2022
Alejandro Sopena; Max Hunter Gordon; Diego García-Martín; Germán Sierra; Esperanza López
Algebraic Bethe Circuits Journal Article
In: arXiv, 2022.
@article{https://doi.org/10.48550/arxiv.2202.04673,
title = {Algebraic Bethe Circuits},
author = {Alejandro Sopena and Max Hunter Gordon and Diego García-Martín and Germán Sierra and Esperanza López},
doi = {10.48550/ARXIV.2202.04673},
year = {2022},
date = {2022-01-01},
urldate = {2022-01-01},
journal = {arXiv},
publisher = {arXiv},
abstract = {The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. Here we bring the ABA to unitary form, for its direct implementation on a quantum computer. This is achieved by distilling the non-unitary R matrices that make up the ABA into unitaries using the QR decomposition. Our algorithm is deterministic and works for both real and complex roots of the Bethe equations. We illustrate our method in the spin-1/2 XX and XXZ models. We show that using this approach one can efficiently prepare eigenstates of the XX model on a quantum computer with quantum resources that match previous state-of-the-art approaches. We run numerical simulations, preparing eigenstates of the XXZ model for systems of up to 24 qubits and 12 magnons. Furthermore, we run small-scale error-mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models in 4 sites. Finally, we derive a new form of the Yang-Baxter equation using unitary matrices, and also verify it on a quantum computer.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. Here we bring the ABA to unitary form, for its direct implementation on a quantum computer. This is achieved by distilling the non-unitary R matrices that make up the ABA into unitaries using the QR decomposition. Our algorithm is deterministic and works for both real and complex roots of the Bethe equations. We illustrate our method in the spin-1/2 XX and XXZ models. We show that using this approach one can efficiently prepare eigenstates of the XX model on a quantum computer with quantum resources that match previous state-of-the-art approaches. We run numerical simulations, preparing eigenstates of the XXZ model for systems of up to 24 qubits and 12 magnons. Furthermore, we run small-scale error-mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models in 4 sites. Finally, we derive a new form of the Yang-Baxter equation using unitary matrices, and also verify it on a quantum computer.
2021
Manuel Campos; German Sierra; Esperanza Lopez
Tensor Renormalization Group for interacting quantum fields Journal Article
In: Quantum, vol. 5, pp. 586, 2021.
@article{Campos_2021,
title = {Tensor Renormalization Group for interacting quantum fields},
author = {Manuel Campos and German Sierra and Esperanza Lopez},
url = {https://arxiv.org/abs/2105.00010},
doi = {10.22331/q-2021-11-23-586},
year = {2021},
date = {2021-11-01},
urldate = {2021-11-01},
journal = {Quantum},
volume = {5},
pages = {586},
publisher = {Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften},
abstract = {We present a new tensor network algorithm for calculating the partition function of interacting quantum field theories in 2 dimensions. It is based on the Tensor Renormalization Group (TRG) protocol, adapted to operate entirely at the level of fields. This strategy was applied in Ref.[1] to the much simpler case of a free boson, obtaining an excellent performance. Here we include an arbitrary self-interaction and treat it in the context of perturbation theory. A real space analogue of the Wilsonian effective action and its expansion in Feynman graphs is proposed. Using a λϕ4 theory for benchmark, we evaluate the order λ correction to the free energy. The results show a fast convergence with the bond dimension, implying that our algorithm captures well the effect of interaction on entanglement.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We present a new tensor network algorithm for calculating the partition function of interacting quantum field theories in 2 dimensions. It is based on the Tensor Renormalization Group (TRG) protocol, adapted to operate entirely at the level of fields. This strategy was applied in Ref.[1] to the much simpler case of a free boson, obtaining an excellent performance. Here we include an arbitrary self-interaction and treat it in the context of perturbation theory. A real space analogue of the Wilsonian effective action and its expansion in Feynman graphs is proposed. Using a λϕ4 theory for benchmark, we evaluate the order λ correction to the free energy. The results show a fast convergence with the bond dimension, implying that our algorithm captures well the effect of interaction on entanglement.
