Magic states are fundamental building blocks on the road to fault-tolerant quantum computing.Calderbank-Shor-Steane (CSS) codes play a crucial role in the construction of magic distillation protocols.Previous work has cast quantum computing with magic states for odd dimension d within a phase-space setting in which universal quantum computing is described by the statistical mechanics of quasiprobability distributions.Here we extend this framework to the important d = 2 qubit case and show that we can exploit common structures in CSS circuits to obtain distillation bounds capable of outperforming previous monotone bounds in regimes of practical interest.Moreover, in the case of CSS-code projections, we arrive at a novel cutoff result on the code length n of the CSS code in terms of parameters characterizing a desired distillation, which implies that for fixed target error rate and acceptance probability, one needs to consider only CSS codes below a threshold number of qubits.These entropic constraints are not due simply to the data-processing inequality but rely explicitly on the stochastic representation of such protocols.Magic states are fundamental building blocks on the road to fault-tolerant quantum computing.Calderbank-Shor-Steane (CSS) codes play a crucial role in the construction of magic distillation protocols.Previous work has cast quantum computing with magic states for odd dimension d within a phase-space setting in which universal quantum computing is described by the statistical mechanics of quasiprobability distributions.Here we extend this framework to the important d = 2 qubit case and show that we can exploit common structures in CSS circuits to obtain distillation bounds capable of outperforming previous monotone bounds in regimes of practical interest.Moreover, in the case of CSS-code projections, we arrive at a novel cutoff result on the code length n of the CSS code in terms of parameters characterizing a desired distillation, which implies that for fixed target error rate and acceptance probability, one needs to consider only CSS codes below a threshold number of qubits.These entropic constraints are not due simply to the data-processing inequality but rely explicitly on the stochastic representation of such protocols.

arXiv:2211.07535PRX QuantumJun 30, 20231 citation

Search for high-threshold qutrit magic-state distillation routines

Shiroman Prakash, Rishabh Singhal

Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We show that the performance of a qudit correcting code for magic state distillation is captured by its complete weight enumerator. For the qutrit strange state -- a maximally magic non-stabilizer state -- the performance of a code is captured by its simple weight enumerator. This result allows us to carry out an extensive search for high-threshold magic state distillation routines for the strange state. Our search covers all $[[n,1]]_3$ qutrit stabilizer codes with a complete set of transversal Clifford gates for $n\leq 23$ , and all $[[n,1]]_3$ stabilizer codes with a transversal $H^2$ gate with $n \leq 9$ qudits. For $n=23$ , we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic.Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We show that the performance of a qudit correcting code for magic state distillation is captured by its complete weight enumerator. For the qutrit strange state -- a maximally magic non-stabilizer state -- the performance of a code is captured by its simple weight enumerator. This result allows us to carry out an extensive search for high-threshold magic state distillation routines for the strange state. Our search covers all $[[n,1]]_3$ qutrit stabilizer codes with a complete set of transversal Clifford gates for $n\leq 23$, and all $[[n,1]]_3$ stabilizer codes with a transversal $H^2$ gate with $n \leq 9$ qudits. For $n=23$, we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic.

arXiv:2408.00436Physical Review AFeb 3, 20260 citations

Borrowed Identities: Malleable Distillation Factories and a Unified Numerical Search

Shraddha Singh, Craig Gidney, Cody Jones

Magic-state distillation is one of the leading overheads in fault-tolerant quantum computation. Existing methods for finding distillation factories require a transversal gate to act correctly on the entire codespace, a constraint that limits both generality and search efficiency. We introduce a strictly weaker borrowed-identity condition, requiring only that the distillation circuit act as the identity on a single input state. It applies uniformly across all levels of the Clifford hierarchy and unifies, within a single level, factories that distill different magic states -- for example, the $|T\rangle$ , $|CS\rangle$ , and $|CCZ\rangle$ factories. A brute-force search over borrowed-identity circuits with two-group symmetry recovers, within the search range, all distance-2 factories known from code-construction approaches, including entangled-output and multi-output factories previously outside the scope of any single numerical search. This unification yields parent circuits that encode multiple factories, so the output magic-state type can be chosen at compile time rather than fixed by a hard-coded design. The framework also extends beyond CSS codes, unifying constructions, including synthillation and non-CSS catalytic factories, previously obtained by disparate approaches.Magic-state distillation is one of the leading overheads in fault-tolerant quantum computation. Existing methods for finding distillation factories require a transversal gate to act correctly on the entire codespace, a constraint that limits both generality and search efficiency. We introduce a strictly weaker borrowed-identity condition, requiring only that the distillation circuit act as the identity on a single input state. It applies uniformly across all levels of the Clifford hierarchy and unifies, within a single level, factories that distill different magic states -- for example, the $|T\rangle$, $|CS\rangle$, and $|CCZ\rangle$ factories. A brute-force search over borrowed-identity circuits with two-group symmetry recovers, within the search range, all distance-2 factories known from code-construction approaches, including entangled-output and multi-output factories previously outside the scope of any single numerical search. This unification yields parent circuits that encode multiple factories, so the output magic-state type can be chosen at compile time rather than fixed by a hard-coded design. The framework also extends beyond CSS codes, unifying constructions, including synthillation and non-CSS catalytic factories, previously obtained by disparate approaches.

