## Reading group: Quantum algorithm for the Laughlin wave function, arXiv:0902.4797

In this post I’d like to experiment by sharing my thoughts on a recent paper as I read through it critically. I’m thinking of trying to emulate something like a reading group presentation.

While this isn’t original research, I’m sure that reading papers certainly does form an integral part of the workflow of any researcher: critically reading papers allows you to learn new ideas and techniques and, crucially, by asking difficult questions while reading a paper you often discover new research directions that, otherwise, would never occur to you. This has often happened to me. Indeed, the main inspiration for several of my papers has come from the critical evaluation of (and, sometimes, the resulting frustrations from reading) a recent paper.

Due mostly to time constraints I don’t really read that many papers these days (I tend to skim quite a few, but actually read only a couple a year). Nevertheless, I hope to do something like this post on a semi-regular/sporadic basis.

I’m going to be rather polite and not actually make any direct criticisms. I don’t see the point of being totally negative anyways: if there is a criticism it probably means there is some aspect of the paper that could be explored further. This, to my mind, equals a new research project. So, let’s be positive and consider every quibble — if there are any — as an opportunity.

The paper I’d like to discuss today is the latest from José-Ignacio Latorre and coworkers. This intriguing paper discusses how quantum computers could help prepare physically interesting quantum states.

As will be evident, I’m not an expert in the area of this paper and it is entirely possible that I’ll make several wrong statements. Any question I raise here is probably due to a misunderstanding on my behalf. Finally, in the true spirit of a reading group discussion, if you think you can answer any of my questions, or clarify the description anywhere then please do not hesitate to comment!

1. Introduction

The authors discuss the idea of developing exact quantum circuits to prepare physically relevant quantum states. They mention a paper by Benenti and Strini and this. Arguably, I think that this paper by König, Reichardt, and Vidal also fits within this category (and there are possibly more examples that I’m not aware of…). The idea here is to present efficient quantum circuits which could prepare, from a product state, say, the ground state of an interesting physical system. These states could be presumably then be studied by unleashing the power of a fully operational quantum computer to study, eg., their correlations etc.

2. Representing the Laughlin wave function

The main purpose of the paper is to study the quantum Hall effect (see here for starters), i.e., to show how a quantum computer could prepare quantum states, Laughlin states, which faithfully represent the physics of the quantum Hall ground state. As the quantum Hall effect pertains to a collection of fermions (electrons) confined in 2D by a strong magnetic field, we need to specify how we’ll represent the state. The way Laughlin did it was to write out the (Nobel prize winning) ground state for the QHE with filling fraction ${\nu = 1/m}$ (roughly speaking: with a density of ${\nu}$ electrons) in the position representation, where it reads

where the ${x}$ and ${y}$ position of an electron are encoded as ${z = x+iy}$. It’s pretty obvious that this state is fully antisymmetric under interchange of particles, so it’s a legal electron state. However, the position representation is pretty inconvenient for quantum circuits (implicit here are several continuous degrees of freedom, namely positions, which are a pain to represent in a quantum register). So what the authors do is identify a convenient basis with respect to which the Laughlin state requires only a linear combination of a finite number to specify. So infinite information is reduced to finite information at the expense of a more complicated basis. (This basis — the Fock-Darwin basis — isn’t that complicated, they are just the eigenstates of an electron in a parabolic potential in a magnetic field, so it’s pretty natural.)

To continue the authors restrict their attention to the ${m=1}$ setting, i.e., the integer quantum Hall effect. Also, since the Fock-Darwin basis is a basis for the single particle sector, they need to find a way to specify the quantum state for ${n}$ electrons with respect to this basis. Here the authors adopt a novel approach. My first guess would have been to write out the Laughlin state in second-quantised formalism where one specifies the occupation number of each possible basis state, i.e.,

where ${n_j}$ is the number of electrons in the ${j}$th Fock-Darwin state. But this would be trivial for ${k}$ electrons in the ${m=1}$ state because it is simply ${n_j=1}$ for all ${j\le k}$. This representation, of course, is not so useful for extracting observables because an antisymmetrisation must be performed before being able to calculate, eg.,

