00mjk / qiskit-surface-codes

QISKit implementation of surface codes

Surface Codes and Error Correction

For a general overview without too many technical requirements, see the

• Medium article

A good introductory video on closely related work at IBM can be found here:

• A graph-based formalism for surface codes and twists

For an introduction on how to use this library see the following notebook:

• Surface Code Tutorial II

Current Capabilities

This package implements general "surface codes" which generalize A. Kitaev's toric code for quantum error correction pictured below (see pg. 67 of [1]).

A more general example of a graph embedded on a torus that gives a surface code can be seen below (taken from [2]),

However, our construction is even more general than this and allows arbitrary graphs on surfaces of arbitrary genus. In particular, any code specified by a graph cellularly embedded in a compact Riemann surface can be constructed using this package. The topological interpretation can be summed up in the following table (pg. 68 [1]):

Obviously error correction is necessary for fault tolerant quantum computing and surface codes are one of the best studied, most robust, and most easily implemented types of error correction. Furthermore, they only require nearest neighbor qubit interaction (gates) for implementation. The surface codes implemented in this package include the toric code as well as the hyperbolic surface codes described in:

• Constructions and Noise Threshold of Hyperbolic Surface Codes
• Homological Quantum Codes Beyond the Toric Code

These codes provide a higher error rate threshold than the standard toric codes, which have an error threshold of around 1%. The codes can be transpiled in QISKit and run on any IBM hardware backend, or they can be run on specialized hardware for hyperbolic surface code such as the hardware constructed with around 150 qubits in:

• Hyperbolic Lattices in Circuit Quantum Electrodynamics

and used in the simulation of hyperbolic spaces in

• Quantum Simulation of Hyperbolic Space with Circuit Quantum Electrodynamics: From Graphs to Geometry

With error rates on current hardware now reaching the 0.1% range and error thesholds well above 1% using surface codes, the implementation of error correction using surface codes is both timely and necessary for near term applications and will provide fault tolerance on most current hardware. The primary reference for implementing error correction protocols and devloping applications to condensed matter physics and spin-glass models is:

• Quantum Computation with Topological Codes: from qubit to topological fault-tolerance

Future Applications

Future applications will involve:

• modeling zeros of L-functions given by partition functions of lattice (or graph) Ising type models in order to understand the behavior of prime numbers and RSA public key cryptography and elliptic curve cryptography,
• understanding the Yang-Mills mass gap problem through modeling lattice gauge theories using surface codes,
• modeling arbitrage and cryptocurrency economies using lattice gauge theory,
• training machine learning models on simulations of these models to predict arbitrage vulnerabilities and potential cryptocurrency instabilities
• training machine learning models on exotic quantum hardware architecture to learn and improve error correction protocols and logical gate implementations on non-standard surface codes that go beyond simple grid qubit layouts on standard quantum hardware,
• modeling quantum gravity(ies) and the AdS/CFT correspondecne of J. Maldecena
• modeling quantum dynamical systems to better understand quantum chaos, quantum complexity, and phase transitions in order to better understand molecular dynamics
• using simulations of molecular dynamics to to train machine learning models for drug discovery and materials discovery
• using molecular dynamics to understand complex quantum phenomena such as protein folding, molecular biology processes, genetics for CRISPR, and synthetic biology

Companion Work

This package will be a part of a larger whole on error correction. The error correction software will also implement graph states and basic applications of graph states such as:

• encoding arbitrary stabilizer codes as graph states
• Measurement Based Quantum Computing (MBQC)
• Bipartite graph states and quantum cryptography
• blind quantum computation
• modeling entanglement entropy of multipartite systems
• computing stabilizer generators of arbitrary stabilizer codes
• weighted graph states and generalized graph states with arbitrary (variational) controlled-U3 gates as entangling gates for quantum machine learning on weighted graphs and graph completion problems in network analysis and knowledge graphs

Currently some basic constructors of graph states based on arbitrary input graphs, as well as bipartite input graphs for MBQC, quantum cryptography, and blind quantum computation have been implemented, along with some basic measurement protocols. This should be ready to use in the very near future as well.

IBM has already implemented simple repetition codes and will likely implement some of the standard examples of stabilizer codes such as Calderbank-Shor-Steane codes (CSS-codes), so we will likely not focus on implementing those in their standard form. However, developing algorithms which convert these codes and arbitrary stabilizer codes efficiently into graph states or surface codes will likely be useful and is something we are likely to focus on in the future.

For supplementary material explaining some of the theoretical aspects of surface codes, see also the following notebook.