Table of Contents
- Introduction
- Quantum Computation Models: A Brief Recap
- What Is Measurement-Based Quantum Computation (MBQC)?
- History and Origins of MBQC
- The One-Way Quantum Computer
- Cluster States and Graph States
- Preparing the Initial Resource State
- Local Measurements Drive Computation
- Adaptive Measurement Strategy
- Entanglement as the Computational Resource
- Classical Control in MBQC
- Universal Gate Sets via MBQC
1. Introduction
Measurement-Based Quantum Computation (MBQC), also known as the one-way quantum computer model, is a paradigm where computation is driven by adaptive measurements on an entangled resource state. It separates quantum state preparation from logical operations, allowing powerful computations to be performed using local measurements and classical control.
2. Quantum Computation Models: A Brief Recap
Quantum computation can be realized in different models:
- Circuit-based (gate model)
- Adiabatic computation
- Topological computation
- Measurement-based computation
While these models are computationally equivalent, MBQC offers practical and theoretical advantages in some settings.
3. What Is Measurement-Based Quantum Computation (MBQC)?
MBQC involves preparing a highly entangled quantum state—typically a cluster state—and performing a sequence of adaptive single-qubit measurements to carry out quantum algorithms. The outcome of one measurement can determine the basis for the next, creating a logical flow of computation.
4. History and Origins of MBQC
MBQC was proposed in the early 2000s by Raussendorf and Briegel. It demonstrated that a universal quantum computer could be realized using only entanglement and measurements, without the need for sequential quantum gates.
5. The One-Way Quantum Computer
In this model, computation is inherently irreversible: the entangled resource state is consumed during computation. Hence the term “one-way.” The computation proceeds via measurements and classical processing, consuming qubits as the algorithm progresses.
6. Cluster States and Graph States
Cluster states are special types of graph states that serve as the universal resource in MBQC. They are constructed by initializing all qubits in \(|+
angle\) and applying Controlled-Z (CZ) gates according to a chosen graph structure.
7. Preparing the Initial Resource State
The initial state preparation involves:
- Placing all qubits in the state \(|+
angle\) - Applying CZ gates between qubits connected in the graph
The resulting entangled cluster state is then used for computation.
8. Local Measurements Drive Computation
Each logical operation corresponds to a sequence of single-qubit measurements in particular bases (e.g., X, Y, or rotated bases). The entanglement in the cluster state allows these measurements to propagate quantum information.
9. Adaptive Measurement Strategy
Measurement outcomes are probabilistic. To maintain deterministic computation, later measurement bases are chosen based on earlier outcomes. This requires classical feed-forward and conditional operations.
10. Entanglement as the Computational Resource
In MBQC, entanglement is used up as the computation proceeds. This is unlike the circuit model, where entanglement can be sustained and reused. The entire computational power is derived from the structure of the initial entangled state.
11. Classical Control in MBQC
MBQC requires an efficient classical processing layer to:
- Record measurement outcomes
- Adaptively update measurement bases
- Apply classical post-processing to derive the final result
12. Universal Gate Sets via MBQC
Any quantum algorithm can be implemented using only:
- Cluster states
- Single-qubit measurements
- Classical control
MBQC is thus universal for quantum computation.
13. Realizing Quantum Gates through Measurement
- Z-rotations are achieved via measurements in rotated X-Y planes
- Hadamard gates via measurements in the X basis
- CZ gates are built into the cluster state’s structure
Logical gates are effectively encoded in the measurement pattern.
14. Flow and Dependency Structures
The temporal order of measurements is governed by the graph’s flow and gflow (generalized flow), which ensure deterministic evolution and correct classical feed-forward.
15. MBQC vs Circuit Model
MBQC trades active gate control for complex entanglement upfront. While the circuit model is more intuitive for algorithms, MBQC is potentially more scalable for systems like photonics and optical lattices.
16. Measurement Patterns and Logical Operations
Different measurement sequences correspond to different quantum operations. These patterns can be compiled from high-level algorithms, similar to compiling gate sequences.
17. Fault Tolerance and Error Correction in MBQC
MBQC supports fault-tolerant computation via:
- Topological error correction
- Magic state distillation
- Code deformation techniques
These integrate naturally with the cluster state formalism.
18. MBQC and Topological Quantum Computing
In surface-code based MBQC, logical qubits are encoded in topological defects or braids in the cluster state. Computation is performed by manipulating these defects through measurement.
19. Physical Implementations of MBQC
MBQC has been experimentally explored using:
- Photonic qubits and linear optics
- Trapped ions
- Optical lattices
- Rydberg atom arrays
Its modular nature makes it appealing for distributed quantum architectures.
20. Resource Overhead and Efficiency
Preparing large cluster states is resource-intensive. MBQC may incur higher overhead than gate-based systems for some algorithms but compensates through parallelism and fault tolerance.
21. MBQC in Photonic Quantum Computing
Photons are naturally suited for MBQC because:
- They interact weakly, preserving coherence
- Single-qubit measurements are easy
- Multi-photon entanglement and fusion gates generate cluster states
This has made MBQC a key approach in optical quantum computing.
22. MBQC with Qudits and Continuous Variables
Extensions of MBQC include:
- Qudit cluster states (d-level systems)
- Continuous-variable MBQC using squeezed states and homodyne detection
These generalizations expand the applicability and scalability of the model.
23. Open Challenges in MBQC
- Scalable cluster state generation
- Robust adaptive measurement control
- Efficient compilation and optimization of measurement patterns
- Integration with NISQ-era hardware
24. Research Trends and Applications
- Measurement-only topological computation
- MBQC for quantum simulation
- Hybrid models combining circuits and measurements
- Compiler design for MBQC-based languages
25. Conclusion
Measurement-Based Quantum Computation offers a radically different approach to quantum computing, focusing on entanglement and local measurements. While experimentally demanding in some aspects, its elegance, universality, and compatibility with photonic systems make it a cornerstone of future quantum technologies.
