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
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 as information carriers, mainly uses
linear optical elements, or
optical instrument
An optical instrument (or "optic" for short) is a device that processes light waves (or photons), either to enhance an image for viewing or to analyze and determine their characteristic properties. Common examples include periscopes, microscopes, ...
s (including
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
mirror
A mirror or looking glass is an object that reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the direction of the im ...
s and
waveplate
A waveplate or retarder is an optical device that alters the polarization state of a light wave travelling through it. Two common types of waveplates are the ''half-wave plate'', which shifts the polarization direction of linearly polarized ligh ...
s) to process
quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both t ...
, and uses photon detectors and
quantum memories to detect and store quantum information.
Overview
Although there are many other implementations 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 ...
(QIP) and quantum computation,
optical quantum systems are prominent candidates, since they link quantum computation and
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 ...
in the same framework. In optical systems for quantum information processing, the unit of light in a given mode—or
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 ...
—is used to represent a
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
.
Superpositions of quantum states can be easily represented,
encrypted
In cryptography, encryption is the process of encoding information. This process converts the original representation of the information, known as plaintext, into an alternative form known as ciphertext. Ideally, only authorized parties can deci ...
, transmitted and detected using photons. Besides, linear optical elements of optical systems may be the simplest building blocks to realize quantum operations and
quantum gate
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, lik ...
s. Each linear optical element equivalently applies a
unitary transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, ...
on a finite number of qubits. The system of finite linear optical elements constructs a network of linear optics, which can realize any
quantum circuit
In quantum information theory, a quantum circuit is a model for quantum computation, similar to classical circuits, in which a computation is a sequence of quantum gates, measurements, initializations of qubits to known values, and possibly o ...
diagram or
quantum network
Quantum networks form an important element of quantum computing and quantum communication systems. Quantum networks facilitate the transmission of information in the form of quantum bits, also called qubits, between physically separated quantum ...
based on the
quantum circuit
In quantum information theory, a quantum circuit is a model for quantum computation, similar to classical circuits, in which a computation is a sequence of quantum gates, measurements, initializations of qubits to known values, and possibly o ...
model. Quantum computing with continuous variables is also possible under the linear optics scheme.
The universality of 1- and 2-bit
gates
Gates is the plural of gate, a point of entry to a space which is enclosed by walls. It may also refer to:
People
* Gates (surname), various people with the last name
* Gates Brown (1939-2013), American Major League Baseball player
* Gates McFadde ...
to implement arbitrary quantum computation has been proven. Up to
unitary matrix operations (
) can be realized by only using mirrors, beam splitters and phase shifters (this is also a starting point of
boson sampling and of
computational complexity analysis for LOQC). It points out that each
operator with
inputs and
outputs can be constructed via
linear optical elements. Based on the reason of universality and complexity, LOQC usually only uses mirrors, beam splitters, phase shifters and their combinations such as
Mach–Zehnder interferometer
The Mach–Zehnder interferometer is a device used to determine the relative phase shift variations between two collimated beams derived by splitting light from a single source. The interferometer has been used, among other things, to measure p ...
s with phase shifts to implement arbitrary
quantum operators. If using a non-deterministic scheme, this fact also implies that LOQC could be resource-inefficient in terms of the number of optical elements and time steps needed to implement a certain quantum gate or circuit, which is a major drawback of LOQC.
Operations via linear optical elements (beam splitters, mirrors and phase shifters, in this case) preserve the photon statistics of input light. For example, a
coherent (classical) light input produces a coherent light output; a superposition of quantum states input yields a
quantum light state output.
Due to this reason, people usually use single photon source case to analyze the effect of linear optical elements and operators. Multi-photon cases can be implied through some statistical transformations.
An intrinsic problem in using photons as information carriers is that photons hardly interact with each other. This potentially causes a scalability problem for LOQC, since nonlinear operations are hard to implement, which can increase the complexity of operators and hence can increase the resources required to realize a given computational function. One way to solve this problem is to bring nonlinear devices into the quantum network. For instance, the
Kerr effect can be applied into LOQC to make a single-photon
controlled-NOT and other operations.
