|\langleϕ_i|ϕ_j\rangle|^2∣⟨ϕi​∣ϕj​⟩∣2. Compared with existing schemes for measuring the overlap of multiple quantum states, our scheme provides higher precision and less consumption of ancillary qubits. In addition, we performed simulation experiments on IBM quantum cloud platform to verify the superiority of the scheme.
Accurately estimating the overlap between quantum states is a fundamental task in quantum information processing. While various strategies using distinct quantum measurements have been proposed for overlap estimation, the lack of experimental benchmarks on estimation precision limits strategy selection in different situations. Here we compare the performance of four practical strategies for overlap estimation, including tomography-tomography, tomography-projection, Schur collective measurement and optical swap test using photonic quantum systems. We encode the quantum states on the polarization and path degrees of freedom of single photons. The corresponding measurements are performed by photon detection on certain modes following single-photon mode transformation or two-photon interference. We further propose an adaptive strategy with optimized precision in full-range overlap estimation. Our results shed new light on extracting the parameter of interest from quantum systems, prompting the design of efficient quantum protocols.
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 ϵ using photon statistics below a Fock basis cutoff O(Eln⁡ϵ−1) . This cutoff is greatly reduced to E+O(Eln⁡ϵ−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.
Short-depth algorithms are crucial for reducing computational error on near-term quantum computers, for which decoherence and gate infidelity remain important issues. Here we present a machine-learning approach for discovering such algorithms. We apply our method to a ubiquitous primitive: computing the overlap between two quantum states ρ and σ . The standard algorithm for this task, known as the Swap Test, is used in many applications such as quantum support vector machines, and, when specialized to ρ = σ , quantifies the Renyi entanglement. Here, we find algorithms that have shorter depths than the Swap Test, including one that has a constant depth (independent of problem size). Furthermore, we apply our approach to the hardware-specific connectivity and gate sets used by Rigetti's and IBM's quantum computers and demonstrate that the shorter algorithms that we derive significantly reduce the error—compared to the Swap Test—on these computers.