Measurement-Based Quantum Computing (MBQC)

In the standard quantum circuit model, quantum information is processed by applying unitary gates to qubits that persist throughout the computation. Measurement-based quantum computing inverts this picture entirely. The computation begins with a fixed resource state, typically a two-dimensional cluster state, which is prepared before any computation-specific information is encoded. The computation itself is then carried out entirely by single-qubit measurements. Each measurement consumes a qubit and cannot be undone, which is why the model is also called one-way quantum computing. The computational result is read from the qubits that remain unmeasured at the end. This separation between resource state preparation and the computation itself is conceptually striking: the entanglement needed for the whole computation is created upfront, and the logical operations are chosen by selecting which basis to measure each qubit in.

Cluster state preparation and adaptive measurement

A cluster state is prepared by starting each qubit in the |+> = (|0> + |1>)/sqrt(2) state and applying controlled-Z (CZ) gates between all nearest-neighbor pairs on a two-dimensional lattice. The result is a highly entangled state in which no individual qubit carries meaningful information, but the correlations between qubits encode the computational resource. Computation proceeds by measuring qubits in the basis {cos(theta)|0> + sin(theta)|1>, sin(theta)|0> - cos(theta)|1>} for chosen angles theta, which effectively teleports the logical state from one qubit to the next while simultaneously applying a rotation. The key feature is adaptivity: the measurement angle for qubit k depends on the classical outcomes of all earlier measurements. This feedforward of classical information is what makes the computation coherent. Without adaptive measurements, the model can perform only Clifford operations.

Universality

The MBQC model is universal for quantum computation, meaning any unitary operation that can be implemented in the quantum circuit model can also be implemented in MBQC on a suitable cluster state. The proof proceeds by showing that arbitrary single-qubit rotations (including the T gate, which generates non-Clifford operations when combined with Clifford gates) and the CNOT gate can both be implemented by measuring appropriate patterns on the cluster state. The Hadamard and CNOT together with T give a universal gate set, so MBQC can approximate any unitary to arbitrary precision. The overhead in cluster state size is polynomial in the number of circuit gates, so the model is polynomially equivalent to the circuit model in resource usage.

Photonic advantages and connections to other models

MBQC is particularly attractive for photonic quantum computing. Generating two-qubit gates directly between photons requires strong photon-photon nonlinearities that are extremely difficult to achieve in linear optics. But preparing an entangled photonic cluster state can be done offline using probabilistic fusion gates and photon sources, and once the cluster state is available, the computation proceeds through single-qubit measurements alone, which linear optics can perform deterministically. This avoids the need for two-photon gates during the actual computation. Companies such as PsiQuantum and Xanadu have built their fault-tolerant architectures around photonic MBQC with cluster states tailored to the surface code, connecting MBQC directly to topological error correction. MBQC also underlies blind quantum computing, where a client can delegate a computation to a remote quantum server by sending qubits in chosen states; the server performs measurements as instructed without learning the computation’s input, output, or algorithm.

See also

• Quantum Teleportation
• Blind Quantum Computing
• Fault-Tolerant Quantum Computing
• Quantum Circuit Model