# Deformed Fredkin model for the $\nu{=}5/2$ Moore-Read state on thin
cylinders

Cristian Voinea1, Songyang Pu1, Ammar Kirmani2, Pouyan Ghaemi3,4, Armin
Rahmani5, and Zlatko Papić1 1School of Physics and Astronomy, University of
Leeds, Leeds LS2 9JT, United Kingdom  2Theoretical Division, T-4, Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  3Physics Department, City
College of the City University of New York, NY 10031, USA  4Physics Program,
Graduate Center of City University of New York, NY 10031, USA  5Department of
Physics and Astronomy and Advanced Materials Science and Engineering Center,
Western Washington University, Bellingham, Washington 98225, USA

###### Abstract

We propose a frustration-free model for the Moore-Read quantum Hall state on
sufficiently thin cylinders with circumferences $\lesssim 7$ magnetic lengths.
While the Moore-Read Hamiltonian involves complicated long-range interactions
between triplets of electrons in a Landau level, our effective model is a
simpler one-dimensional chain of qubits with deformed Fredkin gates. We show
that the ground state of the Fredkin model has high overlap with the Moore-
Read wave function and accurately reproduces the latter’s entanglement
properties. Moreover, we demonstrate that the model captures the dynamical
response of the Moore-Read state to a geometric quench, induced by suddenly
changing the anisotropy of the system. We elucidate the underlying mechanism
of the quench dynamics and show that it coincides with the linearized bimetric
field theory. The minimal model introduced here can be directly implemented as
a first step towards quantum simulation of the Moore-Read state, as we
demonstrate by deriving an efficient circuit approximation to the ground state
and implementing it on IBM quantum processor.

##  I Introduction

The enigmatic fractional quantum Hall (FQH) state observed at filling fraction
$\nu{=}5/2$ [1] stands out as a rare example of an even-denominator state
among the majority of odd-denominator states described by the Laughlin wave
functions [2] and composite fermion theory [3]. One of the leading theoretical
explanations of the $\nu{=}5/2$ state is based on the Moore-Read (MR)
variational wave function [4]. Two unique properties of the MR state are worth
highlighting: (i) it represents a $p$-wave superconductor of composite
fermions [5]; (ii) its elementary charge excitations behave like Ising anyons,
i.e., they carry charge $e/4$ and exhibit non-Abelian braiding statistics [4,
6]. The latter has motivated the use of MR state as a potential resource for
topological quantum computation [7], whereby quantum information is encoded in
the collective states of MR anyons and quantum gates are executed by braiding
the anyons. Such operations would be protected by the topological FQH gap,
avoiding the costly quantum error correction.

On the fundamental side, the understanding of particle-hole symmetry and
collective excitations in the $\nu{=}5/2$ state has recently generated a
flurry of interest. While the numerics [8, 9] provided initial support of the
MR wave function capturing the physical ground state at $\nu{=}5/2$, it has
been realized that preserving (or breaking) particle-hole (PH) symmetry can
lead to distinct phases of matter. For example, by PH-conjugating the MR wave
function one obtains a distinct state known as the “anti-Pfaffian” state [10,
11], while enforcing the PH symmetry leads to another, PH-symmetric Pfaffian
state (“PH-Pf”) [12]. Understanding the relation of these states with the MR
state in light of physical PH symmetry breaking effects, such as Landau level
mixing [13, 14, 15] remains an important task for reconciling numerics [16,
17, 18, 19] with experiment [20, 21].

On the other hand, collective excitations of the $\nu{=}5/2$ state have also
attracted much attention. The pairing in the MR state mentioned above gives
rise to an additional collective mode – the unpaired “neutral fermion” mode –
which has been “seen” in the numerical simulations [22, 23, 24], but so far
not detected in experiment. The gap of the neutral fermion mode is of direct
importance for topological quantum computation, as the former can be excited
in the process of fusion of two elementary anyons. Recently, Ref. [25]
proposed a description of the neutral fermion mode based on an emergent
“supersymmetry” with the more conventional, bosonic density-wave excitation
[26, 27]. The numerics in Ref. [28] suggests that supersymmetry can indeed
emerge in a realistic microscopic model of $\nu{=}5/2$.

Figure 1: (a)-(b): Two types of 3-electron scattering processes present in the
Moore-Read Hamiltonian. The cylinder circumference, $L_{2}$, controls the
spacing $2\pi\ell_{B}^{2}/L_{2}$ between Landau level orbitals (dashed lines).
(c)-(d): Sending $L_{2}/\ell_{B}{\to}0$ suppresses the longer-range hopping
(d) compared to the one in (c). It will be shown that (c), where one electron
is fixed while the other two electrons hop between the nearest-neighbor
orbitals, maps to a controlled-SWAP (Fredkin) gate.

In this paper we develop a framework for studying the MR state in a quasi-one-
dimensional limit, obtained by placing the FQH fluid on a streched cylinder or
torus whose lengths in the two directions obey $L_{2}{\ll}L_{1}$. This so-
called “thin-torus” limit has been fruitful in gaining understanding of the
structure of many FQH ground states and their excitations [29, 30, 31, 32, 33,
34, 35, 36]. The thin-torus limit provides a natural classical “cartoon” of
the complicated physics in the two-dimensional limit: the off-diagonal matrix
elements of the Hamiltonian describing a FQH state become strongly suppressed
${\sim}\exp[-(2\pi\ell_{B}/L_{2})^{2}]$ in the limit $L_{2}/\ell_{B}{\to}0$,
allowing for considerable simplications of the problem – see Fig. 1 for an
illustration.

However, there have been comparatively few studies of the MR state near the
thin-torus limit. Most previous works [37, 38, 39, 40] focused on the
“extreme” thin-torus limit, also known as the Tao-Thouless limit [29], where
the Hamiltonian is reduced to purely classical electrostatic repulsion. It is
therefore important to develop an analytically-tractable model for the MR
state _beyond_ the strict Tao-Thouless limit, where some correlated hopping
terms are present in the model. For the Laughlin and Jain states, such models
were previously formulated in Refs. [41, 42, 43, 44] and one of the goals of
this paper is to work out an analogous model for the MR state. The intrinsic
one-dimensional (1D) structure of such models makes them suitable for
implementation on digital quantum computers, as recently shown for the
$\nu{=}1/3$ Laughlin state [45, 46]. The versatility of such devices allows to
probe questions, such as the real-time dynamics following a quench, that are
challenging for traditional solid-state experiments [47, 48, 49, 50]. In
particular, the implementation on IBM quantum processor allowed to simulate
the “graviton” dynamics induced by deforming the geometry of the Laughlin
state [51, 52, 53, 54, 55].

The remainder of this paper is organized as follows. We start by reviewing the
parent Hamiltonian of the MR state in Sec. II. We make use of the second-
quantization formalism to derive a simplified frustration-free model near the
thin-cylinder limit and we show that its ground state has high overlap with
the MR state, with similar entanglement properties. In Sec. III we show that
the frustration-free model can be expressed as a deformed Fredkin model for
spin-1/2 degrees of freedom. Working in the spin representation, we present an
intuitive picture of the ground state of this deformed Fredkin model and
derive its matrix-product state (MPS) representation. We also demonstrate that
the ground state can be efficiently approximated by a quantum circuit, which
we implement on the IBM quantum processor. In Sec. IV we show that the Fredkin
model also captures the dynamics of the MR state induced by quenching the
anisotropy of the system, and we elucidate the mechanism of this dynamics. Our
conclusions are presented in Sec. V, while Appendices contain technical
details of the derivations, further characterizations of the ground state, and
a generalization of the Laughlin case in Ref. [41] to a closely-related
Motzkin spin chain.

##  II Moore-Read Hamiltonian on a thin cylinder

In this section we formulate a frustration-free model that provides a good
approximation of the MR ground state near the thin-cylinder limit. In the
infinite 2D plane, the parent Hamiltonian of the MR state is a peculiar
interaction potential that penalizes configurations of any three electrons
forming a state with relative angular momentum equal to 3 – the smallest
possible momentum allowed by the Pauli exclusion principle [56, 57]. At the
same time, pairs of electrons do not experience any interaction. The
combination of these two effects gives rise to an exotic many-electron state
with $p$-wave pairing correlations [5].

Concretely, the MR interaction potential can be written in real space using
derivatives of delta functions [9]:

$H_{\mathrm{MR}}=-\sum_{i<j<k}S_{i,j,k}\left[\nabla_{i}^{4}\nabla_{j}^{2}\delta^{2}\left(\mathbf{r}_{i}-\mathbf{r}_{j}\right)\delta^{2}\left(\mathbf{r}_{j}-\mathbf{r}_{k}\right)\right],$
(1)

where $S_{i,j,k}$ is a symmetrizer over the electron indices $i$, $j$, $k$. At
filling $\nu{=}1/2$, the ground state of this Hamiltonian has energy $E{=}0$
and it is unique (on a disk, sphere or cylinder geometry) or six-fold
degenerate on a torus, corresponding exactly to the wave function written down
by Moore and Read [4]. The same state was shown to have high overlap with the
exact ground state of Coulomb interaction in the first-excited Landau level
[9, 58]. Moreover, the Hamiltonian above also captures the collective
excitations of the MR state [22, 23, 59, 24, 60]. Below we first convert the
Hamiltonian (1) into a second-quantized form on the cylinder and torus
geometries. This form allows us to derive a simplified model for the MR state
on a thin cylinder.

###  II.1 Moore-Read Hamiltonian in second quantization

The singularities in Eq. (1) are naturally regularized by projection to the
lowest Landau level (LLL). Assuming Landau gauge $\mathbf{A}=(0,Bx,0)$, the
single-electron wave functions are given by [61]

$\displaystyle\phi_{j}(\mathbf{r})=\frac{1}{\sqrt{L_{2}\sqrt{\pi}\ell_{B}}}e^{i2\pi
jy\ell_{B}^{2}/L_{2}}e^{-(x-2\pi j\ell_{B}^{2}/L_{2})^{2}/2\ell_{B}^{2}},$ (2)

where $L_{2}$ is the cylinder circumference in the $y$-direction and
$\ell_{B}=\sqrt{\hbar c/eB}$ is the magnetic length. The $j$th magnetic
orbital is therefore exponentially localized (in $x$-direction) around $2\pi
j\ell_{B}^{2}/L_{2}$. For simplicity, unless specified otherwise, below we
will work in units $\ell_{B}{=}1$.

