Polyqubit quantum processing Wesley C. Campbell1 2 3and Eric R. Hudson1 2 3 1Department of Physics and Astronomy Los Angeles California 90095 USA

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Polyqubit quantum processing

Wesley C. Campbell1, 2, 3 and Eric R. Hudson1, 2, 3

1Department of Physics and Astronomy, Los Angeles, California 90095, USA

2UCLA Center for Quantum Science and Engineering,

University of California – Los Angeles, Los Angeles, California 90095, USA

3Challenge Institute for Quantum Computation,

University of California – Los Angeles, Los Angeles, California 90095, USA

(Dated: October 28, 2022)

We describe the encoding of multiple qubits per atom in trapped atom quantum processors and

methods for performing both intra- and inter-atomic gates on participant qubits without disturbing

the spectator qubits stored in the same atoms. We also introduce techniques for selective state

preparation and measurement of individual qubits that leave the information encoded in the other

qubits intact, a capability required for qubit quantum error correction. The additional internal

states needed for polyqubit processing are already present in atomic processors, suggesting that the

resource cost associated with this multiplicative increase in qubit number could be a good bargain

in the short to medium term.

Quantum technologies use quantum objects, such

as atoms, photons, phonons, and electrons to house,

transport, and process quantum information. While

some applications, like certiﬁed randomness [1], ex-

plicitly utilize the nonlocality of quantum entangle-

ment as a resource, most quantum computing algo-

rithms do not. For these applications, it is therefore

not necessary that each qubit be encoded in a physi-

cally distinct object. Given that the atoms currently

used to host qubits have many accessible internal

states, it is natural to ask: can the computational

power of atomic processors be increased, at an ac-

ceptable resource cost, by deﬁning multiple qubits

within each atom?

As an example, a problem requiring a Hilbert

space of dimension 2ncould in principle be pro-

cessed by a single atom, a unary encoding, with

a large-enough number of internal states. Though

this has the advantage of requiring only one atom,

both the information storage and system control re-

sources grow exponentially with problem size, and it

has been shown that scalable quantum computing is

not possible with a unary processor [2, 3].

At the other extreme is the current pagadigm, in

which each atom hosts a single qubit, a monoqubit

encoding. Here, the computational Hilbert space can

be factored as a tensor product of n, 2-dimensional

qubit subspaces as H(qubit)

comp =Nn

i=1 H(i)

2. Since

the control resources, which essentially dictate this

decomposition [4], do not grow exponentially with

problem size, such an encoding can potentially be

used for scalable quantum processing. However, at

present, a number of technical considerations, in-

cluding practical bounds on the number of shared

Bosonic modes, laser paths, trap zones, occupied

tweezer sites, frequency modulators, and/or other

FIG. 1. (a-c) A p= 2 polyqubit encoded in four atomic

eigenstates. (a) Each atomic Pauli operator ˆs(i)

m n acts

on only two atomic eigenstates. (b) “Vertical” (ˆσ(i)

and (c) “Horizontal” (ˆσ(i)

H) qubit operators connect qubit

states. (d) A p= 3 polyqubit can be encoded in eight

atomic eigenstates to add a “Depth” (D) qubit, H23=

H(D)

2⊗ H(H)

2⊗ H(V)

2. Each qubit operator is built from

2p−1atomic Pauli operators.

controls [5–7], limit the number of atoms that can

be reliably employed in realized processors to tens

to hundreds. Therefore, in the present qubit-host-

limited (QHL) era it may be beneﬁcial to encode a

small number (p > 1) of qubits per atom, provided

the control resources can be managed.

Here, we show how atomic processors may be built

from a polyqubit encoding with a Hilbert space com-

posed as H(polyqubit)

comp =Nn/p

i=1 H(i)

2p. The main dis-

tinction between this polyqubit processing and non-

binary (qudit) quantum processing is that polyqubit

arXiv:2210.15484v1 [quant-ph] 27 Oct 2022

processing requires the ability to perform state

preparation and measurement (SPAM) and gates

on participant qubits without disturbing spectator

qubits stored in the same atom. This operational ac-

cess allows the atomic subspace, H2p, to be factored

into a tensor product of qubit subspaces as H(i)

2p=

j=1 H(i,j)

2[4], whereas an equally-dimensioned qu-

dit does not in general admit this factorization. This

enables a polyqubit machine to use standard, binary

quantum algorithms without modiﬁcation, including

qubit quantum error correction (QEC).

