Error Correction 2

Abstract: 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.

Abstract: We investigate layer codes, a family of three-dimensional stabilizer codes that can achieve optimal scaling of code parameters and a polynomial energy barrier, as candidates for self-correcting quantum memories. First, we introduce two decoding algorithms for layer codes with provable guarantees for local stochastic and adversarial noise, respectively. We then prove that layer codes are partially self-correcting quantum memories. With memory times scaling exponentially in the linear size of the system, layer codes outperform the previously demonstrated subexponential scaling of the welded solid code. Notably, we argue that partial self-correction without the requirement of efficient decoding is more common than expected, as it arises from a diverging energy barrier. This draws a sharp distinction between partially self-correcting systems, and partially self-correcting memories. Another novel aspect of our work is an analysis of layer codes constructed from random Calderbank–Shor–Steane codes. We show that these random layer codes have optimal scaling (up to logarithmic corrections) of code parameters and a polynomial energy barrier. Finally, we present numerical studies of their memory times and report behavior consistent with partial self-correction.

Abstract: Random classical linear codes are widely believed to be hard to decode, exponentially so at constant coding rate. If the rate vanishes asymptotically sufficiently rapidly, slightly sub-exponential decoding algorithms are known. By contrast, the complexity of decoding a random quantum stabilizer code has remained an open question for quite some time. This work closes the gap in our understanding of the algorithmic hardness of decoding random quantum versus random classical codes. We prove that decoding a random stabilizer code with even a single logical qubit is at least as hard as decoding a random classical code at constant rate—the maximally hard regime. This result suggests that the easiest random quantum decoding problem is at least as hard as the hardest random classical decoding problem, and shows that any sub-exponential algorithm decoding a typical stabilizer code, at any coding rate, would immediately imply a breakthrough in cryptography. More generally, we also characterize many other complexity-theoretic properties of stabilizer codes. While classical decoding admits a random self-reduction, we prove significant barriers for the existence of random self-reductions in the quantum case. This result follows from new bounds on Clifford entropies and Pauli mixing times, which may be of independent interest. As a complementary result, we demonstrate various other self-reductions which are in fact achievable, such as between search and decision. Our work also demonstrates several ways in which quantum phenomena, such as quantum degeneracy, force several reasonable definitions of stabilizer decoding—all of which are classically identical—to have distinct or non-trivially equivalent complexity.

Abstract: The development of quantum codes with good error correction parameters and useful sets of transversal gates is an area of major interest in quantum error correction. Abundant prior works have studied transversal gates which are restricted to acting on all logical qubits simultaneously. In this work, we study codes that support transversal gates which induce addressable logical gates, i.e., the logical gates act only on logical qubits of our choice. As we consider scaling from low-rate to high-rate codes, the study and design of low-overhead, addressable logical operations presents an important problem for both theoretical and practical purposes. In this work, we construct the first quantum codes to support transversally addressable non-Clifford gates. Concretely, given any three logical qubits across one or multiple codeblocks, one can execute the logical CCZ on those qubits via a depth-one physical circuit of CCZ gates. We present a simple, explicit construction based on Reed-Solomon codes that is nearly asymptotically good, and a more involved, asymptotically good construction based on transitive, iso-orthogonal algebraic geometry codes. We go on to develop a powerful theory of quantum codes supporting a rich class of transversally addressable gates in the Clifford hierarchy, going far beyond just the CCZ gate. We call this framework addressable orthogonality, and show that it can be used to construct asymptotically good quantum codes supporting an arbitrary product of multiply-controlled Z gates transversally and addressably, enabling major adaptivity to particular algorithms. Our constructions mark the first quantum codes to support any multi-qubit gate transversally and addressably. Accordingly, our results have major implications for the general addressabilitiy problem in error correction. This is a merged submission based on arXiv:2502.01864 and arXiv:2507.05392.