Robust Encoding of a Qubit in a Molecule

Victor V. Albert ,

1,2

Jacob P. Covey ,

and John Preskill

1,2

Institute for Quantum Information and Matter, California Institute of Technology,

Pasadena, California 91125, USA

Walter Burke Institute for Theoretical Physics, California Institute of Technology,

Pasadena, California 91125, USA

(Received 18 November 2019; revised 27 May 2020; accepted 21 July 2020; published 1 September 2020)

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

dimensional Hilbert 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 non-Abelian groups and develop orthogonality relations for coset spaces, laying the

groundwork for quantum information processing with exotic configuration spaces.

DOI:

10.1103/PhysRevX.10.031050

Subject Areas: Atomic and Molecular Physics,

Chemical Physics,

Quantum Information

I. INTRODUCTION

Quantum systems described by continuous variables

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

wave resonator in a superconducting circuit or the motional

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. However,

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 momentum

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

’

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 intro-

duced by noise is not too large) by performing a compen-

sating shift. These codes are expected to perform well

against realistic noise, including dissipation, which typi-

cally acts locally in phase space

–

. Construction of

GKP grid states has recently been demonstrated exper-

imentally

[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 (continuous)

orientation and (discrete) angular momentum of an asym-

metric rigid body. GKP codes for objects that rotate

FIG. 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 U

, (b) a rigid rotor whose

orientation is an element of the three-dimensional rotation group

, or (c) a linear rotor whose orientation is a point on the two-

sphere S

. A basis state for the code, or code word, is a

superposition of a finite number of orientations.

Published by the American Physical Society under the terms of

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Creative Commons Attribution 4.0 International

license.

Further distribution of this work must maintain attribution to

the author(s) and the published article

’

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about a fixed axis [Fig.

1(a)

] were already discussed in

Ref.

[1]

. In that case, the orientation of the object

corresponds to an element of the two-dimensional rotation

group U

∞

. New issues arise for an object that

rotates freely in three dimensions [Figs.

1(b)

and

1(c)

], with

orientation described as an element of the three-dimen-

sional rotation group SO

(for an object with no sym-

metries) or a point on the two-sphere S

(for an

object with a symmetry axis).

Our work is motivated by recent progress in trapping and

coherently manipulating individual diatomic and polya-

tomic molecules

–

16]

. Molecules offer long coherence

times in both their nuclear and rotational states, have built-

in long-range dipolar interactions, and can be scaled up to

large arrays without compromising on their indistinguish-

ability, coherence time, or interaction fidelity. Furthermore,

couplings between a molecule

’

s internal degrees of free-

dom can be readily engineered and utilized. These features

beg the question of whether it is possible to utilize the

rich yet spatially compact molecular Hilbert space for

quantum error correction; our work shows that such is

indeed the case. Since we consider only a molecule

’

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 rota-

tional 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

[17

–

28]

(see also Refs.

[29,30]

) 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

[31

–

33]

. Other proposals advocate using vibrational or spin

degrees of freedom

[34,35]

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 orientation.

Our molecular codes, inspired by GKP codes, are designed

to protect against both momentum kicks and orientational

diffusion of a single molecule. Here, we develop 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 noisy intermediate-scale

quantum (NISQ) era

[36]

Though our work is partially motivated by advances in

molecular physics, the coding methods we use are best

explained in an abstract group-theoretic framework, which

we summarize in the next section. In Sec.

III

, we enumerate

a variety of physical settings, in molecular physics and

beyond, where our code constructions may be applicable.

The connection with rotational states of the three molecular

rotors from Figs.

1(a)

–

1(c)

is developed in greater detail in

Secs.

–

, respectively. Section

VII

discusses extensions

to more abstract state spaces. Section

VIII

contains con-

clusions 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

⊂

Here, G is a continuous group of shifts in the position of a

physical object. If no nontrivial subgroup of G leaves the

object invariant, and any position can be reached by

applying an element of G 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: (i) The discrete subgroup H of

the continuous group G leaves any state in the code space

invariant, and (ii) the subgroup K acts transitively on a basis

for the code space.

For the standard GKP code, G is the Abelian non-

compact group R, the group of translations in position

space of a particle in one spatial dimension. The subgroup

K is the infinite discrete group containing all translations of

the particle by an integer multiple of

, where

is a fixed

real number. The subgroup H contains all translations by an

integer multiple of

, 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)

∝

∈

¼ð

;

where

is a position state of the oscillator and

∈

;

...

−

. We refer to each such basis element

of the code as a

code word

. Thus, a translation of

leaves the code words invariant, and a translation of

permutes the code words 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 R. Such functions can be

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Fourier-expanded using irreducible representations (irreps)

of R, labeled by the momentum

. The irreps that preserve

the code space are those with

an integer multiple of

, 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 G/K,

and the

“

logical

”

position shift errors that preserve the

code space are labeled by elements of K/H. Undetectable

logical phase errors correspond to representations of G

which represent the subgroup H trivially but represent K

nontrivially.

In Sec.

, we illustrate the concepts underlying gener-

alized GKP codes by discussing the example of a planar

rotor. In this case, G is U

, the infinite compact group of

rotations in a two-dimensional plane, K is the finite

subgroup of U

containing rotations by an angle which

is an integer multiple of

=dN

, and H is the subgroup of K

containing rotations by an angle which is an integer

multiple of

. Here,

and

are positive integers,

and

is the dimension of the code space. This code can

correct a rotation of the planar rotor by any angle less than

=dN

and can correct a shift in angular momentum by any

integer less than

. The structure of this code, for the

case

and

, is depicted in Fig.

2(a)

While these planar-rotor codes were already introduced

in Ref.