The second-quantized representation of the MR Hamiltonian is:

$H_{\mathrm{MR}}=\sum_{j_{1},\dots,j_{6}=0}^{N_{\phi}-1}V_{j_{1}j_{2}j_{3}j_{4}j_{5}j_{6}}\,c_{j_{1}}^{\dagger}c_{j_{2}}^{\dagger}c_{j_{3}}^{\dagger}c_{j_{4}}c_{j_{5}}c_{j_{6}},$
(3)

where the operators $c_{j}^{\dagger}$, $c_{j}$ create or destroy an electron
in the orbital $\phi_{j}(\mathbf{r})$. The matrix elements are derived by
integrating Eq. (1) between the single-electron eigenfunctions (2), which
yields

$\displaystyle V_{j_{1}j_{2}j_{3}j_{4}j_{5}j_{6}}$
$\displaystyle=\delta_{j_{1}+j_{2}+j_{3},j_{4}+j_{5}+j_{6}}$
$\displaystyle\times(j_{1}-j_{2})(j_{1}-j_{3})(j_{2}-j_{3})$
$\displaystyle\times(j_{6}-j_{5})(j_{6}-j_{4})(j_{5}-j_{4})$
$\displaystyle\times\exp\bigg{[}-\frac{\kappa^{2}}{2g_{11}}\bigg{(}\sum
j_{i}^{2}-\frac{1}{6}\big{(}\sum j_{i}\big{)}^{2}$
$\displaystyle+ig_{12}\big{(}j_{1}^{2}+j_{2}^{2}+j_{3}^{2}-j_{4}^{2}-j_{5}^{2}-j_{6}^{2}\big{)}\bigg{)}\bigg{]},$
(4)

where we have dropped the overall normalization constant for simplicity. The
magnitude of the matrix element is controlled by the cylinder circumference
$L_{2}$ in units of magnetic length $\ell_{B}$, which defines the parameter
$\kappa{=}2\pi\ell_{B}/L_{2}$. We have derived the matrix elements by assuming
a general anisotropic band-mass tensor $g_{ab}$, which will be relevant for
the discussion of geometric quench in Sec. IV. Note that the matrix element
$V_{j_{1}\cdots j_{6}}$ is properly antisymmetric, resulting in a minus sign
when any two electrons are exchanged, hence the limits in the sum in Eq. (3)
can be restricted to $j_{1}{>}j_{2}{>}j_{3}$, $j_{6}{>}j_{5}{>}j_{4}$ without
loss of generality. The delta function in Eq. (II.1) encodes momentum
conservation during a scattering process, hence one of the indices, e.g.,
$j_{6}$ can be eliminated in terms of $j_{1},\ldots,j_{5}$.

A few comments are in order. We have denoted by integer $N_{\phi}$ the number
of magnetic orbitals. For a cylinder, $N_{\phi}=2N_{e}{-}2$, where $N_{e}$ is
the number of electrons. The offset $-2$ is a geometric feature of the MR
state called the Wen-Zee shift [62]. The total area of the fluid of dimensions
$L_{1}\times L_{2}$ must be quantized in any FQH state [63], thus we take the
thin-cylinder limit according to

$\displaystyle L_{2}/\ell_{B}\to 0,\;\;\;\text{such
that}\;\;\;L_{1}L_{2}=2\pi\ell_{B}^{2}N_{\phi},$ (5)

which ensures that the number of orbitals, and hence the filling factor,
remains constant.

Although we will focus on cylinder geometry in this paper, we note that the
same Hamiltonian, Eqs.(3)-(II.1), can also be used on a torus, with a few
modifications. On a torus, the shift vanishes and $N_{\phi}=2N_{e}$. However,
because of the periodicity in both $x$ and $y$ directions, the momentum is
only defined modulo $N_{\phi}$. This means that the momentum conservation
takes the form
$j_{1}+j_{2}+j_{3}=j_{4}+j_{5}+j_{6}\;(\mathrm{mod}\;N_{\phi}$). Moreover, the
matrix element (II.1) must be explicitly periodized to make it compatible with
the torus boundary condition, which can be done by replacing
$j_{i}{\to}j_{i}{+}k_{i}N_{\phi}$ and summing over $k_{i}$.

The derivation of the effective Hamiltonian in the thin-cylinder limit
proceeds by noting that, in the limit of $\kappa{\gg}1$ (equivalently,
$L_{2}{\ll}\ell_{B}$), there is a natural hierarchy of matrix elements (II.1),
which are separated by different powers of $\exp(-\kappa^{2})$ [31, 33, 64].
Below we list the first few relevant terms in decreasing order:

$\displaystyle 2^{2}e^{-2\kappa^{2}}n_{p+1}n_{p}n_{p-1};\quad 111$ (6)
$\displaystyle 2^{2}3^{2}e^{-14\kappa^{2}/3}n_{p+2}n_{p+1}n_{p-1};\quad 1011$
(7) $\displaystyle
2^{5}e^{-5\kappa^{2}}c^{\dagger}_{p-1}c^{\dagger}_{p}c^{\dagger}_{p+1}c_{p+2}c_{p}c_{p-2};\quad
10101\to 01110$ (8) $\displaystyle
2^{3}3^{2}e^{-20\kappa^{2}/3}c^{\dagger}_{p}c^{\dagger}_{p+1}c^{\dagger}_{p+4}c_{p+3}c_{p+2}c_{p};\quad
11001\to 10110$ (9) $\displaystyle
2^{8}e^{-8\kappa^{2}}n_{p+2}n_{p}n_{p-2};\quad 10101$ (10) $\displaystyle
2^{3}5e^{-8\kappa^{2}}c^{\dagger}_{p-2}c^{\dagger}_{p+2}c^{\dagger}_{p+3}c_{p+2}c_{p+1}c_{p};\quad
001110\to 100011$ (11) $\displaystyle
2^{4}3^{2}e^{-26\kappa^{2}/3}n_{p+4}n_{p+3}n_{p};\quad 11001$ (12)
$\displaystyle
2^{2}3^{2}5e^{-26\kappa^{2}/3}c^{\dagger}_{p-1}c^{\dagger}_{p}c^{\dagger}_{p+2}c_{p+3}c_{p}c_{p-2};\quad
101001\to 011010.$ (13)

We have included a binary mnemonic to represent the type of process generated
by each Hamiltonian term. A single pattern, e.g., 1011, represents a diagonal
term in the Hamiltonian which assigns energy penalty for the given local
pattern of occupation numbers anywhere in the system. The terms containing an
arrow, such as $10011{\to}01101$, can be visualized as correlated hopping
processes, Fig. 1. In such cases, the Hermitian conjugates of the processes,
corresponding to reflected hoppings with the same amplitude, are implied.

###  II.2 Tao-Thouless limit

The “extreme” thin-torus limit, also known as the Tao-Thouless limit, of the
MR Hamiltonian was discussed in Refs. [38, 37]. In this limit, the only terms
that survive are Eqs. 6 and 7, giving energy penalty to configurations $\dots
111\dots$ and $\dots 1011\dots$. Hence, the ground states at filling
$\nu{=}1/2$ (with zero energy) are

$\displaystyle\dots 110011001100\dots\;\;\mathrm{and}\;\;\dots 10101010\dots,$
(14)

while any other Fock state will be higher in energy by at least an amount
${\sim}\exp(-14\kappa^{2}/3)$, see Eq. 7. This gives the expected 6-fold
ground state degeneracy of the MR state on the torus [5], since the first
state in Eq. (14) is 4-fold and the second is 2-fold degenerate under
translations. These ground states have different momenta on the torus, so they
live in different sectors of the Hilbert space [65]. Similarly, the ground
states in Eq. (14) are the _densest_ zero-energy states one can construct, as
increasing the filling factor would necessarily violate the terms in Eq.
(6)-(7). On the other hand, _decreasing_ the filling factor is allowed,
leading to many more $E{=}0$ states. These correspond to quasihole excitations
and can be interpreted as domain walls between two different types of ground
state patterns in Eq. (14), see Refs. [38, 37].

On a finite cylinder, the densest zero-energy ground state is found instead at
$N_{\phi}{=}2N{-}2$, as expected from the Wen-Zee shift. This coincides with
the root partition of the Jack polynomial corresponding to the MR state [66],
$11001100\dots 0011$, with $11$ at each boundary. The other torus root state
can be similarly adapted to a finite cylinder according to $1010\ldots 101$.
However, this requires an extra orbital, since the flux is now
$N_{\phi}{=}2N{-}1$. Thus, the second type of torus ground state becomes an
excited state on a cylinder. In both cylinder and torus geometries, the Tao-
Thouless ground states are trivial product states without any entanglement. We
next discuss how to go beyond the extreme thin-cylinder limit and generate an
entangled ground state.

###  II.3 Frustration-free model beyond the Tao-Thouless limit

Beyond the extreme limit discussed above, we would like to retain a few more
terms, with smaller powers of $\exp(-\kappa^{2})$, and thereby generate a more
accurate approximation of the MR state over a slightly large range of $L_{2}$.
A natural way to do to this would be to choose a magnitude cut-off and keep
only the Hamiltonian matrix elements that are larger than this cutoff.
However, we would also like to be able to analytically solve for the ground
state of the resulting truncated Hamiltonian. In this sense, it is natural to
look for a Hamiltonian which is frustration-free, i.e., has a ground state
that is simultaneously annihilated by all individual terms in the Hamiltonian.
In such cases it is often possible to find analytically exact ground states
even though the Hamiltonian overall may not be solvable, e.g., as in the case
of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model [67].

Unfortunately, the program outlined above fails for our 3-body Hamiltonian:
keeping the terms in the order they are listed in Eqs. (6)-(13) does not
result in a positive semi-definite operator. This can be seen by considering
the first two correlated hoppings, Eqs. 8 and 9. One would want to include
these hoppings as they naturally act on the two types of ground states in the
extreme thin-torus limit, Eq. (14), and would create some entanglement.
However, the “dressed” ground states would no longer be zero modes and their
degeneracy would be lifted. Inspired by the Laughlin construction [41], one
could try to remedy this by including the terms Eqs. 10 and 12 to create a sum
of _two_ positive semi-definite operators. One quickly realizes that the
hopping term Eq. 13 now becomes a problem, spoiling the frustration-free
property. In Ref. [64], an attempt was made to define a frustration-free model
for a bosonic MR state by dropping the equivalent of hopping Eq. 13 (as well
as the hopping Eq. 11). Unfortunately, upon further inspection, we have found
the claim in Ref. [64] to be inaccurate because the model proposed there does
not yield strictly zero-energy ground states.

Figure 2: Overlaps between the ground state $|\psi_{\mathrm{MR}}\rangle$ of
the full model in Eq. 3 (i.e., the Moore-Read state) and the ground state
$|\psi_{0}\rangle$ of the truncated model in Eq. 15, for different system
sizes. For cylinder circumferences up to $L_{2}{\approx}7\ell_{B}$ (shaded),
the overlap is $95\%$ or higher, indicating this truncation returns a good
approximation for the ground state in the thin-cylinder regime.

