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I am going through Quantum Adiabatic Evolution Algorithms with Different Paths by Farhi et al.

Here, the authors propose to add a special term to the adiabatic Hamiltonian so that the path of the adiabatic evolution is changed but the algorithm still succeeds. On page $5$, the authors consider an ansatz about the ground state of the adiabatic Hamiltonian. I would like to quote them here.

In this symmetric case the total Hamiltonian $\tilde{H}(s)$ given by (11) is a function of the total spin operators $S_x$, $S_y$, and $S_z$. For $n$ large, we make the ansatz that for each $s$ the ground state of $\tilde{H}(s)$ is a state $|\theta, \varphi\rangle$ that is an eigenstate of the total spin in the $(\theta, \varphi)$ direction $$ (\sin \theta \cos \varphi~ S_x + \sin \theta \sin \varphi~ S_y + \cos \theta ~S_z) |\theta, \varphi\rangle = \frac{n}{2}|\theta, \varphi\rangle \tag{20}$$

My question:

  1. What is the intuition basis of this ansatz?
  2. If I have a different adiabatic Hamiltonian, can I use similar intuition to come up with such an ansatz?
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    $\begingroup$ The spin is quite literally rotated in the theta-phi direction. You should be able to find analogous examples. $\endgroup$ – Count Iblis Jun 5 '16 at 20:12
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The given ansatz includes two assumptions.

(1) The approximate ground states is seeked on a subspace of the Hilbert space consisting of rotated versions of single constant vector. (This subspace is a $2$-sphere parametrized by a unit vector in $\mathbb{R}^3$

(2) The value of the spin projection in the direction of the unit vector is half of the number of spins.

The explanation is as follows:

A system of $n$ distiguishable spins lives on a Hilbert space of dimension $2^n$. The observables constitute of the set of Hermitian $2^n \times 2^n$ matrices which generate $U(2^n)$. However, the set of observables has another reprersentation as the universal enveloping algebra of $SU(2)$, which in the multipole basis which includes: The total spin generators: $$S_x = \sum_{i=1}^n \sigma_x^{(i)}$$ $$S_- = \sum_{i=1}^n \sigma_-^{(i)}$$ $$S_+ = \sum_{i=1}^n \sigma_+^{(i)}$$. The 5 quadrupole operators ($Q_0, Q_{\pm1}, Q_{\pm2})$ , for example $$Q_0 = \frac{1}{2}(2S_z^2-S_-S_+)$$ etc. When the number of spins becomes very large, the correlations becomes scaled by powers of $\frac{1}{n}$ with respect to the average values and the system will tend to behave classically. Please see the following review by Yaffe for further details.

The classical limit of the spin system, however, is not unique. It is a coadjoint orbit of the dynamical group generated by the minimal set of operators needed to distinguish between the system states. For example, if the system's spetrum fits into a representation of $SU(2)$, then in the classical limit , the phase space is a coadjoint orbit of $SU(2)$ which is the two spher $S^2$. This is the example given in the question, where the active subspace where the minimum of the Hamiltonian is seeked is $S^2$ and the corresponding states are rotated versions of some vector. The quantum dynamics will be identical to the clasical dynamics on $S^2$. In this case the only "active" operators are the total spins. The values of a commuting set of these operators will be the mean fields. All the higher multipoles will be just classical functions of the mean fields.

If on the other hand, the quadrupole operators are active, the dynamical group in this case will become $SU(3)$ with $3+5=8$ generators and the classical phase space will be a coadjoint orbit of $SU(3)$ which is not unique by itself and can be the complex projective space $\mathbb{C}P^2$ or the flag manifold $Fl_3$. In these cases in general, the quadrupole generators will be independent of the total spins and in general will recieve quantum corrections in addition to the classical contributions.

In general, the low energy limit will prefer smaller coadjoint orbits since then the Hamiltonian will include less quantum corections. In addition, the authors considered a case where the Hamiltonian is linear in the total spin generators which excludes the higher coadjoint orbits.

The second part of the ansatz is just based on the central limit theorem. We can consider a single spin component in the direction of the sphere unit vector as a classical random variable because it is the only commuting variable (classical bit). This random variable has an average of $\frac{0+1}{2} = \frac{1}{2}$ and standard deviation of $\frac{1}{\sqrt{2}}$. The average of $n$ independent spins will have an average of $\frac{1}{2}$ and a standard deviation of $\frac{1}{\sqrt{2n}}$. Thus its value in the large $n$ limit will be fixed at $\frac{1}{2}$. Thus the average of a sum of $n$ spins will be fixed at: $\frac{n}{2}$.

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  • $\begingroup$ is $\sigma_{\pm}$ defined as $\sigma_x \pm i \sigma_y$? Moreover, how are $I_\pm$ defined? $\endgroup$ – Omar Shehab Jun 8 '16 at 18:20
  • $\begingroup$ Sorry, I used different notations for the same variables. They are corrected now. $\endgroup$ – David Bar Moshe Jun 9 '16 at 1:32

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