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arXiv:2205.07171

Multi-state Swap Test Algorithm

Wen Liu, Han-Wen Yin, Zhi-Rao Wang, Wenqin Fan

Published May 15, 2022• 1 citation
arXivHTMLPDF

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 we designed a recursive quantum circuit to measure overlaps of multiple quantum states $|ϕ_1...ϕ_n\rangle$ concurrently with $O(n\log n)$ controlled-swap (CSWAP) gates and $O(\log n)$ ancillary qubits. This circuit enables us to get all pairwise overlaps among input quantum states $|\langleϕ_i|ϕ_j\rangle|^2$. 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.

DOI
10.48550/arxiv.2205.07171

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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.
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.
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.
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 propose the implementation of the swap test using a charge qubit in a double quantum dot. The swap test is a fundamental quantum subroutine in quantum machine learning and other applications for estimating the fidelity of two unknown quantum states by measuring an auxiliary qubit. Our proposal uses a controlled three-qubit gate, which is natural to quantum dot charge qubits. It allows us to implement a swap test with a circuit depth of six layers, and an estimated gate time of less than 3 ns, that is below the coherence time of double quantum dots. This work paves the way for enhancing the toolbox of quantum machine learning developments in semiconductor qubits.
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.
In this paper, we propose a quantum circuit for calculating the squared sum of the inner product of quantum states. The circuit is designed by the multi-qubits controlled-swapping gates, in which the initial state of each control qubit is [Formula: see text] and they are in the equilibrium superposition state after passing through some Hadamard gates. Then, according to the control rules, each basis state in the superposition state controls the corresponding quantum states pair to swap. Finally, the Hadamard gates are applied to the control qubits again, and the squared sum of the inner product of many pairs of quantum states can be obtained simultaneously by measuring only one control qubit. We investigate the application of this method in quantum images matching on a classical computer, and the experimental results verify the correctness of the proposed method.