Quantum Control Landscapes for Generation of <i>H</i> and <i>T</i> Gates in an Open Qubit with Both Coherent and Environmental Drive
An important problem in quantum computation is the generation of single-qubit quantum gates such as Hadamard (<i>H</i>) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-10-01
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| Series: | Photonics |
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| Online Access: | https://www.mdpi.com/2304-6732/10/11/1200 |
| _version_ | 1797458065044275200 |
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| author | Vadim N. Petruhanov Alexander N. Pechen |
| author_facet | Vadim N. Petruhanov Alexander N. Pechen |
| author_sort | Vadim N. Petruhanov |
| collection | DOAJ |
| description | An important problem in quantum computation is the generation of single-qubit quantum gates such as Hadamard (<i>H</i>) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi><mo>/</mo><mn>8</mn></mrow></semantics></math></inline-formula> (<i>T</i>) gates, which are components of a universal set of gates. Qubits in experimental realizations of quantum computing devices are interacting with their environment. While the environment is often considered as an obstacle leading to a decrease in the gate fidelity, in some cases, it can be used as a resource. Here, we consider the problem of the optimal generation of <i>H</i> and <i>T</i> gates using coherent control and the environment as a resource acting on the qubit via incoherent control. For this problem, we studied the quantum control landscape, which represents the behavior of the infidelity as a functional of the controls. We considered three landscapes, with infidelities defined by steering between two, three (via Goerz–Reich–Koch approach), and four matrices in the qubit Hilbert space. We observed that, for the <i>H</i> gate, which is a Clifford gate, for all three infidelities, the distributions of minimal values obtained with a gradient search have a simple form with just one peak. However, for the <i>T</i> gate, which is a non-Clifford gate, the situation is surprisingly different—this distribution for the infidelity defined by two matrices also has one peak, whereas distributions for the infidelities defined by three and four matrices have two peaks, which might indicate the possible existence of two isolated minima in the control landscape. It is important that, among these three infidelities, only those defined with three and four matrices guarantee the closeness of the generated gate to a target and can be used as a good measure of closeness. We studied sets of optimized solutions for the most general and previously unexplored case of coherent and incoherent controls acting together and discovered that they form sub-manifolds in the control space, and unexpectedly, in some cases, two isolated sub-manifolds. |
| first_indexed | 2024-03-09T16:31:44Z |
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| institution | Directory Open Access Journal |
| issn | 2304-6732 |
| language | English |
| last_indexed | 2024-03-09T16:31:44Z |
| publishDate | 2023-10-01 |
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| spelling | doaj.art-c0e64a279f2741f78136bb646c7e5c192023-11-24T15:01:20ZengMDPI AGPhotonics2304-67322023-10-011011120010.3390/photonics10111200Quantum Control Landscapes for Generation of <i>H</i> and <i>T</i> Gates in an Open Qubit with Both Coherent and Environmental DriveVadim N. Petruhanov0Alexander N. Pechen1Department of Mathematical Methods for Quantum Technologies, Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Str., Moscow 119991, RussiaDepartment of Mathematical Methods for Quantum Technologies, Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Str., Moscow 119991, RussiaAn important problem in quantum computation is the generation of single-qubit quantum gates such as Hadamard (<i>H</i>) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi><mo>/</mo><mn>8</mn></mrow></semantics></math></inline-formula> (<i>T</i>) gates, which are components of a universal set of gates. Qubits in experimental realizations of quantum computing devices are interacting with their environment. While the environment is often considered as an obstacle leading to a decrease in the gate fidelity, in some cases, it can be used as a resource. Here, we consider the problem of the optimal generation of <i>H</i> and <i>T</i> gates using coherent control and the environment as a resource acting on the qubit via incoherent control. For this problem, we studied the quantum control landscape, which represents the behavior of the infidelity as a functional of the controls. We considered three landscapes, with infidelities defined by steering between two, three (via Goerz–Reich–Koch approach), and four matrices in the qubit Hilbert space. We observed that, for the <i>H</i> gate, which is a Clifford gate, for all three infidelities, the distributions of minimal values obtained with a gradient search have a simple form with just one peak. However, for the <i>T</i> gate, which is a non-Clifford gate, the situation is surprisingly different—this distribution for the infidelity defined by two matrices also has one peak, whereas distributions for the infidelities defined by three and four matrices have two peaks, which might indicate the possible existence of two isolated minima in the control landscape. It is important that, among these three infidelities, only those defined with three and four matrices guarantee the closeness of the generated gate to a target and can be used as a good measure of closeness. We studied sets of optimized solutions for the most general and previously unexplored case of coherent and incoherent controls acting together and discovered that they form sub-manifolds in the control space, and unexpectedly, in some cases, two isolated sub-manifolds.https://www.mdpi.com/2304-6732/10/11/1200incoherent controlcontrol by environmentopen quantum systemqubitquantum gate generationgradient method |
| spellingShingle | Vadim N. Petruhanov Alexander N. Pechen Quantum Control Landscapes for Generation of <i>H</i> and <i>T</i> Gates in an Open Qubit with Both Coherent and Environmental Drive Photonics incoherent control control by environment open quantum system qubit quantum gate generation gradient method |
| title | Quantum Control Landscapes for Generation of <i>H</i> and <i>T</i> Gates in an Open Qubit with Both Coherent and Environmental Drive |
| title_full | Quantum Control Landscapes for Generation of <i>H</i> and <i>T</i> Gates in an Open Qubit with Both Coherent and Environmental Drive |
| title_fullStr | Quantum Control Landscapes for Generation of <i>H</i> and <i>T</i> Gates in an Open Qubit with Both Coherent and Environmental Drive |
| title_full_unstemmed | Quantum Control Landscapes for Generation of <i>H</i> and <i>T</i> Gates in an Open Qubit with Both Coherent and Environmental Drive |
| title_short | Quantum Control Landscapes for Generation of <i>H</i> and <i>T</i> Gates in an Open Qubit with Both Coherent and Environmental Drive |
| title_sort | quantum control landscapes for generation of i h i and i t i gates in an open qubit with both coherent and environmental drive |
| topic | incoherent control control by environment open quantum system qubit quantum gate generation gradient method |
| url | https://www.mdpi.com/2304-6732/10/11/1200 |
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