Weak non-linearities and cluster states
Weak non-linearities and cluster states
Sebastien G.R. Louis,1, 2, ∗ Kae Nemoto,1 W. J. Munro,3, 1 and T. P. Spiller3
1National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
2Department of Informatics, School of Multidisciplinary Sciences,
The Graduate University for Advanced Studies, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430 Japan
3Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom
(Dated: February 6, 2020)
We propose a scalable approach to building cluster states of matter qubits using coherent states of light.
Recent work on the subject relies on the use of single photonic qubits in the measurement process. These
schemes have a low initial success probability and low detector efficiencies cause a serious blowup in resources.
In contrast, our approach uses continuous variables and highly efficient measurements. We present a two-qubit
scheme, with a simple homodyne measurement system yielding an entangling operation with success probability
1/2. Then we extend this to a three-qubit interaction, increasing this probability to 3/4. We discuss the important
issues of the overhead cost and the time scaling, showing how these can be vastly improved with access to this
new probability range.
PACS numbers: 03.67.Lx, 03.67.Mn, 42.50.Dv, 32.80.-t
I. INTRODUCTION
The intriguing idea of one-way or cluster state quan-
tum computing was initially developed by Briegel and
Raussendorf [1]. They showed that a two-dimensional ar-
ray of qubits, entangled in a particular way (through Condi-
tional Phase gates), combined with single qubit operations,
feed forward and measurements are sufficient for universal
quantum computation. All the required interactions are al-
ready contained inside the system, and the computation pro-
ceeds through a series of local measurements (with classical
feed forward), efficiently simulating quantum circuits. In ef-
fect, the logical gates are prepared off-line and imprinted onto
the qubits as they are transmitted through the cluster.
This approach was quickly applied [2, 3, 4] to linear op-
tics quantum computing [5, 6] (and references therein), both
having been experimentally demonstrated [7, 8]. It pushes the
problem with the probabilistic nature of 2-qubit gates into the
off-line preparation of the cluster [2, 3]. In this context it was
shown that simple parity gates are sufficient for building the
required states. These schemes are then bounded by the single
beam-splitter success probability of 1/2 and in fact this initial
probability is far reduced when the single photon detection in-
efficiencies are taken into account. Supplementing the linear
optical approaches with weak nonlinearities[9, 10, 11, 12, 13]
allows for the construction of parity gates with significantly
higher success probabilities (near unity in some cases). A core
issue however with photonic qubits is their ‘flying’ nature and
the storage requirements this mandates.
A natural way around this issue is to move to solid state
or condensed matter qubits and use single photons for com-
munication between them. Many proposals make use of sin-
gle photons to effectively mediate interactions between mat-
ter qubits [14, 15, 16, 17, 18, 19]. Having interacted with
them, the photons then interact with each other in a linear op-
∗Electronic address: seblouis@nii.ac.jp
tical setup before being measured, thus projecting the matter
qubits into the required state without them having interacted
directly. It has been shown that entanglement and logical op-
erations can be generated in this way. The next step was to
use these probabilistic entangling schemes to prepare cluster
states of matter qubits [20, 21] using techniques like double-
heralding or repeat-until-success. However the schemes are
generally limited by the detection of the single photons (more
than one in some cases). This can severely limit the probabil-
ity of realizing the entangling operation and hence the creation
of the cluster state. An alternative is available and this is what
we will describe in this letter. Instead of using single pho-
tons, we can use coherent states of light (similar to the weak
nonlinearity approach). Homodyne measurements on coher-
ent light fields can be made much more efficient than single
photon detection and so we can achieve entangling operations
with a probability greater than 1/2. In this paper we will show
how this and other factors make continuous variables a very
powerful tool for growing matter qubit based cluster states.
II. GATES
There are quite a number of well studied systems where
one has a natural interaction between the matter qubit and the
electromagnetic field. These include atoms (real and artifical)
in CQED (both at the optical and telecom wavelengths) [22],
NV-centers in diamond [23], quantum dots with a single ex-
cess electron [24], trapped ions [25] and SQUIDs [26] to name
only a few. All these systems are likely to be suitable candi-
dates for what we describe below but to illustrate the details a
little more clearly let us consider a lambda based CQED sys-
tem. One could use cesium atoms or an NV-diamond center
embedded in the cavity. Both of these systems operate in the
optical frequency range and so are well matched to efficient
homodyne measurements. The interaction between the coher-
ent field mode and our matter qubit can generally be described
by the Jaynes-Cummings interaction h̄g(σ−a† + σ+a) and in
the dispersive limit (large detunings) one obtains an effective
http://arxiv.org/abs/0704.1931v1
mailto:seblouis@nii.ac.jp
interaction Hamiltonian of the form [27, 28]:
Hint = h̄χσza
†a. (1)
where a (a†) refers to the annihilation (creation) operator of
an electromagnetic field mode in a cavity and the matter qubit
is described using the conventional Pauli operators, with the
computational basis being given by the eigenstates of σz ≡
|0〉〈0| − |1〉〈1|, with |0〉 ≡ | ↑z〉 and |1〉 ≡ | ↓z〉. The atom-
light coupling strength is determined via the parameter χ =
g2/∆, where 2g is the vacuum Rabi splitting for the dipole
transition and ∆ is the detuning between the dipole transition
and the cavity field. The interaction Hint applied for a time
t generates a conditional phase-rotation ±θ (with θ = χt) on
the field mode dependent upon the state of the matter qubit.
