1911.00099.pdf

Robust encoding of a qubit in a molecule

Victor V. Albert,

Jacob P. Covey,

and John Preskill

Institute for Quantum Information and Matter

and Walter Burke Institute for Theoretical Physics

California Institute of Technology, Pasadena CA 91125, USA

(Dated: November 20, 2019)

We construct quantum error-correcting codes that embed a finite-dimensional code space in the

infinite-dimensional Hilbert state space of rotational states of a rigid body. These codes, which

protect against both drift in the body’s orientation and small changes in its angular momentum, may

be well suited for robust storage and coherent processing of quantum information using rotational

states of a polyatomic molecule. Extensions of such codes to rigid bodies with a symmetry axis are

compatible with rotational states of diatomic molecules, as well as nuclear states of molecules and

atoms. We also describe codes associated with general nonabelian compact Lie groups and develop

orthogonality relations for coset spaces, laying the groundwork for quantum information processing

with exotic configuration spaces.

I. INTRODUCTION

Quantum systems described by continuous variables

arise in many laboratory settings. For example, a mi-

crowave resonator in a superconducting circuit or the mo-

tional degree of freedom of a trapped ion can be viewed as

a harmonic oscillator with an infinite-dimensional Hilbert

space. Such continuous-variable systems have potential

applications to quantum information processing. How-

ever, quantum information encoded in an oscillator can

be easily damaged by ubiquitous noise sources such as

dissipation and diffusive motion in phase space.

Robustness against noise can be achieved more easily

by encoding a protected finite-dimensional system within

the infinite-dimensional Hilbert space of an oscillator.

One method for doing so was proposed some years ago by

Gottesman, Kitaev, and Preskill (GKP) [1]. A GKP code

is a quantum error-correcting code designed to protect

against noise that slightly shifts the position or momen-

tum of an oscillator. The ideal basis states for the code

space are “grid states” supported on periodically spaced

points in position or momentum space. By measuring

the code’s check operators, one can diagnose a shift error

that may have occurred, without disturbing the encoded

quantum information, and then correct the error (if the

shift introduced by noise is not too large) by performing a

compensating shift. These codes are expected to perform

well against realistic noise, including dissipation, which

typically acts locally in phase space [2–4]. Construction

of GKP grid states has recently been demonstrated ex-

perimentally [5, 6].

In this paper, we develop GKP-like codes that protect

against, not noise that shifts the position and momentum

of an oscillator, but rather noise that shifts the (continu-

ous) orientation and (discrete) angular momentum of an

asymmetric rigid body. GKP codes for objects that ro-

tate about a fixed axis [Fig. 1(a)] were already discussed

in [1]. In that case, the orientation of the object cor-

responds to an element of the two-dimensional rotation

group

∞

. New issues arise for an object

that rotates freely in three dimensions [Fig. 1(b-c)], with

orientation described as an element of the 3-dimensional

Figure 1.

Rigid bodies.

A molecular code protects against

errors in the orientation and angular momentum of a rigid

body, which may be

(a)

a planar rotor whose orientation is an

element of the two-dimensional rotation group

(b)

a rigid

rotor whose orientation is an element of the 3-dimensional

rotation group

, or

(c)

a linear rotor whose orientation is

a point on the two-sphere

. A basis state for the code, or

codeword

, is a superposition of a finite number of orientations.

rotation group

(for an object with no symmetries)

or a point on the two-sphere

(for an object

with a symmetry axis).

Our work is motivated by recent progress in trapping

and coherently manipulating individual diatomic and

polyatomic molecules [7–11]. Since we only consider a

molecule’s rotational degrees of freedom, for our purposes

a molecule is equivalent to a rigid body. For that reason,

we refer to quantum codes embedding a protected finite-

dimensional subspace in the infinite-dimensional Hilbert

space of a rigid body as

molecular codes

The rigid rotor Hamiltonian describing molecular ro-

tational motion is inherently anharmonic; because the

energy levels are unevenly spaced, transitions between

levels can be individually addressed using microwave

fields. Hence, proposals for storing quantum information

in molecules [12–23] (see also [24, 25]) typically pick out

two low-lying long-lived energy eigenstates as basis states

for a qubit. One can also introduce an external electric

field, and encode a qubit using the resulting “pendular”

eigenstates [26–28]. Other proposals have advocated us-

ing vibrational or spin degrees of freedom [29, 30].

