Bath-Optimized Minimal-Energy Protection of Quantum Operations from Decoherence

We put forward a general strategy for dynamic control that ensures bath-optimized fidelity of a desired multidimensional quantum operation in the presence of non-Markovian baths and noises with stationary autocorrelations. It benefits from the vast freedom of arbitrary, not just pulsed, time-dependent control. This allows the dramatic reduction of the invested energy and the corresponding error compared to pulsed control.

Figure 1

Spectral overlaps between bath spectra $G_i(\omega )$ (solid red), modulation spectra $F_i^{\mathrm{OPT}}(\omega )$ (dashed green) for an optimized (BOMEC) storage (identity, 0) gate at $E_S=130.5$ [(a),(b),(c)], and $E_S=181.1$ [(d),(e),(f)], respectively, and modulation spectra $F_i^{\mathrm{QDD}}$ for pulse sequences [3] QDD4 [(a),(b),(c)] and QDD3 [(d),(e),(f)] (dotted blue), with $i=1$, 2, 3 corresponding to $X$, $Y$, and $Z$ component, respectively. Graphs (a),(b),(c) represent an Ohmic bath spectrum with softened cutoff, whereas graphs (d),(e),(f) represent a Lorentzian spectrum with a dip. The optimal (BOMEC) modulation spectra $F_i^{\mathrm{OPT}}(\omega )$ are always anticorrelated with the bath spectra $G_i(\omega )$. By contrast, the DD pulse-sequence spectra are only anticorrelated with $G_i$ for Ohmic baths (a),(b),(c) but not for the bath spectra (d),(e),(f). The same is true for CUDD sequences [4].

Figure 2

Qubit gate error $\overline{⟨\Delta \widehat{\varrho }(t)⟩}$ scaled to unmodulated error, as a function of the energy constraint $E_S$ in units of the energy scale of each method (see text). Solid magenta (dashed green): optimized (BOMEC) storage (identity, $B0$) and flip ($B\pi$) gates, Solid blue: periodic $X\mathrm{\text{−}}Z$ “bang bang” ($2,4,\dots ,30$ pulses). Separate points: DD pulse sequences [3, 4]. (a) and (b) correspond to bath spectra shown on the left and right in Fig. 1. The energy scale of DD (top) is more than 2 orders of magnitude higher than that of BOMEC (bottom).

Figure 3

Relative surplus error ${\overline{⟨\Delta \widehat{\varrho }(t)⟩}}_{\mathrm{rel}}=({\overline{⟨\Delta \widehat{\varrho }(t)⟩}}_{\mathrm{with}}-{\overline{⟨\Delta \widehat{\varrho }(t)⟩}}_{\mathrm{without}})/{\overline{⟨\Delta \widehat{\varrho }(t)⟩}}_{\mathrm{without}}$ incurred by the inevitable leakage to the additional level $|3⟩$ (inset) caused by the control as a function of the energy $E_S$, i.e., error with allowance for leakage compared to the error without (disregarding) leakage. Solid magenta: optimized (BOMEC) storage (identity, $B0$) gate. Dashed blue: periodic $X\mathrm{\text{−}}Z$ “bang bang” ($2,4,\dots ,30$ pulses) and QDD as shown in Fig. 2a and a bath spectrum as shown in Fig. 1 (left). A (truncated) $1/\omega$ bath-coupling spectrum describes an amplitude coupling between levels $|2⟩$ and $|3⟩$ (leakage). The scales of $E_S$ are as in Fig. 2.