We now describe the simplest frustration-free truncation of the Hamiltonian in
Eqs. (6)-(13) that we have found. We will focus on the cylinder root state
$|R_{0}\rangle=|11001100\dots 0011\rangle$, which is nondegenerate. In order
to obtain a unique “dressed” ground state on the cylinder, we consider the
Hamiltonian terms that act nontrivially on this root state. The resulting
states will be the first relevant corrections to the ground state. The
effective Hamiltonian contains terms in Eqs. 6, 7, 9 and 12:

$H_{\mathrm{MR}}^{\prime}=\sum_{i=0}^{N_{\phi}-3}A^{{\dagger}}_{i}A_{i}+\sum_{i=0}^{N_{\phi}-4}\big{(}B^{{\dagger}}_{i}B_{i}+C^{{\dagger}}_{i}C_{i}\big{)},$
(15)

where the operators $A$, $B$ and $C$ are given by

$\displaystyle A_{i}$ $\displaystyle=$ $\displaystyle\alpha
c_{i}c_{i+1}c_{i+2},$ (16) $\displaystyle B_{i}$ $\displaystyle=$
$\displaystyle\beta c_{i}c_{i+2}c_{i+3}+\gamma c_{i}c_{i+1}c_{i+4},$ (17)
$\displaystyle C_{i}$ $\displaystyle=$ $\displaystyle\beta
c_{i+1}c_{i+2}c_{i+4}+\gamma c_{i}c_{i+3}c_{i+4}.$ (18)

For brevity, we have introduced the parameters

$\alpha=\sqrt{V_{012210}},\ \beta=\sqrt{V_{023320}},\
\gamma=e^{i\theta}\sqrt{V_{014410}}\ ,$ (19)

given in terms of matrix elements (II.1) and
$\theta=2\kappa^{2}g_{12}/g_{11}$. Amongst the terms omitted, Eqs. 8, 10 and
13 do not act directly on the root state, bringing only subleading
contributions. While the term in Eq. 11 can directly act on the extreme root
state, its contribution is suppressed in the vicinity of the thin-cylinder
limit because of the prefactor being much smaller than the hopping term
retained, Eq. (9).

Therefore, to a first order approximation, $H^{\prime}_{\mathrm{MR}}$ is the
correct effective Hamiltonian that captures the departure of the Pfaffian
state from the root $|R_{0}\rangle$ in this geometry (in Sec. IV we will also
investigate its dynamical properties to show that the model captures the
properties of excited eigenstates). On the torus, our model preserves the
$6$-fold ground state degeneracy of the MR Hamiltonian. Four of those,
corresponding to root unit cell $1001$ and its translations, will become
“dressed”, in analogy to the ground state of the cylinder Hamiltonian. The
other two will remain inert, i.e. equal to $|101010\dots\rangle$ and
$|010101\dots\rangle$ for any value of $L_{2}$ (since the hopping in Eq. 9
cannot produce new configurations).

To confirm the validity of the model in Eq. (15) we performed several tests.
First, we evaluated the overlap of the model’s ground state with the full MR
state, i.e., the ground state of the untruncated Hamiltonian. Fig. 2 shows
that this overlap is very high close to the thin-cylinder regime, with overlap
on the order ${\sim}95\%$ at $L_{2}{\approx}7\ell_{B}$. As we are not exactly
capturing the full state at a finite value of $L_{2}$, the overlaps naturally
decay with system size (and vanish in the thermodynamic limit). Nevertheless,
the fact that they remain very high and weakly dependent on system size in the
range $L_{2}\lesssim 7\ell_{B}$ gives us confidence that the model captures
the right physics, as will be further demonstrated below.

Figure 3: Bipartite entanglement entropy $S_{A}$ of the full MR state [i.e.,
the ground state of Eq. 3] and the ground state of the truncated model, Eq.
15, as a function of cylinder circumference, $L_{2}$. Data is for $N_{e}{=}14$
electrons and $N_{\phi}{=}26$ magnetic orbitals. All types of bipartitions are
considered: choosing subsystem $A$ to contain $N_{\phi}^{A}{=}11$ orbitals,
the boundary looks like $0|0$; it can be seen that entanglement entropy here
starts growing early on. The case $N_{\phi}^{A}{=}12$, corresponding to $1|0$,
behaves similarly. By contrast, choosing $N_{\phi}^{A}{=}13$ corresponds to
the bipartition type $1|1$, where entropy grows much more slowly. The
truncated model accurately captures the behavior of the model in the range of
circumferences $L_{2}\lesssim 7\ell_{B}$ (shaded). The inset shows the
topological entanglement entropy $\gamma_{\mathrm{top}}$, extracted
numerically from $S_{A}$ using a linear fit over the interval
$[5.5\ell_{B},L_{\mathrm{max}}]$. Only the bipartitions $0|0$ and $1|0$ were
used, as those scale correctly near the Tao-Thouless limit. Restricting
ourselves to the range of validity for the truncated model, our
$\gamma_{\mathrm{top}}$ estimate is within $20\%$ of the theoretical value of
$\ln\sqrt{8}$.

An example of a physical quantity that can be meaningfully scaled with system
size and is sensitive to both local and nonlocal correlations is the bipartite
entanglement entropy, $S_{A}$. We compute $S_{A}$ by choosing a bipartition in
orbital space, i.e., the subsystem $A$ contains $N_{\phi}^{A}$ orbitals and
the complementary subsystem $B$ contains the remaining
$N_{\phi}{-}N_{\phi}^{A}$ orbitals [68]. Due to the Gaussian localization of
the magnetic orbitals (2), this roughly corresponds to the more conventional
partitioning of the system in real space [69, 70]. The entanglement entropy is
the von Neumann entropy, $S_{A}=-\mathrm{tr}\rho_{A}\ln\rho_{A}$, of the
reduced density matrix $\rho_{A}=\mathrm{tr}_{B}|\psi\rangle\langle\psi|$ for
the (truncated) MR ground state $|\psi\rangle$. In Fig. 3 we plot $S_{A}$ as a
function of cylinder circumference, contrasting the full MR state with the
ground state of the truncated model (15). The entanglement entropy of the MR
state has been shown to scale linearly with the circumference of the cylinder
[71, 72], which is the “area law” scaling expected in ground states of gapped
systems [73]. We observe that this linear scaling is obeyed also by the
truncated model for $L_{2}\lesssim 7\ell_{B}$. Furthermore, the subleading
correction $\gamma_{\mathrm{top}}$ to the area law,
$S_{A}=cL_{2}-\gamma_{\mathrm{top}}+O(e^{-\xi/\ell_{B}})$, where $c$ is a
constant and $\xi$ is the correlation length, is known to be a sensitive
indicator of topological order as it arises due to fractionalized anyon
excitations [74, 75]. As shown in the inset of Fig. 3, in the range of
validity of the truncated model we obtain $\gamma_{\mathrm{top}}$ within 20%
of the theoretically expected value $\ln\sqrt{8}$ for the MR case [71].

Beyond the area-law regime, the entanglement entropy of the truncated model
saturates, illustrating that the model is no longer able to capture the
relevant correlations in the full MR state. Conversely, near the Tao-Thouless
limit $L_{2}{\lesssim}4\ell_{B}$, there is practically no growth of entropy
with $L_{2}$, as the ground state remains a product state. In the latter
regime, $\gamma_{\mathrm{top}}{=}0$, illustrating that topological order is
completely lost since the system is too narrow in the $y$ direction.

Finally, Fig. 3 illustrates the sensitivity of entanglement scaling with
respect to the location of the bipartition. This is due to the cylinder ground
state being dominated by the pattern $\dots 11001100\dots$. Given this form of
the root state unit cell, there are three distinct types of locations where we
could place the partition. If we partition between two occupied orbitals
(i.e., $1|1$), at first order there will be no correlated hoppings across this
boundary, leading to a very slow growth in entanglement, as indeed seen in
Fig. 3. Instead, if we partition the system next to an empty orbital (i.e., at
$0|0$ or $1|0$), then there will be correlated hopping across the boundary and
the two subsystems can get entangled more easily. Note that this sensitivity
to the location of the partition is not present in the Laughlin case [41].
Moreover, it is an artefact of being near the thin-cylinder limit where the
ground state still possesses a crystalline-like density modulation, which
becomes strongly suppressed in the isotropic 2D limit where the fluid is
spatially uniform. Nevertheless, Fig. 3 shows that our truncated model (15)
successfully captures all the entanglement features of the full MR state in
the regime $L_{2}\lesssim 7\ell_{B}$. In the next section, we show that the
model (15) can be expressed as a well-known spin-1/2 chain model.

##  III Mapping to a deformed Fredkin chain

Our effective Hamiltonian (15) is frustration-free and it has an exact zero-
energy ground state, which is unique on a cylinder. To write down the ground
state wave function and provide its intuitive representation, we map the model
(15) to a deformed Fredkin chain [76, 77, 78]. The Fredkin model is a
spin-$1/2$ analog of the Motzkin chain model [79, 80, 81, 82]. As shown in
Appendix C, the Motzkin chain allows to generalize the construction from Ref.
[41] and describe the $\nu{=}1/3$ Laughlin state over a larger range of
cylinder circumferences.

###  III.1 Spin mapping

The mapping to spin-1/2 degrees of freedom is possible because we are only
interested in the connected component of the ground state, i.e., the Krylov
subspace spanned by states $(H_{\mathrm{MR}}^{\prime})^{n}|11001100\dots
110011\rangle$, for an arbitrary integer $n$ [83]. For our truncated model,
the dimension of such a subspace does not exhaust the full Hilbert space
dimension, allowing the mapping onto a spin-1/2 chain. This is an example of
Hilbert space fragmentation [84] and it has previously been used to perform
mappings of 2-body FQH Hamiltonians onto spin models [85]. Thus, without any
loss of generality, we can investigate both the ground state and dynamics by
restricting to the Krylov subspace built from the Tao-Thouless root pattern.

To perform the mapping, start from the root state $110011\dots 0011$ and pad
it with one fictitious $0$ on each side. This allows us to group sites in
pairs of two, noticing that the only present pairs are $01$ and $10$. These
are mapped to spins:

$|01\rangle\to\,\mid\uparrow\rangle\quad\mathrm{and}\quad|10\rangle\to\,\mid\downarrow\rangle.$
(20)

Thus, the root state maps to the antiferromagnetic (Néel) state
$|(0)110011\dots
0011(0)\rangle\to\,\mid\uparrow\downarrow\uparrow\downarrow\dots\uparrow\downarrow\rangle$.
Since $H_{\mathrm{MR}}^{\prime}$ acts on a maximum of 5 consecutive sites at
once, its equivalent acts on 3 consecutive spins. As discussed below, acting
with $H_{\mathrm{MR}}^{\prime}$ on any product state that can be mapped to
spins, i.e., consists of a sequence of 01 and 10 pairs, generates another
product state that can be similarly mapped to spins. An example is presented
in Fig. 4 which shows the connected sector of the adjacency graph of
$H_{\mathrm{MR}}^{\prime}$ that contains the Néel state. This shows that in
this connected component of the Hilbert space, our model (15) is equivalent to
a spin-$1/2$ model. For simplicity, we will denote the number of spins by $N$,
although it should be kept in mind this is equal to the number of electrons
$N_{e}$.

Figure 4: Adjacency graph of the truncated model $H_{\mathrm{MR}}^{\prime}$,
Eq. (15), in the connected sector containing the root state
$(0)11001100110011(0)$ or, equivalently,
$\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow$ in
the spin representation (marked in red). The graph vertices are product states
that can be reached by repeated application of the Hamiltonian. The edges
denote the nonzero matrix elements of the Hamiltonian between respective
vertices. This example is for $N_{e}{=}8$ electrons, where the connected
component of the root state contains only $14$ out of total $151$ states with
the same momentum quantum number. Any basis configurations not present here
are dynamically disconnected and cannot be reached from the ground state under
the dynamics generated by $H_{\mathrm{MR}}^{\prime}$. The Fredkin moves (swaps
with $\uparrow$ on the first or $\downarrow$ on the last site acting as a
control qubit), as implemented by the Hamiltonian, are also shown in the
bottom panels. Each move changes the area of the path by a constant value.

