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Efficient Decoding of Topological Color Codes
Pradeep Sarvepalli, Robert Raussendorf
TL;DR
Color codes offer transversal encoded gates, while efficient decoding remains an important research direction. This paper presents a recursive decoder that directly decodes the hexagonal-lattice color code and reports a 7.8% error threshold for independent spin and phase flip errors.
Problem
The paper reports no previously established threshold for the hexagonal-lattice color code, in a setting where efficient color-code decoding is an active research direction.
Source: §I · p. 1Method
A recursive decoder applies message passing directly to the color code, using rescaling and soft decisions for syndrome splitting and lattice rescaling.
Source: §I · p. 1, A. Rescaling · p. 2, A. Soft splitting of syndrome · p. 4Results
7.8% error threshold for independent spin and phase flip errors, assuming perfect syndrome extraction.
Source: §V · p. 7Takeaways & Limitations
The work reports a threshold value for the hexagonal-lattice color code, for which none had previously been reported.
Source: §I · p. 1The reported threshold assumes perfect syndrome extraction.
Source: §V · p. 7
Abstract
from arXiv · showhide
Color codes are a class of topological quantum codes with a high error threshold and large set of transversal encoded gates, and are thus suitable for fault tolerant quantum computation in two-dimensional architectures. Recently, computationally efficient decoders for the color codes were proposed. We describe an alternate efficient iterative decoder for topological color codes, and apply it to the color code on hexagonal lattice embedded on a torus. In numerical simulations, we find an error threshold of 7.8% for independent dephasing and spin flip errors.
I. INTRODUCTION
Color codes offer a larger group of transversal or topological gates than surface codes, but their decoding can involve hypergraph matching, for which no polynomial-time algorithm is known. This paper presents a recursive decoder that directly decodes color codes and reports a 7.8% threshold on the hexagonal lattice.
I. INTRODUCTION
- Color codes allow a larger group of transversal or topological gates than surface codes, potentially reducing operational overhead.Surface codes have a small set of gates realizable transversally or by code deformation; other gates invoke magic state distillation.Source: §I · p. 1
- Color-code error correction can be formulated as hypergraph matching, for which no polynomial-time algorithm is known, unlike graph matching for surface codes.Approximate polynomial-time solutions have been proposed.Source: §I · p. 1
- 7.8% is the reported error threshold for the hexagonal-lattice color code under independent dephasing and spin flip errors.The paper notes that no threshold value had previously been reported for this lattice.Source: §I · p. 1
- The proposed recursive decoder uses message passing and decodes the color code directly, without mapping it to two surface codes.An earlier approach maps the square-octagon color code to two surface codes and decodes them recursively [18].Source: §I · p. 1
A. Topological color codes
- A color code uses a trivalent graph embedded on a surface with 3-colorable faces; the paper studies a hexagonal lattice embedded on a torus.This choice gives four encoded qubits, with logical operators associated with nontrivial torus cycles and carrying a color label.Source: A. Topological color codes · pp. 1–2
B. The dual lattice
The decoder represents the color code on a dual lattice, where checks are vertices and qubits are triangular faces. Its rescaling transformation maps the code into a smaller copy for iterative decoding.
B. The dual lattice
- The dual-lattice representation places checks at vertices and qubits on triangular faces, with logical errors corresponding to nontrivial face cycles.The code is considered on a regular hexagonal lattice embedded on a torus.Source: Figure 2 · p. 2, B. The dual lattice · p. 2
A. Rescaling
- Rescaling maps the code into a smaller copy by decoding blocks locally and replacing each block with a logical qubit.Interior and non-corner boundary checks are used at each block; corner checks are retained for later decoding levels.Source: A. Rescaling · p. 2
- A triangular cell containing n^2 qubits and n(n + 1)/2 checks can be rescaled if and only if its side length n is not divisible by 3.When n is divisible by 3, the three corners have the same color and no error pattern supported within the cell can flip all three corners simultaneously.Source: A. Rescaling · p. 3
- For a cell with side length not divisible by 3, errors can connect same-color checks, allowing the corner checks to be flipped by an operator acting as the cell’s logical operator.A single interior-qubit error flips three checks of different colors.Source: A. Rescaling · p. 3
B. Splitting the syndrome
Decoding locally requires partitioning each shared boundary syndrome between adjacent cells. The decoder estimates split probabilities, exchanges messages to make neighboring estimates consistent, and iterates splits until stable or for a predetermined number of rounds.
B. Splitting the syndrome
- Each shared non-corner boundary syndrome is split between adjacent cells so each cell can be decoded locally.For a measured shared syndrome s, the top and bottom cells contribute st and sl such that st + sl = s.Source: B. Splitting the syndrome · p. 3
- The decoder computes conditional probabilities for the shared-syndrome split, including the probability that the top cell’s contribution is 1.It similarly computes the lower cell’s split probability.Source: B. Splitting the syndrome · p. 4
- Neighboring cells exchange messages to make their split-probability estimates consistent while enforcing that the contributions sum to the measured syndrome.Once conditional probabilities are obtained, the decoder chooses a split and repeats until splits stabilize or a predetermined number of rounds is reached.Source: B. Splitting the syndrome · p. 4
C. Decoding the basic cell
The basic cell is decoded locally with a complete lookup table, while corner syndromes are deferred to later levels. Constant-size cells keep the overall decoding complexity at O(n log^2 n).
