wesley7137/quantum · Datasets at Hugging Face

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states in equivalent systems [14, 18, 19]. Any of these methods would be suitable for the
creation of linear cluster states in this proposal. The attractive feature of the method
Quantum Computing Using Crossed Atomic Beams 5
Figure 2. Thebasicexperimentalapparatus,atomictransitionsandtimingpulsesfor
the creation of linear and two dimensional cluster states. The basic two dimensional
entanglement elements consist of two sources of linear cluster states (beams 1 and 2)
incident on a collision cavity. The atoms are timed to collide in this cavity thereby
turning the linear entangled states into a two dimensional cluster state. The single
qubitrotationzonesR1aandR1bproducetheHadamardtransformationontheinput
qubits as required. The single qubit rotation zones R2a and R2b can be controlled to
produce any arbitrary rotation triggered by the state selective detectors, D on either
transition. The lower right hand side shows the sequence of pulses for producing the
atomic sequence Hadamard transformationH1 on alternate atoms. The zone R1a can
be used to introduce single qubit rotations to atoms in the state.
we outline here is its passive nature, the low experimental overhead and the high degree
of integration that we can achieve.
The experimental arrangement in figure 2 represents a subset of the larger device
of figure 1, encompassing just two atomic beams. The three-level atomic structure and
the single qubit rotation sequence for creating the atomic beams is shown. Atoms from
the two beams collide in the central cavity, and this collision can be used to produce
entanglement between pairs of atoms from perpendicular beams.
To perform the cross beam entanglement, the cavity is used in a dispersive regime
bydetuning thecavity fromresonanceby anamount δ fromthe e f transition, but
\| i ↔ \| i
remaining far off-resonance from the g f and e g transitions. The detuning
\| i ↔ \| i \| i ↔ \| i
is chosen so that δ Ω, where Ω is the coupling strength between the e f
≫ \| i ↔ \| i
transition and the cavity mode. Under these circumstances, the following interactions
occur:
e−iλt[cosλt
e f e f isinλt f e ] (1)
1 2 1 2 1 2
\| i \| i → \| i \| i − \| i \| i
e−iλt
e g e g (2)
1 2 1 2
\| i \| i → \| i \| i
f f f f (3)
1 2 1 2
\| i \| i → \| i \| i
f g f g (4)
1 2 1 2
\| i \| i → \| i \| i
Quantum Computing Using Crossed Atomic Beams 6
(5)
2
Where λ = Ω /δ. We choose the basis states of the operationto be e , f for the
1 1
\| i \| i
first atom and f , g for the second. When λt = π the interaction produces a logical
2 2
\| i \| i
CPhase gate [7, 43], and this is the basic building-block operation for cluster state
creation. The interaction has been experimentally demonstrated with good fidelity by
Osnaghi et al [37], and the theoretical fidelity of the gate remains high in the presence
of small timing errors, achieving 99% fidelity for a 1% difference in arrival time relative
to interaction time [43]. The interactions occur between one atom prepared in some
superposition of the states e and f and a second atom in the f and g states. The
\| i \| i \| i \| i
atomic sources produce atoms in each basis alternately. Therefore if source B is delayed
by one atom relative to source A, then when any two atoms meet in the collision cavity
the interaction will occur between atoms prepared in a superposition of the correct basis
states. The arrangement of the entanglement cavities in the full-scale device (figure 1)
allows the generation of linear cluster states by collisional entanglement. If collisions
occur in the highlighted cavities, a linear cluster state is formed, the odd-numbered
atoms travelling perpendicular to the even-numbered atoms. Such collisions can be
guaranteed if thewhole device runs on a clocked scheme, with the atomicsources farther
from the origin being delayed by a small amount relative to those nearer, to allow for
the time of flight of atoms between the entanglement cavities. The size of this linear
cluster state is limited only by the number of atomic beams in the apparatus, and states
of this size can be produced in every clock cycle.
To gain full benefit from the quantum cluster computing model, a state with two-
dimensional entanglement is required. To produce a two dimensional state from the
above architecture, a memory of some sort must be introduced such that successive
atoms from each source are entangled with each other. Tripartite entanglement has
been experimentally produced in a cavity-QED system [38], and this method integrates
well with the collisional entanglement scheme.
The generation of entanglement between clock cycles (and therefore the continuous
generation of a 2D cluster state) is performed using the classical microwave fields and
micromaser cavities in each beamline, prior to the collisional entanglement. An initial
atom A1, prepared in the state f interacts with the cavity, which is resonant on
\| i
the \|e \|f transition, producing the state ψ1 = (1/√2)( \|f \|0 + \|e \|1 iC) -
i ↔ i i1 iC i1
an entangled state of the atom and the cavity photon number. The second atom
A2 to interact with the cavity is prepared in the state (1/√2)( f + g ) by the
2 2
\| i \| i
classical fields R1a and R1b. The interaction time with the cavity is chosen to perform
a CPhase gate between the atomic state and the cavity photon number. For this
choice of initial state, the gate is equivalent to a non-demolition measurement of the
cavity photon number, where the phase of the superposition is flipped in the presence
of a photon, and remains unaffected otherwise. The resulting A1-A2-Cavity state is
ψ2 = (1/2)[ \|f \|f \|0 + \|f \|g \|0 + \|e \|f \|1 \|e \|g \|1 iC]. The state of
i1 i2 iC i1 i2 iC i1 i2 iC − i1 i2
the cavity can be copied onto a third atom, prepared in f , which interacts with
\| i
the cavity so as to completely absorb a photon, if one is present in the cavity. This
Quantum Computing Using Crossed Atomic Beams 7

states in equivalent systems [14, 18, 19]. Any of these methods would be suitable for the

creation of linear cluster states in this proposal. The attractive feature of the method