The moves implemented by the $B$ and $C$ terms in Eq. (15) are
$\mid\uparrow\uparrow\downarrow\rangle\leftrightarrow\
\mid\uparrow\downarrow\uparrow\rangle$ and
$\mid\uparrow\downarrow\downarrow\rangle\leftrightarrow\
\mid\downarrow\uparrow\downarrow\rangle$, i.e., they are controlled swaps of
spins or Fredkin gates [86], as illustrated in Fig. 4. Note that the $A$ term
in our subspace is redundant, as this subspace does not contain any $\dots
111\dots$ patterns. The resulting spin Hamiltonian is a sum of local
projectors and can be written as

$\displaystyle H_{F}$
$\displaystyle=\sum_{i=0}^{N-3}\mathcal{P}^{\uparrow}_{i}P^{\varphi(\tau)}_{i+1,i+2}+\mathcal{P}^{\varphi(\tau)}_{i,i+1}\mathcal{P}^{\downarrow}_{i+2}$
(21)

where the single-spin projector
$\mathcal{P}_{i}^{\sigma}{=}|\sigma_{i}\rangle\langle\sigma_{i}|$ projects
onto a local spin pointing in the direction $\sigma=\uparrow,\downarrow$. The
two-spin projector
$\mathcal{P}^{\varphi(\tau)}_{i,i+1}=|\varphi(\tau)_{i,i+1}\rangle\langle\varphi(\tau)_{i,i+1}|$
projects onto the deformed superposition state

$\displaystyle|\varphi(\tau)_{i,i+1}\rangle$ $\displaystyle=$
$\displaystyle\mid\uparrow_{i}\downarrow_{i+1}\rangle-\tau\mid\downarrow_{i}\uparrow_{i+1}\rangle,$
(22) $\displaystyle\tau=-\gamma/\beta$ $\displaystyle=$
$\displaystyle-2\exp\left(-2\kappa^{2}\frac{1-ig_{12}}{g_{11}}\right).$ (23)

The model (21) is our central result of this section. It can be recognized
that this model corresponds to a (colorless) deformed Fredkin chain from Ref.
[77]. Note that the boundary Hamiltonian terms from Ref. [77], i.e.,
$H_{\partial}=\mathcal{P}^{\uparrow}_{0}+\mathcal{P}^{\downarrow}_{N-1}$, have
been omitted because our subspace has the first $\uparrow$ spin and the last
$\downarrow$ spin frozen.

For convenience, we note that $H_{F}$ can be equivalently expressed in terms
of usual Pauli spin operators:

$H_{\mathrm{F}}=\sum_{j=0}^{N_{e}-3}\biggl{(}\bigl{(}\frac{1}{2}\mathds{1}+S^{z}_{j}\bigr{)}h_{j+1,j+2}(\tau)+h_{j,j+1}(\tau)\bigl{(}\frac{1}{2}\mathds{1}-S^{z}_{j+2}\bigr{)}\biggr{)},$
(24)

where

$\displaystyle h_{j,j+1}(\tau)$
$\displaystyle=\frac{1+|\tau|^{2}}{4}(\mathds{1}-4S^{z}_{j}S^{z}_{j+1})+\frac{1-|\tau|^{2}}{2}(S^{z}_{j}-S^{z}_{j+1})$
$\displaystyle-2\mathrm{Re}(\tau)(S^{x}_{j}S^{x}_{j+1}+S^{y}_{j}S^{y}_{j+1})$
$\displaystyle-2\mathrm{Im}(\tau)(S^{x}_{j}S^{y}_{j+1}-S^{y}_{j}S^{x}_{j+1}).$
(25)

Note that outside the connected component of the ground state, the mapping can
be extended by defining the additional composite degrees of freedom:
$|11\rangle\to|{+}\rangle$, $|00\rangle\to|{-}\rangle$. In this mapping, the
constrained dynamics of the model resembles that of fractonic models in Refs.
[87, 88, 89, 90], which can lead to different thermalization properties in
different Krylov fragments [83].

###  III.2 The ground state

The ground state of the Fredkin chain is a weighted superposition of “Dyck
paths” of length $N$ (the set of which we denote $\mathcal{D}_{N}$). These are
product state configurations with $S^{z}_{\mathrm{total}}=0$ and
$\sum_{i=0}^{k}S^{z}_{i}\geq 0$ for all $k$. The last condition is equivalent
to the number of spin $\uparrow$ sites always being greater than or equal to
the number of spin $\downarrow$ sites, as we go through the chain from left to
right. The paths in $\mathcal{D}_{N}$ can be interpreted graphically, as a
“mountain range” where each $\uparrow$ corresponds to an upward slope, while
$\downarrow$ corresponds to a downward slope. The Dyck constraint is
equivalent to the height of these graphs starting and ending at zero, and
always staying positive. The weight of each path (configuration) $p$ in the
ground state $|\psi_{0}\rangle$ will be determined by the area $A(p)$ under
the mountain range:

$|\psi_{0}\rangle=\mathcal{N}^{-1}\sum_{p\in\mathcal{D}_{N}}\tau^{A(p)/2}\,|p\rangle.$
(26)

According to the phase diagram of the Fredkin spin model, for $|\tau|<1$ the
entanglement entropy of the ground state is bounded (obeys area law), whereas
$|\tau|=1$ is a critical point where the scaling becomes logarithmic in system
size $\sim\log N$.

Ref. [76] also discusses the subtleties of the spin model with periodic
boundary conditions. For $|\tau|=1$, the ground state degeneracy scales
linearly with $N$, with zero modes in every $S^{z}_{\mathrm{total}}$ sector.
However, upon decreasing the deformation away from the critical point we find
that the extensive degeneracy disappears – only 4 zero modes survive. Two of
them are in the sectors $S^{z}_{\mathrm{total}}=\pm N/2$, corresponding to the
inert states $|{\uparrow}\rangle^{\otimes N}$ and
$|{\downarrow}\rangle^{\otimes N}$ (or the Fock states $|101010\dots\rangle$
and $|010101\dots\rangle$). The other two are in the sector
$S^{z}_{\mathrm{total}}=0$ and will correspond to the root unit cells $1001$
and $0110$; the two remaining translations break our spin mapping but can be
obtained by shifting every orbital by one position and then applying the
mapping. All other zero modes disappear because deformed Fredkin ground states
are constructed using the Dyck path area as in Eq. 26, which is only well
defined when $S^{z}_{\mathrm{total}}=0$. These results are in agreement with
the $6$-fold degeneracy found in the fermionic model.

With this understanding, we can map back to the fermionic ground state. All
Dyck paths can be obtained from the root configuration
$\mid\uparrow\downarrow\uparrow\dots\downarrow\uparrow\downarrow\rangle$ by
exchanging a number of $\downarrow$ with an equal number of $\uparrow$ further
along the chain. In the fermionic picture, this is equivalent to performing a
number of “squeezes”, i.e., applications of the operator

$\displaystyle\hat{S}_{k,d}=c^{\dagger}_{2k}c^{\dagger}_{2(k+d)-1}c_{2(k+d)}c_{2k-1},$
(27)

with $d,k>0$. Similar structure exists in the Laughlin state [45, 46]. The
resulting states are in one-to-one correspondence with those in
$\mathcal{D}_{N}$, and we will denote their set by $\mathcal{D}_{N}^{\prime}$.
Every configuration $s$ in $\mathcal{D}^{\prime}_{N}$ can be obtained from a
number of $n(s)$ squeezes applied to the root.

$s\in\mathcal{D}_{N}^{\prime}\Longleftrightarrow|s\rangle=\prod_{i=1}^{n(s)}\hat{S}_{k_{i},d_{i}}\,|110011\dots
0011\rangle$ (28)

The weight of such a basis state in the ground state is now determined by the
total distance squeezed $D(s)=\sum_{i=1}^{n(s)}d_{i}$, which is equivalent to
the previous definition (26) expressed in terms of the area under the Dyck
path.

$|\psi_{0}\rangle=\mathcal{N}^{-1}\sum_{s\in\mathcal{D}_{N}^{\prime}}\tau^{D(s)/2}\,|s\rangle.$
(29)

###  III.3 Matrix-product state representation

Along with the undeformed chain ($\tau{=}1$), Ref. [76] introduced a matrix-
product state (MPS) representation for its ground state. The associated MPS
matrices have bond dimension $\chi=N/2+1$, where $N$ is the number of spins:

$M^{\uparrow}_{jk}=\delta_{j+1,k}\quad\mathrm{and}\quad
M^{\downarrow}=(M^{\uparrow})^{T}.$ (30)

As we are working with open boundary conditions, we use the boundary vectors
$v_{L}=v_{R}^{T}$ with $(v_{L})_{j}=\delta_{j,0}$. This MPS can be directly
extended to the deformed chain, where we need to introduce the deformation
parameter in the following way:

$(M^{\uparrow}_{\tau})_{jk}=\tau^{j/2}\,\delta_{j+1,k}\quad\mathrm{and}\quad
M^{\downarrow}_{\tau}=(M^{\uparrow}_{\tau})^{T}$ (31)

This holds for any $\tau\in\mathbb{C}$ so it can be used for anisotropic
states as well. Therefore the Fredkin ground state can be written as:

$|\psi_{0}\rangle=\mathcal{N}^{-1/2}\,v^{T}M_{\tau,0}M_{\tau,1}\dots
M_{\tau,N-1}\,v,$ (32)

where the MPS tensor is given by

$M_{\tau,j}=\begin{pmatrix}0&|{\uparrow_{j}}\rangle&0&0&\dots\\\
|{\downarrow_{j}}\rangle&0&\tau^{1/2}|{\uparrow_{j}}\rangle&0&\\\
0&\tau^{1/2}|{\downarrow_{j}}\rangle&0&\tau|{\uparrow_{j}}\rangle&\\\
0&0&\tau|{\downarrow_{j}}\rangle&0&\\\ \vdots&&&&\ddots\end{pmatrix}.$ (33)

For e.g. $N{=}6$ this expression gives:

$\displaystyle|\psi_{0}^{N=6}\rangle$
$\displaystyle=|{\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow}\rangle+\tau\big{(}|{\uparrow\downarrow\uparrow\uparrow\downarrow\downarrow}\rangle+|{\uparrow\uparrow\downarrow\downarrow\uparrow\downarrow}\rangle\big{)}$
$\displaystyle+\tau^{2}|{\uparrow\uparrow\downarrow\uparrow\downarrow\downarrow}\rangle+\tau^{3}|{\uparrow\uparrow\uparrow\downarrow\downarrow\downarrow}\rangle$
(34)

which indeed agrees with Eq. (29). We note that alternative tensor network
representations of the Fredkin ground states have been discussed in the
literature [91].

