Phys. Rev. A 97, 033602 (2018) - Global optimization for quantum dynamics of few-fermion systems

Global optimization for quantum dynamics of few-fermion systems

Xikun Li, Daniel Pęcak, Tomasz Sowiński, Jacob Sherson, and Anne E. B. Nielsen

Phys. Rev. A 97, 033602 – Published 1 March 2018

Abstract

Quantum state preparation is vital to quantum computation and quantum information processing tasks. In adiabatic state preparation, the target state is theoretically obtained with nearly perfect fidelity if the control parameter is tuned slowly enough. As this, however, leads to slow dynamics, it is often desirable to be able to carry out processes more rapidly. In this work, we employ two global optimization methods to estimate the quantum speed limit for few-fermion systems confined in a one-dimensional harmonic trap. Such systems can be produced experimentally in a well-controlled manner. We determine the optimized control fields and achieve a reduction in the ramping time of more than a factor of four compared to linear ramping. We also investigate how robust the fidelity is to small variations of the control fields away from the optimized shapes.

Received 30 January 2018

DOI:https://doi.org/10.1103/PhysRevA.97.033602

Physics Subject Headings (PhySH)

Fermi gases Quantum control

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Xikun Li¹, Daniel Pęcak², Tomasz Sowiński², Jacob Sherson¹, and Anne E. B. Nielsen^3,*

¹Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark
²Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland
³Max-Planck-Institut für Physik komplexer Systeme, D-01187 Dresden, Germany

^*On leave from Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark.

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Vol. 97, Iss. 3 — March 2018

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Figure 1
Energy spectrum of the system of three fermions for three scenarios, namely, 1Li-2Li, 2K-1Li, and 1K-2Li. The thick orange line gives the ground-state energy. The noninteracting ground state $g = 0$ and the strongly correlated target state $g = 10$ are shown by the red circles. For the 1Li-2Li and the 2K-1Li systems, there are, respectively, two and one other states that have energies close to the ground-state energy in the strong coupling limit. These states do, however, have different symmetries than the ground state, and the population of these states remains 0 throughout the dynamics. The energies and the interaction strengths are measured in units of $ℏ ω$ and ${(ℏ^{3} ω / m_{↓})}^{1 / 2}$ , respectively.
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Figure 2
Fidelity versus duration for the 1K-2Li system using different control fields: optimized (solid blue line), exponential (dashed red line), and linear (dash-dotted black line). The curve for the optimized control field is obtained using the CMA-ES algorithm. The stars mark the shortest durations for which a fidelity of $F = 0.99$ is reached: $T_{Opt} = 2.7$ , $T_{Exp} = 6.5$ , and $T_{Lin} = 11$ (the time unit is $ω^{- 1}$ ).
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Figure 3
Control field for different durations in a 1K-2Li system. (a) $T = 0.5$ , (b) $T = 2.7$ , (c) $T = 5$ , and (d) $T = 15$ (the time unit is $ω^{- 1}$ ). Solid blue lines are the optimized control fields obtained using the CMA-ES method; dashed red lines, exponential rampings; and dash-dotted black lines, linear rampings.
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Figure 4
Comparing the CMA-ES and SaDE. Top left: The infidelity ( $IF = 1 - F$ ) as a function of the duration $T$ using the CMA-ES (squares) and SaDE (asterisks) for 1Li-2Li (solid blue line), 1K-2Li (dashed red line), and 2K-1Li (dash-dotted black line), where the vertical axis is on the logarithm scale. Top right: Red squares represent numerical data, which are obtained using the CMA-ES method, for the infidelity IF as a function of the duration $T$ for the 1K-2Li system, while solid blue lines show the fitting obtained using the function $k {cos}^{2} (\frac{T π}{2 β})$ for the numerical results, where $k$ and $β$ are free fitting parameters. Bottom: Comparisons of $log$ -infidelity versus iterations in the 1K-2Li system are shown between the CMA-ES (solid cyan line) and the SaDE (dashed magenta line) for durations $T = 2.6$ (bottom left) and $T = 5$ (bottom right), with the time unit $ω^{- 1}$ .
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Figure 5
Effects of deviations from the optimized control fields, which are obtained by mixing the optimized control field and linear ramping with a certain weight. Top: The deviation between the fidelity of the modified optimized control field $F_{w}$ and that of the optimized control field $F_{o}$ are shown as a function of the weight for $T = 2.6$ (left) and $T = 5$ (right), with the time unit $ω^{- 1}$ . Bottom: Both the optimized control field (solid blue line) and its modification whose weight is $10 %$ (dash-dotted red line) are shown for $T = 2.6$ (left) and $T = 5$ (right). The fidelity of the optimized control field $F_{o}$ and that of the modified one $F_{w}$ are also shown, and the difference between them is of order $10^{- 3}$ for both durations.
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Physical Review A

covering atomic, molecular, and optical physics and quantum science

Global optimization for quantum dynamics of few-fermion systems

Xikun Li, Daniel Pęcak, Tomasz Sowiński, Jacob Sherson, and Anne E. B. Nielsen

Phys. Rev. A 97, 033602 – Published 1 March 2018

Abstract

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Physical Review A

covering atomic, molecular, and optical physics and quantum science

Global optimization for quantum dynamics of few-fermion systems

Xikun Li, Daniel Pęcak, Tomasz Sowiński, Jacob Sherson, and Anne E. B. Nielsen

Phys. Rev. A 97, 033602 – Published 1 March 2018

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