C. Decoding the basic cell
- Each basic cell is decoded optimally with a complete lookup table using its internal syndrome and split boundary syndrome.This accounts for interior and boundary checks, but not corner checks.Source: C. Decoding the basic cell · p. 4
- Corner checks are carried to the next level to account for errors in the current-stage estimates.The basic-cell decoder does not account for these checks.Source: C. Decoding the basic cell · p. 4
- O(n log^2 n) is the overall decoding complexity, with O(n) work at each level.The number of cells is O(n/x^2) for code length n = 18·x^2m and cell size x.Source: C. Decoding the basic cell · p. 4
D. Rescaling the lattice
Rescaling establishes a reference split of shared syndromes, applies the corresponding canonical estimate, and passes an error probability to the next decoding level.
D. Rescaling the lattice
- The reference splitting can be chosen randomly or by selecting the most likely split for each shared syndrome.The most-likely choice is deterministic: identical split probabilities return the same reference splitting.Source: D. Rescaling the lattice · p. 4
- For a given syndrome, the rescaling algorithm applies its corresponding canonical estimate.The block decoder also returns an estimate and its error probability.Source: D. Rescaling the lattice · p. 4
- The rescaled-qubit error probability compares the probabilities of the complementary error e + X and estimate e, conditioned on the syndrome.This probability is passed as the prior to the next level.Source: D. Rescaling the lattice · p. 4
A. Soft splitting of syndrome
Soft syndrome splitting retains split probabilities across iterations instead of immediately rounding them to hard choices. The probabilities are conditioned on the measured syndrome and initialized from local error probabilities.
A. Soft splitting of syndrome
- The soft-decision method uses a recursion for split probabilities instead of rounding them to 0 or 1 in each iteration.The hard-decision heuristic discards information by immediately rounding the computed probabilities.Source: A. Soft splitting of syndrome · p. 4
- For measured syndrome s = 0, the top split probability is updated while enforcing consistency with the measured syndrome.The update uses the current top and lower split probabilities.Source: A. Soft splitting of syndrome · p. 5
- Initial split probabilities come from odd-error probabilities in the relevant cell regions, then are conditioned on the measured syndrome.The initial probabilities are computed for the upper region and lower cell.Source: A. Soft splitting of syndrome · p. 5
B. Soft rescaling of the lattice
Soft rescaling combines information across possible syndrome splittings rather than relying only on a hard choice. Stabilizer operations relate estimates across splittings, allowing the rescaled error probability to be updated with that uncertainty in view.
B. Soft rescaling of the lattice
- The method uses all available splitting information because the correct splitting is not known for sure.Hard-decision rescaling is described as non-optimal.Source: B. Soft rescaling of the lattice · p. 5
- Stabilizer generators canonically relate estimates across splittings because applying a stabilizer can flip the corresponding check splitting.The estimate for a changed splitting is obtained by applying the half-stabilizers that flip one splitting into the other.Source: B. Soft rescaling of the lattice · p. 5
- The rescaled-qubit error probability for a candidate splitting is conditioned on both that splitting and the estimate and correction made under the reference splitting.The relevant stabilizer restriction to the cell determines whether the erroneous splitting requires a correction at the next level.Source: B. Soft rescaling of the lattice · p. 5
- The rescaled error probability is independent of the reference splitting.The rule combines contributions associated with different splittings.Source: B. Soft rescaling of the lattice · p. 5
C. Looking ahead at the corners
The decoder looks ahead at corner syndromes to adjust qubit error priors in small cells. This addresses malicious error patterns that could otherwise persist through higher recursion levels.
C. Looking ahead at the corners
- In small cells, malicious error patterns could otherwise persist with appreciable probability at higher recursion levels.The algorithm therefore requires a slight modification for these cells.Source: C. Looking ahead at the corners · p. 6
- Taking corner syndromes into account ahead of time modifies the qubit priors to direct the decoder toward correcting these errors.Each qubit in a corner check belongs to a different cell.Source: C. Looking ahead at the corners · p. 6
- The corner-qubit error probability is updated using its interior probability, the corner syndrome, and the external qubits’ contribution.The passage denotes the interior-qubit probability as p_i and the external contribution as p_e.Source: C. Looking ahead at the corners · p. 6
D. Belief propagation for prior update
The decoder can refine qubit-error priors with belief propagation, but the color-code Tanner graph’s many 4-cycles argue against excessive iterations. Its recursive procedure initializes and reconciles split probabilities, decodes locally, and propagates estimates across levels.
D. Belief propagation for prior update
- Belief propagation updates qubit-error priors, but the color-code Tanner graph’s smaller girth and more 4-cycles make too many iterations inadvisable.The passage describes this as a recommendation about updating the priors.Source: D. Belief propagation for prior update · p. 6
- Figure 10 plots logical error rate against physical error probability for independent spin-flip errors across codes with m = 2, 3, 4, and 5.The caption identifies the code family as [[18 · 4m, 4, 2m+2]].Source: Figure 10 · p. 6
- The algorithm takes a syndrome and channel error rate, updates initial qubit probabilities, and computes boundary split probabilities from qubits in each cell.It sends p or 1 − p to a neighboring cell according to whether the shared syndrome is 0 or 1.Source: D. Belief propagation for prior update · p. 6
- Adjacent cells enforce consistency by combining a cell’s split-probability estimate with the neighboring cell’s message.The stated expression is p_l p_r / (p_l p_r + (1 − p_l)(1 − p_r)).Source: D. Belief propagation for prior update · p. 6
- The recursive procedure updates corner-qubit probabilities, sends messages, splits syndromes, decodes locally, and estimates errors at the next level.It repeats to level 0, uses a lookup table for the final error estimate, and propagates the error to lower levels.Source: D. Belief propagation for prior update · p. 6
V. CONCLUSIONS
The paper presents a recursive decoder for the color code on a hexagonal lattice. Under perfect syndrome extraction, it reports a 7.8% threshold for independent spin- and phase-flip errors.