Furthermore, the MPS representation above is able to capture the critical
point at $|\tau|=1$, which is precisely the reason behind $\chi$ increasing
linearly with system size. This limits our ability to extract thermodynamic
limit behaviour in this phase using the MPS tensors from Eqs. 30 and 31.
However, the regime of interest for this paper is $|\tau|\lesssim 0.4$ (i.e.,
$L_{2}\lesssim 7\,\ell_{B}$), which is far from the critical point. Hence, it
is possible to describe the ground state with high accuracy by truncating to a
finite bond dimension. Consider the following tensors:

$M_{\tau,j}^{(\chi=3)}=\begin{pmatrix}0&|{\uparrow_{j}}\rangle&0\\\
|{\downarrow_{j}}\rangle&0&\tau^{1/2}|{\uparrow_{j}}\rangle\\\
0&\tau^{1/2}|{\downarrow_{j}}\rangle&0\end{pmatrix}.$ (35)

For a chain with even number of spins, this MPS yields the following simple
ground state:

$|\psi_{0}^{(\chi=3)}\rangle=|{\uparrow}\rangle\big{(}|{\downarrow\uparrow}\rangle+\tau|{\uparrow\downarrow}\rangle\big{)}^{\otimes\frac{N-2}{2}}|{\downarrow}\rangle.$
(36)

With a fixed $\chi$, it is straightforward to analytically calculate the
behavior of relevant quantities in the thermodynamic limit by using the MPS
transfer matrix. The average orbital density takes the form:

$\langle\hat{n}_{4j/4j+1}\rangle=\frac{1}{1+\tau^{2}}\,,\quad\langle\hat{n}_{4j+2/4j+3}\rangle=\frac{\tau^{2}}{1+\tau^{2}}.$
(37)

As expected, this resembles a CDW pattern, which in this approximation (and
also in the full Fredkin ground state state) is predicted to disappear at
$|\tau|=1$, corresponding to $L_{2}{\approx}10.7\ell_{B}$ (outside the range
of validity of the truncated model). Figure 5 shows a comparison of orbital
density between the MR state, the Fredkin state and the $\chi{=}3$
approximation above. At $L_{2}{=}7\ell_{B}$ the two truncated states still
capture the CDW pattern, with the Fredkin state showing more accurate results.
Since this approximate state can be written in the tensor product form above,
the density-density correlations decay to zero with a finite correlation
length.

Figure 5: Comparison between the average orbital density of the MR state and
different truncated states: the Fredkin state and the state obtained by
truncating the Fredkin MPS at $\chi{=}3$. The system has $N_{e}{=}14$
electrons and the cylinder circumference is $L_{2}{=}7\ell_{B}$. The truncated
states deviate slightly from the charge density wave pattern of the MR state.

###  III.4 Quantum algorithm for preparing the ground state and its
implementation on IBM quantum processor

The simple structure of the MPS wave function in the above approximation (for
$\chi{=}3$) is amenable to implementation on noisy intermediate-scale quantum
devices. Indeed, all states in the superposition can be obtained from a
direct-product root pattern by only one layer of one- and two-qubit gates.
Furthermore, the parameters of the circuit can be determined analytically
without the need for any classical or hybrid optimization, which allows for
direct implementation on a large number of qubits.

Figure 6: The structure of the quantum circuit for six qubits. The $X$ gates
create the root patterns, and the rotations and CNOTs implement the action of
MPS matrices in Eq. (35).

The structure of the quantum circuit is shown in Fig. 6. If we choose the
angle $\theta$ to be equal to

$\theta=2\arctan(\tau),$ (38)

the $y$-rotation creates a superposition
$|{\uparrow}\rangle+\tau|{\downarrow}\rangle$ and the CNOT then changes the
state of the two qubits to
$|{\downarrow\uparrow}\rangle+\tau|{\uparrow\downarrow}\rangle$. As a quick
check, we executed the circuit on the ibmq_mumbai device, a 27-qubit processor
with quantum volume 128. This implementation was carried out using the Qiskit
package. In this similuation, we used $N=26$, with the initial and final
qubits held in trivial up and down states. Notably, we refrained from
employing any error mitigation techniques, and we deliberately incorporated
qubits and couplings from the device with lower quality. Our simulation
utilized a mere couple of thousand shots to ascertain bitstring probabilities
in the computational basis. We found very good agreement of the measured
orbital densities with the analytical results of Eq. (37), save for a few
instances where gate calibrations during the simulation were imperfect.

Figure 7: The orbital density from the quantum circuit. There is good
agreement between the results of Eq. (37), classical simulation of the circuit
using IBM Aer simulator and quantum implementation on the ibmq_mumbai device.
The IBM data was obtained on 31/8/2023 at 2:02 PM.

The results for the orbital density are shown in Fig. 7. We used 2048 shots in
both the quantum execution and the simulation of the circuit with IBM’s Aer
simulator. The Aer simulation is in excellent agreement with Eq. 37, with
slight differences due to the finite number of shots. There is also good
agreement with the quantum device, except for a few qubits. Despite its
simplicity, the approximate ground state prepared above can serve as a
valuable starting point for exploring the dynamics of the MR phase on quantum
computers.

##  IV Quench dynamics

Given that our Fredkin model (21) was constructed by focusing on the root
state and it does not represent a truncation of the MR Hamiltonian according
to a decreasing order of magnitude, it is not obvious that the excited
spectrum necessarily matches that of the full MR Hamiltonian. To demonstrate
the correspondence of key physical properties between the two spectra, in this
section we focus on the dynamical response of the Fredkin model. In
particular, we study geometric quench [52] to probe the compatibility between
the two models, which was previously used to a similar effect in the
$\nu{=}1/3$ Laughlin case [46].

Geometric quench is designed to elicit the dynamical response of the Girvin-
MacDonald-Platzman (GMP) collective mode [26, 27], which is present in all
known gapped FQH states, including the MR state [22, 23, 92, 24, 93]. In the
long-wavelength limit $k\ell_{B}{\to}0$, the GMP mode forms a quadrupole
degree of freedom that carries angular momentum $L{=}2$ and can be represented
by a quantum metric [51]. In this respect, the $k\ell_{B}{\to}0$ limit of the
GMP mode has formal similarity with the fluctuating space-time metric in a
theory of quantum gravity [94, 95] and it is sometimes referred to as “FQH
graviton” [92, 96]. It was shown that the quantum metric fluctuations can be
exposed by introducing anisotropy which breaks rotational symmetry of the
system [52, 53]. Such geometric quenches induce coherent dynamics of the FQH
graviton, even though the latter resides in the continuum of the energy
spectrum, making it a useful probe of physics beyond the ground state
considered thus far.

###  IV.1 Spectral function

The GMP mode, to a high accuracy, can be generated by a simple ansatz called
the “single-mode approximation” [26, 27]: the state belonging to the mode with
momentum $\mathbf{k}$ is obtained by acting with the projected density
operator, $\bar{\rho}_{\mathbf{k}}$, on the ground state, i.e.,
$|\psi_{\mathbf{k}}^{\mathrm{GMP}}\rangle=\bar{\rho}_{\mathbf{k}}|\psi_{0}\rangle$.
Thus, the GMP states are automatically orthogonal to the ground state as they
live in different momentum sectors. However, in practice, it is more
convenient to study dynamics within the $\mathbf{k}=0$ sector of the ground
state. This is the case with the geometric quench setup, described in Sec.
IV.2 below. Thus, in order to identify the relevant GMP state in the
$\mathbf{k}=0$ sector, possibly hidden in the continuum of the energy
spectrum, we need a different tool. We identify the long-wavelength limit of
the GMP mode using the following spectral function [52, 54]:

$\displaystyle I(\omega)$
$\displaystyle=\sum_{n}|\langle\psi_{n}|\hat{O}|\psi_{0}\rangle|^{2}\delta(\omega-\omega_{n}),$
(39)

where $\hat{O}$ is a 3-body operator with quadrupolar $x^{2}-y^{2}$ symmetry,
given in Ref. [54], and the sum runs over (in principle, all) energy
eigenstates $|\psi_{n}\rangle$ with energies $\omega_{n}$, measured relative
to the ground state energy $\omega_{0}$. In second quantization, the matrix
element of $\hat{O}$ is

$\displaystyle O_{j_{1}j_{2}j_{3}j_{4}j_{5}j_{6}}$
$\displaystyle=\delta_{j_{1}+j_{2}+j_{3},j_{4}+j_{5}+j_{6}}\left(\sum
j_{i}^{2}-\frac{1}{6}\big{(}\sum j_{i}\big{)}^{2}\right)$
$\displaystyle\times(j_{1}-j_{2})(j_{2}-j_{3})(j_{1}-j_{3})$
$\displaystyle\times(j_{4}-j_{5})(j_{5}-j_{6})(j_{4}-j_{6})$
$\displaystyle\times\exp\left[-\frac{\kappa^{2}}{2}\left(\sum
j^{2}_{i}-\frac{1}{6}(\sum j_{i})^{2}\right)\right],$ (40)

which allows to readily evaluate Eq. (39). Note that in this section we
consider the spectral function for an _isotropic_ system, hence there is no
metric dependence in Eq. (IV.1). As before, the matrix element given here is
derived for cylinder geometry and appropriate modifications are needed to make
it compatible with torus boundary conditions, as explained in Sec. II.1.

Figure 8: Evolution of the spectral function $I(\omega)$ in Eq. 39 in the
Fredkin model (top) and the full MR model (bottom), as a function of cylinder
circumference $L_{2}$. The peak(s) in the spectral function are identified
with the long-wavelength limit of the GMP mode, i.e., the FQH graviton. System
size is $N_{e}{=}10$ electrons, $N_{\phi}{=}18$ flux quanta.

In Fig. 8 we plot the evolution of the spectral function $I(\omega)$ as the
cylinder circumference is varied from the Tao-Thouless limit towards the
isotropic 2D limit, in both the untruncated and Fredkin models. We see there
is good agreement between the two models for $L_{2}{\lesssim}7\ell_{B}$, i.e.,
across the same range where we previously established high overlap between the
ground states of the two models. For larger circumferences, it becomes
impossible to adiabatically track the evolution of the graviton peak in
$I(\omega)$ due to multiple avoided crossings in Fig. 8. The graviton resides
in the continuum of the spectrum and it is not protected by a symmetry of the
Hamiltonian, hence its support over energy eigenstates may undergo complicated
“redistribution” as the geometry of the system is varied. In particular, away
from the Tao-Thouless limit, there is also a clear splitting of spectral
weight between several energy eigenstates, suggesting that the graviton degree
of freedom may not correspond to a single eigenstate in this regime.

###  IV.2 Geometric quench

Given the complex evolution of the spectral function in Fig. 8 when
interpolating between the isotropic 2D limit and the thin-cylinder limit we
are interested in, it is natural to inquire if the graviton oscillations
observed in the Laughlin case in Refs. [52, 46] persist in the MR case and
what their origin may be. In this section we analyze the geometric quench
dynamics in the thin-cylinder limit and establish that it corresponds to a
linearized bimetric theory of Gromov and Son [97]. This shows that, despite
the simplicity of our model (21), it is successful at capturing a nontrivial
many-body effect of a 2D FQH system away from equilibrium.

