Quantum error correction with metastable states of trapped ions using erasure conversion Mingyu Kang1 2Wesley C. Campbell3 4 5and Kenneth R. Brown1 2 6 7

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Quantum error correction with metastable states of trapped ions using erasure

conversion

Mingyu Kang,1, 2, ∗Wesley C. Campbell,3, 4, 5 and Kenneth R. Brown1, 2, 6, 7, †

1Duke Quantum Center, Duke University, Durham, NC 27701, USA

2Department of Physics, Duke University, Durham, NC 27708, USA

3Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA

4Challenge Institute for Quantum Computation, University of California, Los Angeles, CA 90095, USA

5Center for Quantum Science and Engineering, University of California, Los Angeles, CA 90095, USA

6Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA

7Department of Chemistry, Duke University, Durham, NC 27708, USA

(Dated: July 3, 2023)

Erasures, or errors with known locations, are a more favorable type of error for quantum error-

correcting codes than Pauli errors. Converting physical noise into erasures can signiﬁcantly improve

the performance of quantum error correction. Here we apply the idea of performing erasure con-

version by encoding qubits into metastable atomic states, proposed by Wu, Kolkowitz, Puri, and

Thompson [Nat. Comm. 13, 4657 (2022)], to trapped ions. We suggest an erasure-conversion

scheme for metastable trapped-ion qubits and develop a detailed model of various types of errors.

We then compare the logical performance of ground and metastable qubits on the surface code under

various physical constraints, and conclude that metastable qubits may outperform ground qubits

when the achievable laser power is higher for metastable qubits.

I. INTRODUCTION

The implementation of quantum error correction

(QEC) is a necessary path toward a scalable and fault-

tolerant quantum computer, as quantum states are often

inherently fragile and physical operations on quantum

states have limited ﬁdelities. QEC protects quantum in-

formation from errors by encoding a logical qubit into

entangled states of multiple physical qubits [1].

There have been exciting eﬀorts on manipulating and

exploiting the type of physical error such that the per-

formance of QEC is improved. One example is the engi-

neering of qubits and operations that have strong bias be-

tween the Xand ZPauli noise [2–5] and designing QEC

codes that beneﬁt from such bias by achieving higher

thresholds [6–13].

Another example is converting physical noise into era-

sures, i.e., errors with known locations [14–16]. It is clear

that erasures are easier to correct than Pauli errors for

QEC codes, as a code of distance dis guaranteed to cor-

rect at most d−1 erasures but only ⌊(d−1)/2⌋Pauli

errors.

Erasure conversion is performed by cleverly choosing

the physical states encoded as qubits, such that physical

noise causes leakage outside the qubit subspace. Cru-

cially, such leakage should be detectable using additional

physical operations [17–20]. Typically, undetected leak-

age errors can have even more detrimental eﬀects on

QEC than Pauli errors, as traditional methods for cor-

recting Pauli errors do not apply [21] and methods such

as leakage-reducing operations [22–24] and circuits [25–

30] require signiﬁcant overhead. However, when leakage

∗mingyu.kang@duke.edu

†ken.brown@duke.edu

errors are detectable, they can be converted to erasures

by resetting the leaked qubit to a known state, e.g., the

maximally mixed state, within the qubit subspace. Era-

sure conversion is expected to achieve signiﬁcantly higher

QEC thresholds for hardware platforms such as super-

conducting qubits [20] and Rydberg atoms [19].

Trapped ions are leading candidate for a scalable

quantum computing platform [31]. In particular, QEC

has been demonstrated in various trapped-ion experi-

ments [32–39], which include fault-tolerant memory [35,

36] and even logical two-qubit gates [37–39] on distance-3

QEC codes. Here we address the question of whether the

idea of erasure conversion can be applied to trapped-ion

systems.

In fact, the erasure-conversion method in Ref. [19], de-

signed for Rydberg atoms, can be directly applied to

trapped ions. In Ref. [19], a qubit is encoded in the

metastable level, such that the majority (approximately

98%) of the spontaneous decay of the Rydberg states dur-

ing two-qubit gates does not return to the qubit subspace.

