We show how to realize a general quantum circuit involving gates between arbitrary pairs of qubits by means of geometrically local quantum operations and efficient classical computation. We prove that circuit-level local stochastic noise modeling an imperfect implementation of our derived schemes is equivalent to local stochastic noise in the original circuit. Our constructions incur a constant-factor increase in the quantum circuit depth and a polynomial overhead in the number of qubits. To execute an arbitrary quantum circuit on n qubits, we give a three-dimensional quantum fault-tolerance architecture involving O ( n 3 / 2 log 3 n ) qubits and a quasi-two-dimensional architecture using O ( n 2 log 3 n ) qubits. Applied to recent fault-tolerance constructions, this gives a fault-tolerance-threshold theorem for universal quantum computations with local operations, a polynomial qubit overhead, and a quasipolylogarithmic depth overhead. More generally, our transformation dispenses with the need for considering the locality of operations when designing schemes for fault-tolerant quantum information processing.
We show how to realize a general quantum circuit involving gates between arbitrary pairs of qubits by means of geometrically local quantum operations and efficient classical computation. We prove that circuit-level local stochastic noise modeling an imperfect implementation of our derived schemes is equivalent to local stochastic noise in the original circuit. Our constructions incur a constant-factor increase in the quantum circuit depth and a polynomial overhead in the number of qubits. To execute an arbitrary quantum circuit on n qubits, we give a three-dimensional quantum fault-tolerance architecture involving O ( n 3 / 2 log 3 n ) qubits and a quasi-two-dimensional architecture using O ( n 2 log 3 n ) qubits. Applied to recent fault-tolerance constructions, this gives a fault-tolerance-threshold theorem for universal quantum computations with local operations, a polynomial qubit overhead, and a quasipolylogarithmic depth overhead. More generally, our transformation dispenses with the need for considering the locality of operations when designing schemes for fault-tolerant quantum information processing.
Fault-tolerant quantum computation traditionally incurs substantial resource overhead, with both qubit and time overheads scaling polylogarithmically with the size of the computation. While prior work by Gottesman showed that constant qubit overhead is achievable under stochastic noise using quantum low-density parity-check (QLDPC) codes, it has remained an open question whether similar guarantees hold under more general, non-stochastic noise models. In this work, we address this question by considering a general circuit-level noise model defined via the diamond norm, which captures both stochastic and non-stochastic noise, including coherent and amplitude damping noise. We prove that constant qubit overhead fault-tolerant quantum computation is achievable in this general setting, using QLDPC codes with constant rate and linear minimum distance. To establish our result, we develop a fault-tolerant error correction scheme and a method for implementing logic gates under general circuit noise. These results extend the theoretical foundations of fault-tolerant quantum computation and offer new directions for fault-tolerant architectures under realistic noise models.
We study fault-tolerant quantum computation using a fixed, local error-correction rule and a geometrically simple code: the cubical (d,d′)-Toric code with d′,d−d′≥2, including the 4D Toric code. Under sufficiently weak local-stochastic Pauli noise, our construction uses noiseless, bounded-depth classical processing per physical time step. Its correction cycle acts only on the current syndrome, has constant depth, and is repeated unchanged during idle logical computation; it requires neither multiscale decoder states nor scale-dependent communication delays.
Our proof treats errors and syndromes as chains in a cubical complex, combining geometric shrinking under the sweep rule with a space--time explanation-tree argument inspired by Toom's theorem. It does not rely on expansion. We establish three results:
1.Explicit higher-dimensional sweep rule. We give a local algorithm for choosing sweep corrections on cubical (d,d′)-Toric complexes, resolving the local-existence question of Kubica and Preskill.
2. Fault-tolerant computation. We prove a circuit-level threshold theorem for noisy sweep cycles interleaved with logical gates. In particular, the 4D Toric code admits a fixed, constant-depth syndrome-based correction cycle without a multiscale decoder or expansion-based encoding. We do not claim constant total-depth overhead.
3. Ideal decoding under local-stochastic noise. For an initial local-stochastic Pauli error of rate p below threshold, nd+1 noiseless sweep cycles recover the encoded state on a side-length-n(d,d′)-Toric code with probability at least 1−O(poly(n))pΩ(n), extending earlier guarantees for independent errors.
We prove a fault-tolerance theorem for quantum computation against adversarial noise. For every quantum circuit on Nˉ logical qudits of depth Tˉ, we construct a fault-tolerant circuit on N=poly(Nˉ) physical qudits of depth Tˉ⋅Nˉo(1), which is robust against an adversary who may arbitrarily choose and corrupt an almost-linear number N1−o(1) of physical qudits at each time step. This robustness significantly improves upon prior fault-tolerance theorems, which assumed corruptions were either local and stochastic, or else only act on a polynomially vanishing fraction of qudits.