[1]

, generalized GKP codes where G is non-Abelian

have not been previously discussed to our knowledge. In

Sec.

, we introduce

molecular codes

, which can protect an

asymmetric rigid body from rotational shift errors and

angular-momentum kicks. In this case, G is SO

, the

infinite compact group of proper rotations in 3D space.

The finite subgroups H

⊂

can be chosen in

various ways. By choosing H

⊂

to be

discrete cyclic groups of rotations about one axis (for

chemists, Z

), we obtain codes that can correct small

rotations of the body about

any

axis and can also correct

momentum kicks that change the total angular momentum

of the body by

l

<N=

. For a pictorial representation of

this code in the case

, see Fig.

2(b)

. Each ideal

code word is not normalizable, a super-position of a finite

number of position eigenstates, but there are normalizable

approximate code words which maintain good error-cor-

recting properties. We also discuss examples where

and

are finite non-Abelian subgroups of the rotation group.

We generalize the code construction further in Sec.

VII

where we allow G to be any finite group, compact Lie

group, or sufficiently well-behaved noncompact 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. This formulation provides a unified

treatment that encompasses molecular codes (G

Calderbank-Shor-Steane (CSS) codes (G

), and GKP

codes for qudits (Z

), planar rotors (U

or Z), and

oscillators (R).

FIG. 2. Code word constructions. (a) Left: Sketch of the planar-

rotor state space U

. Black and white points represent the positions

present in the two code words

(10)

of the Z

⊂

GKP rotor code.

Correctable shifts

−

;

are highlighted in blue. Right: U

angular-momentum ladder

l

∈

Z. Black and white squares re-

present momentum states present in the logical-

code words

(11)

(b) Sketch of the same features of the Z

⊂

molecular code

(45)

Left: Position space is drawn as a ball of radius

with antipodal

points identified, and each SO

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: Momentum space is a 3D square

pyramid with the height labeled by

l

and the base by

≤

l

We plot only the

part, where the code words

(49)

have

support. (c) Sketch of similar features of the Z

⊂

linear-rotor

code

(102)

, whose states are equal superpositions of equidistant

orientations along an equator. Left: The blue spherical lune

contains all points that are closer to the enclosed black point than

to any other black or white point. Right: Momentum space is a 2D

pyramid with base

≤

l

, showing states participating in the

logical-

code words

(105)

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In Sec.

, we discuss the

linear rotor

, a rigid body with

a symmetry axis, such as a heteronuclear diatomic mol-

ecule. 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 SO

. Code words of a linear-rotor code are

uniform superpositions of antipodal points on S

, which lie

in the same

orbit

of H acting on S

, where H is a finite

subgroup of SO

. See Fig.

2(c)

for the case H

and

code space dimension two.

The linear-rotor codes can also correct small rotations

about any axis, and analyzing correction of momentum

kicks follows closely the corresponding discussion for

molecular codes. However, for correction of

combinations

of rotations and momentum kicks, there are complications

which arise, because each SO

rotation acting on S

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 correspondence with

elements of the group G, and the position shifts detected by

the code are labeled by elements of G/K. In linear-rotor

codes, the position basis states themselves are in one-to-one

correspondence with elements of the coset space SO

Since coset spaces play a central role in both settings, we

formulate position and momentum bases, shift operators,

and orthogonality relations for general G/H in the

Appendix

. These are applicable to H-symmetric mole-

cules when G

(see Sec.

III C

) and may be of

independent interest for general G.

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 setting,

where the orientations of a molecule correspond to ele-

ments of SO

(in the case of an asymmetric polyatomic

molecule) or S

(in the case of a heteronuclear diatomic

molecule). In addition, other physical systems, including

atomic or molecular hyperfine, vibrational, and electronic

states, as well as atomic ensembles and levitated nano-

particles, realize similar configuration spaces.

A. Molecular rotors

GKP codes were realized experimentally

[5,6]

nearly

20 years after the initial proposal

[1]

, and full-fledged error

correctionfor molecularqubits may still bemany yearsaway

(see Ref.

[37]

, Sec. V D). Nevertheless, significant steps

toward the realization of molecular codes may be feasible

during the NISQ era

[36]

as the technology for trapping and

controlling molecules

[38

–

43]

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

[44,45]

, trapping and imaging in tweezer

arrays

[10,12]

and magnetic traps

[46,47]

, preparation of

pure quantum states

[48]

, and the first quantum degenerate

gas of polar molecules

[49]

. Recent efforts have succeeded

in controlling rotational states of CaH

[15]

as well as

coupling them to a neighboring ion

[16]

. Coherence times

of approximately 100 ms to approximately 1 s in angular-

momentum states of diatomic polar molecules have already

been observed in several experiments

[50

–

52]

Laser cooling and quantum control of polyatomic

molecules continues to be a rapidly progressing field

[11,37,53,54]

. The possibility of angular-momentum

state-resolved detection has recently been considered

[13,37]

. In addition, quantum gates of optically trapped

symmetric top molecules have recently been analyzed

[26]

Symmetric top molecules also hold promise for simulating

quantum magnetism

[55,56]

. Moreover, specific classes of

polyatomic linear polar molecules that feature more than

one optically active metal atom have recently been pro-

posed for laser cooling and trapping

[57]

. Prospects for

cooling other complex polyatomic molecules have also

been analyzed

[58,59]

Here, we highlight a few techniques that could help

realize aspects of our codes in real systems.

