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Wikipedia / One-way quantum computer

<noinclude></noinclude> 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. It is "one-way" because the resource state is destroyed by the measurements.

The outcome of each individual measurement is random, but they are related in such a way that the computation always succeeds. In general, the choices of basis for later measurements need to depend on the results of earlier measurements, and hence the measurements cannot all be performed at the same time.

The implementation of MBQC is mainly considered for photonic devices, due to the difficulty of entangling photons without measurements, and the simplicity of creating and measuring them. However, MBQC is also possible with matter-based qubits. The process of entanglement and measurement can be described with the help of graph tools and group theory, in particular by the elements from the stabilizer group.

Definition

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. Another key feature for quantum computing relies on the entanglement between the qubits.

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. ancillary qubits of them. output qubits will be affected by the measurements because of the entanglement with the measured qubits. The one-way computer has been proved to be a universal quantum computer, which means it can reproduce any unitary operation over an arbitrary number of qubits.

General procedure

The standard process of measurement-based quantum computing consists of three steps: 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: in order to carry on the computation in a deterministic way, some correction operators, called byproducts, are introduced.

Preparing the source state

state:

bits:

gate operation. The matrix representation of such two-qubits operator is given by

gate over two qubits can be described by the following system:

state, the overall state

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. gates between photons are difficult to realize. Therefore, probabilistic entangling gates such as Bell state measurements are typically considered. Furthermore, quantum emitters such as atoms or quantum dots can be used to create deterministic entanglement between photonic qubits.

Measuring the qubits

observable:

belong to the Pauli matrices-th qubit.

Correcting the output

. In order to perform a deterministic computation, some corrections must be introduced. The correction operators, or byproduct operators, are applied to the output qubits after all the measurements have been performed..-th qubit. Eventually, even though the outcome of a measurement is not deterministic in quantum mechanics, the results from measurements can be used in order to perform corrections, and carry on a deterministic computation.

CME pattern

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. Such pattern of computation is referred to as CME standard pattern. In the CME byproducts.

When performing a computation following the CME.

An example: Euler rotations

As an illustrative example, consider the Euler rotation basis: such operation, in the gate model of quantum computation, is described as

defines a global phase which is irrelevant for the computation. To perform such operation in the one-way computing frame, it is possible to implement the following CME pattern:

gates.

. Eventually the fifth qubit (the only one not to be measured) figures out to be the output state.

returns the identity.

As seen in such example, in the measurement-based computation model, the physical input qubit (the first one) and output qubit (the third one) may differ each other.

Equivalence between quantum circuit model and MBQC

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.

single-qubit operator is defined as follows:

can be converted into a CME pattern as follows, with qubit 1 being the input and qubit 2 being the output:

gate has been decomposed into the CME corrections. In order to lead the whole flow of computation to a CME pattern, some rules are provided.

Standardization

entanglements at the beginning of the process, some rules of commutation must be pointed out:

-th qubits.

Pauli simplification

commute with the corrections in the following manner:

angles, some simplifications, called Pauli simplifications, can be introduced:

Signal shifting

can be explained through its rules of commutation:

sequence, no substitution will occur. To perform a correct CME must be translated at the end of the pattern.

Stabilizer formalism

is an abelian subgroup from the Pauli group. stabilizer group:

whose vertices correspond to the qubits, while the edges., defined as

generators from the stabilizer group:

may differ, with respect to the connections implemented by the edges in the graph.

The Clifford group

(controlled phase gate):

from the Clifford group on such state, the following equalities hold:

generators of the stabilizer group.

The Gottesman–Knill theorem from the circuit.

Hardware and applications

Topological cluster state quantum computer

Measurement-based computation on a periodic 3D lattice cluster state can be used to implement topological quantum error correction. 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.

Implementations

One-way quantum computation has been demonstrated by running the 2 qubit Grover's algorithm on a 2x2 cluster state of photons. A linear optics quantum computer based on one-way computation has been proposed.

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

AKLT state as a resource

It has been shown that the (spin) AKLT state on a 2D honeycomb lattice can be used as a resource for MBQC. More recently it has been shown that a spin-mixture AKLT state can be used as a resource.

References

General
Non-cluster resource states
Measurement-based quantum computation, quantum carry-lookahead adder