Additional operations can detect such decay, thereby re-

vealing the locations of errors. For trapped ions, the

spontaneous decay of the excited states during laser-

based gate operations is also the fundamental source of

errors, which we aim to convert to erasures in this paper.

Note that an earlier work [40] has proposed a method of

detecting a diﬀerent type of error for trapped ions using

qubits in the metastable level.

While the most popular choice of a trapped-ion qubit

is the ground qubit encoded in the S1/2manifold, the

metastable qubit encoded in the D5/2or Fo

7/2manifold

is also a promising candidate [41]. Recently, high-ﬁdelity

coherent conversion between ground and metastable

qubits has been experimentally demonstrated using Yb+

ions [42]. Also, it has been proposed that when ground

and metastable qubits are used together, intermediate

arXiv:2210.15024v4 [quant-ph] 30 Jun 2023

state measurement and cooling can be performed within

the same ion chain [41]. Therefore, it is a timely task

to add erasure conversion to the list of functionalities of

metastable qubits.

A careful analysis is needed before concluding that

metastable qubits will be more advantageous in QEC

than ground qubits. As discussed below, the erasure

conversion relies on the fact that the excited states are

more strongly coupled to the ground states than the

metastable states. However, this fact may also cause

the Rabi frequency of metastable qubits to be signiﬁ-

cantly lower than that of ground qubits, which leads to

longer gate time required for metastable qubits. Also,

most metastable states decay to the ground manifold

after a ﬁnite lifetime, while ground qubits have practi-

cally inﬁnite lifetime. Whether the advantage of having

a higher threshold overcomes these drawbacks needs to

be veriﬁed.

This paper is organized as follows. In Sec. II, we in-

troduce the method of laser-based gate operation and

erasure conversion on metastable qubits. In Sec. III, we

show the model of various types of errors for ground

and metastable qubits and discuss the criteria of com-

parison. In Sec. IV, we brieﬂy introduce the surface

code and the simulation method. In Sec. V, we present

the results of comparing the QEC performance between

ground and metastable qubits. Speciﬁcally, we conclude

that metastable qubits may outperform ground qubits

when metastable qubits allow higher laser power than

ground qubits, which is reasonable considering the ma-

terial loss due to lasers. In Sec. VI, we compare the

erasure-conversion scheme on trapped ions and Rydberg

atoms, as well as discuss future directions. We conclude

with a summary in Sec. VII.

II. ERASURE-CONVERSION SCHEME

In this paper, we denote the hyperﬁne quantum state

as |L, J;F, M⟩, where L,J,F, and Mare the quantum

numbers in the standard notation. Also, we denote a set

of all states with the same Land Jas a manifold.

We deﬁne the ground qubit as hyperﬁne clock qubit en-

coded in the S1/2manifold as |0⟩g:= |0,1/2; I−1/2,0⟩

and |1⟩g:= |0,1/2; I+ 1/2,0⟩, where Iis the nuclear

spin [43]. Similarly, we deﬁne the metastable qubit as

hyperﬁne clock qubit encoded in the D5/2manifold as

|0⟩m:= |2,5/2; F0,0⟩and |1⟩m:= |2,5/2; F0+ 1,0⟩,

which is suggested for Ba+, Ca+, and Sr+ions [41].

Here, F0can be chosen as any integer that satisﬁes

|J−I| ≤ F0< J +I. Both qubits are insensitive to

magnetic ﬁeld up to ﬁrst order as M= 0.

Unlike ground qubits, metastable qubits are suscepti-

ble to idling error due to their ﬁnite lifetime. As a D5/2

state spontaneously decays to the S1/2manifold, such an

error is a leakage outside the qubit subspace. The proba-

bility that idling error occurs during time duration tafter

state initialization is given by

p(idle)(t) = 1 −e−t/τm,(1)

where τmis the lifetime of the metastable state. Typi-

cally, τmis in the order of a few to tens of seconds for

D5/2states [41].

