Constant-Overhead Quantum Error Correction with Thin Planar Connectivity

High-performance fault-tolerant quantum computing with many-hypercube codes

Standard approaches to quantum error correction for fault-tolerant quantum computing are based on encoding a single logical qubit into many physical ones, resulting in asymptotically zero encoding rates and therefore huge resource overheads. To overcome this issue, high-rate quantum codes, such as quantum low-density parity-check codes, have been studied over the past decade. In this case, however, it is difficult to perform logical gates in parallel while maintaining low overheads. Here, we propose concatenated high-rate small-size quantum error-detecting codes as a family of high-rate quantum codes. Their simple structure allows for a geometrical interpretation using hypercubes corresponding to logical qubits. We thus call them many-hypercube codes. They can realize both high rates, e.g., 30% (64 logical qubits are encoded into 216 physical ones), and parallelizability of logical gates. Developing dedicated decoder and encoders, we achieve high error thresholds even in a circuit-level noise model. Thus, the many-hypercube codes will pave the way to high-performance fault-tolerant quantum computing.

High-threshold and low-overhead fault-tolerant quantum memory

The accumulation of physical errors1-3 prevents the execution of large-scale algorithms in current quantum computers. Quantum error correction4 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 algorithm5. 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 codes6. Our approach achieves an error threshold of 0.7% for the standard circuit-based noise model, on par with the surface code7-10 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.

Quantum LDPC Codes Based on Cocyclic Block Matrices

Motivated by a family of binary cocyclic block matrices over GF(2), we proposed a construction method to gain the stabilizer of long-length quantum error-correction codes (QECCs). Stabilizer quantum codes (SQCs) can be obtained by the different rows of the yielded circulant permutation matrices; hence, the quantum codes have the virtue of a fast construction algorithm. The recursive relation of a block matrix is employed in the proposed approach, so that the generator matrix of quantum cocyclic codes with long length can be constructed easily. Furthermore, the obtained quantum codes have the low-density advantage of there being no 4-cycles in the Tanner graph.

Quantum Coding via Quasi-Cyclic Block Matrix

An effective construction method for long-length quantum code has important applications in the field based on large-scale data. With the rapid development of quantum computing, how to construct this class of quantum coding has become one of the key research fields in quantum information theory. Motivated by the block jacket matrix and its circulant permutation, we proposed a construction method for quantum quasi-cyclic (QC) codes with two classical codes. This simplifies the coding process for long-length quantum error-correction code (QECC) using number decomposition. The obtained code length N can achieve O(n2) if an appropriate prime number n is taken. Furthermore, with a suitable parameter in the construction method, the obtained codes have four cycles in their generator matrices and show good performance for low density codes.

Quantifying Nonlocality: How Outperforming Local Quantum Codes Is Expensive

Quantum low-density parity-check (LDPC) codes are a promising avenue to reduce the cost of constructing scalable quantum circuits. However, it is unclear how to implement these codes in practice. Seminal results of Bravyi et al. [Phys. Rev. Lett. 104, 050503 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.050503] have shown that quantum LDPC codes implemented through local interactions obey restrictions on their dimension k and distance d. Here we address the complementary question of how many long-range interactions are required to implement a quantum LDPC code with parameters k and d. In particular, in 2D we show that a quantum LDPC code with distance d∝n^{1/2+ϵ} requires Ω(n^{1/2+ϵ}) interactions of length Ω[over ˜](n^{ϵ}). Further, a code satisfying k∝n with distance d∝n^{α} requires Ω[over ˜](n) interactions of length Ω[over ˜](n^{α/2}). As an application of these results, we consider a model called a stacked architecture, which has previously been considered as a potential way to implement quantum LDPC codes. In this model, although most interactions are local, a few of them are allowed to be very long. We prove that limited long-range connectivity implies quantitative bounds on the distance and code dimension.

Low-overhead fault-tolerant quantum computing using long-range connectivity

Vast numbers of qubits will be needed for large-scale quantum computing because of the overheads associated with error correction. We present a scheme for low-overhead fault-tolerant quantum computation based on quantum low-density parity-check (LDPC) codes, where long-range interactions enable many logical qubits to be encoded with a modest number of physical qubits. In our approach, logic gates operate via logical Pauli measurements that preserve both the protection of the LDPC codes and the low overheads in terms of the required number of additional qubits. Compared with surface codes with the same code distance, we estimate order-of-magnitude improvements in the overheads for processing around 100 logical qubits using this approach. Given the high thresholds demonstrated by LDPC codes, our estimates suggest that fault-tolerant quantum computation at this scale may be achievable with a few thousand physical qubits at comparable error rates to what is needed for current approaches.