User:Yaniv.N/sandbox/linear optical quantum computing

Linear Optical Quantum Computing or Linear Optics Quantum Computation (LOQC) is a paradigm of quantum computation, allowing (under certain conditions that are described below such as in the KLM protocol) universal quantum computation. LOQC uses photons as information carriers, mainly uses linear optical elements including beam splitters, phase shifters, and mirrors to process quantum information, and uses photon detectors and quantum memories to detect and store quantum information.

Overview of linear optical quantum computation
Although there are many other implementations for quantum information processing (QIP) and computation, optical quantum systems are prominent candidates for QIP, since they link quantum computation and quantum communication in the same framework. Among the optical systems for quantum information processing, the unit of light in a given mode—or photon—is used to represent a qubit. Superpositions of quantum states can be easily represented, encrypted, transmitted and detected using photons. Besides, linear optical elements of optical systems may be the simplest building blocks to realize quantum operations and quantum gates. Each linear optical element equivalently applies a unitary transformation 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 diagram or quantum network based on the quantum circuit model. Quantum computing with continuous variables is also possible under the linear optics scheme. The universality of 1- and 2-bit gates to implement arbitrary quantum computation has been proven. Up to $$N\times N$$ unitary matrix ($$U(N)$$) operations can be realized by only using mirrors, beam splitters and phase shifters (footnote: it is also a starting point of Boson sampling and computational complexity analysis for LOQC). It points out that each $$U(N)$$ operator with $$N$$ inputs and $$N$$ outputs can be constructed via $$\mathcal{O}(N^2)$$ 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 interferometers 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 the sense 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 optics 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 optics 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 the scalability problem of LOQC, since nonlinear operations are hard to implement which can increase the complexity of operators and hence can reduce the resources required to realize a given computational function. There are basically two ways to solve this problem. One is to bring in 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. 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 optics 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 optics elements.

A more limited model of LOQC is the boson sampling model, this model was suggested and analyzed by Aaronson and Arkhipov in 2013. it is not believed that the boson sampling model is universal, but this model is still able to solve problems that are believed to be beyond the ability of regular computers, such as the boson sampling problem.

The KLM protocol has some advantages over boson sampling, the KLM Protocol is an universal model and the qubit state can be directly represented in this model while the Boson sampling model is not believed to be universal and only the qudit state of the entire quantum system can be represented.

It seems However 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, the measurement must be of all the modes at the end of the computation, the only scalability problem in this model arise from the requirement that all the photons must arrives 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. In order to utlize those non-deterministic quantum gates we make use of Gates teleportation in which multiple probabilistic gate are prepared offline and additional measurement mid circuit are also necessary (not just a single measurement at the end of the computation). those 2 factors are the cause for additional scalability problems in the KLM protocol.

Elements of LOQC
The basic building blocks for LOQC are introduced below.

DiVincenzo's criteria for quantum computation and QIP give that a system for QIP should satisfy at least the following requirements: 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 show that LOQC is a good candidate for QIP.
 * 1) a scalable physical system with well characterized qubits,
 * 2) the ability to initialize the state of the qubits to a simple fiducial state, such as $$|000\cdots\rangle$$,
 * 3) long relevant decoherence times, much longer than the gate operation time,
 * 4) a "universal" set of quantum gates,
 * 5) a qubit-specific measurement capability; if the system is also aiming for quantum communication, it should also satisfy at least the following two requirements:
 * 6) the ability to interconvert stationary and flying qubits, and
 * 7) the ability faithfully to transmit flying qubits between specified location.

Qubits and modes
A qubit is one of the fundamental QIP units. A qubit state which can be represented by $$\alpha |0\rangle + \beta|1\rangle$$ is a superposition state with probability $$|\alpha|^2$$ of being in the $$|0\rangle$$ state and probability $$|\beta|^2$$ of being in the $$|1\rangle$$ state, where $$|\alpha|^2+|\beta|^2=1$$ is the normalization condition. The states $$|0\rangle$$ and $$|1\rangle$$ could correspond to 0-photon and 1-photon in a given mode channel. In general, there could be $$|n\rangle, n=0,1,2,\cdots$$ photon states for existing $$n$$-photon cases. An optical mode is a physically 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 channels of light which can be picked out with linear optics elements, various frequency channels, or a combination of the two cases above.

In the KLM protocol it is usually the case that each of the photons are 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 we can distinguish between the photons since they are in different modes and therefore we can represent a qubit state using a single photon in two modes, e.g. $$|0\rangle \equiv |01\rangle _{VH}\equiv |0\rangle _V |1\rangle _H$$, $$|1\rangle \equiv |10\rangle _{VH}\equiv |1\rangle _V |0\rangle _H$$. It is common to refer to the states defined via occupation of modes as Fock states. In Boson sampling we do not distinguish between the photons, and therefore we cannot directly represent the qubit state, instead we represent the qudit state of the entire quantum system. the qudit state is represented using the Fock states of M modes which are occupied by N indistinguishable single photons (this is a $$\tbinom {M+N-1} {M} $$-level quantum system)

State measurement/readout
In the KLM protocol, a quantum state can be readout or measured using photon detectors along selected modes, but in the Boson sampling model in order to use the photon detectors we require that all the photons must arrives at them within a short enough time interval and with close enough frequencies, if this requirement is not met then the photons effectively become distinguishable particles. this difficulty is the source of the only scalability problem in this model, and in order manage this problem only a single measurement of all the modes is allowed at the end of the computation

State preparation
To prepare a desired quantum state for LOQC, usually a single-photon state, single-photon generators and some optical modules will be employed. For example, optical parametric down-conversion can be used to conditionally generate the $$|1\rangle$$ state in the vertical polarization channel at time $$t$$ (subscripts are ignored for this single qubit case). By using conditional single-photon source the output state is guaranteed, although there is a cost associated with the success rate. A joint multi-qubit state can be prepared in a similar (possibly more sophisticated) way. In general, an arbitrary quantum state can be generated for QIP with a proper set of photon sources.

In Boson sampling the initial state of the computation is always the state in which the first N modes are occupied by a single photon (N is the number of photons and M ≥ N is the number of modes), we write this state as: $$|1^{N}\rangle=|1,1, ...,1,0,0,...,0 \rangle.$$

Implementations of elementary quantum gates
To achieve universal quantum computing, LOQC should be capable of realizing a complete set of universal gates (please refer to the quantum gate article for the universality of quantum gates). this can be achieved in the KLM protocol but not in the Boson Sampling model.

Ignoring error correction and other issues, implementations of elementary quantum gates using only mirrors, beam splitters and phase shifters have been summarized in some early publications. See, for example, Ref. The basic principle is that using these linear optics elements, one can construct an arbitrary (at least) 2-qubit unitary operation which links 2 or dual-rail qubits; in other words, those linear optical elements support a complete set of $$SU(2)$$ operators.

However, in the Boson Sampling model the photons are non-interacting and the way each of the optical elements work on N photons is uniquely defined by the way they work on a single photon.Therefore there exist a natural homomorphism which maps an m × m unitary transformation acting on a single photon to, the corresponding M × M unitary transformation acting on n photons (M$$ = \tbinom {m+n-1} {m} $$) since this homomorphism exist the operators which are supported are spanned by no more than $$m^2$$ operators and not all operators are supported (a complete basis is of size $$M^2$$).

Integrated photonic circuits for LOQC
In reality, assembling a whole bunch (possibly on the order of $$10^4$$ ) 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 platform, single photon sources and photon detectors can be easily integrated. To separate modes, there have been integrated arrayed waveguide grating (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 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.