One-way quantum computer

The one-way quantum computer, also known as 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.

The outcome of each individual measurement is random, but they are related in such a way that the computation always succeeds.

The purpose of quantum computing focuses on building an information theory with the features of quantum mechanics: instead of encoding a binary unit of information (bit), which can be switched to 1 or 0, a quantum binary unit of information (qubit) can simultaneously turn to be 0 and 1 at the same time, thanks to the phenomenon called superposition.

[3][4][5] Another key feature for quantum computing relies on the entanglement between the qubits.

[6][7][8] In the quantum logic gate model, a set of qubits, called register, is prepared at the beginning of the computation, then a set of logic operations over the qubits, carried by unitary operators, is implemented.

[9][13][14][15] The standard process of measurement-based quantum computing consists of three steps:[16][17] entangle the qubits, measure the ancillae (auxiliary qubits) and correct the outputs.

In the first step, the qubits are entangled in order to prepare the source state.

In the second step, the ancillae are measured, affecting the state of the output qubits.

However, the measurement outputs are non-deterministic result, due to undetermined nature of quantum mechanics:[17] in order to carry on the computation in a deterministic way, some correction operators, called byproducts, are introduced.

When entangling two ancillae, no importance is given about which is the control qubit and which one the target, as far as the outcome turns to be the same.

gates are represented in a diagonal form, they all commute each other, and no importance is given about which qubits to entangle first.

Photons are the most common qubit system that is used in the context of one-way quantum computing.

Therefore, probabilistic entangling gates such as Bell state measurements are typically considered.

[28] Depending on the outcome of the measurement, a byproduct operator can be applied or not to the output state: a

The operations of entanglement, measurement and correction can be performed in order to implement unitary gates.

Such operations can be performed time by time for any logic gate in the circuit, or rather in a pattern which allocates all the entanglement operations at the beginning, the measurements in the middle and the corrections at the end of the circuit.

To perform such operation in the one-way computing frame, it is possible to implement the following CME pattern:[27][30] where the input state

The one-way quantum computer allows the implementation of a circuit of unitary transformations through the operations of entanglement and measurement.

At the same time, any quantum circuit can be in turn converted into a CME pattern: a technique to translate quantum circuits into a MBQC pattern of measurements has been formulated by V. Danos et al.[16][17][31] Such conversion can be carried on by using a universal set of logic gates composed by the

gate has been decomposed into the CME pattern, the operations in the overall computation will consist of

In order to lead the whole flow of computation to a CME pattern, some rules are provided.

operator is called signal shifting, whose action will be explained in the next paragraph.

To perform a correct CME pattern, every signal shifting operator

When preparing the source state of entangled qubits, a graph representation can be given by the stabilizer group.

:[19][33][37] The Clifford group requires three generators, which can be chosen as the Hadamard gate

The Gottesman–Knill theorem states that, given a set of logic gates from the Clifford group, followed by

[19][33][39][40][41] Measurement-based computation on a periodic 3D lattice cluster state can be used to implement topological quantum error correction.

[42] Topological cluster state computation is closely related to Kitaev's toric code, as the 3D topological cluster state can be constructed and measured over time by a repeated sequence of gates on a 2D array.

[43] One-way quantum computation has been demonstrated by running the 2 qubit Grover's algorithm on a 2x2 cluster state of photons.

[46] Cluster states have also been created in optical lattices,[47] but were not used for computation as the atom qubits were too close together to measure individually.

The measurement-based techniques consist of entangling a cluster of qubits and performing a set of measurements. Thanks to the correlation between the entangled qubits, the flow of information (from left to right) is carried on by the measurements on the physical qubits in the cluster.
A quantum circuit implementing the Bernstein-Vazirani algorithm: and represent the logic gates (unitary operators) which act on the register of qubits. In the MBQC frame, the logic gates are performed by entangling the qubits and measuring the auxiliary ones.
The CZ operation in the circuit diagrams.
Implementation of X and Z gates over two qubits in the circuit diagrams.
The Euler rotation (with respect to the XZX basis) in the MBQC computation. The lines describe the entanglement between the qubits. The first qubit corresponds to the input state , the fifth one to the output state. The qubits from 2 to 4 are the ancillae. All the states, except for the input, are prepared in the state. All the qubits, except for the output, are measured by the observable with a specific angle. After the measurements have been carried on, implementing the unitary, the and corrections are performed with respect to the outcomes.
A math graph defined by three vertices and three edges. Each vertex is connected with the other ones by an edge. In the MBQC frame, the vertices represent the qubits, while the links between them the entanglements. In the stabilizer formalism, such graph is represented by the generators, which all of them commute each other with.