Short answer: Fault-tolerant quantum computing needs a universal set of logical gates, but no single topological code provides one with simple operations. Code switching and gauge fixing are techniques for moving between codes or changing a code during computation to get around this. A 2017 paper proposes subsystem lattice surgery between surface and color codes, a 2019 paper unifies lattice surgery and code deformation as gauge fixing, and a 2024 paper designs switching between two- and three-dimensional color codes. These are protocol designs and simulations.
The gate-set problem
The 2019 New Journal of Physics paper explains that large-scale algorithms need basic operations implemented fault-tolerantly. The popular approach uses stabilizer codes embedded in a planar layout, but the set of fault-tolerant operations using unitary gates is typically very limited. This led to measurement-based schemes, called lattice surgery and code deformation, and separately to gauge fixing, a measurement-based method for universal gate sets in subsystem stabilizer codes.
The 2024 PRX Quantum paper states the color-code version of the problem: neither a two-dimensional nor a three-dimensional color code provides the universal set of h, t and cnot, as the t gate is missing in 2D and the h gate in 3D. Switching between a 2D and 3D code while preserving the logical state may overcome both shortcomings.
Gauge fixing as a unifying view
The 2019 paper shows that lattice surgery and code deformation can be expressed as special cases of gauge fixing. This gives a simple and rigorous test for fault tolerance and simple guiding principles for implementing these operations. The authors check the method numerically with surface code examples, some of which are novel.
Subsystem lattice surgery
The 2017 Nature Communications paper notes that some topological codes have high noise thresholds suited to quantum memory, while others allow straightforward gate implementation. To use both, one must be able to switch between them. The authors propose subsystem lattice surgery, which needs only two-body nearest-neighbour interactions in a fixed layout in addition to error correction. It transfers quantum information fault-tolerantly between topological subsystem codes in two dimensions and beyond, and can create a quantum bus between surface code memories and color code processors. It is built on a formalisation of lattice surgery using operator quantum error correction and measurement-based computation, and can be understood as initialising and gauge fixing.
Color-code switching protocols
The 2024 paper builds resource-optimised deterministic and nondeterministic switching protocols for distance-three color codes using fault-tolerant circuits with flag qubits. Deterministic protocols implement logical gates on an encoded state, while nondeterministic ones can prepare magic states. Using error rates of state-of-the-art trapped-ion processors as a reference, they find a logical failure probability of 3 percent for deterministic gates that cannot be done transversally. Replacing the three-dimensional code in magic state preparation with a morphed code reduces logical failure rates by two orders of magnitude, which the authors call a viable method for near-term magic state preparation.
What this does not tell you
The failure rates are simulation results at distance three under assumed error rates. They do not describe an operating machine.
FAQ
Why switch codes?
Different codes offer different gates, and no single one gives a simple universal set.
What is gauge fixing?
A measurement-based method for universal gate sets in subsystem stabilizer codes.
How does it relate to lattice surgery?
A 2019 paper shows lattice surgery and code deformation are special cases of it.
What does subsystem lattice surgery connect?
Surface code memories and color code processors, per a 2017 paper.
Has this been built?
These papers give protocols and numerical results.
All field notes