KLM protocol
It was believed that adding nonlinearity to the linear optical network was sufficient to realize efficient quantum computation. However, to implement nonlinear optical effects is a difficult task. In 2000, Knill, Laflamme and Milburn proved that it is possible to create universal quantum computers solely with linear optical tools.
Their work has become known as the "KLM scheme" or "
KLM protocol", which uses linear optical elements, single photon sources and photon detectors as resources to construct a quantum computation scheme involving only
ancilla resources,
quantum teleportations and
error corrections. It uses another way of efficient quantum computation with linear optical systems, and promotes nonlinear operations solely with linear optical elements.
At its root, the KLM scheme induces an effective interaction between photons 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 states prepared non-deterministically and
quantum teleportation with single-qubit operations
Otherwise, without a high enough success rate of a single quantum gate unit, it may require an exponential amount of computing resources. Meanwhile, 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 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 ...
. As a result, 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 QIP as other known implementations.
Boson sampling
The more limited
boson sampling model was suggested and analyzed by Aaronson and Arkhipov in 2010.
It is not believed to be universal,
but can still solve problems that are believed to be beyond the ability of classical computers, such as the
boson sampling problem.
On Dec 3 2020 a team led by Chinese Physicist
Pan Jianwei (潘建伟) and
Lu Chaoyang (陆朝阳) from
University of Science and Technology of China
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, the ...
in
Hefei,
Anhui
Anhui , (; formerly romanized as Anhwei) is a landlocked province of the People's Republic of China, part of the East China region. Its provincial capital and largest city is Hefei. The province is located across the basins of the Yangtze River ...
Province submitted their results to Science in which they solved a problem that is virtually unassailable by any classical computer; thereby proving
Quantum supremacy of their photon-based
quantum computer called
Jiu Zhang Quantum Computer (九章量子计算机). The boson sampling problem was solved in 200 seconds, they estimated that China's
Sunway TaihuLight Supercomputer would take 2.5 billion years to solve - a quantum supremacy of around 10^14. Jiu Zhang was named in honor of China's oldest surviving mathematical text (Jiǔ zhāng suàn shù)
The Nine Chapters on the Mathematical Art
Ingredients
DiVincenzo's criteria
The DiVincenzo criteria are conditions necessary for constructing a quantum computer, conditions proposed in 2000 by the theoretical physicist David P. DiVincenzo, as being those necessary to construct such a computer—a computer first proposed b ...
for quantum computation and QIP
give that a universal system for QIP should satisfy at least the following requirements:
# a scalable physical system with well characterized qubits,
# the ability to initialize the state of the qubits to a simple fiducial state, such as
,
# long relevant decoherence times, much longer than the gate operation time,
# a "universal" set of quantum gates (this requirement cannot be satisfied by a non-universal system),
# a qubit-specific measurement capability;
if the system is also aiming for quantum communication, it should also satisfy at least the following two requirements:
# the ability to interconvert stationary and
flying qubits, and
# the ability to faithfully transmit flying qubits between specified location.
As a result of using photons and linear optical circuits, in general LOQC systems can easily satisfy conditions 3, 6 and 7.
The following sections mainly focus on the implementations of quantum information preparation, readout, manipulation, scalability and error corrections, in order to discuss the advantages and disadvantages of LOQC as a candidate for QIP
Qubits and modes
A
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
is one of the fundamental QIP units. A
qubit state which can be represented by
is a
superposition state which, if
measured in the
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
, has probability
of being in the
state and probability
of being in the
state, where
is the normalization condition. An optical mode is a distinguishable optical communication channel, which is usually labeled by subscripts of a quantum state. There are many ways to define distinguishable optical communication channels. For example, a set of modes could be different
polarization of light which can be picked out with linear optical elements, various
frequencies, or a combination of the two cases above.
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.
In boson sampling, photons are not distinguished, and therefore cannot directly represent the qubit state. Instead, we represent the
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
state of the entire quantum system by using the Fock states of
modes which are occupied by
indistinguishable single photons (this is a
-level quantum system).
