Inspired by random walks on graphs, the diffusion map (DM) is a class of unsupervised machine learning that offers automatic identification of low-dimensional data structure hidden in a high-dimensional data set. In recent years, among its many applications, the DM has been successfully applied to discover relevant order parameters in many-body systems, enabling automatic classification of quantum phases of matter. However, a classical DM algorithm is computationally prohibitive for a large data set, and any reduction of the time complexity would be desirable. With a quantum computational speedup in mind, we propose a quantum algorithm for the DM, termed the quantum diffusion map (qDM). Our qDM takes as an input $N$ classical data vectors, performs an eigendecomposition of the Markov transition matrix in time $O({log}^{3}N)$ , and classically constructs the diffusion map via the readout (tomography) of the eigenvectors, giving a total expected runtime proportional to ${N}^{2}\text{polylog}\phantom{\rule{0.16em}{0ex}}N$ . Finally, quantum subroutines in the qDM for constructing a Markov transition matrix and for analyzing its spectral properties can also be useful for other random-walk-based algorithms.Inspired by random walks on graphs, the diffusion map (DM) is a class of unsupervised machine learning that offers automatic identification of low-dimensional data structure hidden in a high-dimensional data set. In recent years, among its many applications, the DM has been successfully applied to discover relevant order parameters in many-body systems, enabling automatic classification of quantum phases of matter. However, a classical DM algorithm is computationally prohibitive for a large data set, and any reduction of the time complexity would be desirable. With a quantum computational speedup in mind, we propose a quantum algorithm for the DM, termed the quantum diffusion map (qDM). Our qDM takes as an input $N$ classical data vectors, performs an eigendecomposition of the Markov transition matrix in time $O({log}^{3}N)$, and classically constructs the diffusion map via the readout (tomography) of the eigenvectors, giving a total expected runtime proportional to ${N}^{2}\text{polylog}\phantom{\rule{0.16em}{0ex}}N$. Finally, quantum subroutines in the qDM for constructing a Markov transition matrix and for analyzing its spectral properties can also be useful for other random-walk-based algorithms.

arXiv:2106.07302Physical Review ANov 10, 202114 citations

Dynamical Theory of Spectral Diffusion

Игорь Сергеевич Осадько

Springer series in chemical physicsJan 1, 20030 citations

Algebraic theory of quantum diffusions

R. L. Hudson

Lecture notes in mathematicsJan 1, 198812 citations

Transport and spectral properties of the XX+XXZ diode and stability to dephasing

Kang Hao Lee, Vinitha Balachandran, Chu Guo +1 more

We study the transport and spectral property of a segmented diode formed by an $XX+XXZ$ spin chain. This system has been shown to become an ideal rectifier for spin current for large enough anisotropy. Here we show numerical evidence that the system in reverse bias has signatures pointing toward the existence of three different transport regimes depending on the value of the anisotropy: ballistic, diffusive, and insulating. In forward bias we observe two regimes, ballistic and diffusive. The system in forward and reverse bias shows significantly different spectral properties, with distribution of rapidities converging toward different functions. In the presence of dephasing the system becomes diffusive, rectification is significantly reduced, the relaxation gap increases, and the spectral properties in forward and reverse bias tend to converge. For large dephasing the relaxation gap decreases again as a result of quantum Zeno physics.We study the transport and spectral property of a segmented diode formed by an $XX+XXZ$ spin chain. This system has been shown to become an ideal rectifier for spin current for large enough anisotropy. Here we show numerical evidence that the system in reverse bias has signatures pointing toward the existence of three different transport regimes depending on the value of the anisotropy: ballistic, diffusive, and insulating. In forward bias we observe two regimes, ballistic and diffusive. The system in forward and reverse bias shows significantly different spectral properties, with distribution of rapidities converging toward different functions. In the presence of dephasing the system becomes diffusive, rectification is significantly reduced, the relaxation gap increases, and the spectral properties in forward and reverse bias tend to converge. For large dephasing the relaxation gap decreases again as a result of quantum Zeno physics.

