Magic state distillation enables universal fault-tolerant quantum computation by implementing non-Clifford gates via the preparation of high-fidelity magic states. However, it comes at the cost of substantial logical-level overhead in both space and time. In this work, we propose a very low-cost magic state distillation scheme for biased-noise qubits. By leveraging the noise bias, our scheme enables the preparation of a magic state with a logical error rate of 3 × 10 −7 , using only 53 qubits and 5.5 error correction rounds, under a noise bias of η ≳ 5 × 10 6 and a phase-flip noise rate of 0.1%. This reduces the circuit volume by more than one order of magnitude relative to magic state cultivation for unbiased-noise qubits and by more than two orders of magnitude relative to standard magic state distillation. Moreover, our scheme provides three key advantages over previous proposals for biased-noise qubits. First, it only requires nearest-neighbor two-qubit gates on a 2D lattice. Second, the logical fidelity remains nearly identical even at a more modest noise bias of η ≳ 80, at the cost of a slightly increased circuit volume. Third, the scheme remains effective even at high physical phase-flip rates, in contrast to previously proposed approaches whose circuit volume grows exponentially with the error rate. Our construction is based on unfolding the X stabilizer group of the Hadamard 3D quantum Reed-Muller code in 2D, enabling distillation at the physical level rather than the logical level, and is therefore referred to as unfolded distillation.
Magic state distillation enables universal fault-tolerant quantum computation by implementing non-Clifford gates via the preparation of high-fidelity magic states. However, it comes at the cost of substantial logical-level overhead in both space and time. In this work, we propose a very low-cost magic state distillation scheme for biased-noise qubits. By leveraging the noise bias, our scheme enables the preparation of a magic state with a logical error rate of 3 × 10 −7 , using only 53 qubits and 5.5 error correction rounds, under a noise bias of η ≳ 5 × 10 6 and a phase-flip noise rate of 0.1%. This reduces the circuit volume by more than one order of magnitude relative to magic state cultivation for unbiased-noise qubits and by more than two orders of magnitude relative to standard magic state distillation. Moreover, our scheme provides three key advantages over previous proposals for biased-noise qubits. First, it only requires nearest-neighbor two-qubit gates on a 2D lattice. Second, the logical fidelity remains nearly identical even at a more modest noise bias of η ≳ 80, at the cost of a slightly increased circuit volume. Third, the scheme remains effective even at high physical phase-flip rates, in contrast to previously proposed approaches whose circuit volume grows exponentially with the error rate. Our construction is based on unfolding the X stabilizer group of the Hadamard 3D quantum Reed-Muller code in 2D, enabling distillation at the physical level rather than the logical level, and is therefore referred to as unfolded distillation.
Magic state distillation is a resource intensive subroutine that consumes noisy input states to produce high-fidelity resource states that are used to perform logical operations in practical quantum-computing architectures. The resource cost of magic state distillation can be reduced by improving the fidelity of the raw input states. To this end, we propose an initialization protocol that offers a quadratic improvement in the error rate of the input magic states in architectures with biased noise. This is achieved by preparing an error-detecting code which detects the dominant errors that occur during state preparation. We obtain this advantage by exploiting the native gate operations of an underlying qubit architecture that experiences biases in its noise profile. We perform simulations to analyze the performance of our protocol with the XZZX surface code. Even at modest physical parameters with a two-qubit gate error rate of 0.7 and total probability of dominant errors in the gate O(103) larger compared to that of nondominant errors, we find that our preparation scheme delivers magic states with logical error rate $O({10}^{\ensuremath{-}8})$ after a single round of the standard 15-to-1 distillation protocol, two orders of magnitude lower than using conventional state preparation. Our approach therefore promises considerable savings in overheads with near-term technology.
Zero-level CCZ Distillation
Magic state distillation is a key component of fault-tolerant quantum computation, as it enables the implementation of non-Clifford gates such as the T gate and the CCZ gate via gate teleportation. However, conventional distillation protocols require a large number of logical qubits and introduce substantial spatial and temporal overhead, posing a significant bottleneck for scalable fault-tolerant quantum computation. In this work, we propose a zero-level distillation protocol that efficiently generates a high-fidelity logical CCZ magic state using only physical qubits on a two-dimensional square lattice with nearest-neighbor interactions. Our method leverages the transversal T/T† operation of the [[8,3,2]] code to fault-tolerantly encode the state CCZ∣+++⟩, which is subsequently teleported to three surface-code logical qubits via lattice surgery. To enable teleportation between codes with different distances, we introduce adaptively initialized teleportation (AIT), a tailored initialization procedure for the surface code. Numerical simulations demonstrate that the logical error rate scales as pL≃300×p2 with respect to the physical error rate p. For example, the proposed method improves the logical error rate by approximately one and two orders of magnitude at p=10−3 and p=10−4, respectively, compared to conventional seven-T-gate approaches. The distillation circuit requires only 22 physical qubits, 3 logical qubits, and a circuit depth of 24, reducing the space-time overhead by a factor of approximately 5-10 compared to previous methods. This result highlights the practicality of CCZ-state distillation in early fault-tolerant quantum computation and offers a new direction toward resource-efficient physical-level magic state distillation beyond conventional T-state generation.
