Universal quantum computation in decoherence-free subspaces with hot trapped-ions

Abstract

We consider interactions that generate a universal set of quantum gates on logical qubits encoded in a collective-dephasing-free subspace, and discuss their implementations with trapped ions. This allows for the removal of the by-far largest source of decoherence in current trapped-ion experiments, collective dephasing. In addition, an explicit parametrization of all two-body Hamiltonians able to generate such gates without the system's state ever exiting the protected subspace is provided.

Full text

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arXiv:0706.0203v1 [quant-ph] 1 Jun 2007
Universal quantum computation in decoherence-free subspaces with hot trapped-ions
Leandro Aolita,1,2Luiz Davidovich,1Kihwan Kim,3and Hartmut H¨ affner4
1Instituto de F´ ısica, Universidade Federal do Rio de Janeiro. Caixa Postal 68528, 21941-972 Rio de Janeiro, RJ, Brazil.
2Max-Planck-Institute f¨ ur Physic Komplexer Systeme, N¨ othnitzerstrasse 38, D-01187, Dresden, Germany
3Institut f¨ ur Experimentalphysik, Universit¨ at Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria
4Institut f¨ ur Quantenoptik und Quanteninformation der¨Osterreichischen
Akademie der Wissenschaften, Technikerstraße 21a, A-6020 Innsbruck, Austria
(Dated: February 1, 2008)
We consider interactions that generate a universal set of quantum gates on logical qubits encoded in a
collective-dephasing-free subspace, and discuss their implementations with trapped ions. This allows for the
removal of the by-far largest source of decoherence in current trapped-ion experiments, collective dephasing.
In addition, an explicit parametrization of all two-body Hamiltonians able to generate such gates without the
system’s state ever exiting the protected subspace is provided.
PACS numbers: 03.67.Lx, 03.67.Pp, 32.80.Qk
I.INTRODUCTION
In quantum information processing tasks decoherence can
be overcome either by an active approach or by a passive one.
The formerconsists, in analogywith classical computation,of
encoding information in a redundant fashion by means of the
so-called error-correctingcodes. In this approach information
is encoded in subspaces of the total Hilbert space of the sys-
temin suchaway that“errors”inducedbytheinteractionwith
the environment can be detected and corrected without gain-
ing information about the actual state of the system prior to
corruption [1].
The passive approach, on the other hand, is an error pre-
venting scheme, in which logical qubits are encoded within
decoherence-free subspaces (DFS), which do not decohere
because of symmetry [2]. A simple example is provided by
a system of N spins collectively interacting with the same
reservoir, for which the interaction is mediated by the collec-
tive angular momentum raising and lowering operatorsˆS+≡
?N
correspondingraising and lowering operators, respectively, of
the i-th particle. The collective operators have no support on
theeigenstatesofthetotalsquaredangularmomentumˆS2cor-
responding to zero eigenvalue. The evolution of these eigen-
states is therefore unitary because they simply do not couple
to the reservoir; and they can be used as a logical-qubit basis
for decoherence-freequantum computation [3, 4].
When the coupling to the environment is mediated by the
collective z-angular-momentumoperatorˆSz≡?N
type of noise is called collective dephasing. The interaction
Hamiltonian between the system and the bath is then pro-
portional toˆSz⊗ˆB, whereˆB is an arbitrary operator act-
ing on the Hilbert space associated to the bath. The action of
this type of bath is equivalent to that of randomly-fluctuating
fields: a general qubit-state |Ψ? ≡ a|0? + b|1? transforms
as |Ψ? → a|0? + beiζ|1?, which leads to the loss of coher-
ence of the state for ζ is a random fluctuating phase. By
using one pair of physical qubits, whose members are la-
beled by the subindexes i1 and i2, to encode logical qubit
i, one can protect information from the detrimental action
i=1ˆS+
iandˆS−≡?N
i=1ˆS−
i, whereˆS+
iandˆS−
iare the
i=1ˆSz
i, the
of decoherence. In fact, the well-known [5, 6] logical ba-
sis BLi≡ {|0Li? ≡ |0i11i2?;|1Li? ≡ |1i10i2?} spans a
DFS protected against collective dephasing, which we call
VDFS2i. That is, the logical state |ΨLi? ≡ a|0Li?+b|1Li? =
a|0i1?|1i2?+b|1i1?|0i2? evolves as |ΨLi? → a|0i1?eiζ|1i2?+
beiζ|1i1?|0i2? = eiζ(a|0Li? + b|1Li?) and is thus invariant up
to an irrelevant global phase factor.