The geometric quench setup assumes that electrons are described by an
arbitrary mass tensor $g_{ab}$, with $a,b=1,2$. The mass tensor must be
symmetric and unimodular ($\mathrm{det}\;g=1$) [51], hence we can generally
write it as $g=\exp(\hat{Q})$ where
$\hat{Q}=Q(2\hat{d}_{a}\hat{d}_{b}-\delta_{a,b})$ is a Landau-de Gennes order
parameter and $\hat{\mathbf{d}}=(\cos(\phi/2),\sin(\phi/2))$ is a unit vector
[98]. Parameters $Q$ and $\phi$ intuitively represent the stretch and rotation
of the metric, respectively, with $Q{=}\phi{=}0$ corresponding to the
isotropic case. Under Landau-level projection, the interaction matrix elements
acquire explicit dependence on $g$, as can be seen in Eq. (II.1). For $g$
close to the identity (i.e., at weak anisotropy), the topological gap is
robust and the MR state remains a zero-energy ground state. We assume the
initial state before the quench to be the isotropic MR state with
$g=\mathds{1}$. At time $t=0$, the anisotropy in the Hamiltonian is
instantaneously changed and, for simplicity, we assume the new metric to be
diagonal, $g=\mathrm{diag}[g_{11},g_{22}]$, with $g_{11}\neq g_{22}$. The
deformed $g_{11}$ (and, therefore, $g_{22}$) should be sufficiently close to
unity such that the equilibrium system is still in the MR phase.

Figure 9: Geometric quench dynamics in the Fredkin model (21). The system size
is $N_{e}{=}8$, $N_{\phi}{=}14$, and the circumference is $L_{2}=3.6\ell_{B}$.
The system is initialised in the isotropic ground state and then time-evolved
by the anisotropic Hamiltonian with $Q{=}0.01$. The resulting dynamics is in
excellent agreement with the linearized bimetric theory, shown by dashed
lines. The slight disagreement between the two at late times slow decay comes
from the spectral weight in Fig. 8 spread over more than a single energy
eigenstate.

From Eq. 29 we can directly extract the first order corrections to the root
state, $|R_{0}\rangle\equiv|11001100\ldots 0011\rangle$. These are given by
states where only one squeezing, Eq. (27), is applied at a minimal distance:

$\displaystyle|\psi_{0}\rangle\approx\big{(}1-\tau\sum_{i}\hat{S}_{i,1}\big{)}|R_{0}\rangle.$
(41)

Substituting the deformation parameter in Eq. (22) and assuming
$\exp(-2\kappa^{2})$ and the metric anisotropy $Q,\phi$ to be small, we get

$\displaystyle|\psi_{0}\rangle\approx|R_{0}\rangle-2\exp\big{[}-2\kappa^{2}\big{(}1-Qe^{i\phi}\big{)}\big{]}\sum_{i}\hat{S}_{i,1}|R_{0}\rangle,$
(42)

where we used $g_{11}=\cosh Q$, $g_{12}=\sinh Qe^{i\phi}$ and therefore
$(1-ig_{12})/g_{11}\approx 1-iQe^{i\phi}$. On the other hand, the graviton
state is approximated by:

$\displaystyle|\psi_{g}\rangle$ $\displaystyle=\hat{O}|\psi_{0}\rangle\propto
e^{-14\kappa^{2}/3}\bigg{(}\sum_{i}\hat{S}_{i,1}|R_{0}\rangle+O\big{(}e^{-2\kappa^{2}}\big{)}\bigg{)}.$
(43)

Note that $|\psi_{g}\rangle$ is orthogonal to $|\psi_{\mathrm{0}}\rangle$.
From here, we deduce the graviton root state,

$\displaystyle|R_{g}\rangle$
$\displaystyle=\sum_{i}\hat{S}_{i,1}|R_{0}\rangle$
$\displaystyle=|1011010011\dots\rangle+|1100101101\dots\rangle+\dots,$ (44)

which is proportional to the first order squeezes. This is identical to the MR
ground state first-order correction to the root state and, in some sense, it
is the simplest translationally invariant quadrupole structure that we can
impose on top of it, creating quadrupoles of the form $-++-$ in each unit
cell.

From the graviton root state, we can deduce the geometric quench dynamics up
to first order in $\exp(-2\kappa^{2})$. Assuming, for simplicity, that the
post-quench Hamiltonian has the metric $g_{11}=\exp(A)\approx 1+A$ and
$g_{12}=0$, the initial state is given by

$\displaystyle|\psi(t=0)\rangle=|\psi_{0}^{\mathrm{iso}}\rangle\approx|R_{0}\rangle-2e^{-2\kappa^{2}}|R_{g}\rangle.$
(45)

Denoting by $|\psi_{0}^{\mathrm{aniso}}\rangle$ the ground state of the post-
quench Hamiltonian and using Eq. (42), we get

$\displaystyle|\psi(t=0)\rangle=|\psi_{0}^{\mathrm{aniso}}\rangle-2e^{-2\kappa^{2}}(1-e^{2\kappa^{2}A})|R_{g}\rangle.$
(46)

Very close to the thin-cylinder limit, the graviton root state will be the
correct $O(1)$ approximation to an eigenstate of the Hamiltonian, as confirmed
by the numerics. Thus, to first order, we can treat both
$|\psi_{0}^{\mathrm{aniso}}\rangle$ and $|R_{g}\rangle$ as eigenstates and
write the time-evolved state as

$\displaystyle|\psi(t)\rangle$
$\displaystyle=|\psi_{0}^{\mathrm{aniso}}\rangle-2e^{-2\kappa^{2}}(1-e^{2\kappa^{2}A})e^{-iE_{\gamma}t}\,|R_{g}\rangle,$
(47)

with $E_{\gamma}$ being the energy of the graviton state. Assuming that the
combined anisotropy, coming from the metric deformation and the stretching of
the cylinder, is still small, $\kappa^{2}A\ll 1$, we can rewrite the above
expression

$\displaystyle|\psi(t)\rangle$
$\displaystyle\approx|R_{0}\rangle-2e^{-2\kappa^{2}}(1+2\kappa^{2}A(1-e^{-iE_{\gamma}t}))|R_{g}\rangle.$
(48)

The expression in the bracket can be rewritten

$\displaystyle 1+2\kappa^{2}A(1-e^{-iE_{\gamma}t})\approx
e^{2\kappa^{2}A(1-e^{-iE_{\gamma}t})}.$ (49)

Substituting into the previous equation,

$\displaystyle|\psi(t)\rangle$
$\displaystyle\approx|R_{0}\rangle-2\exp\left(-2\kappa^{2}[1-A(1-e^{-iE_{\gamma}t})]\right)|R_{g}\rangle.$
(50)

We recognize that this is of the same form as Eq. (42), as the expression in
the square bracket can be written as $1-\tilde{Q}\exp(i\tilde{\phi})$, with

$\displaystyle\tilde{Q}(t)=2A\sin(E_{\gamma}t/2),\quad\tilde{\phi}(t)=\pi/2-E_{\gamma}t/2.\;\;\;\;\;$
(51)

These are nothing but the equations of motion of the linearized bimetric
theory [52]. Thus, we have reproduced the graviton dynamics, which in the
thin-cylinder limit reduces to the above two-level system dynamics. Fig. 9
confirms the existence of very regular metric oscillations at
$L_{2}{=}3.6\ell_{B}$ and their agreement with the analytical expression in
Eq. (51). From Section IV.2 we deduce that the energy of the graviton in the
thin-cylinder limit is $E_{\gamma}=2V_{023320}=72\,e^{-14\kappa^{2}/3}\,$,
which agrees with the frequency of the oscillations seen in Fig. 9.

Figure 10: Comparison of the geometric quench dynamics between the Fredkin and
full model at the cylinder circumference $L_{2}{=}5\ell_{B}$. The system size
is $N_{e}{=}8$, $N_{\phi}{=}14$. The system is initialized in the isotropic
ground state and then time-evolved by an anisotropic Hamiltonian with
$Q{=}0.02$. In this case, the dynamics is beyond the simple two-level system
dynamics described by linearized bimetric theory, as the graviton does not
correspond to a single eigenstate. The contribution of additional eigenstates
to the spectral function gives rise to the beating pattern seen here.
Nevertheless, there is still good agreement between the Fredkin and full
model.

Notably, our Fredkin model still accurately captures the dynamics beyond the
regime where it can be analytically treated as a two-level system. For
example, around circumference $L_{2}{\sim}5\ell_{B}$, the graviton peak splits
into a few smaller peaks close in energy. The resulting metric oscillations
can be seen in Fig. 10. There is now a slowly varying envelope that cannot be
accounted for within the simple linearized bimetric theory in Eqs. (51). At
even larger circumferences, the structure of the graviton state becomes
increasingly complicated, as there are many types of quadrupolar
configurations of the root state. The spectrum undergoes dramatic
transformations at intermediate values of $L_{2}$, as the hierarchy of energy
scales in the Hamiltonian changes. It is expected that close to the 2D limit
and in the thermodynamic limit, the energy of the graviton stabilises, as the
energy hierarchy stabilises too when $\kappa$ is small.

##  V Conclusions and discussion

We have formulated a one-dimensional qubit model that captures the MR state
and its out-of-equilibrium properties on sufficiently thin cylinders with
circumferences $L_{2}{\lesssim}7\ell_{B}$. This was demonstrated by computing
the overlap with the MR wave function and scaling of entanglement entropy with
the size of the subsystem, as well as the dynamics following a geometric
quench. One advantage of the proposed model is that its ground state can be
written down exactly and it is amenable to efficient preparation on the
existing quantum hardware, as we have demonstrated using the IBM quantum
processor. At the expense of noise-aware error mitigation schemes [46], these
results can naturally be extended to probe the dynamics of the MR phase on
quantum computers. This would also require an efficient optimally decomposed
circuit to emulate trotterized evolution with our Hamiltonian (24), which is
left to future work.

There are some notable differences between the model presented here and
previous studies of the $\nu{=}1/3$ Laughlin case [46]. While in the latter
case, the truncated model can be easily adapted to either open or periodic
boundary condition, in our case the torus boundary condition leads to
considerable complications. For example, the hopping term in Eq. (8) should no
longer be neglected as it can act on the root state $1010\cdots 1010$, which
is one of the Tao-Thouless torus ground states. Keeping this hopping term, in
combination with Eq. (9) we considered above, leads to more complicated
models, none of which appears to be frustration-free (i.e., until we exhaust
all the terms in the Hamiltonian for a given finite system size). For this
reason, our truncated model in Eq. (15) applies primarily to the cylinder
geometry.

The previously mentioned caveats of boundary conditions highlight the fact
that defining a truncated model is a nontrivial task. Unlike the Laughlin
case, where the Hamiltonian can be naturally truncated according to the
magnitude of the matrix elements, leading to a frustration-free model, such a
truncation scheme was not possible for the MR case. The requirement of a
frustration-free truncated Hamiltonian involves judiciously neglecting certain
terms, which necessitates an independent demonstration of the model’s
validity. In fact, similar difficulties are encountered even in the Laughlin
case in going beyond the first-order truncation in Ref. [41] (see Appendix C),
and they become progressively more severe in higher members of the Read-Rezayi
sequence [64]. It would be useful if a systematic approach could be developed
for generating frustration-free models in all these examples, which would
allow to controllably approach the isotropic 2D limit. The frustration-free
property of the truncated model at $\nu{=}1/3$ has recently allowed to
rigorously prove the existence of a spectral gap in that case [99, 100, 101].
It would be interesting to see if such an approach could be generalized to the
MR state and, potentially, to longer-range truncations.

