Optical multi-qubit gate operations on an excitation blockaded atomic quantum register Adam Kinos1and Klaus Mlmer2y

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Optical multi-qubit gate operations on an excitation blockaded atomic quantum

Adam Kinos1∗and Klaus Mølmer2†

1Department of Physics, Lund University, P.O. Box 118, SE-22100 Lund, Sweden and

2Niels Bohr Insitute, Blegdamsvej 17. 2100 Copenhagen, Denmark.

(Dated: October 13, 2022)

We consider a multi-qubit system of atoms or ions with two computational ground states and an

interacting excited state in the so-called blockade regime, such that only one qubit can be excited

at any one time. Examples of such systems are rare-earth-ion-doped crystals and neutral atoms

trapped in tweezer arrays. We present a simple laser excitation protocol which yields a complex

phase factor on any desired multi-qubit product state, and which can be used to implement multi-

qubit gates such as the n-bit Toﬀoli gates. The operation is performed using only two pulses, where

each pulse simultaneously address all qubits. By the use of complex hyperbolic secant pulses our

scheme is robust and permits complete transfers to and from the excited states despite the variability

of interaction parameters. A detailed analysis of the multi-qubit gate performance is provided.

I. INTRODUCTION

Theoretical and experimental eﬀorts have led to im-

mense progress in the implementation of computation on

quantum systems. Subject to execution of suitable algo-

rithms, these systems make use of the quantum super-

position principle and they may eventually outperform

classical computers for many tasks. In a systematic per-

spective, it has been useful to identify a universal set

of one-bit and two-bit gate operations which serve as

minimal requirements for the physical implementation

of any computational algorithm. But, it has also been

recognized that the interaction mechanisms characteris-

tic of each speciﬁc physical implementation comes with

distinct challenges and advantages. It thus makes sense

to carefully choose among formally equivalent but physi-

cally diﬀerent gate operations and sequences of gates that

minimize physical resources, execution speed, and errors.

This can be done by expert users, and competing auto-

matic and AI inspired strategies are now appearing for

such optimization [1].

An especially challenging, while potentially rewarding

direction of this research concerns the use of physical in-

teractions between more than two qubits for direct imple-

mentation of higher multi-qubit gate operations. This is

challenging because it requires analysis of complex physi-

cal processes and larger state spaces, and it is, ultimately

at variance with the paradigm of breaking computations

down to elementary gates. Still, the rewards may be

large and, when successful, incorporation of system spe-

ciﬁc multi-qubit gates in the elementary set, may provide

substantial shortcuts and robustness and save computing

time. The internal, electronic states that form the qubits

in trapped ions all interact simultaneously with the vibra-

tional modes of motion of the ions, and this thus permits

implementation of all-to-all eﬀective interactions relevant

∗adam.kinos@fysik.lth.se

†moelmer@phys.au.dk

for quantum simulation [2] and multi-qubit conditional

gate operations relevant for quantum computing [3,4].

By the Rydberg excitation blockade mechanism neutral

atoms interact with all atoms within several micrometre

distance and generalization of two-qubit blockade gates

[5] can be employed to make multi-qubit Toﬀoli gates

[6] and implement the conditional phase evolution of the

Grover algorithm by just few laser pulses [7]. Since these

speciﬁc gates are useful for a wide range of algorithmic

tasks and in particular for error correcting codes [8] and

for preparation of pure qubit states [9], it is desirable to

optimize them and exploit them as much as possible in

quantum computing.

In this article we focus on quantum computing us-

ing single rare-earth-ion dopants in inorganic crystals as

qubits [10], but our scheme is also applicable to other sys-

tems. We combine robust schemes previously explored to

enable quantum gates with inhomogeneous ensembles of

dopant ions [11] with the multi-qubit excitation blockade

ideas of Ref. [6], and we assess the expected gate ﬁdelity

by numerical simulations and analytical estimates. Com-

pared to implementing single- and two-qubit gate oper-

ations in these systems [12], our protocol only has the

additional requirement that all qubits are in the blockade

regime and can be addressed simultaneously. In return,

our multi-qubit operation can be faster and have smaller

errors compared to decomposing a multi-qubit operation

into single- and two-qubit operations, while also being

more robust against ﬂuctuations in Rabi frequencies and

uncertainties in the transition frequencies of the qubits.