Two pairs of physical qubits, whose members are labeled
by the subindexes i1and i2, and j1and j2, respectively, are in
turn needed to encode two logical qubits i and j . The direct
product subspace VDFS2i⊗ VDFS2j, spanned by the basis
BLi⊗ BLj, yields a DFS. However, one should note that this
is not the total protectedsubspace supportedby all four qubits
if all four physical qubits experience the same phase fluctua-
tions. In this case the states |0i10i21j11j2? and |1i11i20j10j2?,
which are outside VDFS2i⊗ VDFS2j, are also protected
against collective dephasingfor they have the same amount of
excitations as the states in VDFS2i⊗VDFS2j. In general, any
coherentsuperpositionofstates withthe sameamountofexci-
tationsis immuneagainstcollectivedephasing. Thus, the total
protected subspace, which we call VDFS4ij, is that spanned
by BLi⊗ BLjtogether with the states |0i10i21j11j2? and
|1i11i20j10j2?. Ifpairsiandj arefurtherapartthanthetypical
noise correlation length —but with both qubits from each pair
still subject to to the same fluctuations— VDFS2i⊗ VDFS2j
is the only protected subspace.
On the experimental side, demonstration of immunity of
a DFS of two photons to collective noise was accomplished
in [7] and realizations of DFS’s for nuclear magnetic res-
onance (NMR) systems were carried out in [8].
stration of a collective-dephasing-free quantum memory of
one logical qubit composed of a pair of trapped9Be+ions
was first achieved in [9] and coherent oscillations between
two logical states, encoded into the two Bell states |Ψ±? =
1
√2(|01? ± |10?), by inducing a gradient of the magnetic field
applied to both ions, were reported in [10, 11]. Finally, en-
tanglement lifetimes of more than 7 seconds [11] and ro-
bust entanglement lasting for more than 20 seconds [12] were
attained using ground state hyperfine levels of9Be+ions
and ground state Zeeman sublevels of40Ca+ions, respec-
Demon-

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tively. These experiments demonstrated that for trapped-ions
collective-dephasingis themajorsourceofqubitdecoherence.
We thereforefocus on this type of noise throughoutthe rest of
the paper. Nevertheless, apart from the proof-of-principle ex-
periments mentioned above, demonstrating the robustness of
these subspaces, experimentally accessible implementations
of DFS-encoded gates are still sparse; and in spite of the rich
(but abstract) body of work on DFS’s, a universal set of gates
between two encodedlogical qubits is yet to be demonstrated.
ProposalsforiontrapquantumcomputingwithDFS’s exist,
and they are essentially divided into two families that com-
plement each other. In the first paradigm [6] gates between
two logical qubits are implemented [6, 13] by bringing to-
gether two pairs of ions (each pair encoding a logical qubit),
initially stored in memory regions, to an interaction region
where a simultaneous interaction among all four ions takes
place according to the Sørensen-Mølmer (SM) gate described
in [14, 15]. Individual laser addressing is not necessary for
this scheme, but a reliable ion-shuttlingtechnique is an essen-
tial requirement. In addition, even though this scheme maps
VDFS4ijinto itself, it does not preserve the state inside the
DFS throughout the gate evolution [4]. The second paradigm
[16, 17] works in the individual laser addressing regime and
relaxes the need of ion-shuttling. In this approach, ions are
trapped in a crystal-like effective potential created by arrays
of multi-connectedlinear Paul traps. Each ion is associated to
a neighbor to form a pair that encodes one logical qubit [17].
