Fusion-based quantum computing (FBQC) code

Fusion-based quantum computing (FBQC) code[1]

Description

Code whose codewords are resource states used in an FBQC scheme.

FBQC is a fault-tolerant model of quantum computation built from small constant-sized entangled resource states together with destructive entangling measurements called fusions [1]. Resource states are stabilizer states and can be described, up to local Clifford transformations, by graph states [1]. Unlike standard MBQC, which first prepares a large cluster state and then computes using single-qubit measurements, FBQC integrates entanglement generation, syndrome extraction, and logical processing into the fusion measurements themselves. This makes FBQC particularly natural for photonic platforms, where Bell-type fusion measurements are native operations.

Protection

Protects against erasure, Pauli errors, photon loss, fusion failure from non-determinism, and faults in resource-state preparation. Redundancy in fusion outcomes is captured by the check-operator group. Fusion measurement outcomes form a syndrome that can be decoded to infer the logical Pauli frame, rather than by applying physical recovery operations [1].

Encoding

Resource-state generators, which produce small constant-sized stabilizer states, together with Bell-fusion measurements.

Gates

Clifford gates are performed by introducing and manipulating topological features such as boundaries, defects, or twists through modified fusion bases and, in some constructions, single-qubit measurements. Logical gates can also be performed by code deformation. Non-Clifford gates are performed by magic-state injection.

Decoding

Surface-code-based FBQC schemes often admit a syndrome-graph description, allowing the use of decoders such as minimum-weight matching and union-find [1,2].

Fault Tolerance

Fusion networks can be constructed so that the surviving stabilizers and check operators realize topological surface-code fault tolerance [1].More generally, any three-dimensional cell complex in which each edge has four incident faces defines a surface-code fusion complex, yielding a large family of FBQC fault-tolerant protocols [3].

Threshold

Under the hardware-agnostic fusion error model, pedagogical FBQC schemes have reported thresholds of \(11.98\%\) against erasure in each fusion measurement and \(1.07\%\) against Pauli error [1].For a linear-optical ballistic scheme, reported thresholds include \(43.2\%\) against fusion failure and \(10.4\%\) photon loss per fusion [1].For surface-code logical blocks compiled to FBQC, the threshold for fault-tolerant logical gates was found to agree, within numerical uncertainty, with the bulk memory threshold [2].

Cousins

Topological code— Surface-code-based topological fault-tolerant protocols can be realized in FBQC, including topological features such as boundaries, defects, and twists, by modifying fusion measurements and, in some constructions, adding single-qubit measurements [2,3].
Dual-rail quantum code— FBQC resource states are concatenated with dual-rail codes to increase loss detection.
Dynamically generated QECC— Building a fusion network is done using a measurement-based dynamical process.
Concatenated cat code— The four-component cat code can be concatenated with the XZZX code to yield a fusion-based computation scheme on a 2D lattice [4].
Square-lattice GKP code— GKP states can be used to perform computation in a fusion-based encoding [5].
Concatenated qubit code— Blocklet concatenation uses concatenation and transversal gates in a way that is tailored to FBQC platforms [6].
Cluster-state code— FBQC and MBQC are both computational models in which computation is done by measuring resource states (which are qubit stabilizer states). The difference between the two is in how the states are constructed. FBQC is based exclusively on two-qubit measurements tailored to photonic platforms. These measurements require a foliation with more qubits but one which can be built by fusing smaller modules.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Parents

Qubit stabilizer code

The resource states in FBQC are small stabilizer states, and the surviving stabilizers after fusion determine the encoded output state (conditioned on measurement outcomes).