1. Rotational states

The code words for our codes can be expressed as

coherent superpositions of several different molecular

orientations. Alternatively, each code word can be

expressed as a coherent superposition of eigenstates of

angular momentum (also known as

“

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

l

transform as one-

dimensional irreducible representations of U

; for the case

of a polyatomic molecule, with configuration space SO

, the

basis states

l

correspond to matrix elements of irre-

duciblerepresentations ofSO

; and for thecase ofa diatomic

molecule, with configuration space S

, the basis states

l

correspond to spherical harmonics. In molecular physics

[60

–

63]

l

correspondsto the totalangularmomentumofthe

rotor,

is the

component of the angular momentum in the

lab frame, and

is the

component in the rotor frame. How

code words are expressed as linear combinations of rota-

tional states is illustrated on the right in Figs.

2(a)

–

2(c)

respectively, for three simple rotor codes.

2. Microwave dressing

One can try to stabilize the code words using poly-

chromatic microwave dressing. Note that we consider

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single molecules and neglect any effects due to their

interaction. The inherent rigid-rotor Hamiltonian for SO

and S

is diagonal in the angular-momentum basis, with

eigenvalues

l

l

[64]

. Therefore, transitions between

states with different momenta are individually addressable

(unlike, e.g., the transitions of a harmonic oscillator

Hamiltonian). Selection rules for the internal indices

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-momentum pyramid. That is, the

value of

l

,or

can change by one unit in a single-

photon transition.

However, the angular-momentum states making up each

code word are widely spaced in the internal indices. For

example, in the case of the SO

code depicted in Fig.

2(b)

the code words have support only on

l

states such that

is an integer multiple of 3. For S

[Fig.

2(c)

], a

similar pattern emerges, except that, for even

l

, the

rotational state

l

is populated only if

is an even

multiple of 3, while, for odd

l

l

is populated only if

is an odd multiple of 3.

Because a single microwave tone couples states that

differ by just one unit of

l

, a sequence of virtual

transitions induced by multiple pulses would be needed

to couple states with more widely separated values of

l

. For example, coupling states with

j¼

requires

a three-photon transition that is sufficiently detuned from

the two intermediate states, and coupling states with

l

requires a two-photon transition sufficiently

detuned from the one intermediate state. We outline a

scheme to generate these states in Appendix

. This

scheme requires many pulses, but it is on par with

previously proposed molecular dressing schemes

[65

–

67]

and may even be realized using recent experimental

advancements

[15]

We neglect rotational-state-dependent trapping effects,

which are prominent in optical dipole traps

[50,68

–

70]

These effects are negligible when considering 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 frequency catalog for all

transitions. However, this spread in polarizability poses a

practical problem when considering many molecules, since

one must ensure that they all experience the same optical

intensity. Accordingly, alternative trapping schemes may be

more appropriate for the applications proposed in this

work. Magnetic microtraps

[46]

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

[25,37]

. Such trapping

potentials are substantially less dependent on the rotational

state of the molecule, since they couple to magnetic dipoles

and electric monopoles, respectively.

More generally, one can consider engineering the

desired pulses to generate states or correct errors via

established optimal-control schemes

[43,71]

. It has been

shown that one can control the planar

[72,73]

,linear

[74

–

76]

, and even rigid

[77]

rotors, and it would be useful to

extend these and other efforts

[78,79]

to stabilizing the

required code subspace.

3. Crystal fields

In a class of quantum error-correcting codes called

stabilizer codes

, the code space is the simultaneous eigen-

space with eigenvalue 1 of a set of commuting 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 either

type

; the

-type operators

ˆ

are diagonal in the

computational basis, and the

-type operators

ˆ

permute the computational basis states. The code subspace

may be regarded as the degenerate ground space of the

Hamiltonian

(137)

code

−

ˆ

−

ˆ

Our molecular codes are not stabilizer codes, but, as we

explain in Sec.

, 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 momentum.

Just like its oscillator counterpart (see Ref.

[1]

, Sec.

XIII

this molecular Hamiltonian is gapless, but ground states of

an approximate gapped version would be close to the

approximate code words we introduce in Sec.

The

ˆ

check operators are momentum kicks which

couple well-separated angular-momentum states

l

for

l

for S

. For example,

ˆ

for a linear-rotor code

based on the octahedral group is a superposition of octo-

pole (

l

) spherical harmonics; see Eq.

(116)

. Such

harmonics are, in principle, present in a general interaction

with a bath

[80]

. However, simple laser, dc, or microwave

fields produce only

l

≤

harmonics (see Ref.

[61]

Chaps. 4 and 7).

One way to generate the required higher value of

l

is to

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

on a discrete subgroup K

⊂

, one such

ˆ

is the lowest-

l

function that is symmetric under K. Thus, putting the

rotor into a K-symmetric lattice yields a background field

whose dominant term is exactly this

ˆ

. For example,

putting a linear rotor into an octahedrally symmetric lattice

yields a background potential

[81,82]

that is exactly the

ˆ

(116)

required for the octahedral code. This potential is

minimized at those orientations of the rotor that are

superposed to construct the code words; in fact, these

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degenerate minima were noticed earlier in an experimental

context

[83]

. Similarly, embedding into a two-dimensional

square lattice yields the appropriate

ˆ

(114a)

for a linear-

rotor version of the planar-rotor code introduced in Sec.

To access subgroups of SO

forbidden in crystals, one

could consider embedding a molecule in a quasicrystal.

Crystal symmetries can enforce only the

ˆ

check

operator condition; the

ˆ

check operator condition must

be imposed by some other means. The

ˆ

operators are

trigonometric functions of the angular-momentum opera-

tors

⃖

for SO

ˆ

for S

. These are not naturally available,

as the rigid-rotor Hamiltonian

(75)

and its generalizations

[64]

contain terms that are at most bilinear in the angular-

momentum components. However, there are other terms in

the full rotor-in-lattice Hamiltonian [see Ref.