Physical review. EFeb 14, 20228 citations

Transport and spectral properties of the XX $+$ XXZ diode and stability to dephasingTransport and spectral properties of the XX$+$XXZ diode and stability to dephasing

Kang Hao Lee, Vinitha Balachandran, Chu Guo +1 more

We study the transport and spectral property of a segmented diode formed by an XX+XXZ spin chain. This system has been shown to become an ideal rectifier for spin current for large enough anisotropy. Here we show numerical evidence that the system in reverse bias has signatures pointing toward the existence of three different transport regimes depending on the value of the anisotropy: ballistic, diffusive, and insulating. In forward bias we observe two regimes, ballistic and diffusive. The system in forward and reverse bias shows significantly different spectral properties, with distribution of rapidities converging toward different functions. In the presence of dephasing the system becomes diffusive, rectification is significantly reduced, the relaxation gap increases, and the spectral properties in forward and reverse bias tend to converge. For large dephasing the relaxation gap decreases again as a result of quantum Zeno physics.We study the transport and spectral property of a segmented diode formed by an XX+XXZ spin chain. This system has been shown to become an ideal rectifier for spin current for large enough anisotropy. Here we show numerical evidence that the system in reverse bias has signatures pointing toward the existence of three different transport regimes depending on the value of the anisotropy: ballistic, diffusive, and insulating. In forward bias we observe two regimes, ballistic and diffusive. The system in forward and reverse bias shows significantly different spectral properties, with distribution of rapidities converging toward different functions. In the presence of dephasing the system becomes diffusive, rectification is significantly reduced, the relaxation gap increases, and the spectral properties in forward and reverse bias tend to converge. For large dephasing the relaxation gap decreases again as a result of quantum Zeno physics.

arXiv:2110.15564Oct 29, 20210 citations

Spectral Diffusion Mitigation with a Laser Pulse Sequence

Kilian Unterguggenberger, Alok Gokhale, Aleksei Tsarapkin +6 more

The optical spectrum of a quantum system is jointly determined by the properties of the emitter and the driving field. All-optical spectral control can hence be a promising method to engineer the properties of single photon emitters for quantum technological applications. It was proposed that driving a two-level system with a periodic sequence of optical pi-pulses during the excited state lifetime shifts the emission and absorption maximum to an arbitrarily detuned pulse carrier frequency, enabling the mitigation of spectral diffusion in noisy emitters. In this article, we report on the first experimental observation of this effect. We implement the protocol on a solid-state emitter and reduce its inhomogeneously broadened optical linewidth close to the lifetime limit. By detuning the excitation laser, we are able to concentrate approximately half of the absorption to a freely selectable target frequency. Our approach is solely based on properties of coherently evolving quantum systems, rendering it applicable to a wide range of individual and ensembles of quantum emitters.The optical spectrum of a quantum system is jointly determined by the properties of the emitter and the driving field. All-optical spectral control can hence be a promising method to engineer the properties of single photon emitters for quantum technological applications. It was proposed that driving a two-level system with a periodic sequence of optical pi-pulses during the excited state lifetime shifts the emission and absorption maximum to an arbitrarily detuned pulse carrier frequency, enabling the mitigation of spectral diffusion in noisy emitters. In this article, we report on the first experimental observation of this effect. We implement the protocol on a solid-state emitter and reduce its inhomogeneously broadened optical linewidth close to the lifetime limit. By detuning the excitation laser, we are able to concentrate approximately half of the absorption to a freely selectable target frequency. Our approach is solely based on properties of coherently evolving quantum systems, rendering it applicable to a wide range of individual and ensembles of quantum emitters.

quant-pharXiv:2604.21659Apr 24, 20260 citations

N

classical data vectors, performs an eigendecomposition of the Markov transition matrix in time

O({log}^{3}N)

, and classically constructs the diffusion map via the readout (tomography) of the eigenvectors, giving a total expected runtime proportional to

{N}^{2}\text{polylog}\phantom{\rule{0.16em}{0ex}}N

. Finally, quantum subroutines in the qDM for constructing a Markov transition matrix and for analyzing its spectral properties can also be useful for other random-walk-based algorithms.

We study the transport and spectral property of a segmented diode formed by an

XX+XXZ

spin chain. This system has been shown to become an ideal rectifier for spin current for large enough anisotropy. Here we show numerical evidence that the system in reverse bias has signatures pointing toward the existence of three different transport regimes depending on the value of the anisotropy: ballistic, diffusive, and insulating. In forward bias we observe two regimes, ballistic and diffusive. The system in forward and reverse bias shows significantly different spectral properties, with distribution of rapidities converging toward different functions. In the presence of dephasing the system becomes diffusive, rectification is significantly reduced, the relaxation gap increases, and the spectral properties in forward and reverse bias tend to converge. For large dephasing the relaxation gap decreases again as a result of quantum Zeno physics.

Transport and spectral properties of the XX

+

XXZ diode and stability to dephasing