R.H.M. Smit
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The Cauchy problem on a lightcone differs from the one on a smooth spacelike surface in several ways; the initial surface cannot be smooth everywhere, and the initial surface being characteristic also has implications for the values the initial data may take. We show well-posedness of the vacuum Cauchy problem for a smooth spacelike initial surface in arbitrary dimension. The initial data for this problem is related to initial data for the characteristic Cauchy problem. We derive the Raychaudhuri equation for lightlike geodesics and use it to analyze initial data for the Cauchy problem on a lightcone.
We show how to derive the extrinsic curvature of a lightcone using a field of connection coefficients and the metric. We discuss the Raychaudhuri equation as a constraint equation if one uses the metric as initial data. Alternatively, we show how to use the Raychaudhuri equation to complete initial data when a conformal class of metrics is used as initial data instead. In this case, we give an exact expression for the conformal factor associated with a representative of the conformal class. ...
We show how to derive the extrinsic curvature of a lightcone using a field of connection coefficients and the metric. We discuss the Raychaudhuri equation as a constraint equation if one uses the metric as initial data. Alternatively, we show how to use the Raychaudhuri equation to complete initial data when a conformal class of metrics is used as initial data instead. In this case, we give an exact expression for the conformal factor associated with a representative of the conformal class. ...
The Cauchy problem on a lightcone differs from the one on a smooth spacelike surface in several ways; the initial surface cannot be smooth everywhere, and the initial surface being characteristic also has implications for the values the initial data may take. We show well-posedness of the vacuum Cauchy problem for a smooth spacelike initial surface in arbitrary dimension. The initial data for this problem is related to initial data for the characteristic Cauchy problem. We derive the Raychaudhuri equation for lightlike geodesics and use it to analyze initial data for the Cauchy problem on a lightcone.
We show how to derive the extrinsic curvature of a lightcone using a field of connection coefficients and the metric. We discuss the Raychaudhuri equation as a constraint equation if one uses the metric as initial data. Alternatively, we show how to use the Raychaudhuri equation to complete initial data when a conformal class of metrics is used as initial data instead. In this case, we give an exact expression for the conformal factor associated with a representative of the conformal class.
We show how to derive the extrinsic curvature of a lightcone using a field of connection coefficients and the metric. We discuss the Raychaudhuri equation as a constraint equation if one uses the metric as initial data. Alternatively, we show how to use the Raychaudhuri equation to complete initial data when a conformal class of metrics is used as initial data instead. In this case, we give an exact expression for the conformal factor associated with a representative of the conformal class.
This thesis investigates the optimality of the result achieved in an article by Becker-Slote-Volber-Zhang [1]. The research presented in this article concerns approximating the operator norm of local Hamiltonians.
We consider a Hermitian operator A acting on a complex Hilbert space of dimension 2^n. It has been shown that when A is a local n-qubit Hamiltonian, its operator norm can be approximated independently of n. This is done by maximizing |⟨ψ|A|ψ⟩| over a relatively small collection X_n of product states ψ ∈ (C^2)^(⊗n), called a quantum norm design. More exactly, whenever A is d-local, or deg(A) ≤ d, we have the following inequality:
∥A∥ ≤ C(d) max |⟨ψ|A|ψ⟩|.
Here, the approximation constant C(d) is only dependent on d, and the quantum norm design is independent of A.
In the article by Becker-Slote-Volber-Zhang [1], they prove this holds for two constants, C(d) = (3/2)(3 + 3√2)^d for d-local A and C(d) = 3^d for the case of d-homogeneous A.
We investigate the optimality of these constants using computer simulations. We first simulate using their norm design in the 2-local case for random Hamiltonians. Using optimization methods, we find local optima, giving us a lower bound for the approximation constant C(d), where we obtained the following lower bound:
5 ≤ C(d) ≤ 9,
where the upper bound of 9 comes from the constant achieved in [1].
Afterwards, we investigate the possibility of further improvement by adding entangled states to the norm design. We specifically add multiple variations of the maximally entangled Bell states to the norm design. We again use computer simulations and optimization methods in order to find local optima, where we obtained the following lower bound: ...
We consider a Hermitian operator A acting on a complex Hilbert space of dimension 2^n. It has been shown that when A is a local n-qubit Hamiltonian, its operator norm can be approximated independently of n. This is done by maximizing |⟨ψ|A|ψ⟩| over a relatively small collection X_n of product states ψ ∈ (C^2)^(⊗n), called a quantum norm design. More exactly, whenever A is d-local, or deg(A) ≤ d, we have the following inequality:
∥A∥ ≤ C(d) max |⟨ψ|A|ψ⟩|.
