# Noise Resilience

Magnetic resonance systems typically consist of a stable magnetic field, a tunable coil that excites and detects a magnetic resonance signal and a console containing a transmitter/receiver to deliver/capture magnetic resonance pulses. In real experimental apparatus, the control channels will include some level of noise as most electrical systems do. A resistance producing electrical fluctuations will generate heat  and noise resulting from these thermal fluctuations [2,3]. In simulating this noise, the solution is to create an ensemble of systems, each with their own instance of noise affecting the control channels. Ignoring relaxation the cost functional for this ensemble becomes $J=\frac{1}{N}\sum_{n=1}^N\left\langle \hat{\sigma}\right|\exp_{(0)}\left[-i\int_0^T\left(\hat{\hat{\mathcal{H}}}_0+\sum_k\zeta_{k,n}\hat{\hat{\mathcal{H}}}_k\right)dt\right]\left|\hat{\rho}(0)\right\rangle -K$

where $\zeta_{k,n}(t)=c_k(t)+\xi_{k,n}(t)$, with $\xi(t)$ being the delta-correlated noise requiring $\left\langle\xi(t)\xi(t^{\prime})\right\rangle=\eta\delta(t-t^{\prime})$, and $\eta$ is the variance of the noise, indicating noise strength relative to a nominal maximum power level . The optimality criterion for this ensemble becomes $\frac{1}{N}\sum_{i=1}^N\frac{\partial}{\partial c_n(t)} \left\langle\hat{\sigma}\right|\exp_{(0)}\left(-i\int\limits_0^T\left[\hat{\hat{\mathcal{H}}}+\sum_k\zeta_{k,n}(t)\hat{\hat{\mathcal{H}}}_k\right]dt\right)\left|\hat{\rho}_0\right\rangle=0$

## References

 C. Gardiner, Handbook of stochastic methods for physics, chemistry and the natural sciences, 3rd ed. (Springer-Verlag, 2003).
 J. B. Johnson, Phys. Rev. 32, 97 (1928).
 H. Nyquist, Phys. Rev. 32, 110 (1928).
 S. Kallush, M. Khasin, and R. Kosloff, New Journal of Physics 16, 015008 (2014).