State preparation
To prepare a desired multi-photon quantum state for LOQC, a single-photon state is first required. Therefore,
non-linear optical elements, such as
single-photon generators and some optical modules, will be employed. For example,
optical parametric down-conversion can be used to conditionally generate the
state in the vertical polarization channel at time
(subscripts are ignored for this single qubit case). By using a conditional single-photon source, the output state is guaranteed, although this may require several attempts (depending on the success rate). A joint multi-qubit state can be prepared in a similar way. In general, an arbitrary quantum state can be generated for QIP with a proper set of photon sources.
Implementations of elementary quantum gates
To achieve universal quantum computing, LOQC should be capable of realizing a complete set of
universal gates. This can be achieved in the KLM protocol but not in the boson sampling model.
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 mirrors 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.
Using integrated photonic circuits
In reality, assembling a whole bunch (possibly on the order of
) of beam splitters and phase shifters in an optical experimental table is challenging and unrealistic. To make LOQC functional, useful and compact, one solution is to miniaturize all linear optical elements, photon sources and photon detectors, and to integrate them onto a chip. If using a
semiconductor
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
platform, single photon sources and photon detectors can be easily integrated. To separate modes, there have been integrated
arrayed waveguide grating
{{Unreferenced, date=April 2019
Arrayed waveguide gratings (AWG) are commonly used as Optical add-drop multiplexer, optical (de)multiplexers in wavelength division multiplexing, wavelength division multiplexed (WDM) systems. These devices are capa ...
(AWG) which are commonly used as optical (de)multiplexers in
wavelength division multiplexed (WDM). In principle, beam splitters and other linear optical elements can also be miniaturized or replaced by equivalent
nanophotonics
Nanophotonics or nano-optics is the study of the behavior of light on the nanometer scale, and of the interaction of nanometer-scale objects with light. It is a branch of optics, optical engineering, electrical engineering, and nanotechnology. ...
elements. Some progress in these endeavors can be found in the literature, for example, Refs. In 2013, the first integrated photonic circuit for quantum information processing has been demonstrated using photonic crystal waveguide to realize the interaction between guided field and atoms.
Implementations comparison
Comparison of the KLM protocol and the boson sampling model
The advantage of the KLM protocol over the boson sampling model is that while the KLM protocol is a universal model, boson sampling is not believed to be universal. On the other hand, it seems that the scalability issues in boson sampling are more manageable than those in the KLM protocol.
In boson sampling only a single measurement is allowed, a measurement of all the modes at the end of the computation. The only scalability problem in this model arises from the requirement that all the photons arrive at the photon detectors within a short-enough time interval and with close-enough frequencies.
In the KLM protocol, there are non-deterministic quantum gates, which are essential for the model to be universal. These rely on gate teleportation, where multiple probabilistic gates are prepared offline and additional measurements are performed mid-circuit. Those two factors are the cause for additional scalability problems in the KLM protocol.
In the KLM protocol the desired initial state is one in which each of the photons is in one of two modes, and the possibility that a mode is occupied by more than one photon is zero. In boson sampling, however, the desired initial state is specific, requiring that the first
modes are each occupied by a single photon
(
is the number of photons and
is the number of modes) and all the other states are empty.
Earlier models
Another, earlier model which relies on the representation of several qubits by a single photon is based on the work of C. Adami and N. J. Cerf.
By using both the location and the polarization of photons, a single photon in this model can represent several qubits; however, as a result,
CNOT-gate can only be implemented between the two qubits represented by the same photon.
The figures below are examples of making an equivalent
Hadamard-gate and
CNOT-gate using beam splitters (illustrated as rectangles connecting two sets of crossing lines with parameters
and
) and phase shifters (illustrated as rectangles on a line with parameter
).
In the optical realization of the CNOT gate, the polarization and location are the control and target qubit, respectively.
References
External links
*
*
{{Quantum computing
Quantum information science
Quantum optics
Quantum gates