Linear-optical Quantum Computing
Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern
Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
Outline
•
Why Linear-optical quantum computing?
•
•
•
•
•
What is quantum computation?
Photonic qubits
Why linear-optical quantum computing? (advantages and
disadvantages)
Manipulating photonic qubits with linear optics
Linear optical logic gates
•
•
A theoretical idea
Experimental realization
•
•
Cluster states vs one-way quantum computing
Linear-optical one-way quantum computing
•
single photons + linear optics + measurements
An interface of photons and atomic ensembles: Quantum memory for
polarization qubits
•
Summary and outlook
Why linear-optical quantum computing?
What is quantum computation, first of all?
A quantum computation can be considered as a physical process that transforms an input
quantum state into an output state and as such, it respects quantum laws. The "information
flow" in quantum computing process is carried by qubits which are subject to a designed unitary
evolution.

U 12n
 ' 12n
out
…
12n
…

in
To perform general transformations relies on the ability to engineer arbitrary interactions
between the qubits. Fortunately, this task can be greatly simplified by the following
powerful theorem for universal quantum computation (Barenco et al., 1995; Lloyd, 1995):
Any unitary transformation of an N-qubit system can be
implemented with single-qubit operations and quantum
controlled-NOT (CNOT) gates or equivalent two-qubit gates.
Why linear-optical quantum computing?
Qubits: the building block of a quantum computer
Coherent superposition
Entanglement
 0  1

1
0 1 0 2 111 2
2
“0” and “1”

