Recent work on fault-tolerant quantum computation making use of topological error correction shows great potential, with the 2d surface code possessing a threshold error rate approaching 1\%. However, the 2d surface code requires the use of a complex state distillation procedure to achieve universal quantum computation. The color code of is a related scheme partially solving the problem, providing a means to perform all Clifford group gates transversally. We review the color code and its error correcting methodology, discussing one approximate technique based on graph matching. We derive an analytic lower bound to the threshold error rate of 6.25\% under error-free syndrome extraction, while numerical simulations indicate it may be as high as 13.3\%. Inclusion of faulty syndrome extraction circuits drops the threshold to approximately 0.10 \pm 0.01\%.Recent work on fault-tolerant quantum computation making use of topological error correction shows great potential, with the 2d surface code possessing a threshold error rate approaching 1\%. However, the 2d surface code requires the use of a complex state distillation procedure to achieve universal quantum computation. The color code of is a related scheme partially solving the problem, providing a means to perform all Clifford group gates transversally. We review the color code and its error correcting methodology, discussing one approximate technique based on graph matching. We derive an analytic lower bound to the threshold error rate of 6.25\% under error-free syndrome extraction, while numerical simulations indicate it may be as high as 13.3\%. Inclusion of faulty syndrome extraction circuits drops the threshold to approximately 0.10 \pm 0.01\%.

Quantum Information and ComputationSep 1, 201045 citations

Threshold error rates for the toric and surface codes

D. S. Wang, Austin G. Fowler, Ashley M. Stephens +1 more

The surface code scheme for quantum computation features a 2d array of nearest-neighbor coupled qubits yet claims a threshold error rate approaching 1% (NJoP 9:199, 2007). This result was obtained for the toric code, from which the surface code is derived, and surpasses all other known codes restricted to 2d nearest-neighbor architectures by several orders of magnitude. We describe in detail an error correction procedure for the toric and surface codes, which is based on polynomial-time graph matching techniques and is efficiently implementable as the classical feed-forward processing step in a real quantum computer. By direct simulation of this error correction scheme, we determine the threshold error rates for the two codes (differing only in their boundary conditions) for both ideal and non-ideal syndrome extraction scenarios. We verify that the toric code has an asymptotic threshold of p = 15.5% under ideal syndrome extraction, and p = 7.8 10^-3 for the non-ideal case, in agreement with prior work. Simulations of the surface code indicate that the threshold is close to that of the toric code.The surface code scheme for quantum computation features a 2d array of nearest-neighbor coupled qubits yet claims a threshold error rate approaching 1% (NJoP 9:199, 2007). This result was obtained for the toric code, from which the surface code is derived, and surpasses all other known codes restricted to 2d nearest-neighbor architectures by several orders of magnitude. We describe in detail an error correction procedure for the toric and surface codes, which is based on polynomial-time graph matching techniques and is efficiently implementable as the classical feed-forward processing step in a real quantum computer. By direct simulation of this error correction scheme, we determine the threshold error rates for the two codes (differing only in their boundary conditions) for both ideal and non-ideal syndrome extraction scenarios. We verify that the toric code has an asymptotic threshold of p = 15.5% under ideal syndrome extraction, and p = 7.8 10^-3 for the non-ideal case, in agreement with prior work. Simulations of the surface code indicate that the threshold is close to that of the toric code.

arXiv:0905.0531May 5, 200917 citations

Efficient fault-tolerant decoding of topological color codes

Ashley M. Stephens

Topological color codes defined by the 4.8.8 semiregular lattice feature geometrically local check operators and admit transversal implementation of the entire Clifford group, making them promising candidates for fault-tolerant quantum computation. Recently, several efficient algorithms for decoding the syndrome of color codes were proposed. Here, we modify one of these algorithms to account for errors affecting the syndrome, applying it to the family of triangular 4.8.8 color codes encoding one logical qubit. For a three-dimensional bit-flip channel, we report a threshold error rate of 0.0208(1), compared with 0.0305(4) previously reported for an integer-program-based decoding algorithm. When we account for circuit details, this threshold is reduced to 0.00143(1) per gate, compared with 0.00672(1) per gate for the surface code under an identical noise model.Topological color codes defined by the 4.8.8 semiregular lattice feature geometrically local check operators and admit transversal implementation of the entire Clifford group, making them promising candidates for fault-tolerant quantum computation. Recently, several efficient algorithms for decoding the syndrome of color codes were proposed. Here, we modify one of these algorithms to account for errors affecting the syndrome, applying it to the family of triangular 4.8.8 color codes encoding one logical qubit. For a three-dimensional bit-flip channel, we report a threshold error rate of 0.0208(1), compared with 0.0305(4) previously reported for an integer-program-based decoding algorithm. When we account for circuit details, this threshold is reduced to 0.00143(1) per gate, compared with 0.00672(1) per gate for the surface code under an identical noise model.