By inducing a ˆ σz-dependent force (see [18, 19, 20] and refer-
ences therein) on two ions, each from different pairs, it is in
principle possible to implement a (geometric phase) ˆ σz-gate
between the logical qubits encoded into both pairs. Particular
advantagesof these ˆ σz-gates are that they can be considerably
fast and robust. It has been conjectured [19, 20, 21] though,
that these ˆ σz-gates are very ineffecient with magnetic-field-
insensitive (or “clock”)states, which possess such remarkable
coherence properties [11, 22]. However, it would be very ad-
vantageous to combine clock states with DFSs as this would
lead to very long coherence times and minimize the overhead
due to quantum error correction.
In our present paper we assess different possible interac-
tions involving only two physical qubits at a time that gener-
ate universal quantum gates on DFS-encoded qubits, and de-
scribe feasible experimental demonstrations of each of them
with trapped-ions. The work is conceptually divided into two
parts. The first one (Sec. II) is devoted to the general for-
mal classification of all two-body dynamics able to generate
universal quantum gates inside the DFS without the system’s
state ever leaving it. The aim here is not to establish the set of
formal conditions for a given Hamiltonian to generate univer-
sal DFS quantum computation, as in [3, 4]; but rather to ex-
plicitly construct the allowed Hamiltonians in a simple way in
terms of the Pauli operators associated to each physical qubit.
This is to serve as a simple “classification table” for exper-
imentalists to rapidly check whether the type of interactions
present in their given system qualifies as a candidate for gen-
erating universal DFS quantum computation or not. In partic-
ular,weintroducethemostgeneraltwobodyHamiltonianthat
generates universal quantum computation while guaranteeing
the evolution to take place entirely inside VDFS2i⊗VDFS2j.
Furthermore, we show that the only possible interaction be-
tween two logcial qubits which obeys the previous assump-
tions is of the type ˆ σz⊗ ˆ σz. For the cases where leakage
out of VDFS2i⊗ VDFS2jinto VDFS4ijis allowed, we con-
sider the encoding re-coupling scheme originally introduced
in [25] for NMR systems. There, a maximally entangling
gate is implemented on the DFS through a sequence of trans-
formations that momentarily takes the composite state out of
VDFS2i⊗ VDFS2jbut never out of VDFS4ij.
The second part (Sec. III) describes the technical details
of the implementation on trapped-ions of the ideas presented
in Sec. II. Our implementations work in the individual laser
addressing regime and require no ion-shuttling. We show
that for the realization of local and conditional gates inside
VDFS2i⊗VDFS2j, the SM-gateandthe ˆ σz-gate,respectively,
can be used. For the realization of the encoded re-coupling
scheme in turn, an alternative two-physical-qubit gate is re-
quired. The latter is based on bichromatic Raman fields and
applies to all states in general regardless of their magnetic
properties, includingclock states connected via dipole Raman
transitions. Furthermore, this gate does not require the ions to
beintheirmotionalgroundstate, providedthattheyalwaysre-
main in the Lamb-Dicke regime. Therefore, it is a potentially
useful alternative to the SM-gate and the ˆ σz-gate also outside
the context of DFS’s. Our conclusions are finally summarized
in section IV.