Our fault-tolerance scheme addresses a key bottleneck towards constructing quantum PCPs via the circuit-to-Hamiltonian mapping of Anshu, Breuckmann, and Nguyen (STOC'24). More fundamentally, our result demonstrates that fault-tolerant quantum computation remains possible under noise models that are global, worst-case, and non-Markovian over the full duration of the computation, directly countering concerns that correlated noise could fundamentally undermine quantum fault tolerance.
Our construction is based on a new family of subsystem product codes we develop, which have large dimension and distance along with low-weight parity-checks, and which support transversal non-Clifford gates. We show how to perform single-shot fault-tolerant error correction on these codes using a Floquet-like procedure based on the local testability of classical tensor codes. We then obtain a universal fault-tolerance scheme using repeated code switching in a hypercubic qudit architecture. Finally, we recursively compose our scheme with itself to reduce an initially exponential qudit dimension down to a constant.
Usual scenarios of fault-tolerant computation are concerned with the fault-tolerant realization of quantum algorithms that compute classical functions, such as Shor's algorithm for factoring. In particular, this means that input and output to the quantum algorithm are classical. In contrast to stand-alone single-core quantum computers, in many distributed scenarios, quantum information might have to be passed on from one quantum information processing system to another one, possibly via noisy quantum communication channels with noise levels above fault-tolerant thresholds. In such situations, quantum information processing devices will have quantum inputs, quantum outputs or even both, which pass qubits among each other. Working in the fault-tolerant framework of [Kitaev, 1997], we show that any quantum circuit with quantum input and output can be transformed into a fault-tolerant circuit that produces the ideal circuit with some controlled noise applied at the input and output. The framework allows the direct composition of the statements, enabling versatile future applications. We illustrate this with a concrete application, namely, communication over a noisy channel with faulty encoding and decoding operations [Christandl and M{ü}ller-Hermes, 2024]. For communication codes with linear minimum distance, we construct fault-tolerant encoders and decoders for general noise (including coherent errors). For the weaker, but standard, model of local stochastic noise, we obtain fault-tolerant encoders and decoders for any communication code that can correct a constant fraction random errors.
We construct the first known fault-tolerant scheme for injecting states into quantum error-correcting codes with constant space and time overhead. That is, we construct a family of constant-rate quantum error-correcting codes for which a set of bare physical qubits can be injected, i.e. fault-tolerantly encoded, into a code block. Similarly, a code state can be ejected, i.e. fault-tolerantly decoded, back into bare physical qubits. We show that these injection and ejection procedures succeed under circuit-level locally stochastic noise, while incurring just a small constant probability of corrupting each qubit, which is unavoidable for bare physical qubits. We also show how to perform fault-tolerant error correction and code-state preparation under locally stochastic noise. All of our gadgets can be implemented with constant quantum circuit depth (i.e. are single-shot), and with a number of physical qubits growing linearly with the number of logical qubits, assuming the ability to run polynomial-sized noiseless classical circuits on the side.
We construct our quantum codes by taking a high-dimensional hypergraph product of classical LDPC codes, which in turn are a simplified version of Spielman's linear-time encodable codes (STOC'95). As our resulting product codes are only resilient to physical errors occurring with non-uniform probabilities across qubits, we then show how to concatenate with inner codes of various sizes to obtain fault-tolerance against uniform noise.
Usual scenarios of fault-tolerant computation are concerned with the fault-tolerant realization of quantum algorithms that compute classical functions, such as Shor’s algorithm for factoring. In particular, this means that input and output to the quantum algorithm are classical. In contrast to stand-alone single-core quantum computers, in many distributed scenarios, quantum information might have to be passed on from one quantum information processing system to another one, possibly via noisy quantum communication channels with noise levels above fault-tolerant thresholds. In such situations, quantum information processing devices will have quantum inputs, quantum outputs or even both, which pass qubits among each other. Working in the fault-tolerant framework of [Kitaev, 1997], we show that any quantum circuit with quantum input and output can be transformed into a fault-tolerant circuit that produces the ideal circuit with some controlled noise applied at the input and output. The framework allows the direct composition of the statements, enabling versatile future applications. We illustrate this with a concrete application, namely, communication over a noisy channel with faulty encoding and decoding operations [Christandl and Müller-Hermes, 2024]. For communication codes with linear minimum distance, we construct fault-tolerant encoders and decoders for general noise (including coherent errors). For the weaker, but standard model of local stochastic noise, we obtain fault-tolerant encoders and decoders for any communication code that can correct a constant fraction of random errors.