arXiv:1402.3037Feb 13, 201415 citations

Threshold error rates for the toric and planar codes

D.S. Wang, Austin G. Fowler, Ashley M. Stephens +1 more

The planar code scheme for quantum computation features a 2d array of nearest-neighbor coupled qubits yet claims a threshold error rate approaching 1\%. This result was obtained for the toric code, from which the planar code is derived, and surpasses all other known codes restricted to 2d nearest-neighbor architectures by several orders of magnitude. We describe in detail an error correction procedure for the toric and planar codes, which is based on polynomial-time graph matching techniques and is efficiently implementable as the classical feed-forward processing step in a real quantum computer. By applying one and two qubit depolarizing errors of equal probability $p$ , we determine the threshold error rates for the two codes (differing only in their boundary conditions) for both ideal and non-ideal syndrome extraction scenarios. We verify that the toric code has an asymptotic threshold of $p_{\textrm{th}} = 15.5\%$ under ideal syndrome extraction, and $p_{\textrm{th}} = 7.8 \times 10^{-3}$ for the non-ideal case, in agreement with \cite{Raus07d}. Simulations of the planar code indicate that the threshold is close to that of the toric code.The planar code scheme for quantum computation features a 2d array of nearest-neighbor coupled qubits yet claims a threshold error rate approaching 1\%. This result was obtained for the toric code, from which the planar code is derived, and surpasses all other known codes restricted to 2d nearest-neighbor architectures by several orders of magnitude. We describe in detail an error correction procedure for the toric and planar codes, which is based on polynomial-time graph matching techniques and is efficiently implementable as the classical feed-forward processing step in a real quantum computer. By applying one and two qubit depolarizing errors of equal probability $p$, we determine the threshold error rates for the two codes (differing only in their boundary conditions) for both ideal and non-ideal syndrome extraction scenarios. We verify that the toric code has an asymptotic threshold of $p_{\textrm{th}} = 15.5\%$ under ideal syndrome extraction, and $p_{\textrm{th}} = 7.8 \times 10^{-3}$ for the non-ideal case, in agreement with \cite{Raus07d}. Simulations of the planar code indicate that the threshold is close to that of the toric code.

Quantum Information and ComputationMay 1, 201090 citations

Low-Overhead Magic State Distillation with Color Codes

Seok-Hyung Lee, Felix Thomsen, Nicholas Fazio +2 more

Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty T measurement, achieving infidelities of O ( p 3 ) for physical noise strength p . To achieve lower infidelities, our second scheme integrates distillation with “cultivation” (a distillation-free approach to fault tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., approximately 2 × 10 − 16 at p = 10 − 3 ), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately 2 orders of magnitude fewer resources than the previous best magic-state-distillation schemes for color codes.Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty T measurement, achieving infidelities of O ( p 3 ) for physical noise strength p . To achieve lower infidelities, our second scheme integrates distillation with “cultivation” (a distillation-free approach to fault tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., approximately 2 × 10 − 16 at p = 10 − 3 ), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately 2 orders of magnitude fewer resources than the previous best magic-state-distillation schemes for color codes.

