Variational Monte Carlo

We begin our description of the Quantum Monte Carlo variants with the Variational Monte Carlo (VMC), the most transparent application of the ideas described in previous sections. This algorithm is in the borderline that divides the classical methods from the genuine quantum simulations. Although quantum in nature, it is not the action of the model that is sampled, but a trial wave function.

The main ingredient is a trial wave function $\vert\psi _{T}(\overline{\eta }% )\rangle$, that depends on a set of parameters $\overline{\eta }$. This wave function is represented in terms of a basis of orthogonal states $% \vert x\rangle$

$$\vert\psi _{T}(\overline{\eta })\rangle =\sum_{x}\langle x\v... ...vert x\rangle =\sum_{x}C_{x}(\overline{\eta })\vert x\rangle ,$$

where the coefficients of the parametrization are known functions of $% \overline{\eta }$. We would like the wave function to be a good representation of the actual ground state of a model. Finding the best wave function means finding the right set of parameters $\overline{\eta }$ that maximize the overlap with the actual ground state. In practice this is impossible since we do not know the groud state a priori, and some physical insight is needed to derive a good analytical approximation. Then we apply the variational principle, that states that the variational energy of the trial state is always greater or equal to the exact energy of the ground state:

$$\langle E\rangle _{T}=\frac{\langle \psi _{T}\mid H\mid \psi... ...rangle }{% \langle \psi _{T}\mid \psi _{T}\rangle }\geq E_{0},$$ (278)

and we use the criterium of minimizing the variational energy. In order to do that we require to calculate this quantity for different sets of parameters $\overline{\eta }$, and once we found a proper wave function, we can calculate the physical quantities of interest.

The expectation value of an arbitrary operator $O$ is

$$\langle O\rangle _{Var} = \frac{\langle \psi _{T}\mid O\mid \psi _{T}\rangle }{\langle \psi _{T}\mid \psi _{T}\rangle }$$

$$= \frac{\sum_{x}\langle \psi _{T}\mid x\rangle \langle x\mid O\mid ... ...angle }{\sum_{x}\langle \psi _{T}\mid x\rangle \langle x\mid \psi _{T}\rangle }$$ (279)

$$= \frac{\sum_{x}\left\vert \langle \psi _{T}\mid x\rangle \right\ve... ...\rangle }{\sum_{x}\left\vert \langle \psi _{T}\mid x\rangle \right\vert ^{2}% }$$

$$= \sum_{x}P_{x}O_{x},$$ (280)

with

$$P_{x}=\frac{\left\vert \langle \psi _{T}\mid x\rangle \right... ...}\left\vert \langle \psi _{T}\mid x\rangle \right\vert ^{2}},$$ (281)

and

$$O_{x}=\frac{\langle x\mid O\mid \psi _{T}\rangle }{\langle x\mid \psi _{T}\rangle }.$$ (282)

The equation (282) has precisely the form of a mean value in statistical mechanics, with $P_{x}$ as the Boltzmann factor:

$$P_{x}\geq 0;\,\sum_{x}P_{x}=1.$$

The first step in order to calculate it consists of generating a collection of configurations distributed according to this probability. For that purpose we employ the Metropolis algorithm: starting from a configuration $% \vert x\rangle$, we accept a new configuration $\vert x^{\prime }\rangle$ with probability $R=\left\vert \langle \psi _{T}\mid x^{\prime }\rangle \right\vert ^{2}/\left\vert \langle \psi _{T}\mid x\rangle \right\vert ^{2}$, or $1/(1+R)$ if we use the heat bath approach.

The variational simulations are simple to perform and very stable. Since the probabilities do not deppend on the statistics of the particles involved, they do not suffer from the sign problem. However, the results deppend decisively of the quality of the variational wave function, because they are completely pre-determined by it, and the physical arguments that define it. In the particular situation in which the trial function coincides with the exact ground state, the matrix elements (284) for $O=H$ are all equal to $% E_{0}:$

$$E_{x}=\frac{\langle x\mid H\mid \psi _{T}\rangle }{\langle x\mid \psi _{T}\rangle }=E_{0}.$$

This is the property called ``zero variance'': the more the wave function resembles the actual ground state, the more rapidly the variational enery converges with the number of iterations.

In general, the computation results more complicated, or numerically more expensive, when the number of variational parameters $\overline{\eta }$ that define the trial state is large. Thus, we always try to keep the form of the wave function simple enough and with only a few variational degrees of freedom.