We call this a conditional rotation and it is very similar to the
cross-Kerr interaction between photons. This time dependent
interaction implicitly requires a pulsed probe.
Now the interaction in (1) forms the basis for an entangling
operation. A two-qubit gate has been proposed [13] based on
controlled bus rotations and a subsequent measurement. The
probe field coherent state |α〉 interacts with both qubits, so
an initial state of the system |Ψi〉 = 12 (|00〉 + |01〉 + |10〉 +
|11〉)|α〉 evolves to
|Ψf 〉 =
(|00〉|αe2iθ〉+ |11〉|αe−2iθ〉)
(|01〉+ |10〉)|α〉. (2)
Here we quickly observe that the probe field has evolved
into three potentially distinct states and appropriate measure-
ments can project our two qubits into a number of interest-
ing states. At this stage we can choose from different types
of measurements on the probe beam. The first and simplest
option we have is to perform a homodyne measurement of
some field quadrature X(φ) = (a†eiφ + ae−iφ) which for
a sufficiently strong local oscillator (compared to the signal
strength) implements a projective measurement |x(φ)〉〈x(φ)|
on the probe state [29]. The key advantages with homodyne
measurement, at least in the optical regime are that it is highly
efficient (99% plus [30]) and is a standard tool of continuous
variable experimentalists. The simplest homodyne measure-
ment to perform is the momentum (P = X(π/2)) quadra-
ture. In this case the measurement probability distribution
has three peaks with the overlap error between them given
by Perr =
erfc(α sin θ/
2). As long as αθ ∼ π this over-
lap error is small (< 10−3) and the peaks are well separated.
If our P quadrature measurement projects us onto the cen-
tral peak |α〉, our two matter qubits are conditioned into the
entangled state (|01〉 + |10〉)/
2. This occurs with a proba-
bility of 1/2. Detecting either of the other two side peaks will
project the qubits to the known product states |00〉 or |11〉.
The probability of entangling the two qubits is interesting in
that we have already reached the limits of conventional linear
optical implementations. When realistic detector efficiencies
(η ∼ 70%) are taken into account, their optimal success prob-
ability of 1/2 decreases dramatically (proportional to η or η2
depending on the implementation) and so the probability of
the operation succeeding is now significantly less than 1/2. In
contrast homodyne measurements are highly efficient and so
our success probability will remain very close to 1/2. This
limit may be fundamental to the linear optical schemes but in
our case we can exceed it by changing the nature of our mea-
surement. In principle we could achieve a near deterministic
gate if we measured the the position quadrature (X = X(0)),
however the requirements to ensure the distinguishability of
the probe beam states are much more severe. We could also in
principle use a photon number measurement after displacing
the probe beam [13], but we would fall back into the issues
affecting the linear optical schemes. By restricting ourselves
to P = X(π/2) quadrature measurements and single interac-
tions between the qubits and the probe, we are opting for the
most robust weak-nonlinear approach so far proposed.
Within the same framework of conditional rotations and
P measurements, one can envisage three qubits interact-
ing with a single probe beam. GHZ states are for in-
stance one particularly useful state [3]. One way of pro-
jecting the qubits onto GHZ-type states is to vary the
strength of the interactions between the qubits and the probe
beam [12]. Let us represent a rotation of the coherent
probe beam by R(θσz) = exp(iθa
†aσz). The sequence
R(θσz1)R(θσz2)R(−2θσz3)|α〉 which we depict in Fig (1)
will give the optimal paths and end points in phase space.
The peak centered on the origin will then correspond to the
Qubit 1
Qubit 2
Qubit 3
Probe
(b)(a) P
011+101
000+111
010+100
FIG. 1: Schematic diagram (a) of a three qubit entangling operation.
In (b) the possible probe trajectories caused by the three conditional
rotations. There are five different end-states. Upon measurement,
three of these will project the qubits to entangled states of interest.
