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Quantum Information Processing

Fault-tolerant quantum computation using low-cost joint measurements

Yujin Kang, Jonghyun Lee, Jinyoung Ha, Jun Heo

Published May 14, 2024• 3 citations
Publisher

Abstract

Abstract We introduce a new method to implement joint measurements using a 4-qubit twist defect on a rotated surface code. The proposed method enables us to perform logicalS(Phase gate),T( $$\pi /8$$ π/8 gate), andH(Hadamard) with low overhead. Combined with other universal quantum gates, we can implement fault-tolerant quantum computation at the lattice surgery level beyond the gate level while saving considerable resources. We compare our method with previous methods using benchmark circuits by calculating the space and time costs. The proposed method requires additional lines of physical qubits for each encoded patch. Although it slightly increases the space cost for logicalHcompared to the previous work, it reduces the time cost. In addition, the proposed method decreases the space cost and time cost by introducing a 4-qubit twist defect for logicalSandT. Therefore, the overall space-time cost is reduced.

DOI
10.1007/s11128-024-04389-8

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Abstract We introduce a new method to implement joint measurements using a 4-qubit twist defect on a rotated surface code. The proposed method enables us to perform logicalS(Phase gate),T( π/8\pi /8π/8 π/8 gate), andH(Hadamard) with low overhead. Combined with other universal quantum gates, we can implement fault-tolerant quantum computation at the lattice surgery level beyond the gate level while saving considerable resources. We compare our method with previous methods using benchmark circuits by calculating the space and time costs. The proposed method requires additional lines of physical qubits for each encoded patch. Although it slightly increases the space cost for logicalHcompared to the previous work, it reduces the time cost. In addition, the proposed method decreases the space cost and time cost by introducing a 4-qubit twist defect for logicalSandT. Therefore, the overall space-time cost is reduced.
Lattice surgery is a measurement-based technique for performing fault-tolerant quantum computation in two dimensions. When using the surface code, the most general lattice surgery operations require lattice irregularities called twist defects. However, implementing twist-based lattice surgery may require additional resources, such as extra device connectivity, and could lower the threshold and overall performance for the surface code. Here we provide an explicit twist-based lattice surgery protocol and its requisite connectivity layout. We also provide new stabilizer measurement circuits for measuring twist defects which are compatible with our chosen gate scheduling. We undertake the first circuit-level error correction simulations during twist-based lattice surgery using a biased depolarizing noise model. Our results indicate a slight decrease in the threshold for timelike logical failures compared to lattice surgery protocols with no twist defects in the bulk. However, comfortably below threshold (i.e., with CNOT infidelities below $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$), the performance degradation is mild and in fact preferable over proposed alternative twist-free schemes. Lastly, we provide an efficient scheme for measuring YYY operators along boundaries of surface codes, which bypasses certain steps that were required in previous schemes.
One of the most promising routes towards fault-tolerant quantum computation utilizes topological quantum error correcting codes, such as the Z2{\mathbb{Z}}_{2}Z2​ surface code. Logical qubits can be encoded in a variety of ways in the surface code, based on either boundary defects, holes, or bulk twist defects. However, proposed fault-tolerant implementations of the Clifford group in these schemes are limited and often require unnecessary overhead. For example, the Clifford phase gate in certain planar and hole encodings has been proposed to be implemented using costly state injection and distillation protocols. In this paper, we show that within any encoding scheme for the logical qubits, we can fault tolerantly implement the full Clifford group by using joint measurements involving a single appropriately encoded logical ancilla. This allows us to provide low overhead implementations of the full Clifford group in surface and color codes. It also provides the first proposed implementations of the full Clifford group in hyperbolic codes. We further use our methods to propose state-of-the art encoding schemes for small numbers of logical qubits; for example, for code distances d=3,5,7d=3,5,7d=3,5,7, we propose a scheme using 60,160,30860,160,30860,160,308 (respectively) physical data qubits, which allow for the full logical Clifford group to be implemented on two logical qubits. To our knowledge, this is the optimal proposal to date, and thus may be useful for demonstration of fault-tolerant logical gates in small near-term quantum computers.