Each individual qubit carries no information!
The fact that qubits can be in coherent superposition and entangled states gives
the extraordinary power to a quantum computer to outperform its classical
counterpart.
Any two-level quantum system with an SU(2) symmetry represents a qubit. There
are many different ways of implementing qubits. Examples include electron’s spin,
atoms with two relevant energy levels, superconducting Josephson junctions, and
photon’s polarization or spatial modes, etc.
Photonic realization of qubits is one of the most promising candidates not only
for quantum communication but also for quantum computing.
Why linear-optical quantum computing?
An individual photon possesses a few physical degrees of freedom (DoF), each of which can
in principle be used to carry information under appropriate experimental arrangements.
These DoF of photons include the internal polarization, orbital angular momentum, spatial
mode, emission time and frequency, etc. Two most important realizations of photonic qubits:
Polarization encoding:
polarization qubits
Path encoding:
spatial qubits
 H  V
“H” and “V”
H ( V ) : horizontal (vertical) polarizati on
 a  b
a and b : two different spatial modes
“here” and “there”
Why linear-optical quantum computing?
Why photonic encoding, or why linear-optical quantum computing?
•First of all, the question is interesting in its own right.
•Quantum states of photons can be easily manipulated by simple linear-optical elements with a high
precision at about 99.9% level.
•Easy realization of any single-qubit rotations.
•Robust to the environmental noises (photons have no charge!)
•Photons are the fastest information carriers, which is important for quantum communication and
distributed quantum information processing
Challenges for linear-optical quantum computing
•Difficulty of realizing two-qubit gates for photons due to the lack of photon-photon interaction
(except for photons in certain nonlinear medium which can induce tiny optical nonlinearity)
•Difficulty of storing photons for a reasonable long time
Is linear-optical quantum computing possible, anyway?
Why linear-optical quantum computing?
Yes, in principle!
•Knill, Laflamme and Milburn (KLM), using the dual-rail encoding
(spatial qubits), has shown that nondeterministic quantum logic
operations can be performed using linear optical elements, additional
photons (ancilla) and postselection based on the output of single-photon
detectors.
•They further demonstrated that the success rate of the quantum logic
can be arbitrarily close to one by using more additional ancilla and
detectors.
•This implies that the nondeterministic quantum logic gate based on
linear optics can be used as a basic block for quantum information
protocols, even for efficient quantum computation.
For details, see E. Knill, R. Laflamme, and G. J.
Milburn, Nature 409, 46 (2001); Pieter Kok et al.,
quant-ph/0512071 (a review).
Why linear-optical quantum computing?
•The KLM scheme provides the surprising possibility.
•The scheme itself is complicated in its use of complex interferometers,
resource-consuming, …
•Yet, it is a real breakthrough and motives many subsequent study on
linear-optical quantum information protocols.
•For many of these later development in the context of quantum
computing, please refer to a nice review (Pieter Kok et al., quantph/0512071)
•For pedagogical purposes, this lecture will show you how to implement
•a linear-optical CNOT gate
•linear-optical one-way quantum computing
Some useful notions
Universal set of quantum logic gates
Graphical representations of Hadamard
and CNOT gates. Here, a+b denotes
addition modulo 2.
Some useful notions
A simple network for the Bell-state preparation and measurement
The four Bell state
The reversed quantum network (right-hand
side of the figure) can be used to implement
the so-called Bell-state measurement on the
two qubits by disentangling the Bell states.
•Currently, photonic Bell states can be
prepared via SPDC. – previous lecture
•Partial Bell-state measurement can be
done with linear optics. – previous
lecture
•The Bell-state preparation and
measurement are doable if we have
photonic CNOT (though
nondeterministic) – see below
Manipulating photonic qubits with linear optics
Linear-optical elements include beam splitter (BS), polarizing BS (PBS), halfand quarter-wave plates and phase shifters and respect the conservation of total
photon number
unitarity
50:50 beam splitter
the reflection and
transmission coefficients
beam splitter
Manipulating photonic qubits with linear optics
PBS is used to separate horizontal and
vertical polarizations: it transmits only
photon's horizontal polarization component
and reflects the vertical component.
The function of a PBS. (a) The PBS transmits horizontal, and reflects
vertical, polarization. (b) If the two photons incident onto the PBS
have identical polarization, then they will always go out along different
directions, so there will be one photon in each of the two output
modes. (c) On the other hand, if the two incident photons have
opposite polarization, then they will always go out along the same
direction, so there will be two photons in one of the two outputs and
none in the other. In essence, a PBS can thus be used as a polarization
comparer (Pan and Zeilinger, 1998; Pan et al., 2001).
Polarizing beam splitter
Linear optical logic gates
For optical quantum computing, single-qubit rotations are trivial. The
problem is how to realize the nontrivial two-qubit gates.
Solution: ancilla photons + measurements → measurement-induced
nonlinearity that is able to entangle two input photonic qubits
For scalable optical quantum computing, a crucial requirement is the
classical feed-forwardability:
•It must be in principle possible to detect when the gate has
succeeded by measurement of ancilla photons in some appropriate
states. This information can then be feed-forward in such a way as to
condition future operations on the photon modes.
•A non-destructive CNOT gate for two independent photons is
sufficient for this purpose.
T.B. Pittman, B.C. Jacobs, and J.D. Franson,
Phys. Rev. A 64, 062311 (2001); Z. Zhao et al.,
Phys. Rev. Lett. 94, 030501 (2005).
Linear optical logic gates
T.B. Pittman, B.C. Jacobs, and J.D. Franson,
Phys. Rev. A 64, 062311 (2001); Z. Zhao et al.,
Phys. Rev. Lett. 94, 030501 (2005).
Linear optical logic gates
The calculation:

25

1
[ H ' 3' H
4
+ H ' 3' V
+V'
+V'
+
3'
3'
3

2
H
V
34
=( H
4'
( H
2'
4'
( H
2'
4'
( H
2'
4'
The cases in which there is one
and only one photon in each of
the output modes:
( H
2'
2
V
H
V
H
H 5 + H
5'
+ H
2'
5'
+ H
2'
5'
+ H
2'
5'
+ H
2'
2
H
V
H
V
V
5
+ V
5'
 V
2'
5'
 V
2'
5'
+ V
2'
5'
+ V
2'
H 5 + V
2
H
V
H
V
5'
 V
2
2'
5'
 V
2'
5'
+ V
2'
5'
+ V
2'
V
5
V
V
H
5'