Laser-based gate operations on ground (metastable)

qubits are performed using the two-photon Raman tran-

sition, where the laser frequencies are detuned from the

S1/2(D5/2)→P3/2transition, as described in Fig. 1.

Here we deﬁne the detuning ∆g(∆m) as the laser fre-

quency minus the frequency diﬀerence between the S1/2

(D5/2) and P3/2manifolds. Apart from the “technical”

sources of gate error due to noise in the experimental

system, a fundamental source of gate error is the sponta-

neous scattering of the atomic state from the short-lived

Pstates. During ground-qubit gates, both the P1/2and

the P3/2states contribute to the two-photon transition as

well as gate error, while for metastable-qubit gates, only

the P3/2states contribute, as transition between the D5/2

and P1/2states is forbidden.

When an ion that is initially in the P3/2state de-

cays, the state falls to one of the S1/2,D3/2, and D5/2

manifolds with probability r1,r2, and r3, respectively

(r1+r2+r3= 1), where these probabilities are known as

the resonant branching fractions. Typically, r1is several

times larger than r2and r3.

For ground (metastable) qubits, if the atomic state de-

cays to either qubit level of theS1/2(D5/2) manifold, the

resulting gate error can be described as a Pauli error. On

the other hand, if the atomic state decays to the D3/2

manifold, or the D5/2(S1/2) manifold, or the hyperﬁne

states of the S1/2(D5/2) manifold other than the qubit

states, the resulting gate error is a leakage.

We now describe how the majority of the leakage can

be detected when metastable qubits are used, similarly

to the scheme proposed in Ref. [19]. Speciﬁcally, when-

ever the atomic state has decayed to either the S1/2or

the D3/2manifold, the state can be detected using lasers

that induce ﬂuorescence on cycling transitions resonant

to S1/2→P1/2and to D3/2→P1/2, as described in

Fig. 1(b). Unlike a typical qubit-state detection scheme

where the |1⟩state is selectively optically cycled be-

tween |1⟩and appropriate sublevels in the P1/2manifold,

this leakage detection can be performed using broadband

lasers (such as in hyperﬁne-repumped laser cooling) such

that all hyperﬁne levels in the S1/2and D3/2manifolds

are cycled to P1/2.

In the rare event of detecting leakage, the qubit is re-

set to either |0⟩mor |1⟩m, with probability 1/2 each.

This eﬀectively replaces the leaked state to the max-

imally mixed state I/2 in the qubit subspace, which

completes converting leakage to erasure. Resetting the

metastable qubit can be performed by the standard

ground-qubit state preparation followed by coherent elec-

tric quadrupole transition. This has recently been exper-

imentally demonstrated with high ﬁdelity in less than

1µs using Yb+ions [42].

(b)

(a)

FIG. 1. (a) The laser-based gate operation (blue) on the S1/2ground qubit. (b) The laser-based gate operation (blue) and

leakage detection (red) on the D5/2metastable qubit. The orange curvy arrows show the paths of spontaneous decay. The

gate operation on the ground (metastable) qubit uses two Raman beams detuned from the S1/2(D5/2)→P3/2transition by

−∆g(−∆m). When a P3/2state decays, the state decays to one of the S1/2,D3/2, and D5/2manifolds with probability r1,r2,

and r3, respectively. For the metastable qubit, leakage detection detects the decay to the S1/2and D3/2manifolds using lasers

that are resonant to the S1/2→P1/2and D3/2→P1/2transitions, which cause photons to scatter from the P1/2state and get

collected.

Note that as transition between the P1/2and D5/2

states is forbidden, the photons ﬂuoresced from the P1/2

manifold during leakage detection and ground-qubit state

preparation are not resonant to any transition with the

metastable-qubit states involved. This allows the erasure

conversion to be performed on an ion without destroying

the qubit states of the nearby ions with high probabil-

ity. For ground qubits, an analogous erasure-conversion

scheme of detecting leakage to the D3/2and D5/2man-

ifolds will destroy the ground-qubit states of the nearby

ions, as both the P1/2and the P3/2states decay to S1/2

states with high probability.