quant-pharXiv:2606.28518Jun 30, 20260 citations

Quantum CSS Duadic and Triadic Codes: New Insights and Properties

Reza Dastbasteh, Olatz Sanz Larrarte, Josu Etxezarreta Martínez +3 more

arXiv:2407.07753Lecture notes in computer scienceJan 1, 20253 citations

Fragility of Magic State Distillation under Imperfect Measurements

Yunzhe Zheng, Yuanchen Zhao, Dong E. Liu

Magic state distillation (MSD) is the leading approach for producing the non-Clifford resources required for universal fault-tolerant quantum computation. Most prior analyses of MSD assume ideal projective stabilizer measurements, but this assumption becomes questionable on near-term hardware, where measurement fidelity is limited and large quantum error-correcting codes are unavailable. We consider imperfect measurements that are modeled as noisy projector controlled by measurement strength , which can be regarded as the inverse of measurement noise, and develop a general framework for analyzing MSD under imperfect stabilizer measurements. We focus on MSD protocols that are based on CSS codes with transversal non-Clifford gates, and show that MSD exhibits a measurement-strength threshold that is distinct from the previously known threshold on input-state error. When measurement strength is below a critical threshold, MSD loses its distillation power entirely; When measurement strength is above this threshold, distillation under imperfect measurements remains possible, but the asymptotic target states may deviate from the ideal magic states. In particular, this deviation is at most first-order biased. Moreover, we show that imperfect measurements with finite measurement strength reduce the distillation efficiency to linear, implying exponentially larger distillation overheads than in the ideal-measurement case. We further show that this fragility can be mitigated generally up to the code capacity of MSD protocols by choosing the stabilizer generators to measure in the standard form . Our results reveal fundamental constraints imposed by imperfect measurements on MSD and provide guidance for designing more robust distillation protocols in realistic quantum hardware.Magic state distillation (MSD) is the leading approach for producing the non-Clifford resources required for universal fault-tolerant quantum computation. Most prior analyses of MSD assume ideal projective stabilizer measurements, but this assumption becomes questionable on near-term hardware, where measurement fidelity is limited and large quantum error-correcting codes are unavailable. We consider imperfect measurements that are modeled as noisy projector controlled by measurement strength , which can be regarded as the inverse of measurement noise, and develop a general framework for analyzing MSD under imperfect stabilizer measurements. We focus on MSD protocols that are based on CSS codes with transversal non-Clifford gates, and show that MSD exhibits a measurement-strength threshold that is distinct from the previously known threshold on input-state error. When measurement strength is below a critical threshold, MSD loses its distillation power entirely; When measurement strength is above this threshold, distillation under imperfect measurements remains possible, but the asymptotic target states may deviate from the ideal magic states. In particular, this deviation is at most first-order biased. Moreover, we show that imperfect measurements with finite measurement strength reduce the distillation efficiency to linear, implying exponentially larger distillation overheads than in the ideal-measurement case. We further show that this fragility can be mitigated generally up to the code capacity of MSD protocols by choosing the stabilizer generators to measure in the standard form . Our results reveal fundamental constraints imposed by imperfect measurements on MSD and provide guidance for designing more robust distillation protocols in realistic quantum hardware.

arXiv:2503.01165npj Quantum InformationSep 22, 20260 citations

[[n,1]]_3

qutrit stabilizer codes with a complete set of transversal Clifford gates for

n\leq 23

, and all

[[n,1]]_3

stabilizer codes with a transversal

H^2

gate with

n \leq 9

qudits. For

n=23

, we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic.

|T\rangle

|CS\rangle

, and

|CCZ\rangle

factories. A brute-force search over borrowed-identity circuits with two-group symmetry recovers, within the search range, all distance-2 factories known from code-construction approaches, including entangled-output and multi-output factories previously outside the scope of any single numerical search. This unification yields parent circuits that encode multiple factories, so the output magic-state type can be chosen at compile time rather than fixed by a hard-coded design. The framework also extends beyond CSS codes, unifying constructions, including synthillation and non-CSS catalytic factories, previously obtained by disparate approaches.