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Fault-tolerant quantum computing with three-dimensional surface codes

Michael Vasmer

Published Dec 28, 2019• 2 citations
OpenAlex

Abstract

Quantum computers are far more error-prone than their classical counterparts. Therefore, to build a quantum computer capable of running large-scale quantum algorithms, we must use the techniques of quantum error correction to ensure that the computer produces the correct output even when its components are unreliable. However, the resource requirements of building such a fault-tolerant quantum computer are currently prohibitive. Here, we examine the utility of using three-dimensional (3D) surface codes in a fault-tolerant quantum computer. This family of topological error-correcting codes is a generalization of the well-known 2D surface code to three spatial dimensions. We show that certain 3D surface codes have a transversal logical non-Clifford gate. In a quantum computing architecture, a non-Clifford gate is required to achieve computational universality. Transversal gates do not entangle qubits in different codes, so they are naturally fault tolerant because they do not spread errors. Next, we consider the problem of decoding 3D surface codes. In a quantum error-correcting code, we cannot observe the qubits directly, so we measure parity-check operators to gain information about the state of the code. Decoding is the problem of estimating what error has occurred given a list of unsatisfied parity checks. We observe that 3D surface codes offer asymmetric protection against bit-flip and phase-flip errors, but in both cases, we find that a threshold error rate exists below which we can suppress logical errors by increasing the size of the code. We use our results about logical gates and decoding to propose two fault-tolerant quantum computing architectures that utilize 3D surface codes. Finally, we compare the resource requirements of our architectures with the requirements of leading quantum computing architectures based on topological codes. We find that one of our architectures may be competitive with the leading architectures, depending on the properties of the physical systems used to build the qubits.