Ap-polyqubit encoded in an atomic subspace, i.e.

pqubits stored in one atom, can be visualized as a

p-dimensional hypercube graph with atomic eigen-

states on the vertices and atomic Pauli operations

on each edge (see Fig. 1). From the number of edges,

it is clear that a polyqubit encoding requires p2p−1

sets of atomic Pauli operators that act on only two

atomic eigenstates. Though this control parameter

cost is exponential in p, with ﬁxed pthe proces-

sor itself scales with problem size by increasing the

number of polyqubits as n/p. Thus, by keeping p

small enough to manage control costs, polyqubit en-

coding provides a multiplicative boost to QHL pro-

cessors. In what follows, we describe how pqubits

can be encoded into 2pstates of each single atom

in a trapped ion quantum processor and used with

currently available technology.

As an example of p= 2 polyqubit processing, we

consider a linear chain of atomic ions whose mo-

tion in a particular normal mode serves as a bus

[5]. The atomic ions are assumed to have four, long-

lived internal states appropriate for quantum infor-

mation storage, labeled with underscores for clarity

{|0i,|1i,|2i,|3i} (see Fig. 1). These could be hyper-

ﬁne or Zeeman levels of a ground or metastable elec-

tronic state or some combination thereof [8]. We as-

sume that the transitions between all pairs of states

occur with unique frequencies, and that at least four

of them can be driven to achieve quadrilateral con-

nectivity, as shown in Fig. 1a. On this support, we

deﬁne two qubits that we dub “Horizontal” (H) and

“Vertical” (V), with the mapping:

|0i≡|0iH⊗ |0iV

|1i≡|0iH⊗ |1iV

|2i≡|1iH⊗ |0iV

|3i≡|1iH⊗ |1iV(1)

so that each polyqubit state |x0iH⊗|xiVis the atomic

state index expressed in two-digit (x0x) binary. As

this user-deﬁned designation of states is arbitrary,

results for H and V qubits are always interchange-

able and we require no particular physical diﬀerence

FIG. 2. State detection of a qubit in a polyqubit-encoded

ion (green) by laser shaking and subsequent motion read-

out using a co-trapped ancilla ion (red). An individually-

addressed laser beam with a beatnotes at a normal mode

frequency (purple) used for state detection of a (b) ver-

tical or (c) horizontal qubit. This interaction can also be

leveraged to perform inter-atomic zz gates.

between the types.

SPAM of polyqubits proceeds as follows. If all of

the qubits in a polyencoded atom are to be initialized

or read out, techniques from binary processing can

be adopted with only slight modiﬁcation. For exam-

ple, optical pumping with polarization- or frequency-

controlled light can produce a single atomic state

with high-purity [9], which can be subsequently ma-

nipulated by microwave or optical radiation to pre-

pare any desired polyqubit state. State detection of

all the qubits in an atom can be accomplished by

transferring the polyqubit to a metastable manifold

and serially transferring each atomic state into the

ground state, where laser-induced ﬂuorescence (LIF)

is used for detection [10]. Finding the atom in a sin-

gle state, which is heralded by LIF, fully determines

the value of all of the polyencoded qubits. This type

of state detection is well known [11].

For SPAM of an individual qubit within a polyen-

coded atom, the participant qubit must be measured

without disturbing the spectator qubits. In general,

this can be accomplished by a measurement that

leaves the measured qubit in an eigenstate of the

measurement (a quantum nondemolition measure-

ment) and will typically require an ancilla ion.

As an example, state detection of a single qubit

in a p= 2 polyencoded ion using a co-trapped an-

cilla ion could proceed as follows. First, a desired

mode of motion of the trapped ion crystal is cooled

near its ground state using, for example, the an-

cilla ion. Next, a laser is used to add energy to this

mode if and only if the participant qubit is in the

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PolyqubitquantumprocessingWesleyC.Campbell1,2,3andEricR.Hudson1,2,31DepartmentofPhysicsandAstronomy,LosAngeles,California90095,USA2UCLACenterforQuantumScienceandEngineering,UniversityofCalifornia{LosAngeles,LosAngeles,California90095,USA3ChallengeInstituteforQuantumComputation,UniversityofCalifornia...

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Polyqubit quantum processing Wesley C. Campbell1 2 3and Eric R. Hudson1 2 3 1Department of Physics and Astronomy Los Angeles California 90095 USA

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