Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0702240 · Physical Review A

Optimal two-qubit gate for generation of random bipartite entanglement

Marko Žnidarič

Published Jul 17, 2007• 38 citations
arXivHTMLPDF

Abstract

We numerically study protocols consisting of repeated applications of two qubit gates used for generating random pure states. A necessary number of steps needed in order to generate states displaying bipartite entanglement typical of random states is considered. Numerics indicates that for a generic two qubit entangling gate the decay of purity is exponential with the decay time scaling as $\ensuremath{\sim}n$, implying that of order $\ensuremath{\sim}{n}^{2}$ steps are needed to reach random bipartite entanglement. We also numerically identify the optimal two qubit gate for which the convergence is the fastest. Perhaps surprisingly, applying the same good two qubit gate in addition to a random single qubit rotations at each step leads to a faster generation of entanglement than applying a random two qubit transformation at each step.

DOI
10.1103/physreva.76.012318

Related papers

Efficient generation of random multipartite entangled states using time-optimal unitary operations

Antoni Borràs, A. P. Majtey, M. Casas

We review the generation of random pure states using a protocol of repeated two-qubit gates. We study the dependence of the convergence to states with Haar multipartite entanglement distribution. We investigate the optimal generation of such states in terms of the physical (real) time needed to apply the protocol, instead of the gate complexity point of view used in other works. This physical time can be obtained, for a given Hamiltonian, within the theoretical framework offered by the quantum brachistochrone formalism, the quantum analogue to the brachistochrone problem in classical mechanics [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006)]. Using an anisotropic Heisenberg Hamiltonian as an example, we find that different optimal quantum gates arise according to the optimality point of view used in each case. We also study how the convergence to random entangled states depends on different entanglement measures.

arXiv:0805.2070•Physical Review A•Aug 19, 2008•4 citations

Formation of multipartite entanglement using random quantum gates

Yonatan Most, Yishai Shimoni, Ofer Biham

The formation of multipartite quantum entanglement by repeated operation of one- and two-qubit gates is examined. The resulting entanglement is evaluated using two measures: the average bipartite entanglement and the Groverian measure. A comparison is made between two geometries of the quantum register: a one-dimensional chain in which two-qubit gates apply only locally between nearest neighbors and a nonlocal geometry in which such gates may apply between any pair of qubits. More specifically, we use a combination of random single-qubit rotations and a fixed two-qubit gate such as the controlled-phase gate. It is found that in the nonlocal geometry the entanglement is generated at a higher rate. In both geometries, the Groverian measure converges to its asymptotic value more slowly than the average bipartite entanglement. These results are expected to have implications on different proposed geometries of future quantum computers with local and nonlocal interactions between the qubits.

arXiv:0708.3481•Physical Review A•Aug 23, 2007•15 citations

Exact convergence times for generation of random bipartite entanglement

Marko Žnidarič

We calculate exact convergence times to reach random bipartite entanglement for various random protocols. The eigenproblem of a Markovian chain governing the process is mapped to a spin chain, thereby obtaining exact expression for the gap of the Markov chain for any number of qubits. For protocols coupling nearest-neighbor qubits and a controlled-NOT (CNOT) gate the mapping goes to the $XYZ$ model while for a U(4) gate it goes to an integrable $XY$ model. For coupling between a random pair of qubits the mapping is to an integrable Lipkin-Meshkov-Glick model. In all cases the gap scales inversely with the number of qubits, thereby improving on a recent bound [Phys. Rev. Lett. 98, 130502 (2007)].

arXiv:0809.0554•Physical Review A•Sep 22, 2008•72 citations

Generic Entanglement Can Be Generated Efficiently

Roberto I. Oliveira, Oscar Dahlsten, Martin Bodo Plenio

We find that generic entanglement is physical, in the sense that it can be generated in polynomial time from two-qubit gates picked at random. We prove as the main result that such a process generates the average entanglement of the uniform (unitarily invariant) measure in at most O(N3) steps for N qubits. This is despite an exponentially growing number of such gates being necessary for generating that measure fully on the state space. Numerics furthermore show a variation cutoff allowing one to associate a specific time with the achievement of the uniform measure entanglement distribution. Various extensions of this work are discussed. The results are relevant to entanglement theory and to protocols that assume generic entanglement can be achieved efficiently.

arXiv:quant-ph/0605126•Physical Review Letters•Mar 30, 2007•125 citations

Fastest Local Entanglement Scrambler, Multistage Thermalization, and a Non-Hermitian Phantom

Jaš Bensa, Marko Žnidarič

We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied.The problem is mapped to a Markovian process, and we prove that there are large spectral equivalence classes-different configurations have the same spectrum.Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified.Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity.In particular, non-Hermiticity can cause the rate to be either faster or, even more interestingly, slower than predicted by the matrix eigenvalue gap.This result is caused by expansion coefficients that grow exponentially with system size, resulting in a "phantom" eigenvalue and is due to nonorthogonality of non-Hermitian eigenvectors.We numerically demonstrate that the phenomenon also occurs in random circuits with nonoptimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance in other non-Hermitian settings, including correlations.

arXiv:2101.05579•Physical Review X•Jul 23, 2021•36 citations

Fastest local entanglement scrambler, multistage thermalization, and a non-Hermitian phantom

Jaš Bensa, Marko Žnidarič

We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied. The problem is mapped to a Markovian process, and we prove that there are large spectral equivalence classes—different configurations have the same spectrum. Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified. Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity. In particular, non-Hermiticity can cause the rate to be either faster or, even more interestingly, slower than predicted by the matrix eigenvalue gap. This result is caused by expansion coefficients that grow exponentially with system size, resulting in a “phantom” eigenvalue and is due to nonorthogonality of non-Hermitian eigenvectors. We numerically demonstrate that the phenomenon also occurs in random circuits with nonoptimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance in other non-Hermitian settings, including correlations.