$\displaystyle \langle \Psi |A|\Psi\rangle \ \ \ \ \ (3)$

for some hermitian ${A}$, and that is precisely the hard bit (I think?). Anyways, the authors deal with this by representing (2) in the following way: they set up a register of ${n}$ ${n}$-level systems (qudits) and the write, eg.,

$\displaystyle |0,2,3\rangle \ \ \ \ \ (4)$

to mean

$\displaystyle |1,0,1,1\rangle \ \ \ \ \ (5)$

in the second-quantised formalism, i.e., one electron in the ${l=0}$ Fock-Darwin state, and one electron in each of the ${l=2}$ and ${l=3}$ F-D states. (Note this example doesn’t pertain to the QHE.) Thus the IQHE ground state

$\displaystyle |\Psi\rangle = |1,1, \ldots, 1\rangle \ \ \ \ \ (6)$

is represented, in the author’s register basis, as

where ${S_n}$ is the symmetric group on ${n}$ letters and ${\mbox{sgn}}$ is the sign of a permutation.

3. Quantum circuits to prepare the ${m=1}$ Laughlin ground state

The final part of the paper then describes a quantum circuit which takes an initial product state ${|0,1, \ldots, n-1\rangle}$ to (7). The way the authors do this is to antisymmetrise the initial state. This is actually nontrivial because the antisymmetrisation operation

$\displaystyle \mbox{Alt}(|j_0, j_1, \ldots, j_{n-1}\rangle) \sim \sum_{\pi \in S_n} \mbox{sgn}(\pi)|\pi(j_0)\pi(j_1)\cdots\pi(j_{n-1})\rangle \ \ \ \ \ (8)$

is not unitary (think about ${|1,1\rangle}$). So the quantum circuit they describe mustn’t actually be ${\mbox{Alt}}$, but rather, reduce to ${\mbox{Alt}}$ when applied to ${|0,1, \ldots, n-1\rangle}$.

The authors describe a recursive scheme utilising a 2-qudit gate they call ${W}$, defined in equation (6) in the paper. The argument is inductive, and I’ll spare you the technicalities.

The circuit they describe has depth ${n(n-1)/2}$. So, after representing each qudit using ${\log_2(n)}$ qubits, the circuit scales polylogarithmically with the system size.

4. Questions

So, in essence, this paper describes a quantum circuit to antisymmetrise the product state ${|0,1, \ldots, n-1\rangle}$. This is surprisingly subtle: my first guess would have been that it is trivial to do… But this is clearly wrong as the authors go on to show that any circuit which achieves the same effect must have depth ${n(n-1)/2}$.

Interestingly this method actually prepares the exact ground states for another totally different class of models, namely, exchange hamiltonians

$\displaystyle H = \sum_{j} P_{j,j+1}, \ \ \ \ \ (9)$

where ${P_{j,k}}$ exchanges qudit ${j}$ and ${k}$, on ${n}$ ${n}$-level qudits. (See this paper for a detailed description.)

Several questions suggest themselves:

1. How easy is it to generalise this approach to the fractional quantum Hall effect, with, eg., ${m=3}$? The authors suggest that more than ${W}$ gate is required, but it is unclear if it is even possible to do it efficiently at all. (I’m sure it is possible.)
2. Expectation values: the scheme presented doesn’t include a method to calculate expectation values, this is slightly subtle (?) requiring knowledge of the observable in the Fock-Darwin basis.
3. Relationship to string-net models? The FQHE ground state is topological (?) so it presumably can be modelled using a string-net approach…? (I have no idea what I’m talking about here, I’ve just heard people use these words in the same sentence…)

With respect to my first question: I’m put in mind of the paper of Dunne which shows how to write the Laughlin state as a linear combination of slater determinants. This involves Schur functions: maybe just maybe there is a potential application of the Schur transform here? (Again, I have no real idea what I’m talking about…)

That’s it for the moment: any comments, suggestions, corrections, etc., are most welcome!