arXiv:1911.00099v2 [quant-ph] 19 Nov 2019

Rigid rotor energy eigenstates, if spaced sufficiently far

apart in angular momentum, provide protection against

small jumps in angular momentum, but are unprotected

against dephasing in the angular-momentum eigenstate

basis resulting from fluctuations in the rotor’s orienta-

tion. Our molecular codes, inspired by GKP codes, are

designed to protect against both momentum kicks and

orientational diffusion of a single molecule. Here we de-

velop the theory of molecular codes and generalizations

thereof. Laboratory realizations of these coding schemes

that actually improve the coherence times of molecular

qubits may still be far off, but we propose laying the

foundations for molecular quantum error-correction as a

challenging goal for the physicists and chemists of the

NISQ era [31].

Though our work is partially motivated by advances in

molecular physics, the coding methods we use are best ex-

plained in an abstract group-theoretic framework, which

we will summarize in the next section. In Sec. III, we enu-

merate a variety of physical settings, in molecular physics

and beyond, where our code constructions may be appli-

cable. The connection with rotational states of the three

molecular rotors from Fig. 1(a-c) is developed in greater

detail in Secs. IV-VI, respectively. Section VII discusses

extensions to more abstract state spaces. Section VIII

contains conclusions and ideas for future work.

II. SUMMARY OF OUR FRAMEWORK

We describe a family of codes that generalize the

GKP codes [1], which were initially formulated to encode

a finite-dimensional system in the infinite-dimensional

Hilbert space of a bosonic mode, or of many bosonic

modes. Each code in our generalized GKP code family

is associated with a nested sequence of groups

⊂

(1)

Here

is a continuous group of shifts in the position of

a physical object. If no nontrivial subgroup of

leaves

the object invariant, and any position can be reached by

applying an element of

to a standard initial position,

then we may regard the “position eigenstates”

〉

∈

}

as a basis for the Hilbert space of the object. The

generalized GKP code is a subspace of this Hilbert space

defined by two properties: (1) The discrete subgroup

of the continuous group

leaves any state in the code

space invariant, (2) The subgroup

acts transitively on

a basis for the code space.

For the standard GKP code,

is the abelian noncom-

pact group

, the group of translations in position space

of a particle in one spatial dimension. The subgroup

the infinite discrete group containing all translations of

the particle by an integer multiple of

, where

is a fixed

real number. The subgroup

contains all translations by

an integer multiple of

dα

, where

is the dimension of the

code space. In this case, we may choose the basis for the

code space to be (up to normalization)

∣

〉

∝

∑

∈

= (

)

〉

(2)

where

〉

is a position state of the oscillator and

∈

{

···

−

}

. We refer to each such basis element of

the code as a

codeword

. Thus a translation of

dα

leaves the codewords invariant, and a translation of

permutes the codewords according to

→

modulo

. A shift in

due to an error can be detected

by measuring

modulo

In addition to errors that shift the value of

, the

GKP code also protects against errors that introduce

dependent phases. Phase errors which are diagonal in

the

basis are described by functions on

. Such func-

tions can be Fourier-expanded using irreducible represen-

tations (irreps) of

, labeled by the momentum

. The

irreps that preserve the code space are those with

integer multiple of

dα

, and those that act trivially on the

code space have

an integer multiple of

For a generalized GKP code, the detectable position

shifts are labeled by elements of the coset space

and the “logical” position shift errors that preserve the

code space are labeled by elements of

. Undetectable

logical phase errors correspond to representations of

which represent the subgroup

trivially, but represent

nontrivially.

In Sec. IV, we illustrate the concepts underlying gener-

alized GKP codes by discussing the example of a planar

rotor. In this case

, the infinite compact group

of rotations in a two-dimensional plane,

is the finite

subgroup of

containing rotations by an angle which

is an integer multiple of

, and

is the subgroup of

containing rotations by an angle which is an integer

multiple of

. Here

N,d

are positive integers, and

the dimension of the code space. This code can correct

a rotation of the planar rotor by any angle less than

and can correct a shift in angular momentum by any in-

teger less than

. The structure of this code, for the

case

= 3

and

= 2

, is depicted in Fig. 2(a).