GHZ state (|000〉 + |111〉)
2 (after being detected). This
will happen with a probability of 1/4 (all qubits started in per-
fect superpositions). Next the two peaks having been rotated
through ±2θ will correspond to the qubit states (|01〉1,2 +
|10〉1,2)|1〉3/
2 and (|01〉1,2 + |10〉1,2)|0〉3/
2 respectively.
Now in both of these possible outcomes we obtain the same
Bell state on qubits 1 and 2, disentangled from qubit 3. So
overall we obtain a GHZ state with probability of 1/4 and a
Bell state with probability of 1/2 (on two qubits of our choice),
heralded by the probe beam P quadrature measurement out-
come. The other two outcomes project the qubits to known
product states. Consequently, if all we want to do is entangle
a pair of qubits, we can now do this with a probability of 3/4.
This method can be extended to larger numbers of qubits,
but the 3-qubit case minimizes the ratio of operation time over
success probability. We shall use this result in the remainder
of the paper, observing how current work on the generation of
cluster states is simply inadequate for probabilities exceeding
1/2. Until now strategies have been said to be scalable if the
resources don’t scale exponentially with the size of the cluster
(in general they will scale sub-exponentially). This is a purely
theoretical notion which bares little relation to the practical
scalability we obtain in our approach.
We stress that although the 3-qubit operation is a proba-
bilistic entangling operation with different outcomes, these
outcomes are heralded by the measurement of the bus and so
the operation is a very useful entangling primitive for the con-
struction of cluster states. For example, applying it to join
two sections of cluster with a third ancillary qubit works with
probability 3/4, giving (heralded) outcomes of joined clusters
with a new dangling bond (probability 1/2), or joined clusters
and two new dangling bonds (probability 1/4). Applying the
operation to join three sections of cluster gives (heralded) out-
comes of two sections joined and a new dangling bond (prob-
ability 1/2), or all three sections joined and two new dangling
bonds (probability 1/4). All these outcomes contribute to clus-
ter state construction.
III. SCALING
Now let’s turn to the issue of building up linear cluster states
(chains). In order to efficiently grow a chain with probabilis-
tic gates, one needs to first inefficiently build small chains ex-
ceeding a critical length Lc = 1+2(1−p)/p and then try join-
ing them to the main one. This critical length varies between
different entangling operations. If an actual conditional phase
gate can be immediately implemented, then Lc = 2(1− p)/p
for example. Or if this logical gate requires the qubits from
the cluster to interact directly (non-distributive approach) then
Lc = 4(1− p)/p [31]. Starting from this, and adopting a ‘di-
vide and conquer’ approach to building these minimal chains,
scaling relations are obtained for the average number of entan-
gling operations required and the average time taken, to build
a chain of length L. Using our 2-qubit gate (Lc = 3) and these
scaling relations we obtain N [L] = 12L− 38. This is already
the limiting scenario for simple single photon applications. In
the repeat-until-success method [21], for a failure probability
of 0.6 (and equal success and insurance probabilities, on all
results), the scaling is N [L] = 185L− 1115 and for a failure
probability of 0.4 it becomes N [L] ≃ 16.6L − 47.7. Now
if we switch to our 3-qubit gate, then Lc < 2 and our mini-
mal chain is now simply a 2-qubit cluster (locally equivalent
to a Bell state) yielding N [L] = 8L − 44/3. This is a vast
improvement over previous proposals.
For the two-qubit entangling gate, we essentially stand at
the same point as the photonic cluster state approaches. Opti-
mizing the resources boils down to finding the optimal strate-
gies in combining elements of cluster states. This is a very
complex task, which Gross et al. [32] analyzed in great detail.
For higher probabilities however, this critical length insur-
ing average growth is no longer existent. All previously de-
rived strategies become trivial within this probability range.
Additional scalable approaches such as sequential adding are
at hand and we shall go over the obvious ones. From pre-
vious works on generating cluster states [20, 31], we know
that the simplest way to grow short chains with probabilistic
gates is through a ‘divide and conquer’ approach. It also turns
out to be much quicker than a sequential adding, as we allow
for many gates to operate in parallel. This technique links up
chains of equal length on each round, and discards the chains
which failed to do so.
In the context of higher success probabilities this approach
can be extended to growing large chains in the aim of saving
time. The corresponding average number of entangling oper-
ations becomes:
Ndc[L] =
(2/p)log2(L−1) − 1
. (3)
From the initial strategy we reach a value linear in L:
N [L] = (2/p)
L− 1− 2(1− p)/p
1− 2(1− p)/p
− 1/p , (4)
and a sequential adding yields:
Nseq[L] = (L − 1)/(2p− 1) . (5)
Obviously the latter represents a considerable saving, as can
be verified in Fig. (2). Though the divide and conquer method
doesn’t scale linearly, up till lengths of 250 qubits, it requires
less entangling operations than the initial scheme (which in
fact is a full recycling approach). This is due to the fact that
the probabilities we are dealing with are significantly higher
than in previous proposals, which were undertaken in two
steps, the building of minimal elements and then their merg-
ing, in order to be scalable. If we look at the qubit resources
however, the less recycling we do, the more qubits we waste
in the process. But as the success probability of the gate
increases, the recycling strategies all converge with the no-
recycling strategy (in terms of qubit resources), this being par-
ticularly noticeable for success probabilities higher than 1/2.