Alejandro Sopena; Max Hunter Gordon; Germán Sierra; Esperanza López
Simulating quench dynamics on a digital quantum computer with data-driven error mitigation Journal Article
In: Quantum Science and Technology, vol. 6, no. 4, pp. 045003, 2021.
@article{Sopena_2021,
title = {Simulating quench dynamics on a digital quantum computer with data-driven error mitigation},
author = {Alejandro Sopena and Max Hunter Gordon and Germán Sierra and Esperanza López},
url = {https://arxiv.org/abs/2103.12680},
doi = {10.1088/2058-9565/ac0e7a},
year = {2021},
date = {2021-07-01},
urldate = {2021-07-01},
journal = {Quantum Science and Technology},
volume = {6},
number = {4},
pages = {045003},
publisher = {IOP Publishing},
abstract = {Error mitigation is likely to be key in obtaining near term quantum advantage. In this work we present one of the first implementations of several Clifford data regression based methods which are used to mitigate the effect of noise in real quantum data. We explore the dynamics of the 1-D Ising model with transverse and longitudinal magnetic fields, highlighting signatures of confinement. We find in general Clifford data regression based techniques are advantageous in comparison with zero-noise extrapolation and obtain quantitative agreement with exact results for systems of 9 qubits with circuit depths of up to 176, involving hundreds of CNOT gates. This is the largest systems investigated so far in a study of this type. We also investigate the two-point correlation function and find the effect of noise on this more complicated observable can be mitigated using Clifford quantum circuit data highlighting the utility of these methods.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Error mitigation is likely to be key in obtaining near term quantum advantage. In this work we present one of the first implementations of several Clifford data regression based methods which are used to mitigate the effect of noise in real quantum data. We explore the dynamics of the 1-D Ising model with transverse and longitudinal magnetic fields, highlighting signatures of confinement. We find in general Clifford data regression based techniques are advantageous in comparison with zero-noise extrapolation and obtain quantitative agreement with exact results for systems of 9 qubits with circuit depths of up to 176, involving hundreds of CNOT gates. This is the largest systems investigated so far in a study of this type. We also investigate the two-point correlation function and find the effect of noise on this more complicated observable can be mitigated using Clifford quantum circuit data highlighting the utility of these methods.
Ran He; Ming-Zhong Ai; Jin-Ming Cui; Yun-Feng Huang; Yong-Jian Han; Chuan-Feng Li; Guang-Can Guo; G. Sierra; C. E. Creffield
Riemann zeros from Floquet engineering a trapped-ion qubit Journal Article
In: npj Quantum Information, vol. 7, no. 1, 2021.
@article{He_2021,
title = {Riemann zeros from Floquet engineering a trapped-ion qubit},
author = {Ran He and Ming-Zhong Ai and Jin-Ming Cui and Yun-Feng Huang and Yong-Jian Han and Chuan-Feng Li and Guang-Can Guo and G. Sierra and C. E. Creffield},
url = {https://arxiv.org/abs/2102.06936},
doi = {10.1038/s41534-021-00446-7},
year = {2021},
date = {2021-07-01},
urldate = {2021-07-01},
journal = {npj Quantum Information},
volume = {7},
number = {1},
publisher = {Springer Science and Business Media LLC},
abstract = {The non-trivial zeros of the Riemann zeta function are central objects in number theory. In particular, they enable one to reproduce the prime numbers. They have also attracted the attention of physicists working in Random Matrix Theory and Quantum Chaos for decades. Here we present an experimental observation of the lowest non-trivial Riemann zeros by using a trapped ion qubit in a Paul trap, periodically driven with microwave fields. The waveform of the driving is engineered such that the dynamics of the ion is frozen when the driving parameters coincide with a zero of the real component of the zeta function. Scanning over the driving amplitude thus enables the locations of the Riemann zeros to be measured experimentally to a high degree of accuracy, providing a physical embodiment of these fascinating mathematical objects in the quantum realm.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The non-trivial zeros of the Riemann zeta function are central objects in number theory. In particular, they enable one to reproduce the prime numbers. They have also attracted the attention of physicists working in Random Matrix Theory and Quantum Chaos for decades. Here we present an experimental observation of the lowest non-trivial Riemann zeros by using a trapped ion qubit in a Paul trap, periodically driven with microwave fields. The waveform of the driving is engineered such that the dynamics of the ion is frozen when the driving parameters coincide with a zero of the real component of the zeta function. Scanning over the driving amplitude thus enables the locations of the Riemann zeros to be measured experimentally to a high degree of accuracy, providing a physical embodiment of these fascinating mathematical objects in the quantum realm.