Here, the approximation constant C(d) is only dependent on d, and the quantum norm design is independent of A.
In the article by Becker-Slote-Volber-Zhang [1], they prove this holds for two constants, C(d) = (3/2)(3 + 3√2)^d for d-local A and C(d) = 3^d for the case of d-homogeneous A.
We investigate the optimality of these constants using computer simulations. We first simulate using their norm design in the 2-local case for random Hamiltonians. Using optimization methods, we find local optima, giving us a lower bound for the approximation constant C(d), where we obtained the following lower bound:
5 ≤ C(d) ≤ 9,
where the upper bound of 9 comes from the constant achieved in [1].
Afterwards, we investigate the possibility of further improvement by adding entangled states to the norm design. We specifically add multiple variations of the maximally entangled Bell states to the norm design. We again use computer simulations and optimization methods in order to find local optima, where we obtained the following lower bound: ...
This thesis investigates the optimality of the result achieved in an article by Becker-Slote-Volber-Zhang [1]. The research presented in this article concerns approximating the operator norm of local Hamiltonians.
We consider a Hermitian operator A acting on a complex Hilbert space of dimension 2^n. It has been shown that when A is a local n-qubit Hamiltonian, its operator norm can be approximated independently of n. This is done by maximizing |⟨ψ|A|ψ⟩| over a relatively small collection X_n of product states ψ ∈ (C^2)^(⊗n), called a quantum norm design. More exactly, whenever A is d-local, or deg(A) ≤ d, we have the following inequality:
∥A∥ ≤ C(d) max |⟨ψ|A|ψ⟩|.
Here, the approximation constant C(d) is only dependent on d, and the quantum norm design is independent of A.
In the article by Becker-Slote-Volber-Zhang [1], they prove this holds for two constants, C(d) = (3/2)(3 + 3√2)^d for d-local A and C(d) = 3^d for the case of d-homogeneous A.
We investigate the optimality of these constants using computer simulations. We first simulate using their norm design in the 2-local case for random Hamiltonians. Using optimization methods, we find local optima, giving us a lower bound for the approximation constant C(d), where we obtained the following lower bound:
5 ≤ C(d) ≤ 9,
where the upper bound of 9 comes from the constant achieved in [1].
Afterwards, we investigate the possibility of further improvement by adding entangled states to the norm design. We specifically add multiple variations of the maximally entangled Bell states to the norm design. We again use computer simulations and optimization methods in order to find local optima, where we obtained the following lower bound:
We consider a Hermitian operator A acting on a complex Hilbert space of dimension 2^n. It has been shown that when A is a local n-qubit Hamiltonian, its operator norm can be approximated independently of n. This is done by maximizing |⟨ψ|A|ψ⟩| over a relatively small collection X_n of product states ψ ∈ (C^2)^(⊗n), called a quantum norm design. More exactly, whenever A is d-local, or deg(A) ≤ d, we have the following inequality:
∥A∥ ≤ C(d) max |⟨ψ|A|ψ⟩|.
Here, the approximation constant C(d) is only dependent on d, and the quantum norm design is independent of A.
In the article by Becker-Slote-Volber-Zhang [1], they prove this holds for two constants, C(d) = (3/2)(3 + 3√2)^d for d-local A and C(d) = 3^d for the case of d-homogeneous A.
We investigate the optimality of these constants using computer simulations. We first simulate using their norm design in the 2-local case for random Hamiltonians. Using optimization methods, we find local optima, giving us a lower bound for the approximation constant C(d), where we obtained the following lower bound:
5 ≤ C(d) ≤ 9,
where the upper bound of 9 comes from the constant achieved in [1].
Afterwards, we investigate the possibility of further improvement by adding entangled states to the norm design. We specifically add multiple variations of the maximally entangled Bell states to the norm design. We again use computer simulations and optimization methods in order to find local optima, where we obtained the following lower bound:
Since their theoretical introduction in 1980, quantum computers have progressed significantly in both theory and experimentation. Quantum computers have the potential to solve certain problems significantly faster than their classical counterparts. One particularly promising area is search algorithms. In 1996, Lov Grover introduced a quantum search algorithm that laid the foundation for numerous variants. Among them, the alternating phase-walk stands out as a method for searching an element in an ordered list, which is modeled as a graph. However, a major challenge in practical quantum computing remains quantum decoherence.
The objective of this project was to model decoherence as bond percolation on a star graph during an alternating phase-walk, and to investigate its resulting effects. The introduction of decoherence transformed the search algorithm from a deterministic process into a probabilistic one, as the probability of measuring the marked state is no longer guaranteed to be 100% on every run of the algorithm. The most significant results were observed when varying the walk time in the continuous-time quantum walk (CTQW), both with and without a varying initial state. Here, the optimal time topt and its corresponding maximum average value μmax are presented for each value of p.
