Quantum expander code[1]
Alternative names: Quantum Sipser-Spielman code.
Description
CSS code constructed from a hypergraph product of bipartite expander graphs [2] with bounded left and right vertex degrees. For every bipartite graph there is an associated matrix (the parity check matrix) with columns indexed by the left vertices, rows indexed by the right vertices, and 1 entries whenever a left and right vertex are connected. This matrix can serve as the parity check matrix of a classical code. Two bipartite expander graphs can be used to construct a quantum CSS code (the quantum expander code) by using the parity check matrix of one as \(X\) checks, and the parity check matrix of the other as \(Z\) checks.Rate
\([[n,k=\Theta(n),d=O(\sqrt{n})]]\) code with asymptotically constant rate.Decoding
Small set-flip linear-time decoder, which corrects order \(\Omega(n^{1/2})\) adversarial errors [1].Log-time decoder [3].Constant-time decoder [4].2D geometrically local syndrome extraction circuits acting on a patch of \(N\) physical qubits have to be of depth of order \(\Omega(n/\sqrt{N})\) or deeper. More generally, there is a tradeoff between the depth \(D\) and width \(W\) of a syndrome extraction circuit, namely, \(D \geq n/\sqrt{W}\) [5].Fault Tolerance
Fault-tolerance with constant overhead can be achieved [3].Threshold
Locally stochastic noise: \(2.7 \cdot 10^{-16}\) [6].Cousins
- Expander code
- Topological code— Quantum expander codes realize topological quantum spin glass order [7].
- Self-correcting quantum code— Constant-rate random (quantum) expander codes are self-correcting (quantum) memories, but have no thermodynamic phase transitions [8].
Member of code lists
- Hamiltonian-based codes
- Quantum codes
- Quantum codes based on homological products
- Quantum codes with a rate
- Quantum codes with fault-tolerant gadgets
- Quantum codes with notable decoders
- Quantum codes with other thresholds
- Quantum CSS codes
- Quantum LDPC codes
- Self-correcting quantum codes and friends
- Single-shot codes
- Stabilizer codes
Primary Hierarchy
Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Fiber-bundle codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Hypergraph product (HGP) codeCSS Generalized homological-product Lattice stabilizer QLDPC Stabilizer Hamiltonian-based Qubit QECC Quantum
Parents
Galois-qudit expander codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Quantum expander codes are single-shot [3].
Quantum expander code
References
- [1]
- A. Leverrier, J.-P. Tillich, and G. Zemor, “Quantum Expander Codes”, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science 810 (2015) arXiv:1504.00822 DOI
- [2]
- S. Hoory, N. Linial, and A. Wigderson, “Expander graphs and their applications”, Bulletin of the American Mathematical Society 43, 439 (2006) DOI
- [3]
- O. Fawzi, A. Grospellier, and A. Leverrier, “Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes”, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) 743 (2018) arXiv:1808.03821 DOI
- [4]
- A. Grospellier. Constant time decoding of quantum expander codes and application to fault-tolerant quantum computation. PhD thesis, Inria Paris (2019).
- [5]
- N. Delfosse, M. E. Beverland, and M. A. Tremblay, “Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes”, (2021) arXiv:2109.14599
- [6]
- O. Fawzi, A. Grospellier, and A. Leverrier, “Efficient decoding of random errors for quantum expander codes”, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing 521 (2018) arXiv:1711.08351 DOI
- [7]
- B. Placke, T. Rakovszky, N. P. Breuckmann, and V. Khemani, “Topological Quantum Spin Glass Order and its realization in qLDPC codes”, (2024) arXiv:2412.13248
- [8]
- Y. Hong, J. Guo, and A. Lucas, “Quantum memory at nonzero temperature in a thermodynamically trivial system”, Nature Communications 16, (2025) arXiv:2403.10599 DOI
Page edit log
- Victor V. Albert (2022-09-23) — most recent
- Victor V. Albert (2021-12-16)
- Nolan Coble (2021-12-03)
Cite as:
“Quantum expander code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_expander