Surface codes are versatile quantum error-correcting codes known for their planar geometry, making them ideal for practical implementations. While the original proposal used PauliXor PauliZoperators in a square structure, these codes can be improved by rotating the lattice or incorporating a mix of generators in the XZZX variant. However, a comprehensive theoretical analysis of the logical error rate for these variants has been lacking. To address this gap, we present theoretical formulas based on recent advancements in understanding the weight distribution of stabilizer codes. For example, over an asymmetric channel with asymmetryA= 10 and a physical error ratep→ 0, we observe that the logical error rate asymptotically approachespL→ 10p2for the rotated [[9, 1, 3]] XZZX code andpL→ 18.3p2for the [[13, 1, 3]] surface code. Additionally, we observe a particular behavior regarding rectangular lattices in the presence of asymmetric channels. Our findings demonstrate that implementing both rotation and XZZX modifications simultaneously can lead to suboptimal performance. Thus, in scenarios involving a rectangular lattice, it is advisable to avoid using both modifications simultaneously.
Surface codes are versatile quantum error-correcting codes known for their planar geometry, making them ideal for practical implementations. While the original proposal used PauliXor PauliZoperators in a square structure, these codes can be improved by rotating the lattice or incorporating a mix of generators in the XZZX variant. However, a comprehensive theoretical analysis of the logical error rate for these variants has been lacking. To address this gap, we present theoretical formulas based on recent advancements in understanding the weight distribution of stabilizer codes. For example, over an asymmetric channel with asymmetryA= 10 and a physical error ratep→ 0, we observe that the logical error rate asymptotically approachespL→ 10p2for the rotated [[9, 1, 3]] XZZX code andpL→ 18.3p2for the [[13, 1, 3]] surface code. Additionally, we observe a particular behavior regarding rectangular lattices in the presence of asymmetric channels. Our findings demonstrate that implementing both rotation and XZZX modifications simultaneously can lead to suboptimal performance. Thus, in scenarios involving a rectangular lattice, it is advisable to avoid using both modifications simultaneously.
One of the main challenge for an efficient implementation of quantum information technologies is how to counteract quantum noise. Quantum error correcting codes are therefore of primary interest for the evolution towards quantum computing and quantum Internet. We here analyze the performance of surface codes, one of the most important class for practical implementations, on both symmetric and asymmetric quantum channels. We derive approximate expressions, confirmed by simulations, to evaluate the performance of surface codes and of XZZX codes, and provide a metric to assess the advantage of codes with respect to uncoded systems. Our findings allow to characterize the performance by means of analytical formulas of surface codes, like, for example, the [[13, 1, 3]], the [[23, 1, 3/5]], the [[33, 1, 3/7]], and the [[41, 1, 5]] surface codes.
The code-capacity threshold of a scalable quantum error correcting stabilizer code can be expressed as a thermodynamic phase transition of a corresponding random-bond Ising model. Here we study the XY and XZZX surface codes under phase-biased noise, px=py=pz/(2η) , with η≥1/2 , and total error rate p=px+py+pz . By appropriately formulating the boundary conditions, in the rotated code geometry, we find exact solutions at a special disordered point, p=1+η−12+η−1≳0.5 , for arbitrary odd code distance d , where the codes reduce to one-dimensional Ising models. The total logical failure rate is given by Pf=34−14e−2dZartanh(1/2η) , where dZ=d2 and d for the two codes respectively, is the effective code distance for pure phase-flip noise. As a consequence, for code distances d≪η , and error rates near the threshold, the XZZX code is effectively equivalent to the phase-flip correcting repetition code over d qubits. The large finite size corrections for dZ<η also make threshold extractions, from numerical calculations at moderate code distances, unreliable. We show that calculating thresholds based not only on the total logical failure rate, but also independently on the phase- and bit-flip logical failure rates, gives a more confident estimate. Using this method for the XZZX code with a tensor-network based decoder and code distances up to d≈100 , we find that the thresholds converge to a single value at moderate bias ( η=30,100 ), at an error rate above the hashing bound.