II. GENERAL HAMILTONIANS FOR UNIVERSAL
QUANTUM COMPUTATION IN THE DFS
A. Local operations: the logical SU(2) Lie Algebra
We want to find a complete set of orthogonal operators
mapping VDFS2i(for any i) onto itself. We define then log-
ical identity and Pauli operators, ˆ σ0
ˆ σ2
Li≡ˆILi, ˆ σ1
Li≡ ˆ σx
Li,
Li≡ ˆ σy
Liand ˆ σ3
Li≡ ˆ σz
Liof the i-th logical qubit, as:
ˆ σ0
ˆ σ1
ˆ σ2
ˆ σ3
Li≡ αiˆ σ0
Li≡ βiˆ σ1
Li≡ γiˆ σ2
Li≡ εiˆ σ3
i1⊗ ˆ σ0
i1⊗ ˆ σ1
i1⊗ ˆ σ1
i1⊗ ˆ σ0
i2− (1 − αi)ˆ σ3
i2+ (1 − βi)ˆ σ2
i2− (1 − γi)ˆ σ1
i2− (1 − εi)ˆ σ0
i1⊗ ˆ σ3
i1⊗ ˆ σ2
i⊗ ˆ σ2
i1⊗ ˆ σ3
i2+ˆ0Li,
i2+ˆ0Li,
i2+ˆ0Li,
i2+ˆ0Li;
(1)
where ˆ σp
associated to the n-th (n = 1 or 2) physical qubit of the i-th
pair, and with αi,βi,γi, and εiany real numbers such that
0 ≤ αi,βi,γiand εi ≤ 1. The operatorˆ0Lirepresents the
logical null operator, which is defined as any operatorwithout
support on VDFS2i. The operators in Eq. (1) map VDFS2i
onto itself and their action on BLiis exactly equivalent to
that of the usual identity and Pauli physical operators on the
computational basis. It can be seen that Eq. (1) is the most
general way to construct them from the operators that act on
the physical qubits. For example, if we added the term ˆ σ1
ˆ σ3
inis theidentity(p = 0)orPauli(1 ≤ p ≤ 3)operator
i1⊗
i2tothedefinitionof ˆ σ3
Liin(1)wewouldexitVDFS2i; terms

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as ˆ σ1
instead; and so on. Combinations as ˆ σ1
or ˆ σ1
no support on VDFS2iand can therefore be grouped inside
ˆ0Li. In general there are sixteen possible products between
ˆ σ0
i1⊗ ˆ σ1
i2would not take us out of the DFS but act like ˆ σ1
Li
i1⊗ ˆ σ1
i2− ˆ σ2
i1⊗ ˆ σ2
i2
i1⊗ ˆ σ3
i2−iˆ σ2
i1⊗ ˆ σ0
i2are allowed though, since they have
i1, ˆ σ1
i1, ˆ σ2
i1and ˆ σ3
i1, and ˆ σ0
i2, ˆ σ1
i2, ˆ σ2
i2and ˆ σ3
i2. Each of these
products, or combinations of them, apart from those already
considered in Eq. (1), either takes the state out of VDFS2i,
does not have the desired action, or has no supporton VDFS2i
and is therefore absorbed inside the definition ofˆ0Li. The
most general expression for the logical null operator is given
by
ˆ0Li≡ ρi(ˆ σ0
i1⊗ ˆ σ1
i1⊗ ˆ σ0
i2− iˆ σ3
i2+ ˆ σ3
i1⊗ ˆ σ2
i1⊗ ˆ σ3
i2) + θi(ˆ σ1
i2) + λi(ˆ σ1
i1⊗ ˆ σ0
i1⊗ ˆ σ1
i2− iˆ σ2
i2− ˆ σ2
i1⊗ ˆ σ3
i1⊗ ˆ σ2
i2) + ϑi(ˆ σ1
i2) + ςi(ˆ σ1
i1⊗ ˆ σ3
i1⊗ ˆ σ2
i2− iˆ σ2
i2+ ˆ σ2
i1⊗ ˆ σ0
i1⊗ ˆ σ1
i2) + ζi(ˆ σ3
i2) + ξi(ˆ σ3
i1⊗ ˆ σ1
i1⊗ ˆ σ0
i2− iˆ σ0
i2+ ˆ σ0
i1⊗ ˆ σ2
i1⊗ ˆ σ3
i2) +
κi(ˆ σ0
i2) ,
(2)
with ρi,θi,ϑi,ζi,κi,λi,ςi, and ξiany complex numbers.
The operators in Eq. (1) are orthonormal: Tr[ˆ σp
δpq, with p and q = 0,1,2 or 3, and form therefore a com-
plete orthonormal basis of the space of the complex operators
acting on the two-dimensional subspace VDFS2i. They also
satisfy, inside of VDFS2i, the desired SU(2) usual commuta-
tion relations: [ˆ σp
3; and [ˆ σ0
show this, we calculate explicitly the commutator [ˆ σ1
and obtain
Liˆ σq
Li] =
Li, ˆ σq
Li] = 0, for p = 0,1,2 or 3. As an example to
Li] = 2iǫpqrˆ σr
Li, for p,q and r = 1,2 or
Li, ˆ σp
Li, ˆ σ2
Li]
[ˆ σ1
Li, ˆ σ2
Li] = 2i(1 − βi− γi+ 2βkγk)ˆ σ3
− 2i(βk+ γi− 2βiγi)ˆ σ0
i1⊗ ˆ σ0
i2
i1⊗ ˆ σ3
i2.
