Quantum computation is an inherently noisy process. Scalable quantum computers will require fault-tolerance to implement useful computation. There are many proposed approaches to this, but one promising candidate is the family of topological quantum error correcting codes.
Currently, the qiskit.ignis.verification.topological_codes module provides a general framework for QEC and implements one specific example, the repetition code.
For the hackathon, our team Erwin's Tigers implemented a surface code encoder and decoder for Qiskit Ignis. We hope that this implementation will be useful to other Qiskitters and will inspire others to continue building out the topological_codes module into a diverse family.
Inspired by the Qiskit Textbook, we've written a full set of jupyter notebook tutorials, which are the best way to get up to speed. They detail both the API and the gritty implementation details -- please check them out!
Surface codes are a type of CSS code, consisting of pairwise commuting X and Z stabilizers made of Pauli gates. It defines a logical state on a 2 by 2 lattice made of quantum bits with the stabilizers X1 X2 Z1 Z2.
The code is based on the earlier theoretical idea of a toric code, with periodic boundary conditions instead of open boundary conditions. This has been shown to be largely identical, but embedding a surface code on an actual device is much easier.
In general, we try to follow the existing structure of qiskit.ignis.verification.topological_codes. The code is implemented separately here but is able to easily be merged into Ignis.
There are two main interfaces — corresponding to the encoder and decoder, respectively:
SurfaceCode in surface_code.circuits
SurfaceCode(d, T) generates a QuantumCircuit for creating a logical state and measuring stabilizers. The class is parameterized with the code distance d (which should be odd) and the number of syndrome measurement rounds T (usually T = d). This class also handles parsing of the physical device readout into a form suitable for decoding. Please see the encoder tutorial for a full walkthrough.
GraphDecoder in surface_code.fitters
GraphDecoder(d, T) implements minimum-weight perfect matching (MWPM) on the syndrome measurements of the physical circuit. This class is similar to the existing GraphDecoder for repetition codes, but introduces a new framework to handle the 2D lattice.
Parsed readout from the device is used to generate graphs of error chains in time and space, which decode syndrome measurements into the most likely sequence of qubit flips over time. Please see the decoder tutorial for a full walkthrough.
The scope of the project is quite large, so we focused on completing a "minimum viable product" during the hackathon. However, there are many areas which we'd like to explore going forward. A few immediate ones:
- Expand
SurfaceCode(d, T).circuitsa full set of logical states (1, +, -) -- and ultimately logical gate operations for computation. - Full benchmark of the physical-error to logical-error probabilities to determine the error correction threshold
- More simulation runs: different X/Z error probabilities, more limited
coupling_map, etc. - Our MWPM matching already has the below improvements to the basic algorithm, but are there more?
- For a given pair of syndromes, there may be many possible error chains through space and time. We compute this "path degeneracy" and use it to re-weight the error probabilities.
- We cross-match X and Z errors to produce an overall Y error. However, this doesn't exactly match a depolarizing channel, so ideally the weights would be re-adjusted with conditional probabilities.
- Other approaches to error-chain matching (e.g. neural networks or tensor networks)?
- Our
GraphDecoderimplements two different approaches to syndrome graph generation. One is a "analytic" approach (much faster), and the other uses simulation to insert errors into the circuit. These produce slightly different syndrome graphs, but we get the same decoding results in our tests.
We would like to thank James Wootton for valuable suggestions and feedback. Our code closely follows his RepetitionCode structure in qiskit.ignis.verification.topological_codes, and his tutorials closely guided our initial explorations.
We'd also like to thank Doug McClure for advising us on helpful details of the IBM hardware.