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Journal of Mathematical Physics

Bounding the distance of quantum surface codes

Ethan Fetaya

Published Jun 1, 2012• 14 citations
Publisher

Abstract

Homological quantum codes (also called topological codes) are low density parity check error correcting codes that come from surfaces and higher dimension manifolds. Homological codes from surfaces, i.e., surface codes, have also been suggested as a possible way to construct stable quantum memory and fault-tolerant computation. It has been conjectured that all homological codes have a square root bound on there distance and therefore cannot produce good codes. This claim has been disputed in dimension four using the geometric property of systolic freedom. We will show in this paper that the conjecture holds in dimension two due to the negation of systolic freedom, i.e., systolic rigidity.

DOI
10.1063/1.4726034

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Homological quantum codes (also called topological codes) are low density parity check error correcting codes that come from surfaces and higher dimension manifolds. Homological codes from surfaces, i.e., surface codes, have also been suggested as a possible way to construct stable quantum memory and fault-tolerant computation. It has been conjectured that all homological codes have a square root bound on there distance and therefore cannot produce good codes. This claim has been disputed in dimension four using the geometric property of systolic freedom. We will show in this paper that the conjecture holds in dimension two due to the negation of systolic freedom, i.e., systolic rigidity.
Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new homological quantum error correcting codes. They are low density parity check codes with linear rate and distance n [superscript ε]. Their rate is evaluated via Euler characteristic arguments and their distance using Z[subscript 2] -systolic geometry. This construction answers a question of Zémor [“On Cayley graphs, surface codes, and the limits of homological coding for quantum error correction,” in Proceedings of Second International Workshop on Coding and Cryptology (IWCC), (Lecture Notes in Computer Science). Vol. 5557 (2009), pp. 259–273], who asked whether homological codes with such parameters could exist at all.
Quantum computation has shown advantages in several problems over the corresponding classical algorithms. The noisy intermediate-size\n quantum devices with dozens of qubits make it more promising in the\n past decade, as the quantum advantage was demonstrated\n experimentally. Before entering the era of scalable quantum\n computation, one has to resolve the errors in a quantum many-body\n system, which is inevitable due to the environment interaction during\n quantum control processes. The goal of quantum error correction is to\n reduce such errors and increase decoherence time. The most successful candidates of quantum error correction codes are topological codes, especially surface codes, which were discovered by\n Alexei Kitaev. The ordered phase of the 2D\n Ising model on a torus ensures that toric codes are fault-tolerant\n below a critical error probability. Other than the FT threshold,\n the topological codes feature efficient encoding and\n decoding, local measurements, but suffer from asymptotically zero\n code rate. To encode more logical qubits with finite resources, one can loosen the condition on locality and extend to a broader class of quantum\n low-density parity-check (LDPC)codes. There are many known algebraic\n constructions for such codes, but only a precious few of them have\n finite code rates and meet the fault-tolerant condition:\n the stabilizer weight is bounded and the distance scales at least\n logarithmically with the code size. In this thesis, I construct the\n higher-dimensional quantum hypergraph product (HQHP) codes, which\n generalize quantum hypergraph product (QHP) codes and toric codes in\n all dimensions. The HQHP codes projected into a single space gives\n subsystem product codes, which can then be gauge fixed to\n concatenated codes or homological product codes. Those include some\n common CSS codes, like Shor’s codes, Bacon Shor’s codes, and\n subsystem QHP codes. The HQHP codes can be mapped to the tensor\n product of chain complexes, which provides an algebraic framework to\n construct quantum LDPC codes with finite code rates, square root\n distances, FT thresholds and single-shot properties with redundant\n checks. Meanwhile, their rich connection to other codes are very\n instructive and may lead to further optimizations. Regarding the\n remained procedures towards fault-tolerant universal quantum\n computation through quantum LDPC codes, I will discuss the\n fault-tolerant condition for each code, including distance,\n stabilizer weight, decoding, fault-tolerant gates and measurement\n protocols.\n
Quantum codes with low-weight stabilizers known as LDPC codes have been actively studied recently due to their simple syndrome readout circuits and potential applications in fault-tolerant quantum computing. However, all families of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good families of LDPC codes are known that combine constant encoding rate and linear distance. Here we propose the first family of good quantum codes with low-weight stabilizers. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most n\sqrt{n}n​ qubits, where nnn is the code length. For comparison, all previously known families of good quantum codes have stabilizers of linear weight. Our proof combines two techniques: randomized constructions of good quantum codes and the homological product operation from algebraic topology. We conjecture that similar methods can produce good stabilizer codes with stabilizer weight nan^ana for any a>0a>0a>0. Finally, we apply the homological product to construct new small codes with low-weight stabilizers.
Quantum codes with low-weight stabilizers known as LDPC codes have been actively studied recently due to their potential applications in fault-tolerant quantum computing. However, all families of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good families of LDPC codes are known that combine constant encoding rate and linear distance. Here we propose the first family of good quantum codes with low-weight stabilizers. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most O(√n) qubits, where n is the code length. For comparison, all previously known families of good quantum codes have stabilizers of linear weight. Our proof combines two techniques: randomized constructions of good quantum codes and the homological product operation from algebraic topology. We conjecture that similar methods can produce good stabilizer codes with stabilizer weight O(nα) for any α > 0.
Quantum error correction (QEC) is crucial for realizing scalable quantum technologies, and topological quantum error correction (TQEC) has emerged as the most experimentally advanced paradigm of QEC. Existing homological and topological code constructions, however, are largely confined to orientable two-manifolds with simple boundary conditions. In this work, we develop a unified algebraic-topological framework for TQEC based on homology, cohomology, and intersection theory, which characterizes exactly when an arbitrary-dimensional manifold (with or without boundary) can serve as a quantum memory, thereby extending the standard 2D homological-code picture to arbitrary dimension and to manifolds with boundary via Poincaré-Lefschetz duality. Building on this classification, we introduce concrete code families that exploit nontrivial topology beyond the planar and toric settings. These include ``3-torus code'' and higher-dimensional ``volume codes'' on compact manifolds with mixed XXX- and ZZZ-type boundaries. We further give a topological construction of qudit TQEC codes on general two-dimensional cell complexes using group presentation complexes, which unifies and extends several known quantum LDPC and homological-product-like constructions within a single geometric language. Finally, we combine the theoretical framework with numerical simulations to demonstrate that changing only the global topology can yield improved logical performance at fixed entanglement resources. Taken together, our results provide a systematic set of topological design principles for constructing and analyzing TQEC codes across dimensions and boundaries, and they open new avenues for topology-aware fault-tolerant quantum architectures.