Quantum Computing Using Crossed Atomic Beams 5

Figure 2. Thebasicexperimentalapparatus,atomictransitionsandtimingpulsesfor

the creation of linear and two dimensional cluster states. The basic two dimensional

entanglement elements consist of two sources of linear cluster states (beams 1 and 2)

incident on a collision cavity. The atoms are timed to collide in this cavity thereby

turning the linear entangled states into a two dimensional cluster state. The single

qubitrotationzonesR1aandR1bproducetheHadamardtransformationontheinput

qubits as required. The single qubit rotation zones R2a and R2b can be controlled to

produce any arbitrary rotation triggered by the state selective detectors, D on either

transition. The lower right hand side shows the sequence of pulses for producing the

atomic sequence Hadamard transformationH1 on alternate atoms. The zone R1a can

be used to introduce single qubit rotations to atoms in the state.

we outline here is its passive nature, the low experimental overhead and the high degree

of integration that we can achieve.

The experimental arrangement in figure 2 represents a subset of the larger device

of figure 1, encompassing just two atomic beams. The three-level atomic structure and

the single qubit rotation sequence for creating the atomic beams is shown. Atoms from

the two beams collide in the central cavity, and this collision can be used to produce

entanglement between pairs of atoms from perpendicular beams.

To perform the cross beam entanglement, the cavity is used in a dispersive regime

bydetuning thecavity fromresonanceby anamount δ fromthe e f transition, but

| i ↔ | i

remaining far off-resonance from the g f and e g transitions. The detuning

| i ↔ | i | i ↔ | i

is chosen so that δ Ω, where Ω is the coupling strength between the e f

≫ | i ↔ | i

transition and the cavity mode. Under these circumstances, the following interactions

occur:

e−iλt[cosλt

e f e f isinλt f e ] (1)

1 2 1 2 1 2

| i | i → | i | i − | i | i

e−iλt

e g e g (2)

1 2 1 2

| i | i → | i | i

f f f f (3)

1 2 1 2

| i | i → | i | i

f g f g (4)

1 2 1 2

| i | i → | i | i

Quantum Computing Using Crossed Atomic Beams 6

(5)

2

Where λ = Ω /δ. We choose the basis states of the operationto be e , f for the

1 1

| i | i

first atom and f , g for the second. When λt = π the interaction produces a logical

2 2

| i | i

CPhase gate [7, 43], and this is the basic building-block operation for cluster state

creation. The interaction has been experimentally demonstrated with good fidelity by

Osnaghi et al [37], and the theoretical fidelity of the gate remains high in the presence

of small timing errors, achieving 99% fidelity for a 1% difference in arrival time relative

to interaction time [43]. The interactions occur between one atom prepared in some

superposition of the states e and f and a second atom in the f and g states. The

| i | i | i | i

atomic sources produce atoms in each basis alternately. Therefore if source B is delayed

by one atom relative to source A, then when any two atoms meet in the collision cavity

the interaction will occur between atoms prepared in a superposition of the correct basis

states. The arrangement of the entanglement cavities in the full-scale device (figure 1)

allows the generation of linear cluster states by collisional entanglement. If collisions

occur in the highlighted cavities, a linear cluster state is formed, the odd-numbered

atoms travelling perpendicular to the even-numbered atoms. Such collisions can be

guaranteed if thewhole device runs on a clocked scheme, with the atomicsources farther

from the origin being delayed by a small amount relative to those nearer, to allow for

the time of flight of atoms between the entanglement cavities. The size of this linear

cluster state is limited only by the number of atomic beams in the apparatus, and states

of this size can be produced in every clock cycle.

To gain full benefit from the quantum cluster computing model, a state with two-

dimensional entanglement is required. To produce a two dimensional state from the

above architecture, a memory of some sort must be introduced such that successive

atoms from each source are entangled with each other. Tripartite entanglement has

been experimentally produced in a cavity-QED system [38], and this method integrates

well with the collisional entanglement scheme.

The generation of entanglement between clock cycles (and therefore the continuous

generation of a 2D cluster state) is performed using the classical microwave fields and

micromaser cavities in each beamline, prior to the collisional entanglement. An initial

atom A1, prepared in the state f interacts with the cavity, which is resonant on

| i

the |e |f transition, producing the state ψ1 = (1/√2)( |f |0 + |e |1 iC) -

i ↔ i i1 iC i1

an entangled state of the atom and the cavity photon number. The second atom

A2 to interact with the cavity is prepared in the state (1/√2)( f + g ) by the

2 2

| i | i

classical fields R1a and R1b. The interaction time with the cavity is chosen to perform

a CPhase gate between the atomic state and the cavity photon number. For this

choice of initial state, the gate is equivalent to a non-demolition measurement of the

cavity photon number, where the phase of the superposition is flipped in the presence

of a photon, and remains unaffected otherwise. The resulting A1-A2-Cavity state is

ψ2 = (1/2)[ |f |f |0 + |f |g |0 + |e |f |1 |e |g |1 iC]. The state of

i1 i2 iC i1 i2 iC i1 i2 iC − i1 i2

the cavity can be copied onto a third atom, prepared in f , which interacts with

| i

the cavity so as to completely absorb a photon, if one is present in the cavity. This

Quantum Computing Using Crossed Atomic Beams 7