_{1}

In this article the inherent computational power of the quantum entangled cluster states examined by measurement-based quantum computations is studied. By defining a common framework of rules for measurement of quantum entangled cluster states based on classical computations, the precise and detailed meaning of the computing power of the correlations in the quantum cluster states is made. This study exposes a connection, arousing interest, between the infringement of the realistic models that are local and the computing power of the quantum entangled cluster states.

The quantum computation is essentially mastering and using the quantum mechanics laws for information processing. The quantum computers use quantum bits which are called qubits. Both states of the qubits 0 and 1 manifest simultaneously and connectedly superposition and entanglement. For the microqubits we can use only the Schrodinger and Heisenberg equations. The entangled states of groups of quantum particles are the key to understanding the implementation of the super huge set of quantum states which are implemented in the quantum processors. The superposition is essentially the ability of the quantum system to be in several states simultaneously. This study examines the inherent power of the entangled cluster states that are used in the quantum computations model by measurement in an “one-way” or measurement based quantum computer (MBQC) with a register initialized in a multi-partite entangled state. Instead through gates, MBQC processes the information by single-qubit measurements, the results of which determine the selection of the subsequent measurements. The arrangement and the selection itself of the measurements determine the algorithm, which is computed. Due to the decisive role of the measurements, the MBQC is irreversible and it placed the beginning of a new way of measurement-based quantum information processing. Through the determination of a common framework of the importance of the computing power of the entangled cluster states, they are presented more accurately. For the conduct of this study double entanglements of Bell and triple entanglements of Greenberger-Horne-Zeilinger (GHZ) are used; it presents the relations between the breach of the realistic models that are local and the computing power of the output entangled states.

In the following sections it will be shown that the entangled cluster states used in measurement-based quantum computations (MBQC) possess remarkable computational power. MBQC is a method for computations, distinctly different from the model of the conventional quantum circuit, where the logical operators of the network processed the information. Unlike it in the standard one-way MBQC model a series of adaptive single-qubit measurements on a multi-qubit output state that is entangled process the information. It is typical for this model that the computational power is determined by entangled classical resources obtained as a result of the measurement and not by the quantum computations themselves. In order to derive this computing power it is necessary to use a classical computer for control, which performs processing and feeding of the preliminary measurement of the outputs and directs the prospective adaptive measurements. From this point of view using this computational model, through the results of the entangled cluster measurement the classical computer achieves considerable exceedance of its own computational capabilities.

The controlled cluster increment operator increases a value into an additional code, computated by a group of lines [

At first glance the easiest thing is to apply the construction with large controlled cluster NOT-s [

If there is a circuit with n + 1 lines with n incrementing lines and one ancilla line, the goal is the incrementation to be broken up into smaller operations. In this section is not necessary to get all the way to the operators of Toffoli. Instead, the dimension of the implementation simply has to be reduced.

The bottom lines, depending from n/2 top lines, can be avoided, by caching the crossing of these lines in the ancilla bit. In this way the bottom incrementation needs only one control, see

The controlled cluster increment operator is equipollent to an increment operator with a control line as the new inferior bit. Such an absorbing control is a matter of subsequent switching of the former control line, see

It should be noted that the intent control bit is treated as the slightly bit, though the intented line is in “wrong” position.

The last case with a single ancilla bit is the case with the adopted bit, see

The trick here is to use a bit-wise addition. When the bits of a number in an additional code X are switched, they toggle from storing of X to storing of

If the complemented value is incremented, after which the complement is taken again, then finally is obtained

In other words, the surrounding of an increment operator with NOT-s turns it into a decrementing! (and vice versa.)

In the following sections of this report we will give a more precise definition for the

computational power of the correlated sources, we will discuss the natural classical analogue of the measurement-based computation in the context of the quantum non-locality. We will point out that the double and triple qubit entanglements of Bell and Greenberger-Horne-Zeilinger (GHZ) [

Framework of MBQC―In this study the computing power of the correlated sources is examined in a more common framework from those of the specific MBQC models [

The computer for control can preserve classical information, to perform exchanging of the information with it with the qubits and to compute some functions. The only

part of the described model, where the active computation occurs is the classical computer for control. Before the start of the computation is necessary the components of the system to be pre-programmed in order to specify what computation will be performed. The classical computer obtains the functions that it will assess and the separate operators will obtain a certain set of bases for measurement on the basis of which shall be carried out the computations.

This model is comprised only of classical objects-all quantum characteristics are hidden in the non-classical quantum nature of the entangled cluster states. The system uses the most common single classical system model that operates with the entangled cluster states subject to the non-communicational limitation that each particle is processed only once. But the inner cluster structure functions with minimum restrictions. As a matter of fact, the defined system is so common that it allows models where the entangled cluster states between the qubits do not defy strictly to the quantum mechanic.