In the scheme described above, leakage to D5/2states

other than the qubit states remains undetected. Such

leakage can be handled by selectively pumping the D5/2

hyperﬁne states except for |0⟩mand |1⟩mto the S1/2

manifold through P3/2. With high probability, the

atomic state eventually decays to either the S1/2or the

D3/2manifold, which then can be detected as described

above. However, this requires the laser polarization to

be aligned with high precision such that the qubit states

are not accidentally pumped [28]. Therefore, we defer a

careful analysis on whether such process is feasible and

classify leakage to other D5/2states as undetected leak-

age when the erasure-conversion scheme is used.

III. TWO-QUBIT-GATE ERROR MODEL

In this section, we describe the error model that we

use for comparing the logical performance of ground and

metastable qubits. The source of gate errors that we

consider here is the spontaneous decay of excited states,

which can cause various types of errors. When excited

states decay back to one of the qubit states, either a bit

ﬂip or a phase ﬂip occurs. When excited states decay

FIG. 2. The various types of error rates of the Ba+(top)

and Ca+(bottom) qubits as the detuning ∆q(q=g, m) from

the P3/2manifold is varied. ωFis the frequency diﬀerence

between the P1/2and P3/2manifolds. For the metastable

Ba+(Ca+) qubit, I= 1/2 (7/2) and F0= 2 (5), as shown

in Table II. For ground (metastable) qubits, p(l)

g(p(e)

m) most

signiﬁcantly contributes to pg(pm). For Ca+, the pmand p(e)

curves are barely distinguishable. Note also that metastable

qubits require larger detuning than ground qubits in order to

have the same total error rate.

to any other state, a leakage error occurs. Finally, for

metastable qubits, when the state after decay is outside

the D5/2manifold, such leakage can be converted to an

erasure.

Figure 2 shows the various types of error rates of the

Ba+and Ca+qubits as the lasers’ detuning from the

P3/2manifold is varied. Here, subscripts gand mde-

note the ground and metastable qubit, respectively, and

p(xy)

q,p(z)

q,p(l)

q,p(e)

q, and pq(q=g, m) denote the rate

of bit ﬂip, phase ﬂip, leakage, erasure, and total error,

respectively, for each qubit on which a two-qubit gate is

applied. Up to SubSec. III C, we provide qualitative ex-

planations on how these error rates are calculated from

atomic physics, following the discussion in Refs. [44, 45].

The quantitative derivations are deferred to Appendix B.

In our model, the only controllable parameter that de-

termines the error rates is the laser detuning from the

transition to excited states. In reality, the laser power is

also important, as the gate time is determined by both

the detuning and the laser power. In SubSec. III D, we

provide methods of comparing ground and metastable

qubits with a ﬁxed gate time, such that the errors due to

technical noise are upper bounded to the same amount.

A. Deﬁnitions

In order to compare the gate error rates between

ground and metastable qubits, we need to deﬁne sev-

eral quantities. First, we deﬁne the maximal one-photon

Rabi frequency of the transition between a state in

the manifold of the qubit states, denoted with sub-

script q∈ {g=S1/2, m =D5/2}, and an excited Pstate

(L= 1), as

gq:= Eq

2ℏµq,(2)

where

µq:= pkq⟨L= 1||T(1)(d)||Lq⟩.(3)

Here, Eqis the electric ﬁeld amplitude of the laser used

for the qubit in manifold q,µqis the largest dipole-matrix

element of transition between a state in manifold qand a

Pstate, and T(1)(d) is the dipole tensor operator of rank

1. Also, kqis a coeﬃcient that relates gqto the orbital

dipole transition-matrix element, which is calculated in

Appendix A using the Wigner-3jand 6jcoeﬃcients.