xˆ5' zˆ5'CNOT2'5' 
)
zˆ5'CNOT2'5' 
2'5'
)
xˆ5'CNOT2'5' 
2'5'
)
CNOT2'5' 
5'
5'
34
)
5'
H
)  
2'5'
2'5'
non-one and only one
T.B. Pittman, B.C. Jacobs, and J.D. Franson,
Phys. Rev. A 64, 062311 (2001); Z. Zhao et al.,
Phys. Rev. Lett. 94, 030501 (2005).
Linear optical logic gates
Linear optical logic gates
(A) The experimental fidelity of
achieving the CNOT logic table is
estimated to be 0.78±0.05.
(B) A measured interference fringe
with a visibility of 0.58±0.09,
which is consistent with the
prediction of the interference
fringe for   .
Cluster states vs one-way quantum computing
A significant step in quantum computing is the discovery of "one-way quantum
computing" [R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86, 5188 (2001); R.
Raussendorf, D.E. Browne, and H.J. Briegel, Phys. Rev. A 68, 022312 (2003)], which is
based on the preparation of highly entangled multi-qubit states, the so-called "cluster
states" [H.J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86, 910 (2001)] and on simple
adaptive one-qubit measurements. Here we will focus on the following issues:
•What is the cluster states, and how to prepare them?
•Why one-way quantum computing is a universal quantum computing?
•The computational model of one-way quantum computing.
•A linear optical implementation of one-way quantum computing.
Cluster states vs one-way quantum computing
What is the cluster states?
Cluster states vs one-way quantum computing
What is the cluster states?
Cluster states vs one-way quantum computing
How to prepare cluster states?
tunable Ising
interaction
Cluster states vs one-way quantum computing
Examples of cluster states: The cluster states on a chain of
2, 3, and 4 qubits (all with eigenvalue 0)
A Bell state
A three-qubit GreenbergerHorne-Zeilinger (GHZ) state
Not equivalent to four-qubit
GHZ states; entangled even
after destroying a single qubit
Cluster states vs one-way quantum computing
An intuitive graphical representation of cluster states and the properties of linear
cluster under single-qubit measurements
Cluster states vs one-way quantum computing
A cluster state on a two-dimensional cluster of
rectangular shape, say, is a resource that allows
for any computation that fits on the cluster.
Actually, the CNOT gate and general one-qubit
rotations, which form a universal set of
quantum logic gates, can all be implemented
via one-qubit measurements on a cluster state.
That is, one-way quantum computing is a universal
one.
For a proof, see R. Raussendorf, D.E. Browne, and H.J. Briegel,
Phys. Rev. A 68, 022312 (2003)
Cluster states vs one-way quantum computing
How an one-way quantum
computer works?
The computational model of
one-way quantum computing:
Cluster states vs one-way quantum computing
Cluster states vs one-way quantum computing
The entire quantum computation consists only of a sequence one-qubit
projective measurements in a particular pattern of measurement eigenbases and
in a particular order on this entangled state; Different algorithms require only a
different pattern of adapted single-qubit operations on a sufficiently large
cluster state.
The entanglement in the cluster state is destroyed by the one-qubit
measurements and therefore it can only be used once, hence the name “one-way
quantum computer”.
 The computational model of the one-way quantum computer makes no
reference to the concept of unitary evolution.
The information that is processed is extracted from the measurement
outcomes and is thus a purely classical quantity.
The one-way quantum computer is dramatically different from the standard one
based on sequences of unitary quantum logic gates that process qubits.
The very possibility of one-way quantum computing might well change our
understanding of the requirements for quantum computation and more generally
how we think about quantum physics.
Linear-optical one-way quantum computing
M.A. Nielsen, Phys. Rev. Lett. 93, 040503 (2004):
A deterministic entangling quantum gate may be performed using, on average, a few
hundred coherently interacting optical elements (BS, phase shifters, single photon sources,
and photodetectors with feedforward). This scheme combines ideas from the KLM scheme
and the abstract cluster-state model of quantum computation.
D.E. Browne and T. Rudolph, Phys. Rev. Lett. 95, 010501 (2005):
Cluster states may be efficiently generated from pairs of maximally polarization entangled
photons using linear optical elements.
Requirement of stable interferometry over only the coherence length of the photons.
A much greater degree of efficiency and a simpler implementation than previous