(3)
Doing the identification ε′
0 ≤ βi≤ 1 and 0 ≤ γi≤ 1 we see that 0 ≤ ε′
leads us to
i≡ 1 − βi− γi+ 2βiγiand since
i≤ 1, which
[ˆ σ1
Li, ˆ σ2
Li] = 2i?ε′
kˆ σ3
i1⊗ ˆ σ0
i2− (1 − ε′
i)ˆ σ0
i1⊗ ˆ σ3
i2
?.
(4)
This is, inside of VDFS2i, exactly equivalent to 2iˆ σ3
that the logical operator obtained here and the fourth operator
in Eq. (1) are actually not strictly equal, since εiand ε′
necessarily the same number. Their difference however only
shows when applied to states outside VDFS2i, their action on
this subspace is exactly the same. All the other SU(2) funda-
mental commutation relations are straightforwardly obtained
in the same way. We see thus that the logical Pauli operators
defined in (1) are the most generalrepresentationof the SU(2)
Lie algebra on VDFS2iconstructed from the physical-qubit
operators.
We also notice that the logical operators Xi ≡
ˆ σ1
Zi ≡
corresponding to βi= γi= εi= 1/2. These operators gen-
erate the SU(2) group on the whole Hilbert space, but they
have the same action as those defined in (1) on VDFS2i. The
advantage of the logical operators in (1) is that they give the
experimentalist more freedom of choice, as any choice of αi,
βi, γiand εiworks just as well in VDFS2i. As a matter of
fact, we exploit this freedom below to simplify the procedure
for obtaining DFS-encoded gates for trapped-ions.
Li. Note
iare not
1
2(ˆ σ1
i2) and
i1⊗
i2+ ˆ σ2
i1⊗ ˆ σ2
2(ˆ σ3
i2), Yi ≡
i1− ˆ σ3
1
2(ˆ σ2
i1⊗ ˆ σ1
i2− ˆ σ1
i1⊗ ˆ σ2
1
i2) used in [13] are a particular case of (1),
Thesituationisnowcompletelyequivalenttothatofaphys-
ical qubit, with the logical states in BLiand logical operators
in (1) playing the role of the physical ones. The important
thing to keep in mind though is that these logical operators al-
low us to operate on the logical states in the same way as their
physical counterparts without ever exiting VDFS2i. With this
athandwecannowwritedowntheHamiltonianthatgenerates
the most general unitary operation on the i-th logical qubit, it
reads:
ˆHLi≡ B0
iˆ σ0
Li+ B1
iˆ σ1
Li+ B2
iˆ σ2
Li+ B3
iˆ σ3
Li,
(5)
with B0
unitsofenergy)that playtheroleofa“logicalmagneticfield”.
Notice that we are explicitly including the logical identity in
Hamiltonian (5), even though it only introduces an irrelevant
globalphase factor. This is becausewe want to account,in the
most general way, for the possibility of appearance of terms
proportional to ˆ σ3
plementation on physical qubits.
i, B1
i, B2
iand B3
iany real numbers (times arbitrary
i1⊗ ˆ σ3
i2, which are not irrelevant for an im-
B.Computation in VDFS2i⊗ VDFS2j: the two-physical-qubit
interaction Hamiltonian
We proceed now with the interaction Hamiltonian between
logical qubits i and j,ˆHLiLj. Under the action of this Hamil-
tonian there can be no transfer of excitations between both
qubit pairs, so that each logical qubit evolves inside its own
encoded subspace. The only allowed interactions are then
those ones composed of combinations of products of logical
Pauli operators of both logical qubits. Nevertheless, the re-
markable observation is that ˆ σ3
operators that do not involve interactions between the phys-
ical qubits from the same pair. Any product of two logical
Pauli operators from both logical qubits other than ˆ σ3
will necessarily contain products of more than two physical-
qubit (non-identity) operators. We see, therefore, that there
exists only one type of two-body interaction able to gener-
ate non-trivial two logical qubit operations on the DFS and
at the same time preserving the composite state always inside
Liand ˆ σ3
Ljare the only logical
Li⊗ ˆ σ3
Lj

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VDFS2i⊗ VDFS2j. It is given by:
ˆHLiLj∝ ˆ σ3
Li⊗ ˆ σ3
Lj.
