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arXiv:2608.26272

Fault-tolerant quantum computation cannot be achieved with constant spacetime overhead

Kishor Bharti, Tobias Haug, Andrew Tanggara

quant-phcs.ITSubmitted Aug 26, 2026• Announced Aug 28, 2026• 0 citations
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Abstract

The threshold theorem states that quantum computations can be made reliable below a physical error threshold, at the cost of additional physical qubits and circuit depth. Recent work has reduced these space and time overheads to polylogarithmic or nearly logarithmic scalings, but whether the cumulative spacetime overhead can be constant has remained unclear. Here, we show that even for the simplest task of preserving quantum information in a quantum memory, under an optimistic noise model and allowing general adaptive protocols, there is an unavoidable logarithmic contribution to the cumulative spacetime overhead. This additional cost can nevertheless be shared among many logical qubits, so sufficiently wide computations, including standard implementations of Shor's algorithm, may still achieve constant relative overhead. We further give a positive-rate CSS code construction that attains the memory bound, identify sufficient conditions under which the same scaling extends from quantum memory to fault-tolerant circuit implementations, and derive circuit-size bounds for subsystem spacetime codes. Our work establishes fundamental limits on the resources required for quantum fault tolerance.

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The threshold theorem states that quantum computations can be made reliable below a physical error threshold, at the cost of additional physical qubits and circuit depth. Recent work has reduced these space and time overheads to polylogarithmic or nearly logarithmic scalings, but whether the cumulative spacetime overhead can be constant has remained unclear. Here, we show that even for the simplest task of preserving quantum information in a quantum memory, under an optimistic noise model and allowing general adaptive protocols, there is an unavoidable logarithmic contribution to the cumulative spacetime overhead. This additional cost can nevertheless be shared among many logical qubits, so sufficiently wide computations, including standard implementations of Shor's algorithm, may still achieve constant relative overhead. We further give a positive-rate CSS code construction that attains the memory bound, identify sufficient conditions under which the same scaling extends from quantum memory to fault-tolerant circuit implementations, and derive circuit-size bounds for subsystem spacetime codes. Our work establishes fundamental limits on the resources required for quantum fault tolerance.
Abstract A major challenge in fault-tolerant quantum computation is to reduce both the space overhead, that is, the large number of physical qubits per logical qubit, and the time overhead, that is, the long physical gate sequences needed to implement a logical gate. Here we prove that a protocol using non-vanishing-rate quantum low-density parity-check (QLDPC) codes, combined with concatenated Steane codes, achieves constant space overhead and polylogarithmic time overhead, even when accounting for the required classical processing. This protocol offers an improvement over existing constant-space-overhead protocols. To prove our result, we develop a technique that we call partial circuit reduction, which enables error analysis for the entire fault-tolerant circuit by examining smaller parts composed of a few gadgets. With this approach, we resolve a logical gap in the existing arguments for the threshold theorem for the constant-space-overhead protocol with QLDPC codes and complete its proof. Our work establishes that the QLDPC-code-based approach can realize fault-tolerant quantum computation with a negligibly small slowdown and a bounded overhead of physical qubits.
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.
The threshold theorem is a seminal result in the field of quantum computing asserting that arbitrarily long quantum computations can be performed on a faulty quantum computer provided that the noise level is below some constant threshold. This remarkable result comes at the price of increasing the number of qubits (quantum bits) by a large factor that scales polylogarithmically with the size of the quantum computation we wish to realize. Minimizing the space overhead for fault-tolerant quantum computation is a pressing challenge that is crucial to benefit from the computational potential of quantum devices. In this paper, we study the asymptotic scaling of the space overhead needed for fault-tolerant quantum computation. We show that the polylogarithmic factor in the standard threshold theorem is in fact not needed and that there is a fault-tolerant construction that uses a number of qubits that is only a constant factor more than the number of qubits of the ideal computation. This result was conjectured by Gottesman who suggested to replace the concatenated codes from the standard threshold theorem by quantum error-correcting codes