[84]

Eq. (7.2)], and, akin to superconducting circuit schemes

[85]

, one might engineer the molecule

’

s environment (for

example, by embedding the molecule in a liquid helium

nanodroplet

[86]

) to provide the required

ˆ

terms.

4. 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 rota-

tional states of the molecule to nuclear spin states via

nuclear spin-rotation interactions [see Ref.

[60]

, Eq. (1.32)]

[48,51,87]

nsr

⃖

;

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 nuclear

spin. Similar approaches are applied to solid-state systems

in which electronic spins are coupled to nuclear spins

[88]

This approach is roughly analogous to using a super-

conducting 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

l

. Operations

which shift the occupation number of a cavity can be

applied by coupling the cavity to a three-level atom

[89]

by using linear optics

[90]

. Similar schemes could shift the

value of

l

for a U

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 S

and quotient spaces

H from Table

. We list three manifestations:

spin-

systems in a totally symmetric spin state,

spin-

systems in a totally symmetric state, and a pair 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, all three cases apply equally

well to an atom or a molecule with a sufficiently large

nuclear-spin manifold (cf. Ref.

[91]

1. Many small spins

spin-

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

−

cos

sin

−

⊗

for

¼ð

;

∈

[98,99]

. These states are not orthogo-

nal; instead, they form an overcomplete frame for the

collective spin

’

)-dimensional Hilbert space, with

overlap

j¼ð

.As

→

∞

, the states

become orthogonal and correspond to the position states

of S

(Table V B). For finite

, superpositions of these spin-

coherent states can be approximate code words for a linear-

rotor code.

Numerous manifestations of entangled ensembles of

many spin-

atoms have recently been demonstrated

TABLE I. Quotient spaces mentioned in this work

[92,93]

(see

also Ref.

[94]

, Sec. 3.8, and Refs.

[95

–

97]

). Spaces associated

with SO

characterize rotational states of various molecules (see

Sec.

III C

). Z

is the order-

cyclic group, D

is the order-

dihedral group, U

∞

is the circle group, and

⋊

∞

is the group of planar rotations and

reflections. Some of these spaces are shown in Figs.

and

Space X

Group H

Quotient space X/H

Wigner-Seitz unit cell U

Lens space L

Dihedral D

Prism space

Tetrahedral T

Octahedral space

Octahedral O

Truncated cube space

Icosahedral I Poincar ́

e dodecahedral space

Two-sphere S

Projective plane RP

, T, O, I

Spherical two-orbifold

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[100

–

103]

, and the current status of the field is summarized

in Ref.

[104]

2. Many medium spins

Any pure state of a spin-

system is invariant under a

continuous U

subgroup of the rotation group SO

; 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 representations for which the subgroup which

preserves the state is a nontrivial

discrete

subgroup H of

. For example, the spin-2 state

∝

−

iþ

ffiffiffi

invariant under the tetrahedral subgroup T. Therefore,

applying SO

rotations to

generates a manifold of

states

, where the label

is a point in the coset space

∈

T. The spin-coherent states

⊗

, obtained by

taking a tensor product of many identical elements of this

manifold, approximate the position-basis states of SO

in the limit of large

. This idea can be generalized: Spin-

coherent states

⊗

approximate the position-basis

states for the coset space SO

H, if

is a higher-spin

state with invariance group H.

The above T-symmetric and similar H-symmetric states

(see Ref.

[105]

, Table 2)

—

examples of Perelomov coherent

states

[99]

—

are used as a mean-field ansatz for the ground

space of spin-

Bose-Einstein condensates

[106,107]

.We

use such coherent states to extract error syndrome infor-

mation for molecular codes (see Sec.

), which requires

projectively measuring in this basis

[108]

3. Two large spins

Instead of using only the symmetric subspace, one can

consider the entire space of a pair of spin-

systems. Per

the addition rules (see Ref.

[109]

, Chap. 8)

⊗

⊕

the

orthonormal basis states for this system can be

chosen to be the angular-momentum eigenstates

l

with

l

≤

and

≤

l

. These are precisely the rotational

states of a linear rotor, except for the truncation

l

≤

Formally, then, the state space of a pair of spin-

systems

matches the state space on S

in the limit

→

∞

. Since the

normalizable approximate code words of the linear-rotor

code are necessarily truncated for large

anyway, these

approximate code words can be accurately realized using a

pair of spin-

systems for sufficiently large

(see

Sec.

If one instead considers two different spins

and

, one obtains a different band of S

momentum states.

While developing codes for such band-limited subspaces 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

systems, we can consider nuclear spin spaces of molecules

or single atoms. Diatomic molecules such as NaCs

[24]

offer exactly the band-limited subspaces mentioned above.

Concerning single atoms, lanthanide species such as

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

large total spin manifolds in their ground states due to their

large nuclear spins and many unpaired electrons in their

shells. Accordingly, such atoms have already attracted

attention for the possibility of scaling up quantum comput-

ing by collectively encoding in multilevel atoms

[110

–

112]

. Ho, in particular, has the largest hyperfine ground

space of any atom, with 128 ground states

[112]

. Laser

cooling and trapping techniques are well established for Dy

[113]

,Ho

[114]

, and Er

[115]

, as well as other lanthanides.

Moreover, quantum degenerate gases of Dy

[116,117]

and

[118,119]

are widely used for novel quantum simula-

tions based on their large magnetic dipole moments.

C. Other systems

1. Planar rotors

Several systems have the configuration 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

[120]

. Other

possibilities include the phase difference between two

superconductors on either side of a Josephson junction

[121]

, orbital angular momentum of light

[122]

, or simu-

lating rotor position states using phase states of an ordinary

oscillator

[123]

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

l

;

l

≤

of the planar rotor.

