trial wavefunction optimised by VMC was the starting point for DMC [@casino; @qmcmethods; @qmcreview], which treats the Schrödinger equation as a diffusion equation in imaginary time and evolves the wavefunction to project out our best estimate of the ground state [@dmcintro].

We probe the superconducting state by measuring the expectation values of the momentum density and the condensate fraction. The momentum density is the fourier transform of the one-body density matrix and is defined as $n_{{\mathbf{k}},\sigma} \equiv \langle{c^{\dagger}_{{\mathbf{k}}, \sigma}}{c^{}_{{\mathbf{k}}, \sigma}}\rangle$. The condensate fraction is a modified form of the two-body density matrix and is defined explicitly as $f_{{\mathbf{q}}} \equiv \sum_{\mathbf{k}}(\langle{c^{\dagger}_{{\mathbf{k}}, \uparrow}}{c^{\dagger}_{{\mathbf{q}}-{\mathbf{k}}, \downarrow}}{c^{}_{{\mathbf{q}}-{\mathbf{k}}, \downarrow}}{c^{}_{{\mathbf{k}}, \uparrow}}\rangle - n_{{\mathbf{k}},\uparrow}n_{{\mathbf{q}}-{\mathbf{k}},\downarrow})$. For the BCS wavefunction, it evaluates to $f_{{\mathbf{q}}} = \delta_{{\mathbf{q}},{\mathbf{0}}}\tfrac{{\lvert \Delta \rvert}}{8\pi}(\tan^{-1}\tfrac{\mu}{{\lvert \Delta \rvert}}+\tfrac{\pi}{2})\to\delta_{{\mathbf{q}},{\mathbf{0}}}\tfrac{{\lvert \Delta \rvert}}{8}$ for $\mu\gg{\lvert \Delta \rvert}$, providing an estimate of the superconducting gap ${\lvert \Delta \rvert}$.

Simulation setup and convergence {#sec:finitesizeerror}
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With the simulation methodology in place the final step is to set up the system. The major considerations are the size and shape of the simulation cell that could lead to finite size errors. Discussion of other simulation parameters including scattering length, pseudopotential cutoff length, DMC timestep, and number of DMC walkers is deferred to App. \[subsec:simparams\].

We 