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Optimal control theory for a unitary operation under dissipative evolution

Michael H. Goerz, Daniel M. Reich, Christiane P. Koch

Published Jan 1, 2014• 23 citations
OpenAlex

Abstract

We show that optimizing a quantum gate for an open quantum system requires the time evolution of only three states irrespective of the dimension of Hilbert space. This represents a significant reduction in computational resources compared to the complete basis of Liouville space that is commonly believed necessary for this task. The reduction is based on two observations: The target is not a general dynamical map but a unitary operation; and the time evolution of two properly chosen states is sufficient to distinguish any two unitaries. We illustrate gate optimization employing a reduced set of states for a controlled phasegate with trapped atoms as qubit carriers and a $\sqrt{i\text{SWAP}}$ gate with superconducting qubits.

Related papers

Optimal control theory for a unitary operation under dissipative evolution

Michael H Goerz, Daniel M Reich, Christiane P. Koch

We show that optimizing a quantum gate for an open quantum system requires the time evolution of only three states irrespective of the dimension of Hilbert space. This represents a significant reduction in computational resources compared to the complete basis of Liouville space that is commonly believed necessary for this task. The reduction is based on two observations: the target is not a general dynamical map but a unitary operation; and the time evolution of two properly chosen states is sufficient to distinguish any two unitaries. We illustrate gate optimization employing a reduced set of states for a controlled phasegate with trapped atoms as qubit carriers and a gate with superconducting qubits.

arXiv:1312.0111•New Journal of Physics•May 16, 2014•88 citations

Effectiveness of the Krotov method in controlling open quantum systems

Marllos E. Fonseca, F. F. Fanchini, Emanuel F. de Lima +1 more

We apply the Krotov method for open and closed quantum systems with the objective of finding optimized controls to manipulate qubit/qutrit systems in the presence of the external environment. In the case of unitary optimization, the Krotov method is first applied to a quantum system neglecting its interaction with the environment. The resulting controls from the unitary optimization are then used to drive the system along with the environmental noise. In the case of non-unitary optimization, the Krotov method already takes into account the noise during the optimization process. We consider two distinct computational task: target-state preparation and quantum gate implementation. These tasks are carried out in simple qubit/qutrit systems and also in systems presenting leakage states. For the state-preparation cases, the controls from the non-unitary optimization outperform the controls from the unitary optimization. However, as we show here, this is not always true for the implementation of quantum gates. There are some situations where the unitary optimization performs equally well compared to the non-unitary optimization. We verify that these situations corresponds to either the absence of leakage states or to the effects of dissipation being spread uniformly over the system, including non-computational levels. For such cases, the quantum gate implementation must cover the entire Hilbert space and there is no way to dodge dissipation. On the other hand, if the subspace containing the computational levels and its complement are differently affected by dissipation, the non-unitary optimization becomes effective.

arXiv:2208.03114•Aug 5, 2022•0 citations

Forcing a unitary transformation by an external field: comparing two approaches based on optimal control theory

José P. Palao, Ronnie Kosloff

A quantum gate is realized by specific unitary transformations operating on states representing qubits. Considering a quantum system employed as an element in a quantum computing scheme, the task is therefore to enforce the pre-specified unitary transformation. This task is carried out by an external time dependent field. Optimal control theory has been suggested as a method to compute the external field which alters the evolution of the system such that it performs the desire unitary transformation. This study compares two recent implementations of optimal control theory to find the field that induces a quantum gate. The first approach is based on the equation of motion of the unitary transformation. The second approach generalizes the state to state formulation of optimal control theory. This work highlight the formal relation between the two approaches.

arXiv:quant-ph/0208147•Aug 24, 2002•0 citations

Effectiveness of the Krotov method in finding controls for open quantum systems

Marllos E. F. Fernandes, F. F. Fanchini, Emanuel F. de Lima +1 more

Abstract We apply the Krotov method (KM) for open and closed quantum systems to find optimized controls to manipulate qubit/qutrit systems in the presence of the external environment. In the case of unitary optimization, the KM is first applied to a quantum system neglecting its interaction with the environment. The resulting controls from the unitary optimization are then used to drive the system along with the environmental noise. In the case of non-unitary optimization, the KM already takes into account the noise during the optimization process. We consider two distinct computational tasks: target-state preparation and quantum gate implementation. These tasks are carried out in simple qubit/qutrit systems and also in systems presenting leakage states. For the state preparation cases, the controls from the non-unitary optimization outperform the controls from the unitary optimization. However, as we show here, this is not always true for the implementation of quantum gates. There are some situations where the unitary optimization performs equally well compared to the non-unitary optimization. We verify that these situations correspond to either the absence of leakage states or to the effects of dissipation being spread uniformly over the system, including non-computational levels. For such cases, the quantum gate implementation must cover the entire Hilbert space and there is no way to dodge dissipation. On the other hand, if the subspace containing the computational levels and its complement are differently affected by dissipation, the non-unitary optimization becomes effective.