2. Symmetric molecules

A molecule with symmetry group H has an orientation

state space parameterized by SO

H (see Table

). For

example, the methane molecule CH

has the tetrahedral

symmetry group T, and the alkaline earth monomethoxide

(MOCH

) family

—

potentially useful for quantum comput-

ing

[26]

—

has symmetry group Z

. This group is also the

relevant symmetry group of Posner molecules, postulated

to have potentially useful quantum effects

[124,125]

. The

symmetry group of the fullerene molecule is the icosahe-

dral group I, and the 3-manifold SO

I has an exotic shape

that was once proposed as a model for the geometry of the

Universe

[97,126]

. It is interesting that such exotic spaces

are readily accessible in relatively simple molecules.

Completely asymmetric and U

-symmetric molecules cor-

respond, respectively, to rigid and linear rotors from

Sec.

III A

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More generally, if a group G acts transitively on the

states of a quantum system, and the subgroup H of G leaves

the states invariant, then the configuration space of the

system is G/H. In Appendix

, we develop mathematical

tools for parameterizing the position eigenstates and the

dual momentum states of such a system, including ortho-

gonality and completeness relations as well as a Poisson

summation formula

[127]

3. Electronic states

One can consider embedding certain spaces from Table

in the electronic eigenstates of single atoms. The eigen-

states of hydrogen offer a platform for a band-limited

subspace of the linear rotor S

and even the space SU

(closely related to the rigid rotor SO

; see Appendix

). Let

us label the atom

’

s eigenstates by

;

l

, where

≤

l

and the energy

;

l

∝

. For fixed energy

, the manifold of states is the same subspace of S

as that obtained by combining two large spins in Eq.

(6)

.If

we instead consider all values of

l

, and

, we obtain SU

by an appropriate unitary transformation, related to writing

the hydrogen atom in parabolic coordinates

[128]

4. Vibrational states

One can also consider using vibrational states of

atoms or molecules to encode quantum information

[34]

As control over vibrational states improves, it may be

possible to implement bosonic error-correcting codes

[3]

Position-state subspaces of harmonic oscillators also yield

the two rotational spaces of interest. For example, consid-

ering position states

x; y; z

of three oscillators with

constant yields S

. With four oscillators, one

obtains SU

To simulate S

using momentum states, one can take all

Fock states of two oscillators with an even total occupation

number. Such a simulation allows straightforward imple-

mentation of SO

rotations via beam splitters.

5. Levitated nanoparticles

The code words of our SO

and S

codes are coherent

superpositions of different possible orientations for a rigid

body. Though we emphasize the potential applications to

atoms and molecules, the same ideas can be applied to any

quantized three-dimensional rigid body that can be coher-

ently manipulated. While there is a size limitation due to

decoherence, we are on the cusp of entering the quantum

regime for levitated nanoscale particles of helium

[129]

vaterite

[130]

, diamond (alone

[131]

or doped

[132]

and silicon

[133

–

135]

, to name a few. Nanoparticles may

seem to be unlikely candidates for quantum comput-

ing, but it would be interesting nonetheless to try to

stabilize quantum superpositions of their orientational

states (cf.

[136,137]

IV. ERROR-CORRECTION BASICS

FOR THE PLANAR ROTOR

The goal of error correction is to encode quantum

information into a cleverly chosen subspace (the

code

)

such that it is possible to recover said information from

errors caused by physical noise. Before proceeding to

discuss codes which protect against noise acting on a

three-dimensional rigid body, we review a simpler case

which was previously considered in Ref.

[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

constructions in a familiar mathematical setting. The

interested reader can consult Ref.

[138]

and Chap. 7 in

Ref.

[139]

for other introductory material on quantum error

correction, as well as related work on encodings associated

with U

[123,140]

The position-basis eigenstates for a planar rotor are in

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

dimensional rotation group U

∞

. Equivalently,

these are the position eigenstates for a particle moving on a

circle; the basis elements may be denoted

;

∈

;

Þ¼

, with continuum normalization

i¼

−

. A dual basis is provided by the angular-

momentum eigenstates (also known as

“

rotational states

”

)

l

;

l

∈

, where

l

i¼ð

ffiffiffiffiffiffi

l

and, hence,

l

l

i¼

ll

Noise might rotate the system, applying an operator

ˆ

−

ˆ

;

alternatively, noise might kick the angular momentum,

applying some power of the kick operator

ˆ

ˆ

l

∈

l

l

In fact, we can expand an arbitrary noise channel

acting

on the density operator

of the planar rotor in terms of a

complete basis of operators, where each element of the

basis is a product of an

ˆ

operator and an

l

th power of the

ˆ

operator:

Þ¼

l

;

l

∈

ll

φφ

ˆ

ˆ

l

ˆ

l

†

ˆ

†

Above, the expansion coefficients

ll

φφ

are such that

is a

quantum channel. Our goal is to encode a finite-dimen-

sional logical system in the infinite-dimensional Hilbert

space of the rotor, where this logical system is protected

against any error

ˆ

ˆ

l

where both

and

l

are sufficiently

small. In other words, if

consists of states in the logical

(also known as code) subspace, and if

is expanded using

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only such

correctable

ˆ

ˆ

l

, then we are able recover the

original

from

. Otherwise, recovery may not be

possible, and logical information stored in

may become

corrupted.

A. A protected qubit

For example, the two orthonormal basis states of a

protected qubit can be chosen to be [see Fig.