The work is organized as follows. Sec. II presents how

our gate operation is performed in a simpliﬁed setting

and discusses its requirements. The performance of the

operation is studied in Sec. III. In Sec. IV we generalize

the protocol to work with diﬀerent values of the block-

ade shifts and to provide phase factors conditioned on

any separable multi-qubit state, as well as incorporating

single-qubit gates into the execution of the multi-qubit

gate. We present a conclusion and outlook in Sec. V.

arXiv:2210.06212v1 [quant-ph] 12 Oct 2022

…

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Qubit 



  

   

(b) (c)

(a)

 

    

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 



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Figure 1. (a) The multi-qubit sechyp operation is applied to nqubit ions that all interact strongly in their excited state |ei,

e.g., via dipole-dipole or van der Waals interactions. The operation consists of two parts. First, all ions are simultaneously

excited by sechyp pulses as described by Eq. (2). Second, the pulses are applied again, except all driving ﬁelds have an added

phase of π+θcompared to the ﬁrst pulses. Except for a global phase, this operation applies a phase θto the |11...1istate. (b)

For a given multi-qubit ground state |Ψ(n0)icontaining n0|0icomponents, the Hamiltonian eﬀectively causes excitation, with

the interaction strength √n0Ω(t), to a superposition state |Be(n0)ion the form of Eq. (1). If all qubits experience the same

excited state interaction induced detuning ∆ωof its resonance frequency, the state |Be(n0)iwith a single excitation couples

oﬀ-resonantly to the state |Bee(n0)i, containing doubly excited state components. For more information see Appendix B. (c)

Shows trajectories on the Bloch sphere for qubits subjected to sechyp pulses using various Rabi frequencies Ω0. As can be seen,

the sechyp pulse shape can perform complete transfers for diﬀerent Rabi frequencies, as long as Ω0≥µβ and µ≥2.

II. THE MULTI-QUBIT SECHYP OPERATION

The goal of our gate operation is to apply a com-

plex phase θconditioned on the qubit register populat-

ing any separable multi-qubit state. In this section we

ﬁrst present the protocol to apply such a phase θon the

state |11...1i. The case of a general product state and

extension to other gates, e.g., the n-bit Toﬀoli gates, is

discussed in the subsequent sections.

The system we consider consists of nqubits with two

long-lived ground states |0iand |1i. As shown in Fig.

1(a), we ﬁrst assume that for each qubit we can choose

to apply a laser ﬁeld that couples the state |0ito the ex-

cited state |ei. Such individual control can be achieved

if the transition frequencies of diﬀerent qubits are well-

separated due to inhomogeneous broadening [13]. Fur-

thermore, we assume that all excited state qubits inter-

act strongly with each other by dipole-dipole or van der

Waals interactions, so that if one qubit is excited, it shifts

the resonance frequencies of all other qubits and prevents

them from being simultaneously excited. This deﬁnes the

so-called blockade regime.

Our gate operation consists of two applications of the

same pulse that simultaneously act on the |0i→|eitran-

sition for all nqubits, i.e., the incoming pulse consists

of a frequency comb with teeth centered at the tran-

sition frequencies of the diﬀerent qubits that we want

to participate in the gate. When qubits are simultane-

ously addressed like this, a multi-qubit state |Ψ(n0)i=

|0110...10i, containing n0|0icomponents, couples with a

collectively enhanced interaction strength √n0Ω(t) to a

superposition state

|Be(n0)i=1

√n0

(|e110...10i+|011e...10i+... |0110...1ei),

(1)

with a single shared excitation among all the qubits that

were initially in state |0i. As shown in Fig. 1(b), the

state |Be(n0)iis also oﬀ-resonantly coupled with an in-

teraction strength p2(n0−1)Ω(t) to |Bee(n0)i, which

is a superposition of states with two excited state ions.

To make this drive negligible, we assume that the ex-

cited state interaction ∆ωshifts the resonance enough

to suppress excitation of more than a single ion. Thus,

for our operation to work p2(n0−1)|Ω(t)|and the fre-

quency bandwidth of the sechyp must be much smaller

than ∆ω, for any value of n0= 1, ..., n.

The goal of the operation’s ﬁrst part is to take advan-

tage of the blockade eﬀect to excite all 2ncomputational

multi-qubit ground states except |11...1ito diﬀerent su-

perposition states that contain exactly one excitation.

