Reconstructing a quantum state with a variational autoencoder

Quantum state tomography (QST) is an important and challenging task in the field of quantum information, which has attracted a lot of attentions in recent years. Machine learning models can provide a classical representation of the quantum state after trained on the measurement outcomes, which are part of effective techniques to solve QST problem. In this work, we use a variational autoencoder (VAE) to learn the measurement distribution of two quantum states generated by MPS circuits. We first consider the Greenberger–Horne–Zeilinger (GHZ) state which can be generated by a simple MPS circuit. Simulation results show that a VAE can reconstruct 3- to 8-qubit GHZ states with a high fidelity, i.e., 0.99, and is robust to depolarizing noise. The minimum number ([Formula: see text]) of training samples required to reconstruct the GHZ state up to 0.99 fidelity scales approximately linearly with the number of qubits ([Formula: see text]). However, for the quantum state generated by a complex MPS circuit, [Formula: see text] increases exponentially with [Formula: see text], especially for the quantum state with high entanglement entropy.

Phase diagram of solid hints new fundamental constant and sees atomic orbital

Cold and pressure transform gas into liquid and then into solid. Van der Waals understood the phase diagram of liquefiable gas with the molecular volume and intermolecular attraction, however, was silent on how solid behaved1. Unfortunately, solid-state phase diagram have remained uncomprehended mystery; only its straight boundary2,3 was explained by struggle of order vs. chaos. Here we show that the volume of orbital overlap has its own energy, with the universal density 8.941 eV/Å3 announced as new fundamental atomic constant that determines the transition temperature TC. Furthermore, we devised solid-state tomography, valid to 5 TPa, - imaging orbital through the baric dependencies of TC. Triangle-shaped pattern of the diagram is explained by the only possible way, just as only one plane passes through triangle: -inflation of the intersection volume during the transition determines hysteresis, but its disappearance does triple point; -approaching ions, whose orbitals overlap, curves the line from zero-field-cooling (ZFC) TC to triple point; -the straight line between zero-field-heating (ZFH) TC and triple point is a consequence of straightening tilting angle. Diamond melting point, calculated from volumes of the tetrahedral covalent bonds, excellently agrees with real; furthermore, the points up to 2 TPa agree with experiment4. Our findings open up way to interpret antiferromagnetism and steric effect in mono, binary, and ternary transition-metal oxides and sulfides5-11, and advance in unravelling unconventional superconductivity12,13, ascertaining the roles of s- and p-hybridizations. Thereby, the importance of the solid-state tomography for organic conductors12,13 being high-compressible and interior of stars can scarcely be exaggerated.

Quantum state tomography as a numerical optimization problem

We present a framework that formulates the quest for the most efficient quantum state tomography measurement set as an optimization problem which can be solved numerically, where the optimization goal is the maximization of the information gain. This approach can be applied to a broad spectrum of relevant setups including measurements restricted to a subsystem. To illustrate the power of this method we present results for the six-dimensional Hilbert space constituted by a qubit-qutrit system, which could be realized e.g. by the N-14 nuclear spin-1 and two electronic spin states of a nitrogen-vacancy center in diamond. Measurements of the qubit subsystem are expressed by projectors of rank three, i.e., projectors on half-dimensional subspaces. For systems consisting only of qubits, it was shown analytically that a set of projectors on half-dimensional subspaces can be arranged in an informationally optimal fashion for quantum state tomography, thus forming so-called mutually unbiased subspaces. Our method goes beyond qubits-only systems and we find that in dimension six such a set of mutually-unbiased subspaces can be approximated with a deviation irrelevant for practical applications.

Quantum state tomography of molecules by ultrafast diffraction

Ultrafast electron diffraction and time-resolved serial crystallography are the basis of the ongoing revolution in capturing at the atomic level of detail the structural dynamics of molecules. However, most experiments capture only the probability density of the nuclear wavepackets to determine the time-dependent molecular structures, while the full quantum state has not been accessed. Here, we introduce a framework for the preparation and ultrafast coherent diffraction from rotational wave packets of molecules, and we establish a new variant of quantum state tomography for ultrafast electron diffraction to characterize the molecular quantum states. The ability to reconstruct the density matrix, which encodes the amplitude and phase of the wavepacket, for molecules of arbitrary degrees of freedom, will enable the reconstruction of a quantum molecular movie from experimental x-ray or electron diffraction data.