(6)
This interaction between both logical qubits reduces to a sim-
ple Ising interaction between one physical qubit from pair i
and one from j when the non-symmetric choice εi and εj
equal to 0, or 1, is taken. Also, the fact that the operators
in the z direction play such a preferential role is not surpris-
ing, since, for collective dephasing, it is the total z angular
momentum that mediates the coupling of the qubits to the en-
vironment;and ourprotectedsubspaceis precisely that of null
total z angular momentum.
The aim of Hamiltonians (5) and (6), together with expres-
sions (1) for the single logical-qubit operators, is to provide a
tool for the immediate classification of the allowed two-body
dynamics for the implementation of DFS universal quantum
computation. Any system whose Hamiltonian cannot be ex-
pressed as given by equations (5) and (6), together with (1), is
automatically excluded as a candidate for such computation,
except, of course, for the possible appearance of any combi-
nation of physical-qubit operators that can be expressed as in
Eq. (2).
C.Computation in VDFS4ij: the encoded re-coupling scheme
An alternative technique to entangle logical qubits is the
encoded re-coupling scheme, which was originally developed
for NMR systems in [25]. In this scheme, a ˆ σ3⊗ ˆ σ3inter-
action is effectively simulated by a sequence of ˆ σ+⊗ ˆ σ−-
type interactions between different physical qubits from both
pairs. This provokes an actual transfer of excitations be-
tween both pairs, so that the logical qubits momentarily exit
VDFS2i⊗VDFS2jand “loose their encodedlogical identity”.
But the total amount of excitations remains the same, so that
the whole evolution takes place inside VDFS4ij. The tech-
nique is based on the identity
e−i[ˆ σ+
i1⊗ˆ σ−
j1+h.c.]π/4e−i[ˆ σ+
i1⊗ˆ σ−
i2+h.c.]π/2?ˆ σ+
i1⊗ˆ σ−
i2⊗ ˆ σ−
j1
+h.c.?ei[ˆ σ+
=1
2ˆ σ3
i1⊗ˆ σ−
i2+h.c.]π/2ei[ˆ σ+
j1+h.c.]π/4
i2⊗?ˆ σ3
j1− ˆ σ3
i1
?.
(7)
When applying this five-fold sequence of transformations to
states inVDFS2i⊗VDFS2j, theproduct ˆ σ3
hand side can be ignored, since it is proportional to the logi-
cal identity operator, and introduces thus nothing but a global
phase factor. This leaves us with1
alent to −1
choice εi= 0 and εj= 1 in Eq. (1)).
Also here only interactions between two physical qubits at
a time are required, but the technique has the drawbacks that
it requires more pulses and can be used only when pairs i
and j experience the same phase fluctuations. Nevertheless, it
constitutes an alternative to spin-dependent forces, specially
i2⊗ˆ σ3
i1onthe right-
2ˆ σ3
i2⊗ ˆ σ3
j1, which is equiv-
2ˆ σ3
Li⊗ ˆ σ3
Lj≡ −1
2ˆHLiLj(with the non-symmetric
when Ising-like interactions are not readily available, as it ap-
pears to be the case with clock states connected via dipole
Raman transitions.
