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arXiv:2207.03085

Quantum Computing for Rotating, Charged and String Theory Black Holes

Viti Chandra, McGuigan, Michael

Published Jul 7, 2022• 0 citations
arXivHTMLPDF

Abstract

The quantum mechanics of Rotating, Charged, de Sitter and String Theory black holes are of recent interest because of their peculiar thermodynamic properties, as well the mysterious nature of their microstates. A full quantum treatment of the operators involved in this systems could yield valuable information into their nature, similar to how quantum treatment yields valuable insight into atoms, molecules and elementary particles. We study four types of black holes using quantum computing, which include the 3D Rotating Banados-Teitelboim-Zanelli (BTZ) black hole, the 4D charged Reisner-Nordtrom (RN) black hole, the 4D charged Reisner-Nordstrom -de Sitter (RN-dS) black hole and the 2D charged string black hole. In these cases in addition to the Hamiltonian there is a Mass operator which plays an important role in describing the quantum states of the black hole. We compute the spectrum of these operators using classical and quantum computing. For quantum computing we use the Variational Quantum Eigensolver (VQE) which is hybrid classical-quantum algorithm that runs on near term quantum hardware. We perform our calculations using 4 qubits in both a harmonic oscillator and position basis, realizing the quantum operators of the black holes in terms of 16 x 16 matrices. For the 4 qubit case we find highly accurate results for the Mass eigenvalues for different values of the charge and angular momentum. For the 2D Charged String black hole we also use the VQE to compute the expectation value of the Hamiltonian constraint and the commutator of the Hamiltonian constraint with the mass operator and find excellent agreement with theoretical expectations.