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Quantum computers are far more error-prone than their classical counterparts. Therefore, to build a quantum computer capable of running large-scale quantum algorithms, we must use the techniques of quantum error correction to ensure that the computer produces the correct output even when its components are unreliable. However, the resource requirements of building such a fault-tolerant quantum computer are currently prohibitive. Here, we examine the utility of using three-dimensional (3D) surface codes in a fault-tolerant quantum computer. This family of topological error-correcting codes is a generalization of the well-known 2D surface code to three spatial dimensions. We show that certain 3D surface codes have a transversal logical non-Clifford gate. In a quantum computing architecture, a non-Clifford gate is required to achieve computational universality. Transversal gates do not entangle qubits in different codes, so they are naturally fault tolerant because they do not spread errors. Next, we consider the problem of decoding 3D surface codes. In a quantum error-correcting code, we cannot observe the qubits directly, so we measure parity-check operators to gain information about the state of the code. Decoding is the problem of estimating what error has occurred given a list of unsatisfied parity checks. We observe that 3D surface codes offer asymmetric protection against bit-flip and phase-flip errors, but in both cases, we find that a threshold error rate exists below which we can suppress logical errors by increasing the size of the code. We use our results about logical gates and decoding to propose two fault-tolerant quantum computing architectures that utilize 3D surface codes. Finally, we compare the resource requirements of our architectures with the requirements of leading quantum computing architectures based on topological codes. We find that one of our architectures may be competitive with the leading architectures, depending on the properties of the physical systems used to build the qubits.
One of the leading quantum computing architectures is based on the two-dimensional (2D) surface code. This code has many advantageous properties such as a high error threshold and a planar layout of physical qubits where each physical qubit need only interact with its nearest neighbors. However, the transversal logical gates available in 2D surface codes are limited. This means that an additional (resource-intensive) procedure known as magic state distillation is required to do universal quantum computing with 2D surface codes. Here, we examine three-dimensional (3D) surface codes in the context of quantum computation. We introduce a picture for visualizing 3D surface codes which is useful for analyzing stacks of three 3D surface codes. We use this picture to prove that the CZ and CCZ gates are transversal in 3D surface codes. We also generalize the techniques of 2D surface-code lattice surgery to 3D surface codes. We combine these results and propose two quantum computing architectures based on 3D surface codes. Magic state distillation is not required in either of our architectures. Finally, we show that a stack of three 3D surface codes can be transformed into a single 3D color code (another type of quantum error-correcting code) using code concatenation.
Practical large-scale quantum computation requires both efficient error correction and robust implementation of logical operations. Three-dimensional (3D) color codes are a promising candidate for fault-tolerant quantum computation due to their transversal non-Clifford gates, but efficient decoding remains challenging. In this work, we extend previous decoders for two-dimensional color codes [1], which are based on the restriction of the decoding problem to a subset of the qubit lattice, to three dimensions. Including boundaries of 3D color codes, we demonstrate that the 3D restriction decoder achieves optimal scaling of the logical error rate and a threshold value of 1.55(6)% for code-capacity bit- and phase-flip noise, which is almost a factor of two higher than previously reported for this family of codes [2, 3]. We furthermore present qCodePlot3D, a Python package for visualizing 2D and 3D color codes, error configurations, and decoding paths, which supports the development and analysis of such decoders. These advancements contribute to making 3D color codes a more practical option for exploring fault-tolerant quantum computation.
Fault-tolerant logic gates will consume a large proportion of the resources of a two-dimensional quantum computing architecture. Here we show how to perform a fault-tolerant non-Clifford gate with the surface code; a quantum error-correcting code now under intensive development. This alleviates the need for distillation or higher-dimensional components to complete a universal gate set. The operation uses both local transversal gates and code deformations over a time that scales with the size of the qubit array. An important component of the gate is a just-in-time decoder. These decoding algorithms allow us to draw upon the advantages of three-dimensional models using only a two-dimensional array of live qubits. Our gate is completed using parity checks of weight no greater than four. We therefore expect it to be amenable with near-future technology. As the gate circumvents the need for magic-state distillation, it may reduce the resource overhead of surface-code quantum computation considerably.
Quantum computation holds the promise of solving certain complex problems exponentially faster than classical computers. However, the high prevalent noise in current quantum devices impedes the accurate execution of even basic algorithms. This can be remedied by protecting quantum information with a quantum error-correcting code, where the logical information of an algorithmic qubit is spread across multiple physical qubits. Individual quantum errors are then located and corrected by the fault-tolerant measurement of multi-qubit stabilizer operators (parity checks). Unfortunately, error correction and fault tolerance both impose large demands on the qubit overhead: hundreds to thousands of physical qubits per logical qubit. We reduce the spacetime cost of fault tolerance by redesigning key building blocks of an error-corrected quantum computer. First, we develop a combinatorial proof with flag fault tolerance that exponentially reduces the extra qubits needed to measure a stabilizer of any size, while tolerating one fault. We leverage these proofs to then design state preparation circuits for the Steane and Golay codes with 100% yield. Next, we improve error correction on a planar layout by showing that a distance-four code encoding six logical qubits protects information as well as the distance-five surface code, using one-tenth as many physical qubits. Finally, we optimize the time overhead of logical gates in surface code quantum computers by protecting measurement results with a classical code, cutting computation time by a factor of two to six. Our hardware-agnostic optimizations of fault tolerance overheads thus suggest new routes to advance the timeline of error-free quantum computing.
Fault-tolerant quantum computing (FTQC) is required to execute quantum algorithms capable of tackling computationally intractable problems in chemistry, material science, and cryptography. Noisy Intermediate-Scale Quantum (NISQ) devices remain constrained by physical error rates that degrade quantum coherence over short circuit depths. Quantum error correction (QEC) protects fragile logical states by encoding them into non-local degrees of freedom across topological physical qubit arrays. This research presents a comprehensive architectural and theoretical evaluation of three primary QEC paradigms: planar surface codes, topological color codes, and asymptotic quantum low-density parity-check (qLDPC) codes. We examine error thresholds, transversal gate constraints governed by the Eastin-Knill theorem, magic state distillation (MSD) overheads, lattice surgery, and classical decoding latencies across superconducting transmons, trapped ions, and neutral atom platforms.
Quantum fault tolerance is a prerequisite to harness the full potential of quantum computers. However, achieving fault tolerance remains a major challenge since it requires optimizing tradeoffs between several intertwined theoretical and experimental parameters, including the encoding rate, code distance, parity-check weights, code planarity, and routing. Surface codes remain strong candidates for fault tolerance due to their simple geometric structure, established decoding methods, locality, and experimental attainability. In this work, we present finite families of hyperbolic surface codes that leverage the optimal efficiency scaling kd2/n=C(log⁡k)2kd^2/n = C (\log{k})^2kd2/n=C(logk)2 attainable by surface codes with bounded local geometry. We show that optimizing the periodic identifications can double the code distance and thereby increase the code's efficiency by a factor of four, at fixed qubit count, encoding rate, and parity-check weight. We also address the routing problem of hyperbolic surface codes by introducing a topology-aware algorithm that partitions a hyperbolic code into bounded planar modules while reducing long-range communications demand. Circuit-level Monte Carlo simulations yield finite-size threshold estimates for both modular and monolithic layouts, showing that tripling the long-range gate error probability mildly lowers, but does not eliminate, the estimated error-correction thresholds. Together, these results strengthen the case for hyperbolic surface codes as structured and highly efficient large-scale quantum error correction codes for future fault-tolerant architectures.