While these planar rotor codes were already introduced

in [1], generalized GKP codes where

is nonabelian

have not been previously discussed to our knowledge. In

Sec. V, we introduce

molecular codes

, which can protect

an asymmetric rigid body from rotational shift errors and

angular momentum kicks. In this case

, the infi-

nite compact group of proper rotations in 3D space. The

finite subgroups

⊂

can be chosen in various

ways. By choosing

⊂

to be discrete

cyclic groups of rotations about one axis (for chemists,

), we obtain codes that can correct small rota-

tions of the body about

any

axis, and can also correct

momentum kicks that change the total angular momen-

tum of the body by

δ` < N/

. For a pictorial representa-

tion of this code in the case

= 3

= 2

, see Fig. 2(b).

We also discuss examples where

and

are finite

nonabelian subgroups of the rotation group. Guided by

the stabilizer formalism, we show that for each molecular

code there is a Hamiltonian which has the code as its

ground space. Each ideal codeword is not normalizable, a

superposition of an finite number of position eigenstates,

but there are normalizable approximate codewords which

maintain good error-correcting properties.

We generalize the code construction further in Sec. VII,

where we allow

to be any finite group, compact Lie

group, or sufficiently well-behaved non-compact group.

This formulation provides a unified treatment that en-

compasses molecular codes (

), CSS codes (

) and GKP codes for qudits (

), planar rotors (

), and oscillators (

In Sec. VI, we discuss the

linear rotor

, a rigid body

with a symmetry axis, such as a heteronuclear diatomic

molecule. For this case, the quantum codes we construct

are not generalized GKP codes as defined above, because

the position basis states of the linear rotor are indexed

not by elements of a group, but rather by points in the

coset space

. Codewords of a linear rotor

code are uniform superpositions of antipodal points on

, which lie in the same

orbit

acting on

, where

is a finite subgroup of

. See Fig. 2(c) for the case

and codespace dimension two.

The linear rotor codes can also correct small rota-

tions about any axis, and analyzing correction of momen-

tum kicks follows closely the corresponding discussion for

molecular codes. However, for correction of

combinations

of rotations and momentum kicks, there are complica-

tions which arise because each

rotation acting on

has fixed points.

Coset spaces arise in both generalized GKP codes and

linear rotor codes, but for different reasons. In GKP

codes, position basis states are in one-to-one correspon-

dence with elements of the group

, and the position

shifts detected by the code are labeled by elements of

. In linear rotor codes, the position basis states them-

selves are in one-to-one correspondence with elements of

the coset space

. Since coset spaces play a central

role in both settings, we formulate position and momen-

tum bases, shift operators, and orthogonality relations

for general

in Appx. D. These are applicable to

symmetric molecules when

(see Sec. III C), and

may be of independent interest for general

III. EXPERIMENTAL REALIZATIONS

Before proceeding to discuss code constructions in

more detail, in this section we briefly mention some of the

physical settings where these constructions might apply.

The rotational states of a molecule provide one such set-

ting, where the orientations of a molecule correspond to

elements of

(in the case of an asymmetric polyatomic

molecule) or

(in the case of a heteronuclear diatomic

molecule). In addition, other physical systems, includ-

ing atomic or molecular hyperfine, vibrational, and elec-

tronic states, as well as atomic ensembles and levitated

nanoparticles, realize similar configuration spaces.

Figure 2.

Codeword constructions.

(a)

Left panel:

Sketch of the planar rotor state space

. Black/white points

represent the positions present in the two codewords (10) of

the

⊂

GKP rotor code. Correctable shifts

(

−

]

are

highlighted in blue. Right panel:

angular momentum lad-

der

∈

. Black/white squares represent momentum states

present in the logical-

codewords (11).

(b)

Sketch of the

same features of the

⊂

molecular code (45). Left panel:

Position space is drawn as a ball of radius

with antipodal

points identified, and each

rotation by angle

around

axis

∈

corresponds to the vector

on the ball. The

set of correctable rotations is in blue, but part of it is cut

out to show that it contains the origin (meaning that small

rotations around any axis are correctable). Right panel: Mo-

mentum space is a 3D square pyramid with height labeled by

and base by

| ≤

. We plot only the

part,

where the codewords (49) have support.

