Update README.md

README.md

@@ -18,7 +18,7 @@ Figure 2: Statistical expectation of the energy with standard error of the mean
 
 ## Introduction
 
-Quantum mechanics emerges from classical mechanics by the relaxation of the requirement of commutativity among the observables as assumed by classical probability theory. The most immediate and striking consequence of this relaxation is the insufficiency of real-valued probability distributions to encode the interference phenomena observed by experiment. In response, we generalize the notion of probability distributions from real-valued distributions over the set of possible outcomes which combine convexly, to complex-valued distributions over the set of possible outcomes which combine linearly. The complex-valued probabilities are able to encode the observed interference patterns in their relative phases. Such quantum probability distributions do not describe mutually exclusive outcomes in which only one outcome exists prior to measurement, but rather describes outcomes in which all possible outcomes simultaneously exist prior to measurement and which interfere in a wave-like manner.
+Quantum mechanics emerges from classical mechanics by the relaxation of commutativity among the observables as assumed by classical probability theory. The most striking consequence of this relaxation is the insufficiency of real-valued probability distributions to encode the interference phenomena observed by experiment. In response, we generalize the notion of probability distributions from real-valued distributions over the set of possible outcomes which combine convexly, to complex-valued distributions over the set of possible outcomes which combine linearly. The complex-valued probabilities are able to encode the observed interference patterns in their relative phases. Such quantum probability distributions describe our knowledge of possible outcomes of measurements on systems which cannot be said to be in a definite state prior to measurement.
 
 The increase in predictive power offered by quantum mechanics came with the price of computational difficulties. Unlike the classical world, whose dimensionality scales additively with the number of subsystems, the dimensionality scaling of quantum systems is multiplicative. Thus, even small systems quickly become intractable without approximation techniques. Luckily, it is rarely the case that knowledge of the full state space is required to accurately model a given system, as most information may be contained in a relatively small subspace. Many of the most successful approximation techniques of the last century, such as Born–Oppenheimer and variational techniques like Density Functional Theory, rely on this convenient notion for their success. With the rapid development of machine learning, a field which specializes in dimensionality reduction and feature extraction of very large datasets, it is natural to apply these novel techniques for dealing with the canonical large data problem of the physical sciences.
 
@@ -52,7 +52,7 @@ where we define the variational local energies $E_{\text{loc}}(s,\alpha) = \sum_
 
 In this demonstration, we assume the prototypical Ising spin model for a one-dimensional lattice of spin $\frac{1}{2}$ particles, whose Hamiltonian is given by
  $$H = -J \sum_{j=1}^{n-1} \sigma_j^z \sigma_{j+1}^z - B \sum_{j=1}^n \sigma_x$$
-where we use the shorthand notation $\sigma_j^z \sigma_{j+1}^z = I^{(1)} \otimes \cdots \otimes \sigma_z^{(j)} \otimes \sigma_z^{(j+1)} \otimes \cdots \otimes I^{(n)}$ to denote the tensor product of the $2 \times 2$ identity matrix with the Pauli matrix $\sigma_z$ located at positions $j$ and $j+1$, and where $\sigma_x = I^{(1)} \otimes \cdots \otimes \sigma_x^{(j)} \otimes \cdots \otimes I^{(n)}$ denotes $\sigma_x$ at position $j$. The size of $H$ is $2^n \times 2^n$, and so it is impossible to directly diagonalize even for relatively few particles. The constant $J$ represents the nearest neighbor coupling strength, and $B$ represents the strength of the transverse field. When $J>0$, nearest neighbors tend to align parallel (ferromagnetic), and tend to align anti-parallel when $J<0$ (anti-ferromagnetic). The local energy of a configuration $s \in S$ in the Ising model can is seen to be
+where we use the shorthand notation $\sigma_j^z \sigma_{j+1}^z = I^{(1)} \otimes \cdots \otimes \sigma_z^{(j)} \otimes \sigma_z^{(j+1)} \otimes \cdots \otimes I^{(n)}$ to denote the tensor product of the $2 \times 2$ identity matrix with the Pauli matrix $\sigma_z$ located at positions $j$ and $j+1$, and where $\sigma_x = I^{(1)} \otimes \cdots \otimes \sigma_x^{(j)} \otimes \cdots \otimes I^{(n)}$ denotes $\sigma_x$ at position $j$. The size of $H$ is $2^n \times 2^n$, which would require an enormous amount of storage space even for relatively few particles, and would be impossible to diagonalize with finite computational resources. The constant $J$ represents the nearest neighbor coupling strength, and $B$ represents the strength of the transverse field. When $J>0$, nearest neighbors tend to align parallel (ferromagnetic), and tend to align anti-parallel when $J<0$ (anti-ferromagnetic). The local energy of a configuration $s \in S$ in the Ising model can be seen to be
  $$E_{\text{loc}}(s,\alpha) =-J \sum_{j=1}^{n-1} \sigma_j \sigma_{j+1} - B \sum_{s' \in S_f} \frac{\psi(s',\alpha)}{\psi(s,\alpha)}$$
 where $\sigma_j = -2s_j + 1 \in \\{1,-1\\}$ and $S_f$ consists of $n$ configurations in which a single spin of $s$ has been inverted.