arXiv:2409.07707PRX QuantumMay 30, 202511 citations

High-threshold decoding of non-Pauli codes for 2D universality

Julio C. Magdalena de la Fuente, Noa Feldman, Jens Eisert +1 more

Topological codes have many desirable properties that allow fault-tolerant quantum computation with relatively low overhead. A core challenge for these codes, however, is to achieve a low-overhead universal gate set with limited connectivity. In this work, we explore a non-Pauli stabilizer code that can be used to complete a universal gate set on topological toric and surface codes in strictly two dimensions. Fault-tolerant syndrome extraction for the non-Pauli code requires mid-circuit $X$ corrections, a key difference to conventional Pauli codes. We construct and benchmark a just-in-time (JIT) matching decoder to reliably decide these corrections. Under a phenomenological error model with equally likely physical and measurement errors, we find a high threshold of $\approx 2.5\,\%$ , close to the $\approx 2.9\,\%$ of a decoder with access to the full syndrome history. We also perform a finite-size scaling analysis to estimate how the logical error rate scales below threshold and verify an exponential suppression in both physical error rate and in the system size. A second global decoding step for $Z$ errors is required and the non-Clifford gates in the circuit reduce the threshold from $\approx 2.9\,\%$ to $\approx 1.8\,\%$ with a naive decoder. We show how $Z$ decoding can be improved using knowledge of the $X$ corrections, pushing the threshold to $\approx 2.2\,\%$ . Our results suggest non-Clifford logic in 2D codes could perform comparably to 2D quantum memory. Our formalism for efficient benchmarking and decoding directly generalizes to a broader family of CSS codes whose $X$ stabilizers are twisted by diagonal Clifford operators, and spacetime versions thereof, defined by CSS-like circuits enriched by $CCZ$ , $CS$ , and $T$ gates.Topological codes have many desirable properties that allow fault-tolerant quantum computation with relatively low overhead. A core challenge for these codes, however, is to achieve a low-overhead universal gate set with limited connectivity. In this work, we explore a non-Pauli stabilizer code that can be used to complete a universal gate set on topological toric and surface codes in strictly two dimensions. Fault-tolerant syndrome extraction for the non-Pauli code requires mid-circuit $X$ corrections, a key difference to conventional Pauli codes. We construct and benchmark a just-in-time (JIT) matching decoder to reliably decide these corrections. Under a phenomenological error model with equally likely physical and measurement errors, we find a high threshold of $\approx 2.5\,\%$, close to the $\approx 2.9\,\%$ of a decoder with access to the full syndrome history. We also perform a finite-size scaling analysis to estimate how the logical error rate scales below threshold and verify an exponential suppression in both physical error rate and in the system size. A second global decoding step for $Z$ errors is required and the non-Clifford gates in the circuit reduce the threshold from $\approx 2.9\,\%$ to $\approx 1.8\,\%$ with a naive decoder. We show how $Z$ decoding can be improved using knowledge of the $X$ corrections, pushing the threshold to $\approx 2.2\,\%$. Our results suggest non-Clifford logic in 2D codes could perform comparably to 2D quantum memory. Our formalism for efficient benchmarking and decoding directly generalizes to a broader family of CSS codes whose $X$ stabilizers are twisted by diagonal Clifford operators, and spacetime versions thereof, defined by CSS-like circuits enriched by $CCZ$, $CS$, and $T$ gates.

quant-pharXiv:2604.02033Apr 3, 20260 citations

p

, we determine the threshold error rates for the two codes (differing only in their boundary conditions) for both ideal and non-ideal syndrome extraction scenarios. We verify that the toric code has an asymptotic threshold of

p_{\textrm{th}} = 15.5\%

under ideal syndrome extraction, and

p_{\textrm{th}} = 7.8 \times 10^{-3}

for the non-ideal case, in agreement with \cite{Raus07d}. Simulations of the planar code indicate that the threshold is close to that of the toric code.

X

corrections, a key difference to conventional Pauli codes. We construct and benchmark a just-in-time (JIT) matching decoder to reliably decide these corrections. Under a phenomenological error model with equally likely physical and measurement errors, we find a high threshold of

\approx 2.5\,\%

, close to the

\approx 2.9\,\%

of a decoder with access to the full syndrome history. We also perform a finite-size scaling analysis to estimate how the logical error rate scales below threshold and verify an exponential suppression in both physical error rate and in the system size. A second global decoding step for

Z

errors is required and the non-Clifford gates in the circuit reduce the threshold from

\approx 2.9\,\%

\approx 1.8\,\%

with a naive decoder. We show how

Z

decoding can be improved using knowledge of the

X

corrections, pushing the threshold to

\approx 2.2\,\%

. Our results suggest non-Clifford logic in 2D codes could perform comparably to 2D quantum memory. Our formalism for efficient benchmarking and decoding directly generalizes to a broader family of CSS codes whose

X

stabilizers are twisted by diagonal Clifford operators, and spacetime versions thereof, defined by CSS-like circuits enriched by

CCZ

CS

, and

T

gates.