Experimental Measurement of the Hilbert-Schmidt Distance between Two-Qubit States as a Means for Reducing the Complexity of Machine Learning

Abstract

We report on experimental measurement of the Hilbert-Schmidt distance between two two-qubit states by many-particle interference. We demonstrate that our three-step method for measuring distances in Hilbert space is far less complex than reconstructing density matrices and that it can be applied in quantum-enhanced machine learning to reduce the complexity of calculating Euclidean distances between multidimensional points, which can be especially interesting for near term quantum technologies and quantum artificial intelligence research. Our results are also a novel example of applying mixed states in quantum information processing. Usually working with mixed states is undesired, but here it gives the possibility of encoding extra information as coherence between given two dimensions of the density matrix.

Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance

“…Hilbert-Schmidt distance has been one of the prominent and natural choices for quantifying the separation between given two quantum states [33,49,. It provides a direct interpretation as an information distance between quantum states [63].…”

“…A recent example in this context is its implementation in the Gilbert algorithm [81] to construct entanglement witnesses for unextendible product basis bound entangled states [67,68]. Moreover, Hilbert-Schmidt distance has been utilized as a cost function in variational hybrid quantum-classical algorithms in machine learning and other applications [49,[69][70][71][72]. It has been regularly employed as an estimator in the precision quantum-state tomography [73][74][75][76].…”

Quantum-enhanced bosonic learning machine

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Quantum processors enable computational speedups for machine learning through parallel manipulation of high-dimensional vectors [1]. Early demonstrations of quantum machine learning have focused on processing information with qubits [2][3][4][5][6][7][8]. In such systems, a larger computational space is provided by the collective space of multiple physical qubits.

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“…Recently proposed quantum algorithms show evidences of computational speedups, that comes from the exploitation of large Hilbert space of quantum systems [12][13][14][15]. Demonstrations of quantum machine learning algorithms implemented on discrete variables platforms have shown tremendous potential [2][3][4][5][6][7]. However, the number of gates and qubits must be increased for solving practical problems on near-term quantum processors.…”

“…is the overlap between two quantum states |φ(x) and |φ(x ) [3]. The overlap measurement, reported throughout this work, can be efficiently realized with a constant-depth circuit implemented by a SWAP test [20][21][22][23].…”