It is normal to check how the original model fits within this framework. Each particle contains one qubit with a cluster state and a device for measurement, programed with measurement basis sets

The classical computer for controlling the computation that uses a cluster state does not demand the total power of a given universal classical computer. The only operations necessary for control of the measurements are equivalence computations [

In order to take note of the various computational complexity levels [

The notation ?L → BQP points out that the computer with equivalence is developed to full quantum universality when the state of the cluster is utilized as an output state. The remaining families of output states may be classified easily in our framework, see _{✕}? in

We can now look at the reverse question of the classical computation based on measurement-considering the computer with equivalence, what output states may be used

?L → BQP | P → BQP | ?L → P | |
---|---|---|---|

Cluster | ✓ | ✓ | ✓ |

Graph | ✓ | ✓ | ✓ |

CTN | _{✕}? | ✓ | _{✕}? |

GHZ | ✕ | ✕ | ✓^{1} |

^{1}The cross (✕) points out that the source cannot provide the specific computational improvement, with the supposition that the classes of complexity differ―i.e. ?L 6 = P 6 = BQP. ✕? indicate an assumption for this.

for strengthening its computational power? By the addition of any two-bit deterministic operator, which itself is not a NOT and CNOT operations product represents a classical set of universal operator. The output state which turns the computer with equivalence into a classical universality is of class ?L → P [

In order to satisfy the condition for non-signaling and to be possible the computer with equivalence to decode the outcome, the value of the NAND of the α and β input bits have to be encoded in the equivalence of both results m_{1} and m_{2}, see

The theorem of Bell determines the classical upper limitation for that quantity to 0.75 and its limit values are

the state

putation of a NAND operator in this framework will require stronger correlations than those of the quantum physics, i.e. there is no biequivalence quantum state in which the computer with equivalence can act deterministically to reduce NAND to two independent input bits.

A state of GHZ is a three qubits set in the

measuring devices the input bits are

bits, forming the state of GHZ,

two bits, the third c = a ? b that is input is fixed as an equivalence of the first two. It is important that this operation to be able to be carried out on the computer for control with equivalence. The measuring devices, which receive bit 0 measure the observed Pauli’s values σ_{x}, and those receiving 1 measure σ_{y}. The state

It is important to note that in all cases (−1)^{NAND(a,b)} is the eigenvalue. If we couple together binary 0 with the eigenvalue +1 that is measured and binary 1 c − 1 and call the measured outgoing bits n_{1}, n_{2} and n_{3}, accordingly this means that n_{1} ? n_{2} ? n_{3} = NAND (α, β). The computer with equivalence can easily retrieve NAND (a, b) from the results of the measurements m_{j} (i = 1, 2, 3) through a series of CNOT operations. The measurements of a single GHZ state of three qubits, which are controlled by the computer with equivalence, facilitate the deterministic computing of NAND.

From the NAND’s universality and 1 and 2 follows that the polynomial presence of Bell and GHZ states is the foundation of the MBQC with equivalence using deterministic operators, which turns it into a classical universality (?L → P).

Although the superposition of the qubits in a state of GHZ is greater, the Bell pair’s qubits are more strongly entangled. Due to the monogamy of the entanglement, the Bell pair’s qubits are entangled in a greater extent with one another rather than the GHZ state’s qubits. In a GHZ triplet the third qubit has a tendency to be rather unnecessary than useful. Because the Bell pairs can be utilized for certain tasks, which cannot be carried out by GHZ states (e.g. superdense coding), it is good a state of GHZ to be reduced to a Bell pair by removing one of the qubits. Previously it was accepted, that the only means for this is to find the qubit that is not wanted is with a controlled NOT, controlled by one of the remaining participating qubits. This is how the qubit that is not wanted is cleared by reversing its value in the part all-ON of the superposition while remaining it only in the part all-OFF of the superposition. The approach with a controlled NOT works well, but demands the qubit that is not wanted to be in the same location as one of the remaining qubits. But probably the payment of this price of the quantum bandwidth can be avoided, by closing down the third qubit’s value with a gate of Hadamard, performing a measurement on it, and using the result of the measurement to fix the problem with the parity of the phase, for this purpose it is only necessary to be used a classical bandwidth. This is called “erasing” of the qubit. A given qubit may be removed by a state of GHZ via its measurement along the axis of spinning that is perpendicular to the axis of entanglement and with the aid of the result of the measurement to be made a correction of the phase.

An important characteristic of our results is that the equivalent measurements may be performed in parallel. Another variant to the application of the circuit by multiple GHZ states measurements is intended to provide the entire logic circuit in outcomes from the measurements of a single entangled state with multiple qubits. This may require novel methods for parallelization of the circuit by quantum methods. An important specificity in this study is that the measurements must be adaptive. The initiated framework for the computational power classification of the entangled cluster states based on measurement leaves the qubits internal structure completely unrestricted. We have demonstrated that the polynomial supply of two-qubit Bell states presents an optimal source of the classical computation based on measurement by restricting the number of particles that share entangled states. The proposed MBQC model with equivalence combines the two paradoxes of the non-locality that are most important, providing them an interpretation as computing tasks and delivering a simple interpretation for the obvious infringement of the restriction

Raychev, N. (2016) Computational Cluster with Entangled States. Journal of Applied Mathematics and Physics, 4, 1777-1786. http://dx.doi.org/10.4236/jamp.2016.49183