III. IMPLEMENTATION ON TRAPPED-IONS
We consider next N pairs of ions confined in a linear Paul
trap, or in an arrangement of multi-connected linear Paul
traps, where individual laser addressing is available. The col-
lective vibrational mode along the axial direction z, of fre-
quency ν, might be the center-of-mass or stretch mode. The
i-th logical qubit is encoded into a pair i of neighboring ions
i1and i2. We assume each ion in(n = 1 or 2) to have a mass
M and an equilibriumposition z0in. The ions may either pos-
sess three energy levels in a Λ configuration: two long-lived
ground-state levels, and an excited electronic state; or two en-
ergy levels, one of which is a metastable state, and the other
the ground state. In both cases, we label the physical qubit
states as | ↑in? ≡ |0in? and | ↓in? ≡ |1in?, and their in-
ternal transition frequency ω0. For three-level ions the phys-
ical qubit states are encoded in the two long-lived ground-
state levels, ω0is typically in the microwave region, and the
qubit states are typically connected by a dipole Raman tran-
sition through the excited electronic state, driven by two laser
beams A and B, of frequencies ωAand ωBand wave vectors
along the z direction kAzand kBz. For two-level ions, in turn,
the metastable state encodes | ↑in? ≡ |0in?, the ground state
| ↓in? ≡ |1in?, and they are connected by a weak quadrupole
optical transition directly driven by a single laser L, of fre-
quency ωLand wave vector along the z direction kLz.
A.Single-logical-qubit gates: ˆ σ3
L
We show first how to implement Hamiltonian (5) for the
case B0
an AC Stark shift on only one of the members of the pair,
for example ion in, which can be done by the application of
off-resonant fields δ-detuned from the carrier transition. The
interaction Hamiltonian in the interaction picture with respect
to the unperturbed Hamiltonian without the laser field, and in
the rotating wave approximation (RWA), with the condition
ω0≫ ν ≫ δ, then reads:ˆHin= ?Ωinˆ σ+
Here Ωinis the effective Rabi frequency coupling | ↑in? with
| ↓in? and ϕinis the spin phase, the field’s effective optical
phase at position z0in.
From now on we will always work in the dispersive
regime |Ωin| ≪ δ, in which perturbative calculations with
Ωin
δ
as a perturbation parameter are valid. In fact, a time-
dependentsecond-orderperturbativecalculation, yields an ef-
fective time-independent Hamiltonian given by:
i= B1
i= B2
i= 0. In this case it suffices to induce
inei[δt+ϕin]+ h.c.
ˆHin= ?|Ωin|2
δ
ˆ σ3
in
(8)
Since, according to Eq. (1), ˆ σ3
Licoincides with ˆ σ3
i1for the

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non-symmetric choice εi= 1 and with −ˆ σ3
i2for εi= 0, it is
ˆHin= B3
iˆ σ3
Li,
(9)
with B3
n = 1 (n = 2); implementing thus the desired logical Hamil-
tonian.
It is important to notice that in the above derivation, as
well as in the rest of the paper, the resolved-sideband limit,
|Ωin| ≪ ν, is assumed. In this regime, by tuning the laser
frequency, it is always possible to select the stationary terms
of the Hamiltonian and to neglect —in the RWA— all other
terms rotating at the different vibrational modes’ frequencies.
This was exploited here to neglect terms involving any vibra-
tional mode frequency by setting ωL(or ωA− ωB) close to
ω0, and is exploited in the next subsections to select the de-
sired vibrational mode by setting it close to resonance with a
sideband transition to such mode.
i≡ ±?|Ωin|2
δ
, the “+” (“−”) sign corresponding to
B.Single-logical-qubit gates: ˆ σφ
L
We now concentrate on the implementationof Hamiltonian
ˆHi1i2= Ciˆ σφi
ˆ σφi
the operator contained in the equatorial plane of the logical
Bloch sphere with azimuth angle φi. This is equivalent to
Hamiltonian (5) with B0
B2
implement such Hamiltonian, together with Hamiltonian (9),
suffices to generate any SU(2) operation on the i-th logical
qubit.