...
The objective of this project was to model decoherence as bond percolation on a star graph during an alternating phase-walk, and to investigate its resulting effects. The introduction of decoherence transformed the search algorithm from a deterministic process into a probabilistic one, as the probability of measuring the marked state is no longer guaranteed to be 100% on every run of the algorithm. The most significant results were observed when varying the walk time in the continuous-time quantum walk (CTQW), both with and without a varying initial state. Here, the optimal time topt and its corresponding maximum average value μmax are presented for each value of p.
...
Since their theoretical introduction in 1980, quantum computers have progressed significantly in both theory and experimentation. Quantum computers have the potential to solve certain problems significantly faster than their classical counterparts. One particularly promising area is search algorithms. In 1996, Lov Grover introduced a quantum search algorithm that laid the foundation for numerous variants. Among them, the alternating phase-walk stands out as a method for searching an element in an ordered list, which is modeled as a graph. However, a major challenge in practical quantum computing remains quantum decoherence.
The objective of this project was to model decoherence as bond percolation on a star graph during an alternating phase-walk, and to investigate its resulting effects. The introduction of decoherence transformed the search algorithm from a deterministic process into a probabilistic one, as the probability of measuring the marked state is no longer guaranteed to be 100% on every run of the algorithm. The most significant results were observed when varying the walk time in the continuous-time quantum walk (CTQW), both with and without a varying initial state. Here, the optimal time topt and its corresponding maximum average value μmax are presented for each value of p.
The objective of this project was to model decoherence as bond percolation on a star graph during an alternating phase-walk, and to investigate its resulting effects. The introduction of decoherence transformed the search algorithm from a deterministic process into a probabilistic one, as the probability of measuring the marked state is no longer guaranteed to be 100% on every run of the algorithm. The most significant results were observed when varying the walk time in the continuous-time quantum walk (CTQW), both with and without a varying initial state. Here, the optimal time topt and its corresponding maximum average value μmax are presented for each value of p.
This thesis investigates the peeling-off property of zero rest-mass fields in asymptotically flat space-times, as described by Penrose. Unlike other literature, this work is designed to be accessible to undergraduate physics or mathematics students. It provides a more detailed derivation including numerous explicit calculations, which is appropriate for the assumed background knowledge. The thesis is structured to build from fundamental mathematical concepts to advanced applications in general relativity. The mathematical concepts introduced are topology, compactifications and differential geometry. The main body of the work applies these mathematical tools to the Minkowski space-time and extends the analysis to more general asymptotically flat space-times.
A zero rest-mass field of spin s determines at each event in space-time a set of 2s principal null directions. These are related to the radiative behaviour of the field. These directions exhibit the 'peeling-off' behaviour: to order r-k-1 (k = 0, ..., 2s), 2s-k directions coincide radially, where r is an affine parameter on a null geodesic. Criteria for asymptotically simple and asymptotically flat space-times are given, and this peeling-off behaviour is studied in these settings. This involves the introduction of points 'at infinity' through a conformal completion. These points at infinity then become an ordinary hypersurface ℑ to the conformally completed manifold. The conformal transformations of zero rest-mass fields are investigated so that their behaviour at infinity can be studied at this hypersurface. If the transformed field is continuous at ℑ, we find that the peeling-off property holds. If the Einstein empty-space equations without cosmological constant hold near the boundary, the transformed gravitational field is found to be continuous at the boundary, so that the peeling-off property holds. ...
A zero rest-mass field of spin s determines at each event in space-time a set of 2s principal null directions. These are related to the radiative behaviour of the field. These directions exhibit the 'peeling-off' behaviour: to order r-k-1 (k = 0, ..., 2s), 2s-k directions coincide radially, where r is an affine parameter on a null geodesic. Criteria for asymptotically simple and asymptotically flat space-times are given, and this peeling-off behaviour is studied in these settings. This involves the introduction of points 'at infinity' through a conformal completion. These points at infinity then become an ordinary hypersurface ℑ to the conformally completed manifold. The conformal transformations of zero rest-mass fields are investigated so that their behaviour at infinity can be studied at this hypersurface. If the transformed field is continuous at ℑ, we find that the peeling-off property holds. If the Einstein empty-space equations without cosmological constant hold near the boundary, the transformed gravitational field is found to be continuous at the boundary, so that the peeling-off property holds. ...