Fault-tolerant quantum computation requires logical operations that manipulate encoded information while preserving quantum error-correction protection. In planar surface-code architectures, code deformation and lattice surgery provide a local, measurement-based route to such operations. Here we experimentally realize key elements of patch-based surface-code logical processing on a 107-qubit superconducting quantum processor. We first implement a reusable primitive layer comprising merge and split, patch expansion and shrinkage, and deformations mediated by domain walls and twist defects. We then compose these primitives to realize logical state routing, the logical controlled-NOT gate, and the single-qubit Hadamard and phase gates, which together form a Clifford-generating set. All operations are implemented on distance-three rotated surface-code patches with multi-round syndrome extraction and neural-network decoding, without post-selection. Our results advance superconducting surface-code experiments from protected logical memory to active, patch-based fault-tolerant logical operations.
Fault-tolerant quantum computation demands significant resources: large numbers of physical qubits must be checked for errors repeatedly to protect quantum data as logic gates are implemented in the presence of noise. We demonstrate that an approach based on the color code can lead to considerable reductions in the resource overheads compared with conventional methods, while remaining compatible with a two-dimensional layout. We propose a lattice surgery scheme that exploits the rich structure of the color-code phase to perform arbitrary pairs of commuting logical Pauli measurements in parallel while keeping the space cost low. Compared to lattice surgery schemes based on the surface code with the same code distance, and assuming the same amount of time is needed to complete a round of syndrome measurements, our approach yields about a 3 × improvement in the space-time overhead, obtained from a combination of a 1.5 × improvement in spatial overhead together with a 2 × speedup due to the parallelization of commuting logical measurements. Even when taking into account the color code's lower error threshold using current decoders, the overhead is reduced by 10% at a physical error rate of 10 − 3 and by 50% at 10 − 4 . Published by the American Physical Society 2024
A 2d×d×d\boldsymbol{2d \times d \times d}2d×d×d Spacetime Volume Implementation of a Logical S Gate in the Surface Code
The logical S gate implemented via twist defect braiding in the surface code is one of the major sources of overhead in fault-tolerant quantum computing, since an S-gate correction is required in every logical T-gate teleportation. Existing logical S-gate implementations require spacetime volumes of 2d×2d×d2d \times 2d \times d2d×2d×d or 2d×1.5d×d2d \times 1.5d \times d2d×1.5d×d, where ddd is the code distance of the surface code. To the best of our knowledge, their circuit-level implementations have not yet been shown, hindering quantitative comparisons of fault distances and logical error rates. In this work, we provide these missing circuit-level implementations. Additionally, we propose a novel twist defect braiding protocol that reduces the spacetime volume to 2d×d×d2d \times d \times d2d×d×d. First, we construct an implementation of the proposed method using constant-length non-local gates, and then refine it to utilize only nearest-neighbor two-qubit gates on a square grid, without requiring additional two-qubit gate depth beyond that of standard syndrome extraction circuits. Through numerical simulations, we evaluate the fault distances and logical error rates for both existing and proposed methods. Our results show that, although the proposed method reduces the fault distance by one or three, its logical error rates remain comparable to those of existing methods at large code distances (d≥5d \ge 5d≥5) and at physical error rates near p=10−3p = 10^{-3}p=10−3. This demonstrates that the proposed method is promising for near-term fault-tolerant quantum computing.
Modular architectures are a promising approach to scaling quantum computers to fault tolerance. Small, low-noise quantum processors connected through relatively noisy quantum links are capable of fault-tolerant operation as long as the noise can be confined to the interface. Finding protocols that implement the quantum links between modules as efficiently as possible is essential because inter-module entanglement is challenging to produce at a similar rate and fidelity as local entanglement. We introduce a protocol for lattice surgery on surface codes in which all non-local operations are Bell measurements. The protocol simultaneously confines the link noise and requires only half as many module-crossing gates as previously proposed protocols. To mitigate distance-reducing hook errors, we introduce a strategy of alternating the gate sequence between rounds of syndrome measurement, which prevents multiple hooks from simultaneously aligning with a logical operator in the code. We evaluate our protocol's performance when two logical qubits on separate modules are prepared in a logical Bell state. Circuit-level simulations under depolarizing noise show that the logical error suppression for a given entanglement rate between modules is consistently stronger compared to the best-performing alternative protocols for a wide range of link noise, with a typical 40% entanglement resource saving for a constant logical error rate. Our approach to protocol design is applicable to any quantum circuit that must be divided across processor modules and can therefore guide development of resource-efficient modular quantum computation beyond the surface code.