Quantum Computation: An optical approach
E. Çabej, F. Fontana, C. Conti
Introduction
It is known that the dimensions of the physical system needed to encode a bit of information are becoming smaller and smaller, going towards the dimensions of a single atom. At atomic scale the laws of quantum mechanics govern the behaviour of physical systems. It turns out that quantum systems used to encode a bit of quantum information can support new modes of computation that do not have nothing in common with classical analogues.
The classical unit of information is a bit, which can take one of the two values 0 and 1. Thus any macroscopic system, that can take two well-distinguished values is a physical realization of a bit. So an n-bit classical memory register can exist in any of the 2n logical states labelled 00…0 to 11…1.
The quantum unit of information is a qubit (quantum bit). Any quantum mechanical system can be used as a qubit providing that it is possible to define one of his states as |0ñ and another as |1ñ. It is practical to have a fixed pair of reliably distinguishable states of the qubit (for example horizontal and vertical photon polarization). More generally two quantum states are reliably distinguishable if and only if their vector representation are orthogonal [1].
Quantum mechanics tells us that if a qubit can exist in one or other of two distinguishable states, than it can also exist in a coherent superposition of these states. In these new states, which generally have no classical analogues, the microscopic system represents both values at the same time. So an n-bit quantum memory register can exist in a superposition of 2n qubits at the same time.
The difference between a classical and a quantum register is that the classical register encodes one of the 2n logical states at time, while the quantum register encodes all the 2n logical states simultaneously, being so in a superposition of all possible classical states.
Thus if it is performed a mathematical operation on the quantum register, being this register in a superposition state, the same computation is carried out on 2n numbers in a single step, and the result will be a superposition of all corresponding outputs. In other words the Quantum Computer performs a massive parallel computation.
Even though the quantum computer can store all the outcomes of 2n computations, the laws of quantum mechanics tell us that it is possible to have only one of the outputs. It is the quantum interference, which plays an important role to give the right output.
Another requirement imposed from the quantum mechanics laws is that the evolution of quantum systems must be unitary. So the quantum gates that transform the qubits must be unitary.
Till now, it is introduced that Quantum Computation is based on typical quantum phenomena: superposition of quantum states, entanglement and quantum interference.
Quantum computation might be achieved in these fundamental steps:
· the input is evolved in a superposition of some preselected quantum basis states. This superposition is achieved sending the input through unitary transformations (quantum gates).
· through other unitary transformation (that encode the classical function to be evaluated) are introduced non-local correlation on this superposition state (entanglement between different qubits). As long as the mutual coherence among a set of qubits is preserved, they can simultaneously take on more than one value giving rise to a useful effect known as quantum parallelism.
· Finally through another unitary transformation is achieved multi-particle quantum interference (among different computational paths). The quantum interference amplifies the correct outcomes and suppress the incorrect outcomes of computations. So the final step gives the correct outcomes when the measurement is carried out.
As we noted above, quantum computation is achieved through actions of unitary transformations. The unitary transformations can be constructed with a finite number of 4x4 matrices, that is, using only one and two bit quantum gates, which are universal in quantum computation[2]. From this it can be deduced that quantum gates can be constructed with a finite number of 2x2 and 4x4 matrices. The optical realization for any NxN unitary matrixhas also been demonstrated by Reck et al. [3]. So it is possible to construct optical quantum gates. Furthermore it is shown in ref. [4,5] the simulation of universal quantum gates using linear optics for building simple quantum circuits This method of constructing quantum optical circuits is based on non-local superposition of “eventualities” rather than physical objects. “Qubits” are considered as “which path” eventualities in linear optics, implementable on standard optical benches. The wave function at the exit of the optical circuit can be made to coincide arbitrarily well with the outcome of the anticipated computation, thus implementing the quantum circuit. It provides an excellent means for testing small circuits for quantum error correction or quantum algorithms[5].
The purpose of this survey paper is to introduce Optical Quantum Computing for educational purposes.
The paper is arranged according to the following items:
· Optical devices and Boolean logic
· Optical Hadamard gate
· Optical Controlled-NOT gate
· Teleportation as quantum computation
· Teleportation as an optical quantum circuit.
· Possible applications to quantum optical computing of new concepts coming from advanced components technologies.
In particular it is shown a correspondence between quantum networks and systems of linear optical devices, such as beamsplitters and phase shifters, building on an equivalence between them. This equivalence is inspired by the standard two-slit experiment of quantum mechanics, in which a single quantum can interfere with itself to produce fringes on a screen. The idea is to find protocols for translating any quantum circuit diagram into linear optical networks [4,5].
Optical devices and Boolean logic
It is known that the phase shift between the transmitted and the reflected optical fields of a lossless beamsplitter is . We will try to explain it in few words (for more details one can see ref. [6-8]).
Consider a Michelson interferometer (Fig 1), which is composed of a beamsplitter BS and two totally reflecting mirrors M, N, illuminated by a linearly polarized plane wave whose electric field has complex amplitude E0 . The electric field Et of the output beam is the superposition of two contributions E¢t originated by the path A(BS)N(BS)D and E²t originated by the path A(BS)M(BS)D. The interferometer produces also a reflected field Er which is the superposition of two contributions E¢r and E²r following the two paths A(BS)N(BS)A and A(BS)M(BS)A. If fR is the phase shift due to reflecting on the beamsplitter BS and fT the phase shift due to transmitting on the beamsplitter BS and if we assume that no absorption is present in the interferometer (lossless beamsplitter) [6], from energy conservation condition derives:
It is seen from the last expression that the lossless symmetric beamsplitter introduces a phase shift of between the transmitted and reflected field: the reflected one is in a delay of λ/4 respect to the transmitted one. So we can write the reflected field as [(] multiplying the transmitted field.
