#clusterstate

Quantum Computation Based on Measurements: An Entanglement-Driven Substitute for Quantum Circuits

In the rapidly evolving science of quantum computation, a paradigm known as Measurement-Based Quantum Computation (MBQC) is emerging as a powerful alternative to the traditional circuit model. MBQC fundamentally alters the paradigm by using entanglement as a primary resource and relying on local measurements on each qubit to power the entire computation.

From One-Way Computers to Universal Resources: The Blueprint The MBQC concept was inspired by the "one-way quantum computer" developed by Raussendorf and Briegel. Their landmark work demonstrated that any quantum circuit could be fully implemented by performing local measurements on individual qubits that were pre-prepared in a specific, highly entangled state known as the cluster state.

In the typical circuit model, computing is deterministic due to the unitary feature of quantum gates. On the other hand, measuring a quantum state typically yields random results. A major innovation in MBQC is how to deal with this inherent unpredictability. In order to preserve the determinism of the computation as a whole, subsequent measurement axes must be modified using the measurement results (a process called "feedforward"). This adaptation results in a time ordering of the measurements.

The 2D cluster state, defined on a regular lattice like the square lattice, is one type of graph state that is believed to offer a universal resource for quantum computation. A quantum circuit can be realized by mapping this cluster state onto a measurement pattern. The calculation is driven by the process, which also "uses" the entanglement resource. To achieve universality, the MBQC framework has to implement a universal set of gates, typically random one-qubit gates and a two-qubit entangling gate like the Controlled-NOT (CNOT) gate.

Different Frameworks and Entanglement's Function

Insightful variations of the Measurement-Based Quantum Computation framework have been created. These include:

Teleportation-Based Schemes: This technique, first put forth by Nielsen and Chuang, demonstrated that global quantum computation could be complete without the use of an a priori entangled resource state like a cluster state by using only quantum memory and measurements (which are necessary to create and mediate entanglement). The one-way model only utilizes single-qubit measurements, whereas the original teleportation strategy for a CNOT gate required a challenging four-qubit measurement that was ultimately lowered to two-qubit measures (which were determined to be optimal).

State-Transfer-Based Schemes: Perdrix proposed an alternative technique that transfers a quantum state using only single-qubit and two-qubit observables to demonstrate that arbitrary one-qubit gates and CNOT gates may be constructed.

The third perspective is the correlation-space/tensor-network picture, which uses valence-bond states or projected entangled-pair states (PEPS) to explain the cluster state. To generalize resource states, computation takes place in the Hilbert space of "virtual qubits" rather than the physical qubits by changing local tensors.

Although quantum gates in the circuit model produce entanglement that is subsequently removed by measurement, MBQC basically depends on a large amount of initial entanglement in the resource state. The entanglement requirement, which contends that entanglement "must be consumed in moderation" in order to be successful for universal MBQC, has been studied by researchers. A resource state is deemed unusable for computation if its entanglement is unreasonably high (as in random states).

Resource States and Matter's Computational Phases

In addition to the square-lattice cluster state, it has been shown that cluster states are ubiquitous on other 2D regular lattices, such as triangular, hexagonal, and kagome. Whether or not faulty graph states with erratic connections are ubiquitous depends on the lattice's percolation threshold.

A key area of study is the discovery of universal resource states that emerge spontaneously as ground states of short-ranged Hamiltonians and could facilitate preparation through cooling. Cluster states do not exist as unique ground states for any two-body interacting Hamiltonian. However, there has been interest in Affleck-Kennedy-Lieb-Tasaki (AKLT) countries. Recently, numerical studies have demonstrated that this model features a non-zero spectral gap, which is helpful for resource state generation through cooling. The universality of the 2D spin-3/2 AKLT state on the hexagonal lattice for MBQC was previously shown.

More recently, investigations have focused on the development of Symmetry-Protected Topological (SPT) phases and quantum computational phases of matter. It has been shown that certain SPT phases, most notably the 2D cluster phase, have computing capacity throughout the entire phase, implying a "possible general notion of quantum-computational phases of matter."

Breakthrough new research has extended this concept to systems with long-range entanglement, or topological order, which were previously believed to be outside the scope of computational phase analysis. Using this new framework, we have discovered the first examples of topological phases of matter with uniform computing capability, including a new model where the ground states are confirmed to be universal resources for MBQC. Some computational properties are protected by subsystem symmetries.

Applications and Tolerance for Faults

Fault tolerance is necessary for practical quantum processing. Fault tolerance in MBQC can be achieved by using a three-dimensional cluster state, where each 2D slice replicates a surface code (a typical error-correcting code). By using specific measurement patterns to mimic the braiding of anyons, fault-tolerant gates can be produced. This topologically protected MBQC method has achieved a high error threshold, estimated as high as 0.75%, in contrast to previous estimates of 0.01% or less.

Apart from its robustness, MBQC has enabled significant applications like Blind Quantum Computation (BQC). For instance, BQC allows a client to assign a quantum computation to a server in a cloud setting, making it impossible for the server to determine which quantum circuit is being used. This protocol typically needs the client to construct initial product states and supply the server with measurement axes so that the server can do the measurements on the subsequent brickwork lattice state.

Furthermore, MBQC has provided an alternative, resource-efficient approach to implementing the original linear-optic quantum computation developed by Knill, Laflamme, and Milburn (KLM). Furthermore, the MBQC framework is used in quantum communication for entanglement switching and the construction of robust quantum repeaters.

Progress in Experiments

Experimental realizations of MBQC components are being developed on several platforms.

Cold Atoms: The first experimental realization of a cluster state was achieved using cold atoms confined in an optical lattice, despite the initial difficulty of individual addressing. These days, modern techniques like the Rydberg blockade are used to induce the controlled-Z gates needed for cluster state generation.

Photonic Systems: Small-size cluster and graph states are produced by probabilistically merging entangled photon pairs. Recent deterministic systems using solid-state and quantum-dot emitters have been demonstrated for key elements. It has also been possible to generate continuous-variable cluster states of light on a large scale.

Other Systems: Superconducting qubits have also formed graph and cluster states, as demonstrated by tests on publicly available cloud systems and trapped ions, where error correction codes were created.