We prove that constant-space-overhead fault-tolerant quantum computation can be achieved with provably strictly logarithmic time overhead, improving over the best known results with additional subpolylogarithmic factors. Our main construction uses polynomial-subrank transversal logical $\CCZ$ gates on good quantum locally testable codes to implement addressable universal computation by transferring batches of logical qubits between dense storage and active logical subspaces while reusing the same ancillary workspace. Logical $\CCZ$ gates are implemented directly by the transversal operation, so only stabilizer resource states require separate preparation. Furthermore, we give an alternative construction that also achieves purely logarithmic time overhead based on modifying the quantum Reed--Solomon magic-state distillation scheme of Nguyen and Pattison. Recursively applying a fixed distillation circuit protected by qLTCs of increasing block length eliminates the subpolylogarithmic time factor.
We prove that constant-space-overhead fault-tolerant quantum computation can be achieved with provably strictly logarithmic time overhead, improving over the best known results with additional subpolylogarithmic factors. Our main construction uses polynomial-subrank transversal logical \CCZ gates on good quantum locally testable codes to implement addressable universal computation by transferring batches of logical qubits between dense storage and active logical subspaces while reusing the same ancillary workspace. Logical \CCZ gates are implemented directly by the transversal operation, so only stabilizer resource states require separate preparation. Furthermore, we give an alternative construction that also achieves purely logarithmic time overhead based on modifying the quantum Reed--Solomon magic-state distillation scheme of Nguyen and Pattison. Recursively applying a fixed distillation circuit protected by qLTCs of increasing block length eliminates the subpolylogarithmic time factor.
In a model of fault-tolerant quantum computation with quick and noiseless polyloglog-time auxiliary classical computation, we construct a fault tolerance protocol with constant-space and O(logN)-time overhead, where O(⋅) hides sub-polylog factors. Our construction utilizes constant-rate quantum locally testable codes (qLTC), new fault-tolerant gadgets on qLTCs and qLDPC codes, and a new analysis framework. In particular, 1) we develop a new simple and self-contained construction of magic state distillation for qubits using qudit quantum Reed-Solomon codes with (logε1)γ spacetime overhead, where γ→0. 2) We prove that the recent family of almost-good qLTCs of Dinur-Lin-Vidick admit parallel single-shot decoders against adversarial errors of weight scaling with the code distance. 3) We develop logical state preparation and logical gate procedures with O(1)-spacetime overhead on qLTCs. 4) To combine these ingredients, we introduce a new framework of fault tolerance analysis called the weight enumerator formalism. The framework permits easy formal composition of fault-tolerant gadgets, so we expect it to be of independent interest. Our work gives the lowest spacetime overhead to date, which, for the first time, matches that of classical fault tolerance up to sub-polylog factors. We conjecture this is optimal up to sub-polylog factors.
A basic question in the theory of fault-tolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance d through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci Turaev-Viro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggest the possibility of universal fault tolerant quantum computation with constant space overhead and time overhead of O(d/logd) . For quantum circuits that allow parallel gate operations, this yields the optimal scaling of space-time overhead known to date.
Magic state distillation is a crucial yet resource-intensive process in fault-tolerant quantum computation. The protocol's overhead, defined as the number of input magic states required per output magic state with an error rate below ε, typically grows as O(logγ(1/ε)). Achieving smaller overheads, i.e., smaller exponents γ, is highly desirable; however, all existing protocols require polylogarithmically growing overheads with some γ>0, and identifying the smallest achievable exponent γ for distilling magic states of qubits has remained challenging. To address this issue, we develop magic state distillation protocols for qubits with efficient, polynomial-time decoding that achieve an O(1) overhead, meaning the optimal exponent γ=0; this improves over the previous best of γ≈0.678 due to Hastings and Haah. In our construction, we employ algebraic geometry codes to explicitly present asymptotically good quantum codes for 210-dimensional qudits that support transversally implementable logical gates in the third level of the Clifford hierarchy. The use of asymptotically good codes with non-vanishing rate and relative distance leads to the constant overhead. These codes can be realised by representing each 210-dimensional qudit as a set of 10 qubits, using stabiliser operations on qubits. The 10-qubit magic states distilled with these codes can be converted to and from conventional magic states for the controlled-controlled-Z (CCZ) and T gates on qubits with only a constant overhead loss, making it possible to achieve constant-overhead distillation of such standard magic states for qubits. These results resolve the fundamental open problem in quantum information theory concerning the construction of magic state distillation protocols with the optimal exponent.
The preparation of a quantum state using a noisy quantum computer (gate noise strength δ) will necessarily affect an O(δ)-fraction of the qubits, no matter which protocol is used. Here, we show that fault-tolerant quantum state preparation can be achieved with constant space overhead improving on previous constructions requiring polylogarithmic overhead.
To achieve this, we add to the toolbox of fault-tolerant schemes for circuits with quantum input and output. More specifically, we construct fault-tolerant interfaces that decrease the level of protection for quantum low-density parity-check (LDPC) codes. When information is encoded in multiple code blocks, our interfaces have constant space overhead.
In our decoder construction that change the level of protection by an arbitrary amount, we circumvent bottlenecks to error pileup and overhead by gradual lowering of the level of encoding at the same time as we increase the number of blocks on which decoding is carried out simultaneously.
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.
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.