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arXiv:0912.5150 · Physical Review A

Cluster-based architecture for fault-tolerant quantum computation

Keisuke Fujii, Katsuji Yamamoto

Published Apr 27, 2010• 18 citations
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Abstract

We present a detailed description of an architecture for fault-tolerant quantum computation, which is based on the cluster model of encoded qubits. In this cluster-based architecture, concatenated computation is implemented in a quite different way from the usual circuit-based architecture where physical gates are recursively replaced by logical gates with error-correction gadgets. Instead, some relevant cluster states, say fundamental clusters, are recursively constructed through verification and postselection in advance for the higher-level one-way computation, which namely provides error-precorrection of gate operations. A suitable code such as the Steane seven-qubit code is adopted for transversal operations. This concatenated construction of verified fundamental clusters has a simple transversal structure of logical errors, and achieves a high noise threshold $~3%$ for computation by using appropriate verification procedures. Since the postselection is localized within each fundamental cluster with the help of deterministic bare controlled-$Z$ gates without verification, divergence of resources is restrained, which reconciles postselection with scalability.

DOI
10.1103/physreva.81.042324

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We present a detailed description of an architecture for fault-tolerant quantum computation, which is based on the cluster model of encoded qubits. In this cluster-based architecture, concatenated computation is implemented in a quite different way from the usual circuit-based architecture where physical gates are recursively replaced by logical gates with error-correction gadgets. Instead, some relevant cluster states, say fundamental clusters, are recursively constructed through verification and postselection in advance for the higher-level one-way computation, which namely provides error-precorrection of gate operations. A suitable code such as the Steane seven-qubit code is adopted for transversal operations. This concatenated construction of verified fundamental clusters has a simple transversal structure of logical errors, and achieves a high noise threshold  3~3% 3 for computation by using appropriate verification procedures. Since the postselection is localized within each fundamental cluster with the help of deterministic bare controlled-ZZZ gates without verification, divergence of resources is restrained, which reconciles postselection with scalability.
We investigate a scheme of fault-tolerant quantum computation based on the cluster model. Logical qubits are encoded by a suitable code such as the Steane's 7-qubit code. Cluster states of logical qubits are prepared by post-selection through verification at high fidelity level, where the unsuccessful ones are discarded without recovery operation. Then, gate operations are implemented by transversal measurements on the prepared logical cluster states. The noise threshold is improved significantly by making the high fidelity preparation and transversal measurement. It is estimated to be about 3% by a numerical simulation.
We explore basic necessary protocols to achieve fault tolerance for quantum computation in the cluster state and circuit models. For the cluster state model we simulate the implementation of an arbitrary rotation via only measurement on a decohered cluster state. Fidelity is used to quantify the accuracy of the initial cluster state and a gate fidelity for the arbitrary rotation is determined. In the circuit model we compare the accuracy of two methods that can be used to construct a logical zero state appropriate for the [7, 1, 3] Steane quantum error correction code in a non-equiprobable Pauli operator error environment: a fault tolerant method done by applying error correction on seven qubits all in the state zero, and a non-fault tolerant method done by implementing the encoding gate sequence. We find that the latter construction method, in spite of its lack of "fault tolerance," outputs a seven qubit state with a higher fidelity than the first (fault tolerant) method. However, the fidelity of the single qubit of stored information exhibits almost equivalent values between the two construction methods.
Continuous-variable measurement-based quantum computation on cluster states has in recent years shown great potential for scalable, universal, and fault-tolerant quantum computation when combined with the Gottesman-Kitaev-Preskill (GKP) code and quantum error correction.However, no complete fault-tolerant architecture exists that includes everything from cluster-state generation with finite squeezing to gate implementations with realistic noise and error correction.In this work, we propose a simple architecture for the preparation of a cluster state in three dimensions in which gates can be efficiently implemented by gate teleportation.To accommodate scalability, we propose architectures that allow both spatial and temporal multiplexing, with the temporally encoded version requiring as little as two squeezed light sources.Because of its three-dimensional structure, the architecture supports topological qubit error correction, while GKP error correction is efficiently realized within the architecture by teleportation.To validate fault tolerance, the architecture is simulated using surface-GKP codes, including noise from GKP states as well as gate noise caused by finite squeezing in the cluster state.We find a fault-tolerant squeezing threshold of 12.7 dB, with room for further improvement.
We present a scheme to improve the noise threshold for fault-tolerant topological one-way computation with constant overhead. Certain cluster states of finite size, say star clusters, are constructed with logical qubits through an efficient verification process to achieve high fidelity. Then, the star clusters are connected near-deterministically with verification to form a three-dimensional cluster state to implement topological one-way computation. The necessary postselection for verification is localized within the star clusters, ensuring the scalability of computation. By using the Steane seven-qubit code for the logical qubits, this scheme works with a high error rate of 22%2 and reasonable resources comparable to or less than those for the other fault-tolerant schemes. A higher noise threshold would be achieved by adopting a larger code.
We propose a measurement-based fault-tolerant quantum computing (MB-FTQC) architecture for high-connectivity platforms, targeting early-FTQC devices with thousands to tens of thousands of physical qubits. Unlike earlier teleportation-based architectures that rely on code concatenation for scalability, MB-FTQC operates in a zero-concatenation regime with compact fixed-distance codes, keeping the qubit overhead within reach of such devices. MB-FTQC combines offline-verified logical resource states with Knill's error-correcting teleportation, allowing each online step to use single-round syndrome extraction and independent lookup-table decoding. Under circuit-level depolarizing noise, we study a Steane-code implementation with partially fault-tolerant analog RZ(θ)R_Z(θ)RZ​(θ) rotations and a Golay-code implementation with higher-order zero-level distillation for fault-tolerant TTT-state preparation. At a physical error rate of 10−410^{-4}10−4, our estimates give a megaquop-scale equivalent TTT-count of 1.89×1061.89\times10^{6}1.89×106 for the Steane implementation and a gigaquop-scale characteristic TTT-count of 1.87×1091.87\times10^{9}1.87×109 for the Golay implementation. With sufficient parallel resource supply, MB-FTQC achieves a short online critical path, while resource estimates including verification failures and retries show competitive spacetime costs. These results show that compact, fixed-distance codes can support fault-tolerant computation at substantial scales in the early-FTQC regime, without code concatenation.