We can also compare the time scaling of these various
strategies, in units of time t corresponding to a single mea-
surement. For the complete divide and conquer scheme we
simply have:
Tdc[L] = t (1 + log2(L− 1)) . (6)
and for the initial scheme:
T [L] = (t/p)
1 + log2
L− Lc
L0 − Lc
. (7)
For the sequential adding, the cumulative time obeys TL+1 =
TL + t/p, and the general form for T becomes:
Tseq[L] = t(L− 1)/p . (8)
The first two approaches have a logarithmic dependence on
the length L, however Tdc is significantly lower as might have
sequential
Pf=0.6
Pf=0.4
T[L](a) (b)
100 200 300 400 302010
sequential
D&C one 
EO per 
round
initial
scheme
initial
scheme
FIG. 2: a) Comparison of entangling operation requirements for
chain production. We achieve much lower scalings in comparison
with those obtained through the repeat-until-success (RUS) scheme
(Pf being the failure probability). b) Time scaling for the different
strategies. The three-qubit gate is used in both plots.
been expected (see Fig. (2)). Overall we see that there is a
clear advantage to divide the task up and to run parallel en-
tangling operations. The linear time scaling for the sequential
method is due to the fact that operations cannot be undertaken
in parallel during the growth. If we didn’t have access to
simultaneous entangling operations, the time scaling for the
divide and conquer methods would be equivalent to Ndc[L]
which is sub-exponential. One needs to keep in mind that by
adopting a sequential method, the whole procedure is simpli-
fied considerably and would be more accessible to physical
implementations.
IV. DISCUSSION
The cluster state comprises of active regions in which it is
being built or measured in the computation (both can be un-
dertaken simultaneously) and regions in which the qubits are
simply waiting. Now this waiting can be minimized in the
building itself, through the appropriate protocols, and in the
measurement process. That is, the cluster can be built only
a few layers in advance, so that the qubits have less waiting
to do, between the building and the actual measurement. In
any case, there will be some waiting. Therefore the lowest
decoherence realization would be preferred, but it may not
be the easiest to manipulate. Thus we may envisage hav-
ing two different physical realizations constituting the cluster
state. For example, we could use single electron spins ini-
tially in building the cluster. Once the links are made between
one site and its nearest neighbors, the qubit could be switched
into a nuclear spin state which has a significantly longer co-
herence time, via a swap operation or some other coherent
write and read actions. Most of the waiting would be done
in the long-lived state, before being swapped again for the
readout [33, 34]. This follows the same scenario as using a
second physical system to mediate the interaction and make
the measurements, in distributed quantum computing. In the
present proposal, we use a continuous variable bus and homo-
dyne measurements to generate the links. This physical sys-
tem shows itself to be very efficient in this application. Then,
for example, electron spins or superconducting charge qubits
could be the matter realization interacting with the bus and
serving for the final readout. These systems provide the use-
ful static aspect required, they interact well with the mediating
bus and ensure good single qubit measurements. Finally a low
decoherence realization such as nuclear spin could be envis-
aged, mainly as a storage medium. The swapping or write
and read procedure should have a high fidelity for this storage
to be beneficial. On the whole, we see that optimization will
depend directly on the physical realization(s) we have cho-
sen to work with. For systems with long dephasing times we
would give priority to sequential adding approaches, as we
have some freedom in the time scaling and thus we can make
significant savings in resources. But for realizations with short
dephasing times, we would probably want to divide the task
up and run operations in parallel, in order to accelerate the fab-
rication of the cluster state, at the expense of extra resources.
V. CONCLUSION
In summary we have shown how the concept of the quan-
tum bus can be adapted to efficiently generating cluster states
of matter qubits. We can straightforwardly gain access to
entangling probabilities higher than 1/2, removing the need
to break up the building process into inefficient and efficient
parts. This opens up a new class of strategies, for which the
resource consumption and the time scaling are consequently
vastly improved. Clearly, within this class, detailed strategies
can be envisaged and they will depend on the chosen physical
realization and the levels of decoherence present.
VI. ACKNOWLEDGEMENTS
We would like to thank R. van Meter and J. Eisert for help-
ful discussions. This work was supported in part by MEXT in
Japan and the EU project QAP.
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