Abstract Quantum computers require memories that are capable of storing quantum information reliably for long periods of time. The surface code is a two-dimensional quantum memory with code parameters that scale optimally with the number of physical qubits, under the constraint of two-dimensional locality. In three spatial dimensions an analogous simple yet optimal code was not previously known. Here we present a family of three dimensional topological codes with optimal scaling code parameters and a polynomial energy barrier. Our codes are based on a construction that takes in a stabilizer code and outputs a three-dimensional topological code with related code parameters. The output codes are topological defect networks formed by layers of surface code joined along one-dimensional junctions, with a maximum stabilizer check weight of six. When the input is a family of good quantum low-density parity-check codes the output codes have optimal scaling. Our results uncover strongly-correlated states of quantum matter that are capable of storing quantum information with the strongest possible protection from errors that is achievable in three dimensions.
Abstract Quantum computers require memories that are capable of storing quantum information reliably for long periods of time. The surface code is a two-dimensional quantum memory with code parameters that scale optimally with the number of physical qubits, under the constraint of two-dimensional locality. In three spatial dimensions an analogous simple yet optimal code was not previously known. Here we present a family of three dimensional topological codes with optimal scaling code parameters and a polynomial energy barrier. Our codes are based on a construction that takes in a stabilizer code and outputs a three-dimensional topological code with related code parameters. The output codes are topological defect networks formed by layers of surface code joined along one-dimensional junctions, with a maximum stabilizer check weight of six. When the input is a family of good quantum low-density parity-check codes the output codes have optimal scaling. Our results uncover strongly-correlated states of quantum matter that are capable of storing quantum information with the strongest possible protection from errors that is achievable in three dimensions.
Quantum computers require memories that are capable of storing quantum information reliably for long periods of time. The surface code is a two-dimensional quantum memory with code parameters that scale optimally with the number of physical qubits, under the constraint of two-dimensional locality. In three spatial dimensions an analogous simple yet optimal code was not previously known. Here we present a family of three dimensional topological codes with optimal scaling code parameters and a polynomial energy barrier. Our codes are based on a construction that takes in a stabilizer code and outputs a three-dimensional topological code with related code parameters. The output codes are topological defect networks formed by layers of surface code joined along one-dimensional junctions, with a maximum stabilizer check weight of six. When the input is a family of good quantum low-density parity-check codes the output codes have optimal scaling. Our results uncover strongly-correlated states of quantum matter that are capable of storing quantum information with the strongest possible protection from errors that is achievable in three dimensions. Quantum error correcting codes are inefficient when implemented on platforms with local connectivity in two dimensions, which motivates platforms with 3D connectivity and codes applicable in these conditions. Here, the authors show how to transform any CSS stabilizer code into a 3D topological code, leading to codes with optimal asymptotic scaling in 3D.
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.
Abstract One of the biggest obstacles to building a large scale quantum computer is the high qubit cost of protecting quantum information. For two-dimensional architectures, the surface code has long been the leading candidate quantum memory, but can require upwards of a thousand physical qubits per logical qubit to reach algorithmically-relevant logical error rates. In this work, we introduce a hierarchical memory formed from surface codes concatenated into high-density parity check codes. These yoked surface codes are arrayed in a rectangular grid, with parity checks (yokes) measured along each row, and optionally along each column, using lattice surgery. Our construction assumes no additional connectivity beyond a nearest-neighbor square qubit grid operating at a physical error rate of 10 −3 . At algorithmically-relevant logical error rates, yoked surface codes use as few as one-third the number of physical qubits per logical qubit as standard surface codes, enabling moderate-overhead fault-tolerant quantum memories in two dimensions.
The surface code is a quantum error-correcting code for one logical qubit, protected by spatially localized parity checks in two dimensions. Due to fundamental constraints from spatial locality, storing more logical qubits requires either sacrificing the robustness of the surface code against errors or increasing the number of physical qubits. We bound the minimal number of spatially nonlocal parity checks necessary to add logical qubits to a surface code while maintaining, or improving, robustness to errors. We saturate the lower limit of this bound, when the number of added logical qubits is a constant, using a family of hypergraph product codes, interpolating between the surface code and constant-rate low-density parity-check codes. Fault-tolerant protocols for logical gates in the quantum code can be inherited from its classical parent codes. We provide near-term practical implementations of this code for hardware based on trapped ions or neutral atoms in mobile optical tweezers. Long-range-enhanced surface codes outperform conventional surface codes using hundreds of physical qubits, and they represent a practical strategy to enhance the robustness of logical qubits to errors in near-term devices.
We investigate layer codes, a family of three-dimensional stabilizer codes that achieves optimal scaling of code parameters and 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 constitute partially self-correcting quantum memories which outperform previously analyzed models such as the cubic and welded solid codes. Notably, we argue that partial self-correction without the requirement of efficient decoding is more common than expected, as it arises solely 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 energy barrier. Finally, we present numerical studies of their memory times and report behavior consistent with partial self-correction.
Quantum information is fragile and must be protected by a quantum error-correcting code for large-scale practical applications. Recently, highly efficient quantum codes have been discovered which require a high degree of spatial connectivity. This raises the question of how to realize these codes with minimal overhead under physical hardware connectivity constraints. Here, we introduce a general recipe to transform any quantum stabilizer code into a subsystem code that has local interactions, with weight and degree three, on a given graph. We call the subsystem codes produced by our recipe wire codes, and their code parameters depend on the input code and the given graph. Wire codes can be adapted to have a local implementation on any graph that supports a low-density embedding of the input Tanner graph, with an overhead that depends on the embedding. In particular, applying our results to a stabilizer code and a subdivision of its own Tanner graph, yields a quantum weight reduction procedure with a multiplicative qubit overhead and distance reduction that are linear in the input check degree and weight, respectively. Applying our results to hypercubic lattices leads to a construction of local subsystem codes with optimal scaling code parameters in any fixed spatial dimension. Similarly, applying our results to families of expanding graphs leads to local codes on these graphs with code parameters that depend on the degree of expansion. Our results constitute a general method to construct low-overhead subsystem codes on general graphs, which can be applied to adapt highly efficient quantum error correction procedures to hardware with restricted connectivity.