(c)

Sketch of similar

features of the

⊂

linear rotor code (102). Left panel:

the blue spherical lune contains all points that are closer to

the enclosed black point than to any other black or white

point. Right panel: Momentum space is a 2D pyramid with

base

| ≤

, showing states participating in the logical-

codewords (105).

A. Molecular rotors

GKP codes were realized experimentally [5, 6] nearly

20 years after the initial proposal [1], and full-fledged

error correction for molecular qubits may still be many

years away [32, Sec. V.D]. Nevertheless, significant steps

toward the realization of molecular codes may be feasible

during the NISQ era [31] as the technology for trapping

and controlling molecules [33–38] continues to advance.

Laser cooling and trapping techniques have recently

enabled several seminal advances for diatomic polar

molecules, namely: the creation of low-entropy arrays

in an optical lattice [39, 40], trapping and imaging in

tweezer arrays [7, 9] and magnetic traps [41, 42], and

even the first quantum degenerate gas of polar molecules

[43]. Coherence times of

∼

100

ms to

∼

second in an-

gular momentum states of diatomic polar molecules have

already been observed in several experiments [44–46].

Laser cooling and quantum control of polyatomic

molecules continues to be a rapidly-progressing field

[8, 32, 47, 48]. Further, the possibility of angular mo-

mentum state-resolved detection has recently been con-

sidered [10, 32]. In addition, quantum gates of optically-

trapped symmetric top molecules have recently been an-

alyzed [21]. Symmetric top molecules also hold promise

for simulating quantum magnetism [49, 50]. More-

over, specific classes of polyatomic linear polar molecules

that feature more than one optically active metal atom

have recently been proposed for laser cooling and trap-

ping [51]. Prospects for cooling other complex poly-

atomic molecules have also been analyzed [52].

Here we highlight a few techniques that could help re-

alize aspects of our codes in real systems.

Rotational states

The codewords for our codes can

be expressed as coherent superpositions of several differ-

ent molecular orientations. Alternatively, each codeword

can be expressed as a coherent superposition of eigen-

states of angular momentum (

a.k.a.

“rotational states”).

When discussing experimental realization of the codes,

the basis of rotational states is far more convenient than

the position-eigenstate basis, because rotational states

can be directly addressed using experimental tools.

For the case of a planar rotor, with configuration space

, the rotational basis states

〉}

transform as one-

dimensional irreducible representations of

; for the case

of a polyatomic molecule, with configuration space

the basis states

〉}

correspond to matrix elements of

irreducible representations of

; and for the case of a

diatomic molecule, with configuration space

, the ba-

sis states

〉}

correspond to spherical harmonics. In

molecular physics [53–56],

corresponds to the total an-

gular momentum of the rotor,

is the

-component of

the angular momentum in the lab frame, and

is the

-component in the rotor frame. How codewords are ex-

pressed as linear combinations of rotational states is il-

lustrated in the right panels of Fig. 2(a-c), respectively,

for three simple rotor codes.

Microwave dressing

One can try to stabilize

the codewords using polychromatic microwave dressing.

Note that we consider single molecules and neglect any

effects due to their interaction. The inherent rigid-rotor

Hamiltonian for

and

is diagonal in the angular

momentum basis, with eigenvalues

(

+ 1)

Therefore,

transitions between states with different momenta are in-

dividually addressable (unlike, e.g., the transitions of a

harmonic oscillator Hamiltonian). Selection rules for the

internal indices

n,m

are dictated by the polarization of

the microwave field. Thus, the energy and polarization

of a microwave field can be tuned to couple two angular

momentum states that are neighbors in the angular mo-

mentum pyramid. That is, the value of

, or

can

change by 1 unit in a single-photon transition.

However, the angular momentum states making up

each codeword are widely spaced in the internal indices.