2(a)

]

i¼

ffiffiffi

iþ

;

10a

i¼

ffiffiffi

þj

iþ

10b

Both basis states are eigenstates with eigenvalue 0 of

ˆ

modulo

. Suppose that

is an arbitrary state in the

code space spanned by

and

. If an error occurs which

causes

to shift by

δφ

∈

−

;

, we can unambig-

uously diagnose the error by measuring

ˆ

modulo

Once

δφ

is known, we can correct the error by applying a

unitary transformation that shifts

−

δφ

, restoring the

state of the rotor to the initial undamaged state

Alternatively, we may expand the basis states of the code

in the angular-momentum eigenstate basis, finding

ffiffiffi

ðj

iþj

iÞ ¼

ffiffiffi

∈

l

;

11a

ffiffiffi

ðj

−

iÞ ¼

ffiffiffi

∈

l

11b

Both basis states are eigenstates with eigenvalue 0 of

ˆ

modulo 3. Suppose an error occurs which causes the

angular momentum to shift by

l

∈

−

;

. We can

unambiguously diagnose the error by measuring

ˆ

modulo

3. Once

l

is known, we can correct the error by applying a

unitary transformation that shifts

l

−

l

. Furthermore

(see below),

ˆ

modulo

and

ˆ

modulo 3 are compatible

observables that can be measured simultaneously.

Therefore, we can correct any combination of shifts in

and

l

, as long as the shift in

is no larger than

and

the shift in

l

is no larger than 1.

The code basis states in Eqs.

(10)

and

(11)

are not

normalizable and, therefore, unphysical. However, we may

replace the position eigenstates in Eq.

(10)

by narrow wave

packets; then, the sum over

in Eq.

(11)

is modulated by a

broad envelope function. In that case, the code states are

physical, and the nice error-correction properties we note

still hold, up to negligibly small corrections.

Our main task in this paper is to generalize this code

construction, in various directions. For that purpose, it is

convenient to have other ways to describe the code.

Our first alternative description uses the stabilizer

language

[138,141]

1. Stabilizer formalism

stabilizer code

may be characterized as the simulta-

neous eigenspace with eigenvalue 1 of a set of commuting

unitary operators, called the

stabilizer generators

. For the

code specified by Eqs.

(10)

and

(11)

, we may choose these

operators to be

ˆ

≡

ˆ

ˆ

;

ˆ

≡

ˆ

−

ˆ

To check that these operators commute, recall the relation

ˆ

ˆ

−

ˆ

ˆ

−

and the identity

ˆ

ˆ

ˆ

†

ˆ

ˆ

ˆ

†

for

any unitary

ˆ

and scalar

ˆ

and

ˆ

are the code

’

check

operators

, which we can measure to diagnose errors. Note

that measuring

ˆ

is equivalent to measuring

ˆ

modulo

and that measuring

ˆ

is equivalent to measuring

ˆ

modulo 3, as we assert earlier, and that we can perform

these measurements simultaneously because

ˆ

and

ˆ

commute.

Furthermore, we note that the operators

≡

ˆ

ˆ

;

ˆ

−

ˆ

also commute with the stabilizer generators, which means

that these are

logical operators

which preserve the

code space. We see also that

and

anticommute and

that they square to the identity on the code space, where

ˆ

ˆ

. Thus,

and

may be regarded as the logical

Pauli operators acting on the encoded qubit, where

diagonal in the basis

;

and

is diagonal in the

conjugate basis

fð

ffiffiffi

Þðj

iÞg

2. CSS construction

We may also describe our protected qubit using the

language of CSS codes

[138,141]

. In the CSS construction,

a quantum error-correcting code is built from a classical

error-correcting code K and a subcode H

⊂

In the case of the protected qubit with basis states

(10)

the code K is a six-state system embedded in the infinite-

dimensional Hilbert space of the rotor, with the six states

corresponding to six equally spaced angular positions of

the rotor, rotated by

¼ð

∈

;

...

;

, relative

to a standard reference orientation. This classical system is

protected against errors that shift the rotor slightly, rotating it

through an angle

δφ

∈

−

;

. The subcode H has three

states, with orientations

¼ð

∈

;

, and

protects against rotations which are twice as large:

δφ

∈

−

;

. In the associated quantum code, each

of the basis states

(10)

is a uniform superposition of all the

elements of a

coset

of H in K, the trivial coset (the elements

of H) for the basis state

, and the nontrivial coset for the

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basis state

. The protection of this qubit against shifts of

the rotor is inherited from the corresponding property of the

classical code K.

There is a dual description of this quantum code, making

use of the angular-momentum basis of the rotor rather than

its position basis. The classical code H

⊥

, dual to H,

contains all angular-momentum eigenstates where

l

an integer multiple of 3. These two classical codes are

dual in the sense that the representations of the group U

contained in H

⊥

represent the elements of H trivially.

Similarly, the classical code K

⊥

dual to K contains all

angular-momentum eigenstates where

l

is an integer

multiple of 6, those representations which represent K

trivially. Evidently, K

⊥

is a subcode of H

⊥

. For the

quantum code, each basis state in Eq.

(11)

is a uniform

superposition of all the elements of a coset of K

⊥

in H

⊥

the trivial coset for the basis state

ffiffiffi

ðj

iþj

iÞ

, and the

nontrivial coset for the basis state

ffiffiffi

ðj

−

iÞ

. The

classical code H

⊥

protects against shifts of the angular

momentum by

l

∈

−

;

, and the quantum code

inherits this property.