Anne E. B. Nielsen; Benedikt Herwerth; J. Ignacio Cirac; Germán Sierra
Field tensor network states Journal Article
In: Physical Review B, vol. 103, no. 15, 2021.
@article{Nielsen_2021,
title = {Field tensor network states},
author = {Anne E. B. Nielsen and Benedikt Herwerth and J. Ignacio Cirac and Germán Sierra},
url = {https://arxiv.org/abs/2001.07723},
doi = {10.1103/physrevb.103.155130},
year = {2021},
date = {2021-04-01},
urldate = {2021-04-01},
journal = {Physical Review B},
volume = {103},
number = {15},
publisher = {American Physical Society (APS)},
abstract = {We define a class of tensor network states for spin systems where the individual tensors are functionals of fields. The construction is based on the path integral representation of correlators of operators in quantum field theory. These tensor network states are infinite dimensional versions of matrix product states and projected entangled pair states. We find the field-tensor that generates the Haldane-Shastry wave function and extend it to two dimensions. We give evidence that the latter underlies the topological chiral state described by the Kalmeyer-Laughlin wave function.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We define a class of tensor network states for spin systems where the individual tensors are functionals of fields. The construction is based on the path integral representation of correlators of operators in quantum field theory. These tensor network states are infinite dimensional versions of matrix product states and projected entangled pair states. We find the field-tensor that generates the Haldane-Shastry wave function and extend it to two dimensions. We give evidence that the latter underlies the topological chiral state described by the Kalmeyer-Laughlin wave function.
2020
D. García-Martín; E. Ribas; S. Carrazza; J. I. Latorre; G. Sierra
The Prime state and its quantum relatives Journal Article
In: Quantum, vol. 4, pp. 371, 2020.
@article{Garc_a_Mart_n_2020,
title = {The Prime state and its quantum relatives},
author = {D. García-Martín and E. Ribas and S. Carrazza and J. I. Latorre and G. Sierra},
url = {https://arxiv.org/abs/2005.02422},
doi = {10.22331/q-2020-12-11-371},
year = {2020},
date = {2020-12-01},
urldate = {2020-12-01},
journal = {Quantum},
volume = {4},
pages = {371},
publisher = {Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften},
abstract = {The Prime state of n qubits, |P_n⟩, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than 2n. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of |Pn⟩ up to n=30 qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the Prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The Prime state of n qubits, |P_n⟩, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than 2n. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of |Pn⟩ up to n=30 qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the Prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.
Sudipto Singha Roy; Silvia N. Santalla; Javier Rodríguez-Laguna; Germán Sierra
Entanglement as geometry and flow Journal Article
In: Physical Review B, vol. 101, no. 19, 2020.
@article{Singha_Roy_2020,
title = {Entanglement as geometry and flow},
author = {Sudipto Singha Roy and Silvia N. Santalla and Javier Rodríguez-Laguna and Germán Sierra},
url = {https://arxiv.org/abs/1906.05146},
doi = {10.1103/physrevb.101.195134},
year = {2020},
date = {2020-05-01},
urldate = {2020-05-01},
journal = {Physical Review B},
volume = {101},
number = {19},
publisher = {American Physical Society (APS)},
abstract = {We explore the connection between the area law for entanglement and geometry by representing the entanglement entropies corresponding to all 2^N bipartitions of an N-party pure quantum system by means of a (generalized) adjacency matrix. In the cases where the representation is exact, the elements of that matrix coincide with the mutual information between pairs of sites. In others, it provides a very good approximation, and in all the cases it yields a natural entanglement contour which is similar to previous proposals.Moreover, for one-dimensional conformal invariant systems, the generalized adjacency matrix is given by the two-point correlator of an entanglement current operator. We conjecture how this entanglement current may give rise to a metric entirely built from entanglement.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We explore the connection between the area law for entanglement and geometry by representing the entanglement entropies corresponding to all 2^N bipartitions of an N-party pure quantum system by means of a (generalized) adjacency matrix. In the cases where the representation is exact, the elements of that matrix coincide with the mutual information between pairs of sites. In others, it provides a very good approximation, and in all the cases it yields a natural entanglement contour which is similar to previous proposals.Moreover, for one-dimensional conformal invariant systems, the generalized adjacency matrix is given by the two-point correlator of an entanglement current operator. We conjecture how this entanglement current may give rise to a metric entirely built from entanglement.
