Competition of decoherence and quantum speed limits for quantum-gate fidelity in the Jaynes-Cummings model (2024)

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  Figure 1

  Channel eigenerror of the quantum logical system operated by the JC model with Hamiltonian (2), for the reduced interaction time $\tau =\pi /2$. (a)Eigenerror as a function of the average number of photons in the initial state of the drive, for different values of $s\in [0.02,0.25]$ (colored solid lines). The dash-dot line corresponds to a Poisson distribution for the drive state, which has $s=1.$ (b)Eigenerror as a function of the Fano factor ${\mathrm{\Delta }n}^2/\overline{n}$ of the initial state of the drive, for different values of $\overline{n}\in [50,1000]$. The gray dashed line denotes the reference scaling of $1/\overline{n}$ (a)and $1/s$ (b), respectively.

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  Figure 2

  Channel eigenerror in concatenating a quantum operation. In the numerical simulations, the concatenated quantum gates are enabled by a JC Hamiltonian, and the drive is initialized in a binomial state with $\overline{n}=25$ and Fano factor $s={\mathrm{\Delta }n}^2/\overline{n}=0.2$. (a)The channel eigenerror grows monotonically with the number of concatenations, increasing faster for greater values of $\tau$. (b)Channel eigenerror as a function of the channel reduced time $\tau$. Each line corresponds to a fixed number $C$ of concatenations. See the main text for the explanation of the behavior of the channel eigenerror around $\tau =\pi /2$. (c)Channel eigenerror as a function of the concatenations number. Each line corresponds to a constant cumulative evolution time $C\tau$. For the same cumulative evolution time, we achieve a lower channel eigenerror for the concatenation of a larger number of gates (larger $C$) but with a shorter reduced interaction time $\tau$. However, this comes with a larger energetic cost; see Fig.3 for details.

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  Figure 3

  In each level curve, we present the eigenerror of implementing an $X$ gate by concatenating $C$ pulse shots—each implementing $\sqrt[C]{X}$—using the same total amount of energy (measured in total number of used driving photons, $\overline{n}$). Each shot is achieved using the JC interaction, with the driving system initialized in a coherent state with an average of $\overline{n}/C$ photons. Thus, each curve is associated with the drive energy $\overline{n}\hslash \omega$. For the same amount of total energy, using more shots increases on average the channel eigenerror.

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  Figure 4

  Channel eigenerror committed from concatenating $C$ identical logical operations operated by the JC evolution. The initial state of the drive is a binomial distribution with $\overline{n}=25,s={\mathrm{\Delta }n}^2/\overline{n}=1/2$, and the reduced interaction time $\tau$ is made to vary in the interval $[0,\pi /2]$. Solid line: numerical simulation; dashed line: analytical approximation of the channel eigenerror as provided by Eq.(D4).

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  Figure 5

  Comparison between the analytical (dashed lines) and numerical (solid lines) scaling of the channel eigenerror in concatenating $C$ logical operations enabled by the JC evolution. The numerical scaling is exactly the one in Fig.3, and the analytical prediction of the scaling is provided by Eq.(D5).

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