Viewed as an abstract group, the code K is the subgroup

of U

, and H is the subgroup Z

⊂

. The construction

can be easily generalized to K

and H

, where

and

are positive integers, in which case the quantum code

-dimensional. In the stabilizer language, this more

general code has stabilizer generators

ˆ

ˆ

;

ˆ

ˆ

Its logical operators

ˆ

;

ˆ

=dN

are generalized Pauli operators, obeying the Heisenberg-

Weyl commutation relation

. This quan-

tum code protects against position shifts by

δφ

with

δφ

=dN

and momentum kicks by

l

with

l

≤

−

(for odd

). Note the trade-off: Increasing

improves the

protection against angular-momentum kicks but weakens

the protection against rotations.

3. Partial Fourier transform

There is yet another way to describe the code construc-

tion, using the notion of a

partial Fourier transform

, which

is helpful as we seek further generalizations. Recall that the

position and angular-momentum bases for the planar rotor

are related by Fourier transforming:

l

i¼

−

l

i¼

ffiffiffiffiffiffi

−

l

;

16a

i¼

l

∈

l

l

i¼

ffiffiffiffiffiffi

l

∈

l

−

l

16b

It is useful to imagine that the above integral over

carried out in two steps. We write

þð

where

∈

−

=N;

and

∈

;

...

−

; then,

integrating

from

−

is equivalent to integrating

from

−

and summing H from 0 to

−

Likewise, we can do the sum over

l

in two steps; we write

l

, where

∈

Z and

∈

;

...

−

, and

we separate the infinite sum over

from the finite sum over

. When we speak of a

“

partial Fourier transform,

”

we mean

performing one of these two steps without the other.

By performing the sum over H but not the integral over

, we obtain a new orthonormal basis

;

≡

ffiffiffiffi

∈

−

ffiffiffiffiffiffi

∈

−

iNsa

l

;

with normalization

;

i¼

−

λλ

. From now

on, the presence of a semicolon inside a ket declares that ket

to be an element of this basis.

This

;

basis is convenient for our purposes,

because shifts in position or angular momentum affect

only one of the two indices. A shift in angular momentum

l

acts on the basis according to

;

→

;

l

mod

ið

(up to a phase), shifting

→

l

modulo

. A shift in

position by

δφ

shifts

→

δφ

modulo

;

→

δφ

mod

;

To recover our previous code construction, we choose

basis states

with

and

¼ð

=dN

, finding

i¼

ffiffiffiffi

∈

ffiffiffiffiffiffi

∈

−

l

If an error occurs in which

l

≤

−

(for odd

)

and

δφ

=dN

, we diagnose the error by performing a

measurement which determines the value of

and also the

value of

(mod

=dN

). Then, the value of

unambig-

uously identifies the shift in

, and the value of

unambiguously identifies the shift in

l

. Once known, these

shifts can be corrected to recover the initial undamaged

code states.

The orientation label

of the planar rotor can be viewed

as the element of the group U

describing the rotation

which reaches

starting from a standard initial orientation.

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The basis

;

for the rotor

’

s Hilbert space reflects a

decomposition of U

which may be written symbolically as

≅

That is,

labels an element of U

(a coset of Z

in U

and

labels an element of

(an irreducible representation

of Z

). Our error-correction procedure makes use of a finer

decomposition:

≅

×Z

The correctable rotation error is an element of U

, the

correctable angular-momentum kick is an element of

and code basis states correspond to elements of Z

We use similar decompositions in our constructions of

quantum codes for more general groups.

B. Gates, recovery, and initialization

To use the above codes for quantum computation on

multiple encoded rotors, we need to initialize in the code

subspace, execute quantum gates, and perform the meas-

urement-based error correction described above. For these

tasks, we need operators other than the Pauli-type operators

ˆ

(7)

and

ˆ

l

(8)

. As is typical of quantum codes, there is

“

easy

”

subset of all possible operators that aid us in the

above tasks in a reasonably fault-tolerant manner. For U

rotors, such

normalizer

symplectic

operations are gen-

erated by certain quadratic functions of the rotors

’

positions

and momenta

[142,143]

1. Symplectic operations

Single-rotor symplectic operations include unitary

operators generated by Hamiltonians that are polynomials

in angular momentum of at most degree 2. The quadratic-

phase operator

QUAD

−

ˆ

ˆ

(with angle

∈

)

maps

ˆ

→

ˆ

ˆ

while commuting with position shifts

ˆ

(also generated

ˆ

). The analogous two-rotor

“

conditional-phase

”

operator

CPHS

−

ˆ

⊗

ˆ

[cf.

[123]

,Eq.

(23)

] commutes

with

ˆ

⊗

and

⊗

ˆ

but maps

ˆ

⊗

→

ˆ

⊗

ˆ

and

⊗

ˆ

→

ˆ

⊗

ˆ

Another operation is the conditional rotation

CROT

≡

−

ˆ

⊗

ˆ

⊗

ˆ

;

shifting the position of the second rotor by

, conditioned

on the first rotor being at position

. This maps

ˆ

⊗

→

ˆ

⊗

ˆ

and

⊗

ˆ

→

ˆ

†

⊗

ˆ

while acting trivially on

ˆ

⊗

and

⊗

ˆ

The

QUAD

and

CPHS

operations can be realized by

turning on Hamiltonians quadratic in angular momenta

for a specified amount of time [cf. Eq.

(4)

]. The

CROT

operation, however, cannot be obtained from the

“

Hamiltonian

”

ˆ

⊗

ˆ

, because such an H would

not be invariant under

rotations of the first rotor and,

therefore, would not be single valued. (A similar problem

plagues the Hamiltonian

ˆ

, present in the exponent of

ˆ

while

ˆ

is not single valued even when exponentiated.) To

produce such an operator in the lab, one can consider

adapting implementations of the related oscillator phase

operator to rotors

[89,90]

(see Sec.