proposals.
Redundant encoding of qubits + type-I/II two-qubit fusion.
Without proving the overall efficiency of the scheme.
L.-M. Duan and R. Raussendorf, Phys. Rev. Lett. 95, 080503 (2005):
Efficient quantum computation can be constructed even if all the entangling quantum
gates only succeed with an arbitrarily small probability p.
A linear optical one-way quantum computing scheme combining the Browne-Rudolph and
the Duan-Raussendorf schemes, to be explained below. (Z.-B. Chen, B. Zhao, and J.-W. Pan,
unpublished)
Linear-optical one-way quantum computing
D.E. Browne and T. Rudolph, Phys. Rev.
Lett. 95, 010501 (2005)
Linear-optical one-way quantum computing
Redundant encoding of qubits + typeI/II two-qubit fusion.
Linear-optical one-way quantum computing
A strategy of preparing a twodimensional cluster: entangled
photon pairs are resource and
can be created, e.g., using
single photons and linear
optics.
Z.-B. Chen, B. Zhao, and J.-W. Pan,
unpublished)
Linear-optical one-way quantum computing
Linear-optical one-way quantum computing
The above strategy can finally prepare a square-lattice cluster of N qubits with
a temporal overhead scaling logarithmically with N, and with an operational
overhead (i.e., number of fusion operations) scaling as ~NlnN (Chen et al.,
unpublished).
The described protocol is a linear-optical realization of the Duan-Raussendorf
proposal (2005), but combines the advantages of the Browne-Rudolph scheme,
whose overall efficiency is thereby demonstrated.
Following the above protocol, complex multi-party quantum network can in
principle be created very efficiently with the help of quantum memory (to be
briefly explained below). This provides an exciting perspective for manipulating a
huge number of photons applicable to quantum computing.
Linear-optical one-way quantum computing
A linear-optical polarization
entangler with four single
photons: selecting the case
where there is one and only
one photon is each of the four
output modes (a,A;b,B). This
occurs with the probability of
¼. The total success
probability is 25%*50% (two
out of four Bell states can be
distinguished)=1/8
Linear-optical one-way quantum computing
So far, there is only one
experiment, by P. Walther
et al. [Nature 434, 169
(2005)], demonstrating the
basic idea of the one-way
quantum computing.
Quantum memory for polarization qubits
P. Walther et al., Nature
434, 169 (2005).
Quantum memory for polarization qubits
Let us recall
the challenges for linear-optical quantum computing
 Difficulty of realizing two-qubit gates for photons due to the lack of photon-photon
interaction
Solution: linear optics + quantum measurements → nondeterministic two-qubit gates which
are sufficient for efficient quantum computing √
Difficulty of storing photons for a reasonable long time
Solution: an interface of photons and atomic ensembles, namely, quantum memory for
polarization qubits with atomic ensembles, which is relatively easy to implement under current
technology
Why quantum memory?
Photons travel at the speed of light!!!
Photonic two-qubit gates are probabilistic. With memory, we can keep applying the
probabilistic gates until we succeed and do certain operations in parallel. Otherwise,
optical quantum computing cannot be efficient.
Quantum memory for polarization qubits
Atomic-ensemble-based quantum memory is used to
transfer the photonic states to the excitation in atomic
internal states so that it can be stored, and after the storage
of a programmable time, it should be possible to read out
the excitation to photons without change of its quantum
state.
M.D. Lukin et al., Phys. Rev. Lett. 84, 4232 (2000); M.
Fleischhauer and M.D. Lukin, Phys. Rev. Lett. 84, 5094
(2000).
Quantum memory for polarization qubits
purely photon-like state (release of the single photon
purely atom-like state (storageof the single photon)
Quantum memory for polarization qubits
Summary and Outlook

Photons can be a strong candidate not only for long-distance quantum
communication, but also for large-scale quantum computing.

Given several atomic-ensemble-based technologies, linear-optical quantum
information processing has a brilliant future. The interface of photons and atomic
ensembles offers the fascinating playground for integrating the linear-optical and
atomic-ensemble-based techniques for long-distance quantum communication and
scalable optical quantum computing.

This would open up an exciting perspective for manipulating quantum states of a
huge number of photons, with unexpected applications in future.

…
Thanks!