Journal of Physics A Mathematical and Theoretical•Nov 9, 2023•14 citations

Optimal Control for Open Quantum Systems: Qubits and Quantum Gates

R. Roloff, Markus Wenin, W. Pötz

This article provides a review of recent developments in the formulation and execution of optimal control strategies for the dynamics of quantum systems. A brief introduction to the concept of optimal control, the dynamics of of open quantum systems, and quantum information processing is followed by a presentation of recent developments regarding the two main tasks in this context: state-specific and state-independent optimal control. For the former, we present an extension of conventional theory (Pontryagin's principle) to quantum systems which undergo a non-Markovian time-evolution. Owing to its importance for the realization of quantum information processing, the main body of the review, however, is devoted to state-independent optimal control. Here, we address three different approaches: an approach which treats dissipative effects from the environment in lowest-order perturbation theory, a general method based on the time--evolution superoperator concept, as well as one based on the Kraus representation of the time-evolution superoperator. Applications which illustrate these new methods focus on single and double qubits (quantum gates) whereby the environment is modeled either within the Lindblad equation or a bath of bosons (spin-boson model). While these approaches are widely applicable, we shall focus our attention to solid-state based physical realizations, such as semiconductor- and superconductor-based systems. While an attempt is made to reference relevant and representative work throughout the community, the exposition will focus mainly on work which has emerged from our own group.

arXiv:0910.0362•Journal of Computational and Theoretical Nanoscience•Aug 1, 2009•39 citations

Time-optimal unitary operations

A. Carlini, Akio Hosoya, Tatsuhiko Koike +1 more

Extending our previous work on time-optimal quantum state evolution [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006)], we formulate a variational principle for finding the time-optimal realization of a target unitary operation, when the available Hamiltonians are subject to certain constraints dictated either by experimental or by theoretical conditions. Since the time-optimal unitary evolutions do not depend on the input quantum state this is of more direct relevance to quantum computation. We explicitly illustrate our method by considering the case of a two-qubit system self-interacting via an anisotropic Heisenberg Hamiltonian and by deriving the time-optimal unitary evolution for three examples of target quantum gates, namely, the swap of qubits, the quantum Fourier transform, and the entangler gate. We also briefly discuss the case in which certain unitary operations take negligible time.