Next, in order to obtain the scattering rates that lead

to various types of errors, we ﬁrst deﬁne the decay rate

of the manifold of the excited states, denoted with sub-

script e∈ {P1/2, P3/2}, to the ﬁnal manifold, denoted

with subscript f∈ {S1/2, D3/2, D5/2}, as

γe,f := ω3

e,f

3πc3ℏϵ0X

Ff,Mf|⟨Le, Je;Fe, Me|d|Lf, Jf;Ff, Mf⟩|2

=αe,f ω3

e,f

3πc3ℏϵ0⟨L= 1||T(1)(d)||Lf⟩

2,(4)

where ωe,f is the frequency diﬀerence between the man-

ifolds of the excited and ﬁnal states. Also, αe,f is a co-

eﬃcient that relates γe,f to the orbital dipole transition-

matrix element, which is calculated in Appendix A using

the Wigner-3jand 6jcoeﬃcients. Note that γe,f does

not depend on Feand Meas the frequency diﬀerences

between hyperﬁne states of the same manifold are ig-

nored.

TABLE I. The values of the coeﬃcient kq(αf), for the man-

ifolds of various qubit (ﬁnal) states.

q=S1/2q=D5/2f=S1/2f=D3/2f=D5/2

kq1/3 1/5 αf1/3 1/30 3/10

The laser-based gate operations use two-photon Ra-

man beams of frequency ωLthat are detuned from the

transition between manifolds eand qby −∆e,q. In such

case, the decay rate is given by [44]

γ′

e,f := γe,f ωe,f −∆e,q

ωe,f 3

=γe,f ωL−ωf,q

ωe,f 3

,(5)

where ωf,q is the energy of manifold fminus the energy

of manifold q. Note that the numerator of the cubed

factor does not depend on the choice of the manifold eof

the excited states.

While manifold ecan be either P1/2or P3/2, Eqs. (4)

and (5) remove the dependence of γ′

e,f /αe,f on e. This

allows us to calculate the rates of scattering from the

states of both P1/2and P3/2manifolds only using γ′

f/αf,

where we deﬁne

αf:= αP3/2,f , γ′

f:= γ′

P3/2,f .(6)

Speciﬁcally, combining Eq. (5) and the branching frac-

tions of P3/2states, γ′

fis given by

γ′

f=









(1 −∆q/ωP3/2,S1/2)3×r1γ, f =S1/2,

(1 −∆q/ωP3/2,D3/2)3×r2γ, f =D3/2,

(1 −∆q/ωP3/2,D5/2)3×r3γ, f =D5/2,

(7)

where γis the total decay rate of a P3/2state, and

∆qis the detuning deﬁned as the laser frequency mi-

nus the frequency diﬀerence between manifold qand the

P3/2manifold. For the detunings considered in this pa-

per, ∆q/ωP3/2,f is at most approximately 0.1, so riγ

(i= 1,2,3) is a reasonably close upper bound for the

corresponding γ′

Table I shows the values of kq(αf) for the manifolds

of various qubit (ﬁnal) state, and their derivations can

be found in Appendix A.

B. How errors arise from spontaneous scattering

Spontaneous scattering of the short-lived P1/2and P3/2

states is the fundamental source of errors for laser-based

gates. The type of error (phase ﬂip, bit ﬂip, leakage, or

erasure) depends on to which atomic state the short-lived

states decay.

Rayleigh and Raman scattering are the two types of

spontaneous scattering. Rayleigh scattering is the elastic

case where the scattered photons and the atom do not

exchange energy or angular momentum. Raman scatter-

ing is the inelastic case where the photons and the atom

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QuantumerrorcorrectionwithmetastablestatesoftrappedionsusingerasureconversionMingyuKang,1,2,∗WesleyC.Campbell,3,4,5andKennethR.Brown1,2,6,7,†1DukeQuantumCenter,DukeUniversity,Durham,NC27701,USA2DepartmentofPhysics,DukeUniversity,Durham,NC27708,USA3DepartmentofPhysicsandAstronomy,UniversityofCaliforn...

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Quantum error correction with metastable states of trapped ions using erasure conversion Mingyu Kang1 2Wesley C. Campbell3 4 5and Kenneth R. Brown1 2 6 7

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