This thesis investigates the peeling-off property of zero rest-mass fields in asymptotically flat space-times, as described by Penrose. Unlike other literature, this work is designed to be accessible to undergraduate physics or mathematics students. It provides a more detailed derivation including numerous explicit calculations, which is appropriate for the assumed background knowledge. The thesis is structured to build from fundamental mathematical concepts to advanced applications in general relativity. The mathematical concepts introduced are topology, compactifications and differential geometry. The main body of the work applies these mathematical tools to the Minkowski space-time and extends the analysis to more general asymptotically flat space-times.
A zero rest-mass field of spin s determines at each event in space-time a set of 2s principal null directions. These are related to the radiative behaviour of the field. These directions exhibit the 'peeling-off' behaviour: to order r-k-1 (k = 0, ..., 2s), 2s-k directions coincide radially, where r is an affine parameter on a null geodesic. Criteria for asymptotically simple and asymptotically flat space-times are given, and this peeling-off behaviour is studied in these settings. This involves the introduction of points 'at infinity' through a conformal completion. These points at infinity then become an ordinary hypersurface ℑ to the conformally completed manifold. The conformal transformations of zero rest-mass fields are investigated so that their behaviour at infinity can be studied at this hypersurface. If the transformed field is continuous at ℑ, we find that the peeling-off property holds. If the Einstein empty-space equations without cosmological constant hold near the boundary, the transformed gravitational field is found to be continuous at the boundary, so that the peeling-off property holds.
A zero rest-mass field of spin s determines at each event in space-time a set of 2s principal null directions. These are related to the radiative behaviour of the field. These directions exhibit the 'peeling-off' behaviour: to order r-k-1 (k = 0, ..., 2s), 2s-k directions coincide radially, where r is an affine parameter on a null geodesic. Criteria for asymptotically simple and asymptotically flat space-times are given, and this peeling-off behaviour is studied in these settings. This involves the introduction of points 'at infinity' through a conformal completion. These points at infinity then become an ordinary hypersurface ℑ to the conformally completed manifold. The conformal transformations of zero rest-mass fields are investigated so that their behaviour at infinity can be studied at this hypersurface. If the transformed field is continuous at ℑ, we find that the peeling-off property holds. If the Einstein empty-space equations without cosmological constant hold near the boundary, the transformed gravitational field is found to be continuous at the boundary, so that the peeling-off property holds.
This thesis is on characterizing the superconducting properties of NbTiN thin films under cryogenic conditions for potential applications in modular quantum computers. The characterization involves development of a measurement setup using a four-point probe method, which measures resistivityof magnetron sputtered thin films of NbTiN at room temperature and (cryogenic) temperature near ab- solute zero.
The results revealed unexpected resistivity values at 3.3 K, questioning the sample’s elemental com- position. Despite superconductivity not being reached by the setup, the research shows how impor- tant optimal elemental composition is and that the implementation of pulsed currents minimizes Joule Heating.
Despite not having characterized superconducting properties, taking steps back might be essential to take a step forward in developing a quantum computer.
This thesis was carried out at TU Delft as part of the Bachelor Final Project course. ...
The results revealed unexpected resistivity values at 3.3 K, questioning the sample’s elemental com- position. Despite superconductivity not being reached by the setup, the research shows how impor- tant optimal elemental composition is and that the implementation of pulsed currents minimizes Joule Heating.
Despite not having characterized superconducting properties, taking steps back might be essential to take a step forward in developing a quantum computer.
This thesis was carried out at TU Delft as part of the Bachelor Final Project course. ...
This thesis is on characterizing the superconducting properties of NbTiN thin films under cryogenic conditions for potential applications in modular quantum computers. The characterization involves development of a measurement setup using a four-point probe method, which measures resistivityof magnetron sputtered thin films of NbTiN at room temperature and (cryogenic) temperature near ab- solute zero.
The results revealed unexpected resistivity values at 3.3 K, questioning the sample’s elemental com- position. Despite superconductivity not being reached by the setup, the research shows how impor- tant optimal elemental composition is and that the implementation of pulsed currents minimizes Joule Heating.
Despite not having characterized superconducting properties, taking steps back might be essential to take a step forward in developing a quantum computer.
This thesis was carried out at TU Delft as part of the Bachelor Final Project course.
The results revealed unexpected resistivity values at 3.3 K, questioning the sample’s elemental com- position. Despite superconductivity not being reached by the setup, the research shows how impor- tant optimal elemental composition is and that the implementation of pulsed currents minimizes Joule Heating.
Despite not having characterized superconducting properties, taking steps back might be essential to take a step forward in developing a quantum computer.
This thesis was carried out at TU Delft as part of the Bachelor Final Project course.