In this case it is possible to use the SM-gate [14, 19], driven
by one field detuned by δ from the red sideband, plus another
Li, where Ciis a constant and ˆ σφi
Li≡ cos(φi)ˆ σ1
Li, defined as
Li+ eiφiˆ σ−
Li+ sin(φi)ˆ σ2
Li≡ e−iφiˆ σ+
Li, is
i= B3
i= 0, B1
i≡ Cicos(φi) and
i≡ Cisin(φi). For any fixed value of φi the ability to
one detunedby −δ fromthe blue one. Here we show nonethe-
less that only one of these fields suffices as long as one re-
mains in VDFS2i⊗VDFS2j. We extend the ideas of Ref. [27]
and consider a laser field irradiating simultaneously both ions
of the i-th pair. When the laser frequency or laser frequency
difference is close to resonance with a sideband transition,
a coupling between the internal qubit states and the relevant
vibrational mode is possible. We choose the first red side-
band transition for definiteness, but the blue one would work
just as well. That is, we set ωA− ωB = ω0− ν − δ and
∆kz≡ kBz−kAz?= 0 (non-copropagatingbeams is a further
requirement for Raman couplings), or ωL= ω0− ν − δ. All
other vibrational modes can be neglected under the RWA be-
cause we are in the resolved-sideband limit and they give no
stationarycontribution. TheLamb-Dickeparameteris defined
as ην ≡ ∆kz
?
motional ground-state wave packet. We assume next that the
system is in the Lamb-Dicke limit (LDL) η2
with nνthe mean phonon population, meaning that the wave
packet is very localized as compared to the fields’ wave-
lengths2π∆k−1
z
or 2πk−1
tonian in the RWA is given byˆHi1i2 = ?[Ωi1ˆ σ+
iηνˆ aν)ei(δt+ϕi1)+ Ωi2ˆ σ+
where ˆ aν is the annihilation operator of one phonon. No-
tice that here, in spite of being in the resolved sideband limit,
we have not neglected the fast oscillating term proportionalto
eiνt, since for very low values of ην the contribution of the
latter might be comparable to that of the stationary term pro-
portional to ην.
zν
√2N≡
1
√2N∆kz
?
?
Mν, or ην ≡ kLz
zν
√2N≡
1
√2NkLz
?
Mν, wherezνis theroot-mean-squarewidthofthe
ν(nν+ 1/2) ≪ 1,
Lz. In this case the interactionHamil-
i1(eiνt+
i2(eiνt− iηνˆ aν)ei(δt+ϕi2)+ h.c.],
Taking both Rabi frequencies equal, Ωi1= Ωi2≡ Ωi,
yields the time-independent effective Hamiltonian:
ˆHi1i2= ?|Ωi|2
ν + δ(ˆ σz
i1+ ˆ σz
i2) + ?|Ωiην|2
δ
?ˆI + (ˆ σz
i1+ ˆ σz
i2)(ˆ nν+ 1/2) − (ˆ σ+
i1⊗ ˆ σ−
i2ei(ϕi1−ϕi2)+ h.c.)
?
,
(10)
with ˆ nν ≡ ˆ a†
terms proportional ˆ σz
see that in VDFS2iHamiltonian (10) is equivalent to
νˆ aν. The identity operatorˆI can be omitted as it only generates an irrelevant global phase factor; and so can the
i1+ ˆ σz
i2, for they are equivalent toˆ0Li(taking ρi= ζi= ϑi= θi= κi= λi= ςi= 0 in Eq. (2)). We thus
ˆHi1i2=
− ?|Ωiην|2
δ
?ˆ σ+
i1⊗ ˆ σ−
i2eiφi+ h.c.?= −?|Ωiην|2
2δ
?cos(φi)?ˆ σ1
i1⊗ ˆ σ1
i2+ ˆ σ2
i1⊗ ˆ σ2
i2
?+ sin(φi)?ˆ σ2
i1⊗ ˆ σ1
i2− ˆ σ1
i1⊗ ˆ σ2
i2
??, (11)
with φi ≡ ϕi1− ϕi2. A direct exchange of quanta between
both ions through a virtual excitation of the vibrational mode.
It is in turn immediate to express (11) as the desired Hamilto-
nian
ˆHi1i2= Ciˆ σφi
Li,
(12)
with Ci ≡ −?|Ωiην|2
taken in Eq. (1).
δ
, and where βi= γi = 1/2 have been