For example, in the case of the

code depicted in

Fig. 2(b), the codewords have support only on

〉}

states such that

is an integer multiple of 3. For

[Fig. 2(c)], a similar pattern emerges, except that for

even

the rotational state

〉

is populated only if

an even multiple of

, while, for odd

〉}

is populated

only if

is an odd multiple of

Because a single microwave tone couples states that

differ by just 1 unit of

, a sequence of virtual tran-

sitions induced by multiple pulses would be needed to

couple states with more widely separated values of

. For example, coupling states with

δm

= 3

re-

quires a 3-photon transition that is sufficiently detuned

from the two intermediate states, and coupling states

with

δ`

= 2

requires a two-photon transition sufficiently

detuned from the one intermediate state. We outline

a scheme to generate these states in Appendix A. This

scheme requires many pulses, but it is on par with pre-

viously proposed molecular dressing schemes [57–59].

We have neglected rotational-state-dependent trapping

effects, which are prominent in optical dipole traps [44,

60–62]. These effects will be negligible when consider-

ing a single molecule in the motional ground state of the

trap, whose intensity can be robustly stabilized. In this

case, the unique Stark shift for each rotational state due

to the trap simply requires an updated microwave fre-

quency catalog for all transitions. However, this spread

in polarizability poses a practical problem when con-

sidering many molecules, since one must ensure that

they all experience the same optical intensity. Accord-

ingly, alternative trapping schemes may be more appro-

priate for the applications proposed in this work. Mag-

netic micro-traps [41] are compatible with electronic spin

doublet or triplet molecules such as CaF, SrF, YbF, or

YO. Radio-frequency electric traps are compatible with

molecular ions [20, 32]. Such trapping potentials are sub-

The actual Hamiltonian depends on the molecule’s moments of

inertia [53–56]. We use the “spherical top” Hamiltonian to sim-

plify the analysis.

stantially less dependent on the rotational state of the

molecule since they couple to magnetic dipoles and elec-

tric monopoles, respectively.

More generally, one can consider engineering the de-

sired pulses to generate states or correct errors via es-

tablished optimal-control schemes [38, 63]. It has been

shown that one can control the planar [64, 65], linear [66–

68], and even rigid [69] rotors, and it would be useful to

extend these and other efforts [70, 71] to stabilizing the

required code subspace.

Crystal fields

In a class of quantum error-correcting

codes called

stabilizer codes

, the code space is the simul-

taneous eigenspace with eigenvalue 1 of a set of commut-

ing Pauli operators, which are called

check operators

. A

special subclass of stabilizer codes are the CSS codes,

for which each check operator can be chosen to be ei-

ther

-type

; the

-type operators

{

(

)

}

are

diagonal in the computational basis, and the

-type op-

erators

{

(

)

}

permute the computational basis states.

The code subspace may be regarded as the degenerate

ground space of the Hamiltonian

code

−

∑

(

)

−

∑

(

)

(3)

Our molecular codes are not stabilizer codes, but as we

explain in Sec. V, the code space is the degenerate ground

space of a Hamiltonian which is a sum of

-type and

type terms. Here the

-type check operator rotates the

molecule, while the

-type check operator is diagonal in

the position basis but alters the total angular momen-

tum. Just like its oscillator counterpart [1, Sec. XIII],

this molecular Hamiltonian is gapless, but ground states

of an approximate gapped version would be close to the

approximate codewords we introduce in Sec. V C.

The

check operators are momentum kicks which

couple well-separated angular momentum states

〉}

for

〉}

for

. For example,

for a linear ro-

tor code based on the octahedral group is a superposition

of octopole (

= 4

) spherical harmonics; see Eq. (116).

Such harmonics are in principle present in a general in-

teraction with a bath [72]. However, simple laser, DC,

or microwave fields produce only

≤

harmonics [54,

Chs. 4 and 7].

One way to generate the required higher value of

to put the molecule into a crystal lattice. For rotor codes

based on a discrete subgroup

⊂

is the lowest-

function that is symmetric under

. Thus, putting

the rotor into a

-symmetric lattice yields a background

field whose dominant term is exactly

. For example,

putting a linear rotor into an octahedrally symmetric lat-

tice yields a background potential [73, 74] that is exactly

the

(116) required for the octahedral code. This

potential is minimized at those orientations of the ro-

tor that are superposed to construct the codewords; in

fact, these degenerate minima were noticed earlier in an

experimental context [75]. Similarly, embedding into a

two-dimensional square lattice yields the appropriate

(114a) for a linear-rotor version of the planar rotor code

introduced in Sec. IV. To access subgroups of

forbid-

den in crystals, one could consider embedding a molecule

in a quasicrystal.