As mentioned in Introduction, a unique feature of the MR state is the neutral
fermion collective mode and the emergent supersymmetry relating that mode with
the GMP mode we discussed above. It would be worth investigating signatures of
the neutral fermion mode in the proposed Fredkin model or other appropriate
truncations of the MR Hamiltonian near the thin-cylinder limit. Unlike the GMP
mode, which can be directly probed using the geometric quench, it is not known
how to excite the neutral fermion mode. This is because the latter carries
angular momentum $L{=}3/2$ in the long-wavelength limit. Therefore, it does
not couple to the anisotropic deformations of the FQH fluid studied above. We
leave the investigation of such dynamical probes and their implementation on
quantum hardware to future work.

Note added. During the completion of this work, we became aware of a work by
Causer _et al._ [102] which finds evidence of anomalous thermalization
dynamics and quantum many-body scars in a similar type of deformed Fredkin
model. However, the model studied by Causer _et al._ [103] assumes a different
parameter range $|\tau|>1$, which is unphysical from the point of view of FQH
realization considered here.

###### Acknowledgements.

We would like to thank Zhao Liu and Ajit C. Balram for useful discussions.
This work was supported by the Leverhulme Trust Research Leadership Award
RL-2019-015. This research was supported in part by the National Science
Foundation under Grant No. NSF PHY-1748958. PG acknowledges support from NSF
DMR-2130544 and infrastructural support from NSF HRD-2112550 (NSF CREST Center
IDEALS). PG and AK acknowledge support from NSF DMR-2037996. AR acknowledges
support from NSF DMR-2038028 and NSF DMR-1945395. We acknowledge the use of
IBM Quantum services. We also thank the Brookhaven National Laboratory for
providing access to IBM devices.

##  Appendix A Anisotropic interaction matrix elements for the Moore-Read
state

We sketch the derivation of the interaction matrix elements when an
anisotropic band mass is introduced. The interaction Hamiltonian can be
written as:

$\hat{H}=\frac{1}{N_{\phi}}\sum_{\mathbf{p},\mathbf{q}}\bar{V}(\mathbf{p},\mathbf{q},-\mathbf{p}-\mathbf{q})\,:\bar{\rho}(\mathbf{p})\,\bar{\rho}(\mathbf{q})\,\bar{\rho}(-\mathbf{p}-\mathbf{q}):$
(52)

where $\bar{\rho}(\mathbf{q})=e^{-iq_{x}q_{y}/2}\,\sum_{j}e^{iq_{x}\kappa
j}\,c^{\dagger}_{j+q_{y}/\kappa}c_{j}$ is the projected density operator, and
$\bar{V}(\mathbf{p},\mathbf{q},-\mathbf{p}-\mathbf{q})$ is the interaction
potential multiplied by the corresponding form factor:

$\bar{V}(\mathbf{p},\mathbf{q},-\mathbf{p}-\mathbf{q})=F(\mathbf{p},\mathbf{q},-\mathbf{p}-\mathbf{q})\,v(\mathbf{p},\mathbf{q},-\mathbf{p}-\mathbf{q}),$
(53)

where the form factor is

$F(\mathbf{p},\mathbf{q},-\mathbf{p}-\mathbf{q})=e^{-\mathbf{p}^{2}/4-\mathbf{q}^{2}/4-(\mathbf{p}+\mathbf{q})^{2}/4},$
(54)

and the interaction potential

$\displaystyle v(\mathbf{p},\mathbf{q},$
$\displaystyle-\mathbf{p}-\mathbf{q})=\mathbf{p}^{4}\mathbf{q}^{2}+\mathbf{q}^{4}\mathbf{p}^{2}+\mathbf{q}^{4}(\mathbf{q}+\mathbf{p})^{2}+$
$\displaystyle(\mathbf{q}+\mathbf{p})^{4}\mathbf{q}^{2}+\mathbf{p}^{4}(\mathbf{q}+\mathbf{p})^{2}+(\mathbf{q}+\mathbf{p})^{4}\mathbf{p}^{2}$
(55)

is the Fourier transform of Eq. (1).

Anisotropic band mass tensor affects the single-electron wave functions – see,
e.g., Ref. [104]. Thus, it also modifies the matrix elements:

$\displaystyle V_{j_{1}j_{2}j_{3}j_{4}j_{5}j_{6}}$ $\displaystyle\propto
P_{g}(\\{j_{i}\\})\ \mathrm{exp}\biggl{(}-\frac{\kappa^{2}}{2g_{11}}(\sum
j^{2}_{i}-\frac{1}{6}(\sum j_{i})^{2})$
$\displaystyle+\frac{i\kappa^{2}g_{12}}{2g_{11}}(j^{2}_{6}+j^{2}_{5}+j^{2}_{4}-j^{2}_{3}-j^{2}_{2}-j^{2}_{1})\biggr{)}$
(56)

Just as in the isotropic case, the polynomial $P_{g}$ is tightly constrained:
it has to be antisymmetric in the pairs $(j_{1},j_{2})$, $(j_{1},j_{3})$,
$(j_{2},j_{3})$, $(j_{4},j_{5})$, $(j_{5},j_{6})$, $(j_{5},j_{6})$, and its
maximum total degree is $6$. The only such polynomial is the one that appears
in the isotropic case. Therefore the only contribution of the metric in the
prefactor is a constant. The final form will be:

$\displaystyle V_{j_{1}\dots j_{6}}$
$\displaystyle\propto\frac{\kappa^{8}}{g_{11}^{4}}(j_{1}-j_{2})(j_{1}-j_{3})(j_{2}-j_{3})(j_{6}-j_{4})$
$\displaystyle(j_{6}-j_{5})(j_{5}-j_{4})\
\mathrm{exp}\biggl{(}-\frac{\kappa^{2}}{2g_{11}}(\sum
j^{2}_{i}-\frac{1}{6}(\sum j_{i})^{2})$
$\displaystyle+\frac{i\kappa^{2}g_{12}}{2g_{11}}(j^{2}_{6}+j^{2}_{5}+j^{2}_{4}-j^{2}_{3}-j^{2}_{2}-j^{2}_{1})\biggr{)}.$
(57)

##  Appendix B Nonlocal string order in the Fredkin chain

The nonlocal constraint that defines Dyck paths hints that the Fredkin ground
state might have interesting behavior in certain nonlocal order parameters.
This is reinforced by the fact that such nonlocal correlations were found in
the spin-1 analog of our model, the deformed Motzkin chain [105]. The natural
correlations to probe in the Fredkin chain are the string orders discussed in
Ref. [106] in connection to spin-$1/2$ ladders and the Majumdar-Ghosh chain:

$O_{\mathrm{even/odd}}=\lim_{|i-j|\to\infty}\biggl{<}\big{(}S^{z}_{i}+S^{z}_{i+1}\big{)}e^{i\pi\sum_{l=i+2}^{j-1}S^{z}_{l}}\bigl{(}S^{z}_{j}+S^{z}_{j+1}\bigr{)}\biggr{>}$
(58)

where for $O_{\mathrm{even/odd}}$ the sites $i,j$ are both even/odd,
respectively.

Using the MPS representation (31), we test the Fredkin ground state for
nonlocal order, shown in Fig. 11. First, note that nondecaying expectation
values are only found inside the $|\tau|<1$ phase (as the inset shows),
whereas in the $|\tau|>1$ “domain-wall” phase these nonlocal correlations
decay. This suggests that in the $|\tau|<1$ “antiferromagnetic” phase, short
range valence bonds form between consecutive spins. We also notice that
generally $O_{\mathrm{even}}$ is higher in magnitude compared to
$O_{\mathrm{odd}}$. Given that the spin chain always has even length, the
favored arrangement is where all spins are paired (i.e.
$(0,1),(2,3),\dots,(N-2,N-1)$), as opposed to the case where the first and the
last spins remain unpaired. This implies the bonds starting on an even index
will be stronger.

Figure 11: The behavior of $O_{\mathrm{even/odd}}$ as a function of the
deformation parameter $\tau$. String order is not present at $\tau=0$, i.e.,
when the ground state is a product state. For $|\tau|<1$ the string order
parameter increases but drops quickly at $|\tau|=1$ where the gap closes. The
difference in magnitude between $O_{\mathrm{even}}$ and $O_{\mathrm{odd}}$ is
a result of stronger bonds that form between sites $2i$ and $2i+1$, such that
all spins are paired up. The inset shows the behavior of $O_{\mathrm{even}}$
as a function of $|i-j|$, demonstrating that the nonlocal correlations quickly
stabilize to a $\tau$-dependent value. All numerical results are obtained from
a chain with $N=100$ spins, where the values are already converged.

##  Appendix C Motzkin chain as an effective truncated model of the
$\nu{=}1/3$ Laughlin state

In this Appendix, for the sake of completeness, we show that the Motzkin chain
[79, 80, 81, 82] – a closely related spin-1 cousin of the Fredkin chain –
captures the properties of the $\nu{=}1/3$ Laughlin state. This model contains
more terms compared to the model derived in Ref. [41] and hence captures the
physics of the Laughlin state over a slightly larger range of cylinder
circumferences.

The model in Ref. [41] can be derived via a similar method to the one
presented above, but for a 2-body interaction given in terms of $V_{1}$
Haldane pseudopotential [63]. The corresponding matrix elements are now given
by

$\displaystyle
V_{j_{1}j_{2}j_{3}j_{4}}=(j_{1}-j_{2})(j_{4}-j_{3})e^{-\frac{\kappa^{2}}{4}\left[(j_{1}-j_{2})^{2}+(j_{3}-j_{4})^{2}\right]}.\;\;\;\;\;$
(59)

The minimal model beyond the extreme thin-cylinder limit from Ref. [41] can be
written in the positive-semidefinite form

$H^{\prime}_{\mathrm{L}}=\sum_{i}\left(Q^{\dagger}_{i}Q_{i}+P^{\dagger}_{i}P_{i}\right)$
(60)

where

$Q_{i}=\alpha_{i}c_{i+1}c_{i+2}+\gamma_{i}c_{i}c_{i+3}\,,\quad
P_{i}=\beta_{i}c_{i}c_{i+2}$ (61)

and

$\alpha=\sqrt{V_{0110}},\quad\beta=\sqrt{V_{0220}},\quad\gamma=e^{2i\kappa^{2}\frac{g_{12}}{g_{11}}}\sqrt{V_{0330}}$
(62)

The only configurations which are dynamically connected to the root state are
those that can be obtained from applying squeezing operators
$\hat{S}_{i}=c_{i+1}^{\dagger}c_{i+2}^{\dagger}c_{i+3}c_{i}$ to $100100\ldots
001$. This connected component of the Hilbert space can be mapped to a
spin-$1$ model by considering unit cells of three magnetic orbitals, whose
occupations can only take the following patterns:

$|010\rangle\to|\mathrm{o}\rangle\,,\quad|001\rangle\to|+\rangle\,,\quad|100\rangle\to|-\rangle.$
(63)

Thus, we can write the model of Ref. [41] as

$H^{\prime}_{\mathrm{L}}=\sum_{i=0}^{N-2}\mathcal{P}^{\varphi_{\mathrm{L}}(v)}_{i,i+1}$
(64)

where $|\varphi_{\mathrm{L}}(v)\rangle=|{+-}\rangle-v|\mathrm{oo}\rangle$ and
$v=-\sqrt{V_{0330}/V_{0110}}=-3\exp(-2\kappa^{2})$. It is important to notice
there are no boundary conditions – they are not necessary if the mapped
Hilbert space is used (which entails constraints, e.g. configurations with the
first spin $|-\rangle$ and the last spin $|+\rangle$ being disallowed).

