The KLM scheme or KLM protocol is an implementation of
linear optical quantum computing
Linear optical quantum computing or linear optics quantum computation (LOQC) is a paradigm of quantum computation, allowing (under certain conditions, described below) universal quantum computation. LOQC uses photons as information carriers, main ...
(LOQC), developed in 2000 by
Emanuel Knill,
Raymond Laflamme and
Gerard J. Milburn. This protocol makes it possible to create universal
quantum computers
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
solely with
linear optical tools.
The KLM protocol uses linear optical elements,
single-photon source Single-photon sources are light sources that emit light as single particles or photons. These sources are distinct from coherent light sources (lasers) and thermal light sources such as incandescent light bulbs. The Heisenberg uncertainty principl ...
s and photon detectors as resources to construct a quantum computation scheme involving only
ancilla resources,
quantum teleportations and
error corrections.
Overview
The KLM scheme induces an effective interaction between
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s by making projective measurements with
photodetector
Photodetectors, also called photosensors, are sensors of light or other electromagnetic radiation. There is a wide variety of photodetectors which may be classified by mechanism of detection, such as photoelectric or photochemical effects, or ...
s, which falls into the category of non-deterministic
quantum computation. It is based on a non-linear sign shift between two qubits that uses two ancilla photons and post-selection. It is also based on the demonstrations that the probability of success of the quantum gates can be made close to one by using
entangled state
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
s prepared non-deterministically and
quantum teleportation with single-qubit operations.
Without a high enough success rate of a single quantum gate unit, it may require an exponential amount of computing resources. The KLM scheme is based on the fact that proper quantum coding can reduce the resources for obtaining accurately encoded qubits efficiently with respect to the accuracy achieved, and can make LOQC fault-tolerant for
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
loss, detector inefficiency and phase
decoherence
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wa ...
. LOQC can be robustly implemented through the KLM scheme with a low enough resource requirement to suggest practical scalability, making it as promising a technology for
quantum information processing
Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum effects in p ...
as other known implementations.
Elements of the KLM scheme
This section discusses the implementations of the elements of LOQC in the KLM scheme.
Qubits and modes
To avoid losing generality, the discussion below does not limit itself to a particular instance of mode representation. A state written as
means a state with zero
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s in mode
(could be the "vertical" polarization channel) and one photon in the mode
(could be the "horizontal" polarization channel).
In the KLM protocol, each of the photons is usually in one of two modes, and the modes are different between the photons (the possibility that a mode is occupied by more than one photon is zero). This is not the case only during implementations of
controlled quantum gates such as CNOT. When the state of the system is as described, the photons can be distinguished, since they are in different modes, and therefore a qubit state can be represented using a single photon in two modes, vertical (V) and horizontal (H): for example,
and
. It is common to refer to the states defined via occupation of modes as
Fock state
In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an im ...
s.
Such notations are useful in
quantum computing,
quantum communication
Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum effects in p ...
and
quantum cryptography
Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. The best known example of quantum cryptography is quantum key distribution which offers an information-theoretically secure solution ...
. For example, it is very easy to consider a loss of a single
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
using these notations, simply by adding the vacuum state
containing zero photons in those two modes. As another example, when having two photons in two separated modes (e.g. two time bins or two arms of an
interferometer), it is easy to describe an
entangled state of the two photons. The
singlet state
In quantum mechanics, a singlet state usually refers to a system in which all electrons are paired. The term 'singlet' originally meant a linked set of particles whose net angular momentum is zero, that is, whose overall spin quantum number s=0. A ...
(two linked photons with overall
spin quantum number ) can be described as follows:
if
and
describe the basis states of the two separated modes, then the singlet state is
State measurement/readout
In the KLM protocol, a quantum state can be readout or measured using
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
detectors along selected modes. If a photodetector detects a photon signal in a given mode, it means the corresponding mode state is a 1-photon state before measuring. As discussed in KLM's proposal,
photon loss and detection efficiency dramatically influence the reliability of the measurement results. The corresponding failure issue and error correction methods will be described later.
A left-pointed triangle will be used in circuit diagrams to represent the state readout operator in this article.
Implementations of elementary quantum gates
Ignoring error correction and other issues, the basic principle in implementations of elementary quantum gates using only mirrors, beam splitters and phase shifters is that by using these
linear optical elements, one can construct any arbitrary 1-qubit unitary operation; in other words, those linear optical elements support a complete set of operators on any single qubit.
The unitary matrix associated with a beam splitter
is:
:
,
where
and
are determined by the
reflection amplitude and the
transmission amplitude (relationship will be given later for a simpler case). For a symmetric beam splitter, which has a phase shift
under the unitary transformation condition
and
, one can show that
:
,
which is a rotation of the single qubit state about the
-axis by
in the
Bloch sphere
In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.
Quantum mechanics is mathematically formulated i ...
.
A mirror is a special case where the reflecting rate is 1, so that the corresponding unitary operator is a
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\en ...
given by
:
.
For most cases of beam splitters used in QIP, the
incident angle .
Similarly, a phase shifter operator
associates with a unitary operator described by
, or, if written in a 2-mode format
:
,
which is equivalent to a rotation of
about the
-axis.
Since any two
rotations along orthogonal rotating axes can generate arbitrary rotations in the Bloch sphere, one can use a set of symmetric beam splitters and mirrors to realize an arbitrary
operators for QIP. The figures below are examples of implementing a
Hadamard gate
The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogon ...
and a
Pauli-X-gate (NOT gate) by using beam splitters (illustrated as rectangles connecting two sets of crossing lines with parameters
and
) and mirrors (illustrated as rectangles connecting two sets of crossing lines with parameter
).