Crystal symmetries can enforce only the

check op-

erator condition; the

check operator condition must

be imposed by some other means. The

operators

are trigonometric functions of the angular momentum

operators

←−

for

. These are not nat-

urally available, as the rigid rotor Hamiltonian (75) and

its generalizations

contain terms that are at most bi-

linear in the angular momentum components. However,

there are other terms in the full rotor-in-lattice Hamilto-

nian [76, Eq. (7.2)], and, akin to superconducting circuit

schemes [77], one might engineer the molecule’s environ-

ment (for example, by embedding the molecule in a liquid

Helium nanodroplet [78]) to arrange for adiabatic elimi-

nation to provide the required

terms.

Nuclear spin coupling

If an error causes the

molecule to rotate slightly, we recover from the error

by applying a compensating small rotation. The desired

rotation can be executed by turning on a Hamiltonian

which is linear in the angular momentum. But since the

natural rigid rotor Hamiltonian is quadratic, this linear

term is not so easily realized in the laboratory.

One way to provide a Hamiltonian term which is linear

in the molecule’s angular momentum is to couple the ro-

tational states of the molecule to nuclear spin states via

nuclear spin-rotation interactions [53, Eq. (1.32)][45, 79]

nsr

←−

(4)

where

is the nuclear spin. The nuclear spin can serve

as a convenient ancilla system, and the orientation of

the molecule can be controlled by manipulating the nu-

clear spin. Similar approaches have been applied to solid-

state systems in which electronic spins are coupled to

nuclear spins [80]. This is roughly analogous to using a

superconducting Josephson-junction device coupled to a

bosonic mode for manipulating the states of a bosonic

error-correcting code.

We also need to correct momentum kicks by applying

unitary operations that change the value of

. Operations

which shift the occupation number of a cavity can be

applied by coupling the cavity to a 3-level atom [81] or

by using linear optics [82]. Similar schemes could shift

the value of

for a

rotor. Extensions of such schemes

may be helpful for controlling the rotational states of

higher-dimensional rotors.

B. Spin systems

Certain combinations of spins offer another platform

for simulating the linear rotor space

and quotient

spaces

from Table I. We list three manifestations:

spin-

systems in a totally symmetric spin state,

spin-

systems in a totally symmetric state, and a pair

Space

Group

Quotient space

Wigner-Seitz unit cell

lens space

dihedral

prism space

tetrahedral

octahedral space

octahedral

truncated cube space

icosahedral

Poincaré dodecahedral space

two-sphere

projective plane

spherical two-orbifold

Table I. Quotient spaces mentioned in this work [83, 84] (see

also [85, Sec. 3.8][86–88]). Spaces associated with

char-

acterize rotational states of various molecules (see Sec. III C).

is the order-

cyclic group,

is the order-

dihedral group,

∞

is the circle group, and

o

∞

is the group of planar rotations and

reflections. Some of these spaces are shown in Figs. 2 and 4.

of spin-

systems. In the limit of large

, each of these

systems provides a useful approximation to one of the

spaces of interest. While the first two cases are usually

studied in the context of atomic ensembles, the third case

can easily arise in the nuclear spin manifold of an atom

or a molecule.

Many small spins

spin-1/2 particles in a totally

symmetric spin state have a total angular momentum of

. The

→ ∞

limit of this large collective spin is

sometimes said to be a semiclassical limit, meaning that

the spin-

object behaves like a continuous classical

spin when

is large. An intuitive way to understand

this limit is to consider the spin-coherent states

〉

(

−

iφ/

cos

∣

〉

iφ/

sin

∣

−

〉)

⊗

(5)

for

= (

θ,φ

)

∈

[89, 90]. These states are not orthog-

onal; instead they form an overcomplete frame for the

collective spin’s

(

+ 1)

-dimensional Hilbert space, with

overlap

|〈

′

〉

= (

′

)

. As

→ ∞

, the states

become orthogonal and correspond to the position states

〉

(Table V.B). For finite

, superpositions of these

spin-coherent states can be approximate codewords for a

linear rotor code.