2018
J. C. Xavier; F. C. Alcaraz; G. Sierra
Equipartition of the entanglement entropy Journal Article
In: Physical Review B, vol. 98, no. 4, 2018.
@article{Xavier_2018,
title = {Equipartition of the entanglement entropy},
author = {J. C. Xavier and F. C. Alcaraz and G. Sierra},
url = {https://arxiv.org/abs/1804.06357},
doi = {10.1103/physrevb.98.041106},
year = {2018},
date = {2018-07-01},
urldate = {2018-07-01},
journal = {Physical Review B},
volume = {98},
number = {4},
publisher = {American Physical Society (APS)},
abstract = {The entanglement in a quantum system that possess an internal symmetry, characterized by the Sz-magnetization or U(1)-charge, is distributed among different sectors. The aim of this letter is to gain a deeper understanding of the contribution to the entanglement entropy in each of those sectors for the ground state of conformal invariant critical one dimensional systems. We find surprisingly that the entanglement entropy is equally distributed among the different magnetization sectors. Its value is given by the standard area law violating logarithmic term, that depends on the central charge c, minus a double logarithmic correction related to the zero temperature susceptibility. This result provides a new method to estimate simultaneously the central charge c and the critical exponents of U(1)-symmetric quantum chains. The method is numerically simple and gives precise results for the spin-1/2 quantum XXZ chain. We also compute the probability distribution of the magnetization in contiguous sublattices.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The entanglement in a quantum system that possess an internal symmetry, characterized by the Sz-magnetization or U(1)-charge, is distributed among different sectors. The aim of this letter is to gain a deeper understanding of the contribution to the entanglement entropy in each of those sectors for the ground state of conformal invariant critical one dimensional systems. We find surprisingly that the entanglement entropy is equally distributed among the different magnetization sectors. Its value is given by the standard area law violating logarithmic term, that depends on the central charge c, minus a double logarithmic correction related to the zero temperature susceptibility. This result provides a new method to estimate simultaneously the central charge c and the critical exponents of U(1)-symmetric quantum chains. The method is numerically simple and gives precise results for the spin-1/2 quantum XXZ chain. We also compute the probability distribution of the magnetization in contiguous sublattices.
Erik Tonni; Javier Rodríguez-Laguna; Germán Sierra
Entanglement hamiltonian and entanglement contour in inhomogeneous 1D critical systems Journal Article
In: Journal of Statistical Mechanics: Theory and Experiment, vol. 2018, no. 4, pp. 043105, 2018.
@article{Tonni_2018,
title = {Entanglement hamiltonian and entanglement contour in inhomogeneous 1D critical systems},
author = {Erik Tonni and Javier Rodríguez-Laguna and Germán Sierra},
url = {https://arxiv.org/abs/1712.03557},
doi = {10.1088/1742-5468/aab67d},
year = {2018},
date = {2018-04-01},
urldate = {2018-04-01},
journal = {Journal of Statistical Mechanics: Theory and Experiment},
volume = {2018},
number = {4},
pages = {043105},
publisher = {IOP Publishing},
abstract = {Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints. A contour function for the entanglement entropies corresponding to this configuration is also considered, being closely related to the entanglement hamiltonian. The analytic expressions obtained by considering the curved spacetime which characterises the rainbow model have been checked against numerical data for the rainbow chain, finding an excellent agreement.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints. A contour function for the entanglement entropies corresponding to this configuration is also considered, being closely related to the entanglement hamiltonian. The analytic expressions obtained by considering the curved spacetime which characterises the rainbow model have been checked against numerical data for the rainbow chain, finding an excellent agreement.