There exist many computational models that achieve quantum computation. The most commonly used model is the circuit quantum computational model, which involves an initialization process of qubits, a sequence of quantum gates and a readout (measurement) of qubit state in the computational basis (logical 0/1 state). Another common computational model is the adiabatic quantum computational model, which uses time dependent, smoothly or adiabatically varied Hamiltonians instead of gates. However, both rely on the unitary property of either quantum gates or Hamiltonian evolution. In contrast to these models, measurement-based quantum computation (MBQC) utilizes measurement to achieve emulation of unitary circuits. Since measurement destroys coherence in quantum states, unitary operation is achieved via entanglement. This paper is a review of past and current research concerning MBQC along with the theoretical and experimental progress that has been made.
The origins of MBQC can be traced into the so-called one-way quantum computer, which was based on single-qubit measurements and relied on the mapping of a quantum circuit to a measurement pattern in the cluster state. For any quantum circuit in the standard circuit model, a measurement pattern can be obtained on all the qubits of the cluster state, which indicates the theoretical equivalency of the different computational models. Computation is carried out by the execution of the measurement pattern with possible adaptation which consumes entanglement as a resource. Hence, the amount of entanglement decreases with single-qubit measurement, making the computation “one-way”. MBQC continued to evolve mainly through variants arising from the principles of the one-way quantum computer, such as the teleportation-based, state-transfer-based, and correlation-space approaches.
The teleportation-based approach involves the joint measurement of 2 or more qubits at the same time, which in turns creates entanglement. Subsequently, this generated entanglement will be used to perform gate operations. Depending on the measurement outcomes, a correcting operation completes the teleportation and recovers the unknown state. By using this approach, it is possible to perform universal quantum computation using only measurements and quantum memory without the need of a prior entangled resource state.
On the other hand, the state-transfer scheme uses only single-qubit and two-qubit observables with only two possible outcomes (0 and 1). The basic setup consists of an observable which represents a two-outcome measurement that projects onto the +1 and -1 subspace of the observable eigenstates. In contrast to teleportation, this approach uses only two qubits to transfer a one qubit state. Also, arbitrary one-qubit gates can be implemented by rotating the observables in the state transfer.
Lastly, the correlation-space approach started by interpreting the one-way computer in terms of a tensor network of valence bonds or as projected entangled-pairs states. In that scenario, computation takes place at the virtual qubits and uses teleportation. This approach led to the development of the correlation-space MBQC, which exploits the tensor-network structure of the states, such as the one-dimensional matrix-product states as well as the two-dimensional projected-entangled-pair states.
A potential next step is to explore other resource states beyond cluster states. So far, universal clusters have been shown on regular lattices such as the triangular, hexagonal, and kagome lattices. The universality also holds for cases where the lattice is not perfectly regular. In case of the faulty square-lattice cluster state, the universality depends on the connectivity of the lattice. It has been shown that universal MBQC can also be achieved with ground states of short-ranged interacting Hamiltonians as potential resource states and Affleck-Kennedy-Lieb-Tasaki (AKLT) states. Similarly, symmetry-protected topological states, quantum computational phases of matter and thermal states are explored for MBQC. Another important parameter for MBQC is the amount of entanglement in the universal resource states which, if it's too high, can affect the speedup of the computation. An interesting application of MBQC in the one-way computer is blind computation where a server takes the instruction of measurement axes and reports the outcomes to a client that intends to run some quantum computation without disclosing the executed quantum circuit. That could be useful in future secure cloud-based quantum computation for example.
Efforts of incorporating quantum error correction that will lead to fault-tolerant quantum computation with MBQC, have also been investigated. Such fault-tolerant schemes have been achieved by exploiting a three-dimensional cluster state where each two-dimensional slice is used to simulate the surface code. In order to complete the universality, additional gates can be inserted by a process known as magic-state distillation. The error threshold in this estimation is as high as 0.75%, which is much higher (higher tolerance of errors) than other estimates.
Recent experimental progress in the development of cluster states is utilizing cold atoms trapped in an optical lattice. In addition to bosonic cold atoms, the use of cluster-state generation with trapped fermionic atoms using interplay of the spin-orbit coupling and superexchange interaction also increases the coherence time. Furthermore, entanglement generation between two neutral atoms via the Rydberg blockade have been demonstrated experimentally, which can be used to directly create a cluster state of an array of atoms. Small-size cluster and graph states have also been realized experimentally using solid-state and quantum-dot emitters, also in other physical systems, such as in trapped ions and superconducting qubits. However, generation of resource states beyond cluster states is harder. So far, 1-D tensor-network states used in the correlation-space approach have been realized.
Measurement based quantum computation is a useful alternative to other more popular models of quantum computation. It has already been proven that it is equivalent to the other models and that it is feasible to implement for quantum communication problems. Various interesting applications have been examined and error correction protocols have been successfully applied. Also, proof-of-principle experimental implementations such as photonic, continuous-variable, trapped atoms and ions, and superconducting systems have been devised to realize the MBQC, however, each platforms has its own challenges. Hence, more work is needed to benchmark these platforms and overcome physical and technical challenges.