Numerous manifestations of entangled ensembles of

many spin-1/2 atoms have recently been demon-

strated [91–94], and the current status of the field is sum-

marized in Ref. [95].

Many medium spins

Any pure state of a spin-

system is invariant under a continuous

subgroup of

the rotation group

; each pure state corresponds to a

point on the Bloch sphere, and a rotation about the axis

aligned with that Bloch vector leaves the state invariant.

In contrast, there are pure states in higher-spin represen-

tations for which the subgroup which preserves the state

is a nontrivial

discrete

subgroup

. For example,

the spin-

state

〉∝|

−

〉

√

〉

is invariant under the

tetrahedral subgroup

. Therefore, applying

rota-

tions to

〉

generates a manifold of states

〉

}

, where

the label

is a point in the coset space

∈

. The

spin-coherent states

〉

⊗

}

, obtained by taking a tensor

product of many identical elements of this manifold, ap-

proximate the position-basis states of

in the limit

of large

. This idea can be generalized: spin-coherent

states

〉

⊗

}

approximate the position-basis states for

the coset space

, if

〉

is a higher-spin state with

invariance group

The above

-symmetric and similar

-symmetric

states [96, Table 2] — examples of Perelomov coherent

states [90] — have been used as a mean-field ansatz for

the ground space of spin-

Bose-Einstein condensates

[97, 98]. We will use such coherent states to extract error

syndrome information for molecular codes (see Sec. V B).

Two large spins

Instead of using only the symmet-

ric subspace, one can consider the entire space of a pair

of spin-

systems. Per the addition rules [99, Ch. 8],

⊗

2 = 0

⊕

⊕···⊕

(6)

the

(

+ 1)

orthonormal basis states for this system

can be chosen to be the angular-momentum eigenstates

〉}

, with

≤

and

| ≤

. These are precisely the

rotational states of a linear rotor, except for the trunca-

tion

≤

. Formally, then, the state space of a pair of

spin-

systems matches the state space on

in the

limit

→∞

. Since the normalizable approximate code-

words of the linear rotor code are necessarily truncated

for large

anyway, these approximate codewords can be

accurately realized using a pair of spin-

systems for

sufficiently large

(see Sec. V C).

If one instead considers two different spins

and

′

, one obtains a different band of

momentum

states. While developing codes for such band-limited sub-

spaces is outside the scope of this work, it is possible that

our coding strategies may also be useful there.

As a concrete experimental platform for large-spin sys-

tems, we can consider nuclear spin spaces of molecules or

single atoms. Diatomic molecules such as NaCs [19] of-

fer exactly the band-limited subspaces mentioned above.

Concerning single atoms, Lanthanide species such as dys-

prosium (Dy), holmium (Ho), and erbium (Er) have

large total spin manifolds in the their ground states due

to their large nuclear spins and many unpaired elec-

trons in their f-shells. Accordingly, such atoms have al-

ready attracted attention for the possibility of scaling up

quantum computing by collectively encoding in multi-

level atoms [100–102]. Ho in particular has the largest

hyperfine ground space of any atom, with 128 ground

states [102]. Laser cooling and trapping techniques are

well established for Dy [103], Ho [104], and Er [105], as

well as other lanthanides. Moreover, quantum degener-

ate gases of Dy [106, 107] and Er [108, 109] are widely

used for novel quantum simulations based on their large

magnetic dipole moments.

C. Other systems

Planar rotors

Several systems have the configura-

tion space of the planar rotor. The system depicted in

Fig. 1(a) is a diatomic molecule confined to rotate in a

two-dimensional plane, but one can also consider a two-

ion crystal [110]. Other possibilities include the phase

difference between two superconductors on either side of

a Josephson junction [111] and orbital angular momen-

tum of light [112].

One can also embed the first few angular momentum

states of the planar rotor in the linear and rigid rotors.

For fixed angular momentum

, the linear rotor subspace

〉}

with

| ≤

is equivalent to the band-limited

subspace

〉

|≤

}

of the planar rotor.