•Mar 9, 2023•29 citations
We numerically study protocols consisting of repeated applications of two qubit gates used for generating random pure states. A necessary number of steps needed in order to generate states displaying bipartite entanglement typical of random states is considered. Numerics indicates that for a generic two qubit entangling gate the decay of purity is exponential with the decay time scaling as \ensuremath∼n\ensuremath{\sim}n\ensuremath∼n, implying that of order \ensuremath∼n2\ensuremath{\sim}{n}^{2}\ensuremath∼n2 steps are needed to reach random bipartite entanglement. We also numerically identify the optimal two qubit gate for which the convergence is the fastest. Perhaps surprisingly, applying the same good two qubit gate in addition to a random single qubit rotations at each step leads to a faster generation of entanglement than applying a random two qubit transformation at each step.
We review the generation of random pure states using a protocol of repeated two-qubit gates. We study the dependence of the convergence to states with Haar multipartite entanglement distribution. We investigate the optimal generation of such states in terms of the physical (real) time needed to apply the protocol, instead of the gate complexity point of view used in other works. This physical time can be obtained, for a given Hamiltonian, within the theoretical framework offered by the quantum brachistochrone formalism, the quantum analogue to the brachistochrone problem in classical mechanics [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006)]. Using an anisotropic Heisenberg Hamiltonian as an example, we find that different optimal quantum gates arise according to the optimality point of view used in each case. We also study how the convergence to random entangled states depends on different entanglement measures.
The formation of multipartite quantum entanglement by repeated operation of one- and two-qubit gates is examined. The resulting entanglement is evaluated using two measures: the average bipartite entanglement and the Groverian measure. A comparison is made between two geometries of the quantum register: a one-dimensional chain in which two-qubit gates apply only locally between nearest neighbors and a nonlocal geometry in which such gates may apply between any pair of qubits. More specifically, we use a combination of random single-qubit rotations and a fixed two-qubit gate such as the controlled-phase gate. It is found that in the nonlocal geometry the entanglement is generated at a higher rate. In both geometries, the Groverian measure converges to its asymptotic value more slowly than the average bipartite entanglement. These results are expected to have implications on different proposed geometries of future quantum computers with local and nonlocal interactions between the qubits.
We calculate exact convergence times to reach random bipartite entanglement for various random protocols. The eigenproblem of a Markovian chain governing the process is mapped to a spin chain, thereby obtaining exact expression for the gap of the Markov chain for any number of qubits. For protocols coupling nearest-neighbor qubits and a controlled-NOT (CNOT) gate the mapping goes to the XYZXYZXYZ model while for a U(4) gate it goes to an integrable XYXYXY model. For coupling between a random pair of qubits the mapping is to an integrable Lipkin-Meshkov-Glick model. In all cases the gap scales inversely with the number of qubits, thereby improving on a recent bound [Phys. Rev. Lett. 98, 130502 (2007)].
We find that generic entanglement is physical, in the sense that it can be generated in polynomial time from two-qubit gates picked at random. We prove as the main result that such a process generates the average entanglement of the uniform (unitarily invariant) measure in at most O(N3) steps for N qubits. This is despite an exponentially growing number of such gates being necessary for generating that measure fully on the state space. Numerics furthermore show a variation cutoff allowing one to associate a specific time with the achievement of the uniform measure entanglement distribution. Various extensions of this work are discussed. The results are relevant to entanglement theory and to protocols that assume generic entanglement can be achieved efficiently.
We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied.The problem is mapped to a Markovian process, and we prove that there are large spectral equivalence classes-different configurations have the same spectrum.Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified.Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity.In particular, non-Hermiticity can cause the rate to be either faster or, even more interestingly, slower than predicted by the matrix eigenvalue gap.This result is caused by expansion coefficients that grow exponentially with system size, resulting in a "phantom" eigenvalue and is due to nonorthogonality of non-Hermitian eigenvectors.We numerically demonstrate that the phenomenon also occurs in random circuits with nonoptimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance in other non-Hermitian settings, including correlations.
We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied. The problem is mapped to a Markovian process, and we prove that there are large spectral equivalence classes—different configurations have the same spectrum. Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified. Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity. In particular, non-Hermiticity can cause the rate to be either faster or, even more interestingly, slower than predicted by the matrix eigenvalue gap. This result is caused by expansion coefficients that grow exponentially with system size, resulting in a “phantom” eigenvalue and is due to nonorthogonality of non-Hermitian eigenvectors. We numerically demonstrate that the phenomenon also occurs in random circuits with nonoptimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance in other non-Hermitian settings, including correlations.