Thus, the laser pulses must be able to perform complete

transfers from |Ψ(n0)ito |Be(n0)iwith Rabi frequencies

that vary between Ω(t) for a ground state containing only

one |0i, to a maximum value of √nΩ(t), for the state con-

taining n|0i. This is accomplished by using a complex

hyperbolic secant, or sechyp for short, pulse shape

Ω(t)=Ω0sech βt−tg

21+iµ

,(2)

which is robust against variations of the overall Rabi fre-

quency as long as Ω0≥µβ and µ≥2 [14], as indicated in

Figure 2. Panels (a) and (b) show the time dependent Rabi frequency amplitude and frequency detuning, respectively, of the

sechyp pulse described by Eq. (2). Panel (c) Shows the transfer error as a function of the maximum Rabi frequency amplitude,

|Ω(t)|max/Ω0, after performing two consecutive sechyp pulses that, ideally, ﬁrst excite and then return the ion to its initial state

with a θ=πphase shift. For these ﬁgures, µ= 3 and β= Ω0/µ, which gives a fwhm in intensity of tfwhm = 2 ln (1 + √2)/β

and a frequency width of fwidth =µβ/π. The cutoﬀ duration is tg= 6 ×tfwhm in panels (a-b) and varies in panel (c).

Fig. 1(c). An example of the Rabi frequency amplitude

and frequency of a sechyp pulse is shown in Fig. 2(a-b).

An added beneﬁt of using sechyp pulses is that they are

also robust against variations in transition frequencies

[11].

The second part of the operation is identical to the

ﬁrst one, except that all driving ﬁelds are applied with a

phase changed by the constant amount π+θcompared

with the ﬁrst pulse, as shown in Fig. 1(a). This will

thus deexcite all state components excited by the ﬁrst

pulse back to their respective ground state with a phase

shift of −θ. Except for a global phase, this operation is

equivalent to applying a phase of θto the qubit register

state |11...1i.

III. GATE PERFORMANCE

In this section we investigate the performance of the

gate operation and its robustness against three error

sources: the error due to imperfect sechyp transfers; the

error due to the oﬀ-resonant coupling to the doubly ex-

cited states |Bee(n0)i; and the error due to T2dephasing

of the excited state.

We assume here that all qubits interact with the same

interaction shift given by ∆ω, but we shall return to this

issue again in Sec. IV A. Furthermore, we assume that

the initial state is an even superposition of all 2ncompu-

tational ground states, and defer discussion of the general

case and some analytical results to Appendix A. Under

these assumptions, the total error can be estimated as

= 1 −1

22n1 +

n0=1 n

n02e−γRe [A(n0)] +

n0=1

m0=1

min(n0,m0)

k=max(n0+m0−n,0)

Re [A(n0)A∗(m0)] n

n0n0

kn−n0

m0−kk+ (n0m0−k)e−2γ

n0m0,(3)

where γ=αtg/T2represents the dephasing error during

the pulse duration tgdue to the ﬁnite coherence time T2

(α≈1 estimates how large fraction of the pulse duration

the atom spends in the excited state). A(n0) are complex

numbers

A(n0) = T(n0)exp i2(n0−1)Λ

4∆ω,

Λ = Ztg

0|Ω(t)|2dt =2Ω2

βtanh (βtg/2) ,(4)

and represent the eﬀect of imperfect state transfer,

T(n0), and an AC Stark shift of the singly excited state

due to the oﬀ-resonant coupling to higher excited states,

which is discussed further in the subsequent subsections.

Finally, the pulse and interaction parameters Ω0, ∆ω,

and βare all given in angular frequency units.

A. Transfer errors

The transfer of state amplitude to and from the excited

states is not perfect, and errors occur because the sechyp

pulse has a cutoﬀ duration tgwhich leads to small jumps

in the Rabi frequency amplitude at 0 and tg. The transfer

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Opticalmulti-qubitgateoperationsonanexcitationblockadedatomicquantumregisterAdamKinos1andKlausMlmer2y1DepartmentofPhysics,LundUniversity,P.O.Box118,SE-22100Lund,Swedenand2NielsBohrInsitute,Blegdamsvej17.2100Copenhagen,Denmark.(Dated:October13,2022)Weconsideramulti-qubitsystemofatomsorionswithtwoco...

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Optical multi-qubit gate operations on an excitation blockaded atomic quantum register Adam Kinos1and Klaus Mlmer2y

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