Magic state distillation (MSD) is an essential element for universal fault-tolerant quantum computing, which distills a high-fidelity magic state from noisy magic states using ideal (error-corrected) Clifford operations. For ideal Clifford operations, it needs to be performed on the logical qubits and hence incurs a large spatiotemporal overhead, which is one of the major bottlenecks for the realization of fault-tolerant quantum computers (FTQCs). Here we propose zero-level distillation, which prepares a high-fidelity logical magic state at the physical level, namely , using physical qubits and nearest-neighbor two-qubit gates on a square lattice. We develop a zero-level distillation circuit and show that distillation can be made even more efficient than the conventional sophisticated approaches with logical level distillations. The key idea involves the Knill -type distillation using the Steane code and its careful mapping to the square-lattice architecture with error detection. The distilled magic state on the Steane-code state is then teleported or converted to surface codes. We numerically find that the error rate of the logical magic state scales as approximately 100 × p 2 in terms of the physical error rate p . For example, with a physical error rate of p = 10 − 4 ( 10 − 3 ), the logical error rate is reduced to p L = 10 − 6 ( 10 − 4 ), resulting in an improvement of 2 (1) orders of magnitude. This contributes to reducing both space and time overhead for early FTQC as well as full-fledged FTQC combined with conventional multilevel distillation protocols.
We introduce the parity-unfolded architecture, a fault-tolerant quantum computing scheme that relies on direct preparation and teleportation of small-angle rotations $ Z^{1/2^{k}}$ rather than approximating them with the conventional (Clifford + T) gate set. The architecture is enabled by efficient distillation of gates from an arbitrary level of the Clifford hierarchy, which we refer to as parity unfolding. With it, a state ∣Zk⟩=Z1/2k∣+⟩ can be prepared fault-tolerantly using 2k+3+O(2k/2) biased-noise qubits on a planar chip with nearest-neighbour connectivity. For algorithms requiring native Z1/2k gates, such as the Quantum Fourier Transform and phase estimation, the proposed scheme allows to reduce resource overheads for up to k=7, i.e., up to T1/32. Furthermore, when used for the synthesis of arbitrary small-angle rotations, parity-unfolded distillation of (T + T) reduces the minimum achievable logical error rate by 43% while cutting the resource requirements by 26%, when compared to unfolded distillation of only the T gate.
Multiple quantum computing platforms across neutral atoms [1], nitrogen vacancy centres [2, 3], gate-defined dots [4-6] and 14|15 phosphorus atom qubits [7, 8] in silicon are experimentally exploring the use of high connectivity qubits, beyond that of nearest-neighbour planar lattices. Theoretical works consider modifications to fault-tolerant codes to leverage this higher qubit connectivity, such as non-local LDPC codes [9], inspiring superconducting [10] and photonic [11] platforms to also seek non-planar connectivity. In addition, separate theoretical works consider biased noise, where bit- and phase-flip errors are not equally likely. In this work, we present efficient methods for implementing 6.6.6 and 4.8.8 colour codes, as well as bilayer and folded surface codes, using two-qubit registers. We also leverage noise bias to avoid hook errors, demonstrating comparable performance to the surface code. By combining colour and bilayer codes, we show how magic state cultivation procedures [12-14] can be improved, in a procedure we refer to as the Magic Scroll. Not only does the Magic Scroll escape to a standard surface code, it also improves cultivation volumes by 3x and supports magic ∣T⟩ state fidelities as low as 10−9. We show that this technique can be further leveraged to improve distillation performance, showing a 3x improvement to distillation volumes for error rates of ∼10−15. Through these constructions, we demonstrate techniques to leverage noise bias and high qubit connectivity, showing how register-based architectures can improve error rates and reduce quantum volumes in fault-tolerant quantum computing.
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.