III A

2. Logical gates

The above symplectic operations, for certain

, perform

logical Clifford operations on the encoded qudits. The gate

QUAD

=dN

performs a logical qudit rotation mapping

→

(up to a phase), while

CPHS

=dN

and

CROT

act as

entangling gates.

In the case of a logical qubit (

) with logical

operators

ˆ

and

ˆ

(15)

, the symplectic

operations producing the above logical transformations

act on the rotor positions

;

and momenta

l

;

as follows:

QUAD

∶

→

−

l

l

→

l

;

CPHS

∶

→

−

l

;

l

→

l

;

→

−

l

;

l

→

l

;

CROT

∶

→

l

→

l

−

l

;

→

;

l

→

l

;

with constant

¼ð

Þð

−

(cf. Sec. IX in

Ref.

[1]

). We assume in Eq.

(27)

that

and

l

simulta-

neously have definite values, which makes sense for an

encoded state assuming that

and

l

are sufficiently small.

These transformations do not amplify correctable position

and momentum shifts into uncorrectable ones, and a small

overrotation or underrotation in the implementation of one

of the logical gates introduces only correctable errors, not

logical errors. In this sense, the logical gates are fault

tolerant.

The above symplectic operations do not provide a

universal set of logical operations. One way to upgrade

to such a set is to include unitaries generated by the logical

operators

themselves. The gates

and

allow for arbitrary single-qudit rotations, while

⊗

allows for arbitrary logical

rotations.

Such gates are, however, not fault tolerant, as fluctuations

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in the

’

s produce undetectable errors. One can also

consider using Hamiltonians that are cubic (or higher) in

angular momenta.

3. Diagnosis and recovery

A shift in the position by

δφ

and momentum by

l

maps

logical states

→

jð

=dN

δφ

;

l

(up to a phase).

To diagnose the errors, we need to measure

ˆ

mod

=dN

and

ˆ

mod

. Once this

error syndrome

is known, we can

undo the damage by applying

ˆ

†

δφ

(7)

and

ˆ

l

†

(8)

to the

corrupted logical states.

To measure

ˆ

mod

=dN

, we need an ancilla that can

resolve all possible values of this syndrome while revealing

no information about the protected encoded state. One way

to extract the syndrome is to encode the ancilla using the

same code that protects the data

[144]

. Specifically, we

may prepare an ancillary rotor in the logical-

eigenstate

, a uniform superposition of the position eigenstates

¼ð

=dN

;

...

;dN

−

, which is, there-

fore, invariant under the rotation

→

þð

=dN

Applying the

CROT

gate

(25)

to a noisy logical state and

a noiseless ancilla yields

CROT

δφ

;

l

⊗

δφ

;

l

⊗

ˆ

δφ

The ancilla can then be measured in the

basis, and the

measured value modulo

=dN

determines the shift

δφ

.If

the ancilla is noisy or the measurement is imperfect, then

the extracted value of

δφ

is likewise noisy; nevertheless, if a

fresh supply of ancilla rotors is continuously available, this

recovery procedure with high likelihood prevents small

displacements of the data rotor from accumulating to

produce an uncorrectable logical error.

To measure

ˆ

mod

, we need an ancilla that can resolve

the

values of the syndrome. In this case, we could

initialize an ancilla rotor in the state

and apply

CPHS

to the data and ancilla rotors. This gate rotates the

ancilla by

l

, and the value of

l

can, therefore, be

extracted by measuring the ancilla in the position basis.

Since the syndrome takes discrete values, some noise

resilience is built into the procedure

—

l

is determined

by rounding off the measured value of

to the nearest

multiple of

Since we need only to resolve a discrete number of

momentum syndrome values, a discretized version of the

above scheme using a qunit ancilla works just as well. Let

∈

be the position states of the qunit, and

initialize the qunit in the state

. Then apply the

entangling gate

CPHS

≡

l

∈

l

l

⊗

l

;

where

satisfies

i¼j

(modulo

) and

the identity. This process yields

CPHS

δφ

;

l

⊗

i¼

δφ

;

l

⊗

l

;

and measuring the qunit in the position basis then reveals

the syndrome.

4. Initialization

The above error-correction procedures can equivalently

be used to initialize in certain logical states. For example,

consider one rotor initialized in

, coupled to an

ancillary qunit initially in

. Applying

CPHS

yields

CPHS

⊗

∝

∈

⊗

Measuring the ancilla in the

basis to obtain a particular

⋆

collapses the rotor state to

⋆

. Applying a

momentum kick

ˆ

⋆

†

then produces the logical state

i¼j

0; 0

(20)

, thereby completing the initialization.

Analogous initialization schemes use the position syn-

drome measurement.

V. MOLECULAR CODES

By a

“

molecular code,

”

we mean a finite-dimensional

subspace of the infinite-dimensional Hilbert space of a

rigid body in three dimensions (also known as a

“

rigid

rotor

”

). To define a basis for this infinite-dimensional

Hilbert space, we imagine fixing a coordinate system in

the laboratory, pinning the body

’

s center of mass, and

specifying the orientation of the body relative to a standard

initial configuration in this fixed coordinate system. For a

molecule with no symmetries, the possible orientations

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

3D special orthogonal group SO

; thus, we may choose

the

“

position

”

basis

∈

. This correspondence

between group elements and orientations of the body

follows the same logic as in our discussion of the planar

rotor in Sec.

, where we identify position-basis eigen-

states with elements of U

If the body has symmetries, using a group element to

specify the orientation becomes redundant, and the position

basis should be refined accordingly. For example, if there is

an axis of symmetry (as for a diatomic molecule composed

of two distinct nuclei), the body is invariant under the U

subgroup of rotations about the symmetry axis, and the

possible orientations are in one-to-one correspondence

with the coset space SO

, which is equivalent to the

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