with a constant encoding rate. The main challenge was then to find an appropriate family of quantum codes together with an efficient classical decoding algorithm working even with a noisy syndrome. The efficiency constraint is crucial here: bear in mind that qubits are inherently noisy and that faults keep accumulating during the decoding process. The role of the decoder is therefore to keep the number of errors under control during the whole computation. On a technical level, our main contribution is the analysis of the SMALL-SET-FLIP decoding algorithm applied to the family of quantum expander codes . We show that it can be parallelized to run in constant time while correcting sufficiently many errors on both the qubits and the syndrome to keep the error under control. These tools can be seen as a quantum generalization of the BIT-FLIP algorithm applied to the (classical) expander codes of Sipser and Spielman.
Recent work established that, for independent erasures, storing KKK logical qubits for SSS time steps with error at most ε\varepsilonε requires a spacetime cost of Ω(S(K+log⁡(S/ε)))Ω(S(K+\log(S/\varepsilon)))Ω(S(K+log(S/ε))). A matching construction was also given, but assumes ideal error correction. Here we show that the same scaling can be achieved with an explicit noisy error-correction circuit and efficient decoding, counting all state-preparation, gate, measurement, and wait locations, provided that the hardware supports long-range qubit connectivity and fast, reliable classical processing. Thus, reliability adds only a logarithmic overhead shared by all stored qubits. The construction works for sufficiently weak but otherwise general circuit noise, including faults during syndrome extraction and correlated faults, and for independent erasures it matches the best known lower bound up to constant factors. We attain this scaling using quantum Tanner codes, with each memory step consisting of one round of syndrome extraction followed by a fixed number of parallel classical decoder steps that reduce the residual error enough to keep later faults correctable, regardless of memory size, storage time, or target accuracy. Together, this gives exponentially small failure probability per step while keeping the cost of each memory step linear in the code size. The same analysis also extends to certain constant-depth logical Clifford operations.
Abstract The accumulation of physical errors 1–3 prevents the execution of large-scale algorithms in current quantum computers. Quantum error correction 4 promises a solution by encoding k logical qubits onto a larger number n of physical qubits, such that the physical errors are suppressed enough to allow running a desired computation with tolerable fidelity. Quantum error correction becomes practically realizable once the physical error rate is below a threshold value that depends on the choice of quantum code, syndrome measurement circuit and decoding algorithm 5 . We present an end-to-end quantum error correction protocol that implements fault-tolerant memory on the basis of a family of low-density parity-check codes 6 . Our approach achieves an error threshold of 0.7% for the standard circuit-based noise model, on par with the surface code 7–10 that for 20 years was the leading code in terms of error threshold. The syndrome measurement cycle for a length- n code in our family requires n ancillary qubits and a depth-8 circuit with CNOT gates, qubit initializations and measurements. The required qubit connectivity is a degree-6 graph composed of two edge-disjoint planar subgraphs. In particular, we show that 12 logical qubits can be preserved for nearly 1 million syndrome cycles using 288 physical qubits in total, assuming the physical error rate of 0.1%, whereas the surface code would require nearly 3,000 physical qubits to achieve said performance. Our findings bring demonstrations of a low-overhead fault-tolerant quantum memory within the reach of near-term quantum processors.
Fault-tolerant quantum computation is inherently a spacetime problem, requiring not merely good static quantum error-correcting codes but also low-overhead protocols for protecting and manipulating encoded quantum information over time. Fault complexes provide a homological formalism for treating such protocols as single spacetime objects. Here we initiate the study of low-overhead fault complexes by introducing \emph{spacetime lifting}, a method that constructs fault complexes from symmetry-reduced product structures beyond standard foliation. We show that spacetime lifting yields fault complexes and in particular memory experiments with almost-linear fault distance in the total spacetime cost, which substantially outperforms existing constructions. We further endow fault complexes with the operational interpretation of measurement-based cluster state protocols and identify general conditions under which they realize fault-tolerant logical teleportation, showing that spacetime-lifted constructions combine favorable parameter scaling with operational realizations. Our work establishes a systematic route towards more efficient quantum fault tolerance through general complex constructions.