Symmetric molecules

A molecule with symmetry

group

has an orientation state space parameterized by

(see Table I). For example, the methane molecule

has the tetrahedral symmetry group

, and the al-

kaline earth monomethoxide (MOCH

) family — poten-

tially useful for quantum computing [21] — has sym-

metry group

. This is also the relevant symmetry

group of Posner molecules, postulated to have poten-

tially useful quantum effects [113, 114]. The symme-

try group of the fullerene molecule is the icosahedral

group

, and the 3-manifold

has an exotic shape

that was once proposed as a model for the geometry

of the universe [88, 115]. It is interesting that such

exotic spaces are readily accessible in relatively simple

molecules. Completely asymmetric and

-symmetric

molecules correspond, respectively, to rigid and linear ro-

tors from Sec. III A.

More generally, if a group

acts transitively on the

states of a quantum system, and the subgroup

leaves the states invariant, then the configuration space

of the system is

. In Appendix D, we develop mathe-

matical tools for parameterizing the position eigenstates

and the dual momentum states of such a system, includ-

ing orthogonality/completeness relations, and a Poisson

summation formula [116].

Electronic states

One can consider embedding cer-

tain spaces from Table I in the electronic eigenstates of

single atoms. The eigenstates of hydrogen offer a plat-

form for a band-limited subspace of the linear rotor

and even the space

(closely related to the rigid ro-

tor

; see Appx. B). Let us label the atom’s eigen-

states by

∣

ν,

〉

, where

≤ |

| ≤

` < ν

and the energy

ν,`,m

∝

/ν

. For fixed energy

, the manifold

of states is the same subspace of

as that obtained by

combining two large spins in Eq. (6). If we instead con-

sider all values of

ν,`,m

, we obtain

by an appropriate

unitary transformation, related to writing the hydrogen

atom in parabolic coordinates [117].

Vibrational states

One can also consider using vi-

brational states of atoms or molecules to encode quantum

information [29]. As control over vibrational states im-

proves, it may be possible to implement bosonic error-

correcting codes [3]. Position-state subspaces of har-

monic oscillators also yield the two rotational spaces of

interest. For example, considering position states

x,y,z

〉

of three oscillators with

constant yields

With four oscillators, one obtains

. To simulate

using momentum states, one can take all Fock states of

two oscillators with even total occupation number.

Levitated nanoparticles

The codewords of our

and

codes are coherent superpositions of differ-

ent possible orientations for a rigid body. Though we

have emphasized the potential applications to atoms and

molecules, the same ideas can be applied to any quantized

3-dimensional rigid body that can be coherently manip-

ulated. While there is a size limitation due to decoher-

ence, we are on the cusp of entering the quantum regime

for levitated nanoscale particles of helium [118], vaterite

[119], diamond (alone [120] or doped [121]), and silicon

[122–124], to name a few. Nanoparticles may seem to be

unlikely candidates for quantum computing, but it would

be interesting nonetheless to try to stabilize quantum su-

perpositions of their orientational states (cf. [125]).

IV. ERROR CORRECTION BASICS

FOR THE PLANAR ROTOR

The goal of error correction is to encode quantum in-

formation into a cleverly-chosen subspace (the

code

) such

that it is possible to recover said information from er-

rors caused by physical noise. Before proceeding to dis-

cuss codes which protect against noise acting on a 3-

dimensional rigid body, we will review a simpler case

which was previously considered in [1]: encoding a finite-

dimensional system in the infinite-dimensional Hilbert

space of a

planar rotor

. By discussing this case we can

introduce the key concepts underlying our code construc-

tions in a familiar mathematical setting. The interested

reader can consult [126][127, Ch. 7] for other introductory

material on quantum error correction.

The position-basis eigenstates for a planar rotor are

in one-to-one correspondence with the elements of the

two-dimensional rotation group

∞

. Equiv-

alently, these are the position eigenstates for a parti-

cle moving on a circle; the basis elements may be de-

noted

〉

,φ

∈

) =

}

, with continuum normal-

ization

〈

′

〉

(

−

′

)

. A dual basis is provided by

the angular-momentum eigenstates (

a.k.a.

“rotational

states”)

〉

∈

}

, where

〈

〉

√

i`φ

and hence

〈

′

〉

′

Noise might rotate the system, applying an operator

′

−

iφ

′

∫

′

〉〈

;

(7)

alternatively, noise might kick the angular momentum,

applying some power of the kick operator

∑

∈

+ 1

〉〈

(8)