As it is mentioned above the electric field that impinges on the beamsplitter BS is divided into two beams propagating along the two orthogonal directions (BS)M and (BS)N.
Consider a single photon that impinges on BS. It will be registered with equal probability by two photodetectors situated behind the BS [Fig. 2]. The photon doesn’t split in two, it takes both paths at once. The photon will be in a quantistic superposition of its being in two orthogonal directions, so in a superposition of different spatial location. [16].
Let now consider the two input ports of the beamsplitter. In the reference [4,5] is introduced a single-photon representation of several quantum bits (qubits), in order to build on an equivalence between traditional linear optics elements (such as beamsplitters or phase shifters) and one-bit quantum gates. It is established a kind of correspondence between a photon and the qubits describing its state. For example one qubit is involved in the description of the beamsplitter in terms of a quantum circuit: the "location" qubit, which characterizes the information about "which path" is taken by the photon [4,5]. It is used and to represent the two input modes entering the beamsplitter [5,12].
The quantum state of the photon leaving the beamsplitter depends on its input mode. So if its input mode is , its quantum state after leaving BS1 is:
Furthermore if its input mode is , its quantum state after exiting the beamsplitter BS is:
In matrix form:
This optical symmetric beamsplitter acts as a quantum gate. This gate converts a classical bit into a qubit, namely in a coherent superposition of and [15,16].
It can be easily seen that:
And the matrix operator
(1)
corresponds to the quantum NOT gate. The conclusion is that a sequence of two symmetric beamsplitters must implement the quantumNOT gate [16]. If we analyse such sequence will have:
In matrix form:
(2)
The expression (2) is the same as (1). If it is not considered the phase factor i=eiπ/2 (which can be removed by locating two phase shifters of –π/2 at each exit port), it is clear that the quantum gate NOT can be implemented optically by a sequence of two symmetric beamsplitters [Fig.4].
Optical Hadamard gate
As it is known the Hadamard gate sends:
(3)
In matrix form:
In the language of Quantum Computer we can say that the Hadamard gate is used to initialise it, because it transforms the register of n-qubits in a superposition of equally balanced quantum states.
We can have an optical Hadamard gate very simply: placing two phase shifters of value -p/2 at the input and output ports of a beamsplitter [Fig.5].
Indeed:
So it transforms:
(4)
The transformations in (3) and (4) are the same and the conclusion is that it is possible to have an optical Hadamard gate.
The combination of two Hadamard gates yields a balanced Mach-Zehnder interferometer [Fig.6].
In matrix form:
Thus, a balanced MachZehnder interferometer, can be translated in two Hadamard gates one after the other (Fig.6).
Since H2 =1, it is not a surprise that the location qubit returns to the initial basis state after two beamsplitters. This simple quantum circuit describes the fact that the contributions of the two paths interfere destructively in one of the output ports, so that the photon always leaves the interferometer in the same direction as it entered [4,5].
Optical Controlled-NOT gate
Let’s consider now the same interferometer using horizontally polarized photons at the input. If none of the devices acts on the polarization, the photon exits the interferometer with the same polarization as it entered. In the quantum circuit language, this corresponds to introducing a "polarization" qubit after the "location" qubit and the two qubits remain in a product state ().
So:
stands for entering the BS horizontally
stands for entering the BS vertically
stands for horizontal polarization
stands for vertical polarization
The introduction of a polarization rotator in the path flipping the photon polarization from horizontal to vertical, yields a Controlled-NOT gate, with qubit acting as control qubit and qubit acting as target one [Fig.7].
Indeed this behaviour is described by the following relations :
In matrix form:
Now returning to the balanced Mach-Zehnder interferometer: a polarization rotator is placed in one of the arms of the interferometer, flipping the photon polarization from horizontal to vertical , in the arm of the interferometer where it is placed. As it is discussed above this corresponds to placing a 2-bit Controlled-NOT gate between the two Hadamard gates [Fig.8], where the location qubit is the control bit and the polarization is the target bit [4].
In this expression we note that after leaving the polarization rotator (P) and before entering the second beamsplitter (H2) each path is "tagged" with a particular polarization . So the path has the polarization , while the path has the polarization . In this way it is achieved an entanglement between the location qubit and the polarization qubit and it is reflected to the disappearance of interference. One can also say that the polarization of photon is flipped conditionally on its location.
Here it is introduced another basic quantum optical gate using a polarizing beamsplitter. In this case it is achieved a Controlled-NOT gate, where the location qubit is the target qubit and the polarization qubit is the control qubit [Fig.9]. A polarizing beamsplitter leaves an horizontally polarized photon (in polarizing state ) unchanged, while the vertical polarization (in polarizing state ) is reflected.
Thus the behaviour of the location qubit depends on his state of polarization, (it is reflected or not conditionally on his state of polarization)
In principle a universal quantum computation can be simulated using these optical substitutes for the universal quantum gates. The optical set-up is constructed straightforwardly by inspection of the quantum circuit [5].
So far are introduced three optical quantum gates. In this way it becomes possible to describe quantum circuits using these optical gates. On the other hand the optical gates are obtained by nothing more than the use of linear optical devices.
A circuit involving n qubits will require in general n successive splitting stages of the incoming beam, that is 2n optical paths are obtained via 2n-1 beamsplitters. In this way can be simulated quantum networks with only a small number of qubits, because there is an exponential increase between number of qubits and number of devices involved [4,5].