###  C.1 Extension to the Motzkin chain

A natural attempt to improve the model in Eq. 64 would be to extend the
truncation, $P_{i}\to\beta_{i}c_{i}c_{i+2}+\delta_{i}c_{i}c_{i+4}$. The newly
obtained Hamiltonian $H^{\prime}$ would have the following off-diagonal
actions:

$\displaystyle H^{\prime\prime}_{\mathrm{L}}|\dots 100\,010\dots\rangle$
$\displaystyle=\beta\delta|\dots 010\,100\dots\rangle$ $\displaystyle
H^{\prime\prime}_{\mathrm{L}}|\dots 010\,001\dots\rangle$
$\displaystyle=\beta\delta|\dots 001\,010\dots\rangle$ (65)

and the Hermitian conjugates. In the spin-$1$ mapping, these mean

$\displaystyle H^{\prime\prime}_{\mathrm{L}}|\dots-\,\mathrm{o}\dots\rangle$
$\displaystyle=\beta\delta|\dots\mathrm{o}\,{-}\dots\rangle$ $\displaystyle
H^{\prime\prime}_{\mathrm{L}}|\dots\mathrm{o}\,{+}\dots\rangle$
$\displaystyle=\beta\delta|\dots+\,\mathrm{o}\dots\rangle$ (66)

However, the fermionic Hamiltonian also produces hoppigns of the follwoing
kind:

$\displaystyle H^{\prime\prime}_{\mathrm{L}}|\dots 100\,100\dots\rangle$
$\displaystyle=\beta\delta|\dots 011\,000\dots\rangle$ $\displaystyle
H^{\prime\prime}_{\mathrm{L}}|\dots 001\,001\dots\rangle$
$\displaystyle=\beta\delta|\dots 000\,110\dots\rangle$ (67)

These break our spin mapping, and connect the entire Hilbert space. Even
though we only keep 2 types of off-diagonal terms, we no longer obtain a
significant reduction in complexity, and in fact we find numerically that the
zero-mode property is also lost. Thus, we focus only on the spin model
instead. The extension of the Hamiltonian in Eq. 64 therefore takes the form:

$H_{\mathrm{M}}=\sum_{i=0}^{N-2}\mathcal{P}^{\varphi_{\mathrm{L}}(v)}_{i,i+1}+\mathcal{P}^{U(w)}_{i,i+1}+\mathcal{P}^{D(w)}_{i,i+1},$
(68)

where we introduced the projectors on the states
$|U(w)\rangle=|{+}\mathrm{o}\rangle-w|\mathrm{o}+\rangle$ and
$|U(w)\rangle=|\mathrm{o}-\rangle-w|{-}\mathrm{o}\rangle$, where
$w=-\sqrt{V_{0440}/V_{0220}}=-2\exp(-3\kappa^{2})$. These implement the
additional the additional terms in our truncation, while keeping the spin
Hamiltonian $2$-local.

Figure 12: Squared overlaps of the ground states of models in Eq. 64 (the
minimal model $H^{\prime}_{\mathrm{L}}$) and Eq. 68 (the extended spin model
$H_{\mathrm{M}}$) with the ground state of the untruncated $V_{1}$
Hamiltonian. The extended spin model captures the properties of the Laughlin
state up to cylinder circumferences of $L_{2}\approx 8\,l_{B}$, where the
overlaps are $\gtrsim 95\%$.

The Hamiltonian in Eq. 68 represents a particular deformation of the Motzkin
spin chain introduced in [82]. It has a unique, zero-energy ground state which
is equal to an area-weighted sum of Motzkin paths, $p\in\mathcal{M}_{N}$:

$|\psi_{0}^{\mathrm{M}}\rangle=\mathcal{N}^{-1}\sum_{p\in\mathcal{M}_{N}}v^{A_{\square}(p)}w^{A_{\triangle}(p)}|p\rangle.$
(69)

Fig. 12 shows that this ground state has good overlap with the Laughlin over a
larger range of circumferences. Notice that in the Motzkin ground state with
$v=w$, the weights of a path $p$ are $v^{A(p)}$, i.e. only dependent on the
total area and not its shape. Fig. 13 illustrates how this is different from
Eq. 69.

Figure 13: (a): Two types of allowed moves in the Motzkin chain. The upper
move corresponds to $|\mathrm{oo}\rangle\to|{+-}\rangle$, while the bottom one
shows $|\mathrm{o}{+}\rangle\to|{+}\mathrm{o}\rangle$
($|{-}\mathrm{o}\rangle\to|\mathrm{o}{-}\rangle$ is omitted for brevity).
Although each step increases the area of the path by the same amount, the
corresponding weight in Eq. 69 scales differently depending on the move. (b):
One type of allowed configuration in the ground state, that was not present in
the model Eq. 64. The sketch shows how the total area is divided into
$A_{\square}$ and $A_{\triangle}$, determining its weight in the ground state.

Similar to the Fredkin chain discussed in Section III.3, the ground state of
the Motzkin chain also has an exact MPS representation in terms of matrices:

$A^{\mathrm{o}}_{jk}=w^{j-1}\delta_{j,k}\quad
A^{+}_{jk}=v^{1/2}w^{j-1}\delta_{j+1,k}\quad A^{-}=(A^{+})^{T}$ (70)

The Motzkin chain with equal deformation parameters $v=w$ has been studied in
depth in the literature. Its gap for $v<1$ has been proven [107], and based on
our analogy with the Laughlin state we conjecture that the gap survives for
$w\leq v$.

###  C.2 Laughlin graviton root state and geometric quench dynamics

Here we analyze the graviton root state and dynamics following the geometric
quench for the $\nu{=}1/3$ Laughlin state, following a similar approach the MR
state in Sec. IV. As explained in the main text, we can use the SMA ansatz
[26, 27] to identify the GMP state with nonzero momentum $\mathbf{k}$:

$|\phi_{\mathbf{k}}\rangle=\overline{\rho}_{\mathbf{k}}|\psi_{0}\rangle=e^{\frac{ik_{x}k_{y}}{2}}\sum_{j}e^{ik_{x}\kappa
j}c^{\dagger}_{j+k_{y}/\kappa}c_{j}|\psi_{0}\rangle$ (71)

In the thin-torus limit, the ground state is the root state
$|R_{0}\rangle=|1001001\dots\rangle$. Thus, the graviton root state with
momentum $\mathbf{k}=2\kappa\hat{y}$ is given by

$|R_{g}^{(2)}\rangle\propto|1100001001\dots\rangle+|1001100001\dots\rangle+\dots$
(72)

In the extreme thin-cylinder limit, these states are degenerate in energy, and
the first product state is the Jack root state [108], from which all states
that follow can be obtained by applying a sequence of squeezes.

Figure 14: The spectral function $I(\omega)$ for the untruncated Laughlin
Hamiltonian at $\nu{=}1/3$ as the cylinder circumference $L_{2}$ is varied
between the isotropic 2D limit and the thin-cylinder limit. The system size is
$N=9$ electrons, $N_{\phi}=25$.

However, as the geometric quench preserves momentum, to identify the long-
wavelength limit of the graviton state we rely on spectral function $I(w)$
from Eq. (39). In the present case, $\hat{O}$ is a 2-body operator [54] with
matrix elements

$\displaystyle O_{j_{1}j_{2}j_{3}j_{4}}$
$\displaystyle=\delta_{j_{1}+j_{2},j_{3}+j_{4}}(j_{1}-j_{2})(j_{3}-j_{4})$
$\displaystyle\times\left(\sum j_{i}^{2}-\frac{1}{4}(\sum j_{i})^{2}\right)$
$\displaystyle\times\exp\left[-\frac{\kappa^{2}}{2}\left(\sum
j^{2}_{i}-\frac{1}{4}(\sum j_{i})^{2}\right)\right].$ (73)

The spectral function $I(\omega)$ for the $\nu{=}1/3$ Laughlin state is
plotted in Fig. 14. Similar to the MR case in Fig. 8, we see that the graviton
undergoes a nontrivial evolution as the cylinder circumference is varied, with
clear avoided crossings in the evolution. In the thin-cylinder limit, the gap
of the graviton can be accurately estimated from the dominant matrix element
in the Hamiltonian.

Figure 15: Geometric quench dynamics in the $\nu{=}1/3$ Laughlin state. The
system size is $N_{e}{=}6$, $N_{\phi}{=}16$, and the circumference is
$L_{2}=2.8\ell_{B}$. The system is initialised in the isotropic ground state
and then time-evolved by the anisotropic Hamiltonian with $Q{=}0.02$. The
resulting dynamics is in excellent agreement with the linearized bimetric
theory, shown by dashed lines.

The graviton state is given by acting on the ground state with the quadrupole
operator (C.2). From the model in Eq. 64, we also know that the ground state
is approximated by

$\displaystyle|\psi_{0}\rangle$
$\displaystyle=\prod_{i}\big{(}1-\sqrt{V_{0330}/V_{0110}}\,e^{2i\kappa^{2}g_{12}/g_{11}}\hat{S}_{i}\big{)}|R_{0}\rangle$
$\displaystyle\approx|R_{0}\rangle-3e^{-2\kappa^{2}\frac{1-ig_{12}}{g_{11}}}\sum_{i}\hat{S}_{i}|R_{0}\rangle$
$\displaystyle\approx|R_{0}\rangle-3\exp\big{[}-2\kappa^{2}\big{(}1-Qe^{i\phi}\big{)}\big{]}\sum_{i}\hat{S}_{i}|R_{0}\rangle,$
(74)

where we assumed $e^{-2\kappa^{2}}$ and the metric anisotropy $Q,\phi$ to be
small. The graviton is then approximated by:

$\displaystyle|\psi_{g}\rangle$ $\displaystyle=\hat{O}|\psi_{0}\rangle\propto
e^{-\frac{5\kappa^{2}}{2}}\bigg{(}\sum_{i}\hat{S}_{i}|\psi_{0}\rangle+\mathcal{O}\big{(}e^{-2\kappa^{2}}\big{)}\bigg{)}.$
(75)

From here we deduce the graviton root state,

$|\psi_{g}\rangle=\sum_{i}\hat{S}_{i}|\psi_{0}\rangle=|01100010\dots\rangle+|10001100\dots\rangle+\dots$
(76)

Similar to the MR case, the graviton root state here is also proportional to
the first order squeezes and it encodes the simplest quadrupole structure of
the form $-++-$ in each unit cell.

Repeating the same steps as in Eqs. (45)-(51) of the main text, from the
graviton root state we can determine the time-evolved state, showing that it
takes the form (at first order)

$\displaystyle|\psi(t)\rangle\approx|R_{0}\rangle-3e^{-2\kappa^{2}}(1+2\kappa^{2}A(1-e^{-iE_{\gamma}t}))|R_{g}\rangle,$
(77)

which has the identical form to the linearized bimetric theory in Eq. (51).
This agreement is confirmed in Fig. 15.

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