In this work, we derive achievability (upper) bounds for the logical error probability (LEP) of finite-length CSS codes for two asymmetric channels. In other words, we prove the existence of [[n, k]] CSS codes whose LEP does not exceed certain bounds that depend on the code and channel parameters. The bounds are derived with the help of finding the average LEPs of CSS codes from different code ensembles.The obtained bounds allow us to optimize code parameters for specific quantum channels. We show that such optimization leads to drastic performance improvements in asymmetric channels. We further show that expurgating bad codes from the ensemble of all CSS codes also leads to large performance improvements in certain channels.
In this work, we derive achievability (upper) bounds for the logical error probability (LEP) of finite-length CSS codes for two asymmetric channels. In other words, we prove the existence of [[n, k]] CSS codes whose LEP does not exceed certain bounds that depend on the code and channel parameters. The bounds are derived with the help of finding the average LEPs of CSS codes from different code ensembles.The obtained bounds allow us to optimize code parameters for specific quantum channels. We show that such optimization leads to drastic performance improvements in asymmetric channels. We further show that expurgating bad codes from the ensemble of all CSS codes also leads to large performance improvements in certain channels.
In this paper we estimate the fidelity of stabilizer and CSS codes. First, we derive a lower bound on the fidelity of a stabilizer code via its quantum enumerator. Next, we find the average quantum enumerators of the ensembles of finite length stabilizer and CSS codes. We use the average quantum enumerators for obtaining lower bounds on the average fidelity of these ensembles. We further improve the fidelity bounds by estimating the quantum enumerators of expurgated ensembles of stabilizer and CSS codes. Finally, we derive fidelity bounds in the asymptotic regime when the code length tends to infinity. These results tell us which code rate we can afford for achieving a target fidelity with codes of a given length. The results also show that in symmetric depolarizing channel a typical stabilizer code has better performance, in terms of fidelity and code rate, compared with a typical CSS codes, and that balanced CSS codes significantly outperform other CSS codes. Asymptotic results demonstrate that CSS codes have a fundamental performance loss compared to stabilizer codes.
We introduce an adaptive simulated annealing algorithm to search for moderately-sized quantum error-correcting codes with high encoding rates and large distances. Our search targets two classes of stabilizer codes: (1) CSS codes, and (2) a subclass of CSS codes that we call ``self-dual with equivalent logicals'' (SWEL) codes, the latter of which which admit transversal implementations of logical Hadamard and phase gates that can be leveraged to construct fault-tolerant gate sets. The search is guided by an energy function that acts as a surrogate for the logical error rate in a code-capacity noise model, combining code distance with a count of minimum-weight logical operators to resolve the discrete plateaux that impede naïve distance optimization. For block lengths of up to 50 physical qubits, our search finds state-of-the-art CSS and SWEL codes whose distances frequently meet or exceed the variants of the quantum Gilbert-Varshamov bound. In addition to providing favorable seed codes for fault-tolerant architectures based on code concatenation, the codes found in this work are promising candidates for high-rate code demonstrations on near-term quantum computing hardware.
We consider the probability of decoding error, equivalently the fidelity, of finite length quantum codes assuming quantum information transmission via qubit erasure channel. First, we obtain an analytical expression on this probability via combinatorial invariants of quantum codes. Next, we derive a generalization of the MacWilliams identities for those invariants and use them to formulate a linear programming optimization problem, which allows us to obtain lower bounds on the number of unrecoverable sets of i erasures and the probability of decoding error. Our example shows that the obtain bounds can be very tight. Next, we derive upper (achievability) bounds on the probability of decoding error for finite length stabilizer and CSS codes. CSS codes have many advantages for practical applications. Our results show that CSS codes of small lengths, like 100 qubits, visibly lose in performance to generic stabilizer codes.
Abstract We present a new propagation rule for CSS codes. Starting with a CSS code [[n,k,d]]q [ [ n , k , d ] ] q , we construct a CSS code with parameters [[n−2,k,d−1]]q [ [ n - 2 , k , d - 1 ] ] q . In general, one would only obtain a code with parameters [[n−2,k,d−2]]q [ [ n - 2 , k , d - 2 ] ] q . The construction applies to asymmetric quantum codes from the CSS construction as well.
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.
Quantum error-correcting codes are essential for protecting quantum information against decoherence. This work investigates the construction of quantum CSS codes from classical self-orthogonal binary linear codes derived from extended quadratic residue codes. By exploiting the large automorphism groups of these codes, the minimum-weight codewords are partitioned according to the action of maximal subgroups of the automorphism group. Codes generated from these partitions are shown to be self-orthogonal and to retain large automorphism groups, allowing the construction of associated quantum CSS codes with good parameters. The resulting codes include new quantum codes with lengths ranging from 16 to 48 and minimum distances up to 11. Their automorphism groups and permutation-decoding properties are also analyzed, including the number of permutations required for complete decoding. The results provide new families of quantum CSS codes combining good error-correcting parameters, large automorphism groups, and efficient permutation decoding.