DOI
10.48550/arxiv.2207.03085

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The quantum mechanics of Rotating, Charged, de Sitter and String Theory black holes are of recent interest because of their peculiar thermodynamic properties, as well the mysterious nature of their microstates. A full quantum treatment of the operators involved in this systems could yield valuable information into their nature, similar to how quantum treatment yields valuable insight into atoms, molecules and elementary particles. We study four types of black holes using quantum computing, which include the 3D Rotating Banados-Teitelboim-Zanelli (BTZ) black hole, the 4D charged Reisner-Nordtrom (RN) black hole, the 4D charged Reisner-Nordstrom -de Sitter (RN-dS) black hole and the 2D charged string black hole. In these cases in addition to the Hamiltonian there is a Mass operator which plays an important role in describing the quantum states of the black hole. We compute the spectrum of these operators using classical and quantum computing. For quantum computing we use the Variational Quantum Eigensolver (VQE) which is hybrid classical-quantum algorithm that runs on near term quantum hardware. We perform our calculations using 4 qubits in both a harmonic oscillator and position basis, realizing the quantum operators of the black holes in terms of 16 x 16 matrices. For the 4 qubit case we find highly accurate results for the Mass eigenvalues for different values of the charge and angular momentum. For the 2D Charged String black hole we also use the VQE to compute the expectation value of the Hamiltonian constraint and the commutator of the Hamiltonian constraint with the mass operator and find excellent agreement with theoretical expectations.
The quantum mechanics of Schwarzschild-de Sitter black holes is of great recent interest because of their peculiar thermodynamic properties as well as their realization in modern dark energy cosmology which indicates the presence of a small positive cosmological constant. We study Schwarzschild-de Sitter black holes and also the Kantowki-Sachs Cosmology using quantum computing. In these cases in addition to the Hamiltonian there is a Mass operator which plays an important role in describing the quantum states of the black hole and Kantowski-Sachs cosmology. We compute the spectrum of these operators using classical and quantum computing. For quantum computing we use the Variational Quantum Eigensolver which is hybrid classical-quantum algorithm that runs on near term quantum hardware. We perform our calculations using 4, 6 and 8 qubits in a harmonic oscillator basis, realizing the quantum operators of the Schwarzschild-de Sitter black hole and Kantowski-Sachs cosmology in terms of 16 x 16, 64 x 64 and 256 x 256 matrices respectively. For the 4 qubit case we find highly accurate results but for the other cases we find a more refined variational ansatz will be necessary to represent the quantum states of a Schwarzschild-de Sitter black hole or Kantowki-Sachs cosmology accurately on a quantum computer.
The black-hole--qubit correspondence has been proven to be ``useful for obtaining additional insight into one of the string black hole theory and quantum information theory by exploiting approaches of the other"[Phys. Rev. D 82, 026003 (2010)]. Though different classes of stringy black holes can be related to the well-known stochastic local operations and classical communication (SLOCC) entanglement classes of pure states, the string theory requires a more detailed classification than the SLOCC classification of three qubits. In this paper, we derive the entanglement family of three qubits under local unitary operations (LU), and use the black-hole--qubit correspondence to classify \textquotedblleft large\textquotedblright\ black holes into seven inequivalent families. In particular, we show that two black holes with 4 non-vanishing charges (q0q_{0}q0​, p1p^{1}p1, p2p^{2}p2, and p3p^{3}p3) are LU equivalent if their difference is only in the signs of charges. Thus, the classification of black holes is independent of the signs of charges and is only related to the ratio of the absolute values of charges. This observation simplifies the classification task, as one would only need to consider either the classification of non-BPS black holes or the classification of BPS black holes, but not both. Moreover, through the LU classification, the physical basis for this black-hole--qubit correspondence can be observed, and a relation between the black-hole entropy and the von Neumann entanglement entropy is revealed. Therefore, the LU classification offers a more straightforward physical connection than the SLOCC classification. Based on the LU classification, we further study the properties of von Neumann entanglement entropy for each of the seven families, and find the black holes with the maximal von Neumann entanglement entropy.
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Quantum computers have an exponential speed-up advantage over classical computers. One of the most prominent utilities of quantum computers is their ability to study complex quantum systems in various fields using quantum computational algorithms. Quantum computational algorithms can be used to study cosmological systems and how they behave with variations in the different parameters of the system. Here, we use the variational quantum eigensolver (VQE) algorithm to simulate the Hawking radiation phenomenon. VQE algorithm is a combination of quantum and classical computation methods used to obtain the minimum energy eigenvalue for a given Hamiltonian. Three different custom ansatzes are used in the VQE algorithm from which the results for the case with minimum errors are studied. We obtain the plots for temperature and power from the minimum energy eigenvalue recorded for different values of mass and distance from the center of the black hole. The final result is then analyzed and compared against already existing data on Hawking radiation.
Analyzing complex quantum systems using quantum computational algorithms is one of the most promising applications of quantum computers. This study focuses on evaluating the performance of a custom variational ansatz in the Variational Quantum Eigensolver (VQE) algorithm compared to predefined ansatzes. To achieve this, we employ the evaporating black hole model as a test bed for our analysis. Using the VQE approach, which integrates quantum and classical computing techniques, we aim to minimize the energy expectation value of the Hamiltonian. By training the circuit parameters of a trial wave function as a parameterized quantum circuit, we determine the upper bound for the ground state energy and assess the optimal variational form. We define a custom ansatz for the VQE protocol and compare its performance with other predefined ansatzes. Additionally, we test the performance of three different classical optimizers to further understand their impact on the VQE algorithm’s efficiency and accuracy.
We review the recently established relationships between black hole entropy in string theory and the quantum entanglement of qubits and qutrits in quantum information theory. The first example is provided by the measure of the tripartite entanglement of three qubits (Alice, Bob and Charlie), known as the 3-tangle, and the entropy of the 8-charge ST U black hole of N = 2 supergravity, both of which are given by the [SL(2)] 3 invariant hyperdeterminant, a quantity first introduced by Cayley in 1845. Moreover the classification of three-qubit entanglements is related to the classification of N = 2 supersymmetric ST U black holes. There are further relationships between the attractor mechanism and local distillation protocols and between supersymmetry and the suppression of bit flip errors. At the microscopic level, the black holes are described by intersecting D3-branes whose wrapping around the six compact dimensions T 6 provides the string-theoretic interpretation of the charges and we associate the three-qubit basis vectors, |ABC〉 (A, B, C = 0 or 1), with the corresponding 8 wrapping cycles. The black hole/qubit correspondence extends to the 56 charge N = 8 black holes and the tripartite entanglement of seven qubits where the measure is provided by Cartan’s E7 ⊃ [SL(2)] 7 invariant. The qubits are naturally described by the seven vertices ABCDEF G of the Fano plane, which provides the multiplication table of the seven imaginary octonions,