In the above figures, a qubit is encoded using two mode channels (horizontal lines):
represents a
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
in the top mode, and
represents a photon in the bottom mode.
In the KLM scheme, qubit manipulations are realized via a series of non-deterministic operations with increasing probability of success. The first improvement to this implementation that will be discussed is the nondeterministic conditional sign flip gate.
Implementation of nondeterministic conditional sign flip gate
An important element of the KLM scheme is the conditional sign flip or nonlinear sign flip gate (NS-gate) as shown in the figure below on the right. It gives a nonlinear phase shift on one mode conditioned on two ancilla modes.
![NSGate](https://upload.wikimedia.org/wikipedia/commons/4/4b/NSGate.JPG)
In the picture on the right, the labels on the left of the bottom box indicate the modes. The output is accepted only if there is one
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
in mode 2 and zero photons in mode 3 detected, where the ancilla modes 2 and 3 are prepared as the
state. The subscript
is the phase shift of the output, and is determined by the parameters of inner optical elements chosen.
For
case, the following parameters are used:
,
,
,
,
,
, and
. For the
case, the parameters can be chosen as
,
,
,
,
,
, and
. Similarly, by changing the parameters of beam splitters and phase shifters, or by combining multiple NS gates, one can create various quantum gates. By sharing two ancilla modes, Knill invented the following controlled-Z gate (see the figure on the right) with success rate of 2/27.
![NSCZGate](https://upload.wikimedia.org/wikipedia/commons/a/a4/NSCZGate.JPG)
The advantage of using NS gates is that the output can be guaranteed conditionally processed with some success rate which can be improved to nearly 1. Using the configuration as shown in the figure above on the right, the success rate of an
NS gate is
. To further improve successful rate and solve the scalability problem, one needs to use gate teleportation, described next.
Gates teleportation and near-deterministic gates
Given the use of non-deterministic quantum gates for KLM, there may be only a very small probability
that a circuit with
gates with a single-gate success possibility of
will work perfectly by running the circuit once. Therefore, the operations must on average be repeated on the order of
times or
such systems must be run in parallel. Either way, the required time or circuit resources scale exponentially. In 1999, Gottesman and Chuang pointed out that one can prepare the probabilistic gates offline from the quantum circuit by using
quantum teleportation.
The basic idea is that each probabilistic gate is prepared offline, and the successful event signal is teleported back to the quantum circuit. An illustration of quantum teleportation is given in the figure on the right. As can be seen, the quantum state in mode 1 is teleported to mode 3 through a
Bell measurement and an entangled resource
Bell state
The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form o ...
, where the state 1 may be regarded as prepared offline.
The resource
Bell state
The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form o ...
can be generated from the state
by use of a mirror with parameter
![Quantum teleportation circuit](https://upload.wikimedia.org/wikipedia/commons/d/dc/Quantum_teleportation_circuit.svg)
By using teleportation, many probabilistic gates may be prepared in parallel with
-
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
entangled states, sending a control signal to the output mode. Through using
probabilistic gates in parallel offline, a success rate of
can be obtained, which is close to 1 as
becomes large. The number of gates needed to realize a certain accuracy scales polynomially rather than exponentially. In this sense, the KLM protocol is resource-efficient. One experiment using the KLM originally proposed
controlled-NOT gate with four-photon input was demonstrated in 2011, and gave an average fidelity of
.
Error detection and correction
As discussed above, the success probability of teleportation gates can be made arbitrarily close to 1 by preparing larger
entangled state
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
s. However, the asymptotic approach to the probability of 1 is quite slow with respect to the
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
number
. A more efficient approach is to encode against gate failure (error) based on the well-defined failure mode of the teleporters. In the KLM protocol, the teleporter's failure can be diagnosed if zero or
photons are detected. If the computing device can be encoded against accidental measurements of some certain number of photons, then it will be possible to correct gate failures and the probability of eventually successfully applying the gate will increase.
Many experimental trials using this idea have been carried out (see, for example, Refs
). However, a large number of operations are still needed to achieve a success probability very close to 1. In order to promote the KLM protocol as a viable technology, more efficient quantum gates are needed. This is the subject of the next part.
Improvements
This section discusses the improvements of the KLM protocol that have been studied after the initial proposal.
There are many ways to improve the KLM protocol for LOQC and to make LOQC more promising. Below are some proposals from the review article Ref.
and other subsequent articles:
* Using
cluster states in optical
quantum computing.
* Circuit-based optical
quantum computing revisited.
* Using one-step deterministic multipartite entanglement purification with linear optics to generate entangled photon states.
There are several protocols for using
cluster states to improve the KLM protocol, the computation model with those protocols is an LOQC implementation of the
one-way quantum computer
The one-way or measurement-based quantum computer (MBQC) is a method of quantum computing that first prepares an entangled ''resource state'', usually a cluster state or graph state, then performs single qubit measurements on it. It is "one- ...
:
* The Yoran-Reznik protocol - this protocol makes use of cluster-chains in order to increase the success probability of teleportation.
* The Nielsen protocol - this protocol improves the Yoran-Reznik protocol by first using teleportation to add qubits to cluster-chains and then uses the enlarged cluster chains to further increase the success probability of teleportation.
* The Browne-Rudolph protocol - this protocol improves the Nielsen protocol by using teleportation not only to add qubits to cluster-chains but also to fuse them .
See also
*
Boson sampling
Boson sampling is a restricted model of non-universal quantum computation introduced by Scott Aaronson and Alex Arkhipov after the original work of Lidror Troyansky and Naftali Tishby, that explored possible usage of boson scattering to evaluat ...
References
{{Quantum computing
Quantum information science
Quantum optics
Quantum gates