arXiv:quant-ph/0608039•Physical Review A•Apr 10, 2007•126 citations
We show that optimizing a quantum gate for an open quantum system requires the time evolution of only three states irrespective of the dimension of Hilbert space. This represents a significant reduction in computational resources compared to the complete basis of Liouville space that is commonly believed necessary for this task. The reduction is based on two observations: The target is not a general dynamical map but a unitary operation; and the time evolution of two properly chosen states is sufficient to distinguish any two unitaries. We illustrate gate optimization employing a reduced set of states for a controlled phasegate with trapped atoms as qubit carriers and a iSWAP\sqrt{i\text{SWAP}}iSWAP​ gate with superconducting qubits.
We show that optimizing a quantum gate for an open quantum system requires the time evolution of only three states irrespective of the dimension of Hilbert space. This represents a significant reduction in computational resources compared to the complete basis of Liouville space that is commonly believed necessary for this task. The reduction is based on two observations: the target is not a general dynamical map but a unitary operation; and the time evolution of two properly chosen states is sufficient to distinguish any two unitaries. We illustrate gate optimization employing a reduced set of states for a controlled phasegate with trapped atoms as qubit carriers and a gate with superconducting qubits.
We apply the Krotov method for open and closed quantum systems with the objective of finding optimized controls to manipulate qubit/qutrit systems in the presence of the external environment. In the case of unitary optimization, the Krotov method is first applied to a quantum system neglecting its interaction with the environment. The resulting controls from the unitary optimization are then used to drive the system along with the environmental noise. In the case of non-unitary optimization, the Krotov method already takes into account the noise during the optimization process. We consider two distinct computational task: target-state preparation and quantum gate implementation. These tasks are carried out in simple qubit/qutrit systems and also in systems presenting leakage states. For the state-preparation cases, the controls from the non-unitary optimization outperform the controls from the unitary optimization. However, as we show here, this is not always true for the implementation of quantum gates. There are some situations where the unitary optimization performs equally well compared to the non-unitary optimization. We verify that these situations corresponds to either the absence of leakage states or to the effects of dissipation being spread uniformly over the system, including non-computational levels. For such cases, the quantum gate implementation must cover the entire Hilbert space and there is no way to dodge dissipation. On the other hand, if the subspace containing the computational levels and its complement are differently affected by dissipation, the non-unitary optimization becomes effective.
A quantum gate is realized by specific unitary transformations operating on states representing qubits. Considering a quantum system employed as an element in a quantum computing scheme, the task is therefore to enforce the pre-specified unitary transformation. This task is carried out by an external time dependent field. Optimal control theory has been suggested as a method to compute the external field which alters the evolution of the system such that it performs the desire unitary transformation. This study compares two recent implementations of optimal control theory to find the field that induces a quantum gate. The first approach is based on the equation of motion of the unitary transformation. The second approach generalizes the state to state formulation of optimal control theory. This work highlight the formal relation between the two approaches.
Abstract We apply the Krotov method (KM) for open and closed quantum systems to find optimized controls to manipulate qubit/qutrit systems in the presence of the external environment. In the case of unitary optimization, the KM is first applied to a quantum system neglecting its interaction with the environment. The resulting controls from the unitary optimization are then used to drive the system along with the environmental noise. In the case of non-unitary optimization, the KM already takes into account the noise during the optimization process. We consider two distinct computational tasks: target-state preparation and quantum gate implementation. These tasks are carried out in simple qubit/qutrit systems and also in systems presenting leakage states. For the state preparation cases, the controls from the non-unitary optimization outperform the controls from the unitary optimization. However, as we show here, this is not always true for the implementation of quantum gates. There are some situations where the unitary optimization performs equally well compared to the non-unitary optimization. We verify that these situations correspond to either the absence of leakage states or to the effects of dissipation being spread uniformly over the system, including non-computational levels. For such cases, the quantum gate implementation must cover the entire Hilbert space and there is no way to dodge dissipation. On the other hand, if the subspace containing the computational levels and its complement are differently affected by dissipation, the non-unitary optimization becomes effective.
This article provides a review of recent developments in the formulation and execution of optimal control strategies for the dynamics of quantum systems. A brief introduction to the concept of optimal control, the dynamics of of open quantum systems, and quantum information processing is followed by a presentation of recent developments regarding the two main tasks in this context: state-specific and state-independent optimal control. For the former, we present an extension of conventional theory (Pontryagin's principle) to quantum systems which undergo a non-Markovian time-evolution. Owing to its importance for the realization of quantum information processing, the main body of the review, however, is devoted to state-independent optimal control. Here, we address three different approaches: an approach which treats dissipative effects from the environment in lowest-order perturbation theory, a general method based on the time--evolution superoperator concept, as well as one based on the Kraus representation of the time-evolution superoperator. Applications which illustrate these new methods focus on single and double qubits (quantum gates) whereby the environment is modeled either within the Lindblad equation or a bath of bosons (spin-boson model). While these approaches are widely applicable, we shall focus our attention to solid-state based physical realizations, such as semiconductor- and superconductor-based systems. While an attempt is made to reference relevant and representative work throughout the community, the exposition will focus mainly on work which has emerged from our own group.
Extending our previous work on time-optimal quantum state evolution [Carlini et al., Phys. Rev. Lett. 96, 060503 (2006)], we formulate a variational principle for finding the time-optimal realization of a target unitary operation, when the available Hamiltonians are subject to certain constraints dictated either by experimental or by theoretical conditions. Since the time-optimal unitary evolutions do not depend on the input quantum state this is of more direct relevance to quantum computation. We explicitly illustrate our method by considering the case of a two-qubit system self-interacting via an anisotropic Heisenberg Hamiltonian and by deriving the time-optimal unitary evolution for three examples of target quantum gates, namely, the swap of qubits, the quantum Fourier transform, and the entangler gate. We also briefly discuss the case in which certain unitary operations take negligible time.