Abstract To date, the surface code has become a promising candidate for quantum error correcting codes because it achieves a high threshold and is composed of only the nearest gate operations and low-weight stabilizers. Here, we have exhibited that the logical failure rate can be enhanced by manipulating the lattice size of surface codes that they can show an enormous improvement in the number of physical qubits for a noise model where dephasing errors dominate over relaxation errors. We estimated the logical error rate in terms of the lattice size and physical error rate. When the physical error rate was high, the parameter estimation method was applied, and when it was low, the most frequently occurring logical error cases were considered. By using the minimum weight perfect matching decoding algorithm, we obtained the optimal lattice size by minimizing the number of qubits to achieve the required failure rates when physical error rates and bias are provided .
Practical quantum computers will require resource-efficient error-correcting codes. The rotated surface code uses approximately half the number of qubits as the unrotated surface code to create a logical qubit with the same error-correcting distance. However, instead of distance, a more useful qubit-saving metric would be based on logical error rates. In this work we find the well-below-threshold scaling of logical to physical error rates under circuit-level noise for both codes at high odd and even distances and then compare the number of qubits used by each code to achieve equal logical error rates. We perform Monte Carlo sampling of memory experiment circuits with all valid CNOT orders using the stabilizer simulator Stim and the uncorrelated minimum-weight perfect matching decoder PyMatching 2. We find that the rotated code uses about 74 % the number of qubits used by the unrotated code to achieve a logical error rate of p L = 10 − 12 at the operational physical error rate of p = 10 − 3 . The ratio remains ≈ 75 % for p values within a factor of two of p = 10 − 3 for all useful p L . Our work finds the low- p L scaling of the surface code and clarifies the qubit savings provided by the rotated surface code, providing numerical justification for its use in future implementations of the surface code.
In the implementation of quantum information systems, one type of Pauli error, such as phase-flip errors, may occur more frequently than others, like bit-flip errors. For this reason, quantum error-correcting codes that handle asymmetric errors are critical to mitigating the impact of such impairments. To this aim, several asymmetric quantum codes have been proposed. These include variants of surface codes like the XZZX and ZZZY surface codes, tailored to preserve quantum information in the presence of error asymmetries. In this work, we propose two classes of Calderbank, Shor and Steane (CSS) topological codes, referred to as cylindrical and Möbius codes, particular cases of the fiber bundle family. Cylindrical codes maintain a fully planar structure, while Möbius codes are quasi-planar, with minimal non-local qubit interactions. We construct these codes employing the algebraic chain complexes formalism, providing theoretical upper bounds for the logical error rate. Our results demonstrate that cylindrical and Möbius codes outperform standard surface codes when using the minimum weight perfect matching (MWPM) decoder.
We analyze the performance of quantum stabilizer codes, one of the most important classes for practical implementations, on both symmetric and asymmetric quantum channels. To this aim, we first derive the weight enumerator (WE) for the undetectable errors based on the quantum MacWilliams identities. The WE is then used to evaluate tight upper bounds on the error rate of CSS quantum codes with minimum weight decoding. For surface codes we also derive a simple closed form expression of the bounds over the depolarizing channel. We introduce a novel approach that combines the knowledge of WE with a logical operator analysis, allowing the derivation of the exact asymptotic error rate for short codes. For example, on a depolarizing channel with physical error rate ρ → 0 , the logical error rate ρ L is asymptotically ρ L ≈ 16 ρ 2 for the [ [ 9 , 1 , 3 ] ] Shor code, ρ L ≈ 16.3 ρ 2 for the [ [ 7 , 1 , 3 ] ] Steane code, ρ L ≈ 18.7 ρ 2 for the [ [ 13 , 1 , 3 ] ] surface code, and ρ L ≈ 149.3 ρ 3 for the [ [ 41 , 1 , 5 ] ] surface code. For larger codes our bound provides ρ L ≈ 1215 ρ 4 and ρ L ≈ 663 ρ 5 for the [ [ 85 , 1 , 7 ] ] and the [ [ 181 , 1 , 10 ] ] surface codes, respectively. Finally, we extend our analysis to include realistic, noisy syndrome extraction circuits by modeling error propagation throughout gadgets. This enables estimation of logical error rates under faulty measurements. The performance analysis serves as a design tool for developing fault-tolerant quantum systems by guiding the selection of quantum codes based on their error correction capability. Additionally, it offers a novel perspective on quantum degeneracy, showing it represents the fraction of non-correctable error patterns shared by multiple logical operators.