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arXiv:1906.04864 · Physical Review Letters

Resource-Efficient Topological Fault-Tolerant Quantum Computation with Hybrid Entanglement of Light

S. Omkar, Yong Siah Teo, Hyunseok Jeong

Published Aug 4, 2020• 49 citations
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Abstract

We propose an all-linear-optical scheme to ballistically generate a cluster state for measurement-based topological fault-tolerant quantum computation using hybrid photonic qubits entangled in a continuous-discrete domain. Availability of near-deterministic Bell-state measurements on hybrid qubits is exploited for this purpose. In the presence of photon losses, we show that our scheme leads to a significant enhancement in both tolerable photon-loss rate and resource overheads. More specifically, we report a photon-loss threshold of $\ensuremath{\sim}3.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, which is higher than those of known optical schemes under a reasonable error model. Furthermore, resource overheads to achieve logical error rate of ${10}^{\ensuremath{-}6}({10}^{\ensuremath{-}15})$ is estimated to be $\ensuremath{\sim}8.5\ifmmode\times\else\texttimes\fi{}{10}^{5}(1.7\ifmmode\times\else\texttimes\fi{}{10}^{7})$, which is significantly less by multiple orders of magnitude compared to other reported values in the literature.

DOI
10.1103/physrevlett.125.060501

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We propose an all-linear-optical scheme to ballistically generate a cluster state for measurement-based topological fault-tolerant quantum computation using hybrid photonic qubits entangled in a continuous-discrete domain. Availability of near-deterministic Bell-state measurements on hybrid qubits is exploited for this purpose. In the presence of photon losses, we show that our scheme leads to a significant enhancement in both tolerable photon-loss rate and resource overheads. More specifically, we report a photon-loss threshold of \ensuremath∼3.3\ifmmode×\else\texttimes\fi10\ensuremath−3\ensuremath{\sim}3.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\ensuremath∼3.3\ifmmode×\else\texttimes\fi10\ensuremath−3, which is higher than those of known optical schemes under a reasonable error model. Furthermore, resource overheads to achieve logical error rate of 10\ensuremath−6(10\ensuremath−15){10}^{\ensuremath{-}6}({10}^{\ensuremath{-}15})10\ensuremath−6(10\ensuremath−15) is estimated to be \ensuremath∼8.5\ifmmode×\else\texttimes\fi105(1.7\ifmmode×\else\texttimes\fi107)\ensuremath{\sim}8.5\ifmmode\times\else\texttimes\fi{}{10}^{5}(1.7\ifmmode\times\else\texttimes\fi{}{10}^{7})\ensuremath∼8.5\ifmmode×\else\texttimes\fi105(1.7\ifmmode×\else\texttimes\fi107), which is significantly less by multiple orders of magnitude compared to other reported values in the literature.
We investigate a scheme for topological quantum computing using optical hybrid qubits and make an extensive comparison with previous all-optical schemes. We show that the photon loss threshold reported by Omkar et al. [Phys. Rev. Lett. 125, 060501 (2020)] can be improved further by employing postselection and multi-Bell-state-measurement-based entangling operations to create a special cluster state, known as Raussendorf lattice for topological quantum computation. In particular, the photon loss threshold is enhanced up to $5.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, which is the highest reported value given a reasonable error model. This improvement is obtained at the price of consuming more resources by an order of magnitude compared with the scheme in the aforementioned reference. Nevertheless, this scheme remains resource-efficient compared with other known optical schemes for fault-tolerant quantum computation.
Fault-tolerant photonic quantum computing schemes have suffered low photon-loss thresholds and heavy resource requirements particularly when using single-photon qubits. Recently, several approaches utilizing both discrete and continuous variables of light have been proposed to reach high photon-loss thresholds with relatively low resource requirements. Here we present two schemes that outperform known previous schemes in terms of photon-loss thresholds and resource overheads. The first one is a hybrid scheme that combines discrete-variable (DV) and continuous-variable (CV) states of light. This scheme is nearly ballistic in the sense that passive measurements with hybrid entangled states are sufficient with little feedforwards. The second scheme is based on the parity-based encoding and Bayesian error tracking that enables one to reach a significantly high photon-loss threshold of ~8.5% with moderate resource requirements compared to previous known schemes.
Quantum computation with light, compared with other platforms, offers the unique benefit of natural high-speed operations at room temperature and large clock rate, but a big obstacle of photonics is the lack of strong nonlinearities which also makes loss-tolerant or generally fault-tolerant quantum computation (FTQC) complicated in an all-optical setup. Typical current approaches to optical FTQC that aim at building suitable large multi-qubit cluster states by linearly fusing small elementary resource states would still demand either fairly expensive initial resources or rather low loss and error rates. Here we propose reintroducing weakly nonlinear operations, such as a weak cross-Kerr interaction, to achieve small initial resource cost and high error thresholds at the same time. More specifically, we propose an approach to generate a large-scale cluster state by hybridizing Gottesman-Kitaev-Preskill (GKP) and single-photon qubits. Our approach enables us to implement FTQC based on GKP squeezing of 7.4 and 8.4 dB and a photon loss rate of 1.0 and 5.0 %, respectively. In addition, our scheme has a reduced resource cost, i.e., number of physical qubits/photons per logical qubit or initial entanglement, compared to high-threshold FTQC with optical GKP qubits or fusion-based quantum computation with encoded single-photon-qubit states, respectively. Furthermore, our approach, when assuming very low photon loss, allows to employ GKP squeezing as little as 3.8 dB, which cannot be achieved by using GKP qubits alone.
Hybrid discrete- and continuous-variable (DV-CV) encodings have been developed to overcome operational drawbacks of each individual encoding, while fusion-based quantum computation (FBQC) has emerged to address the practical challenges of scalability toward fault-tolerant photonic quantum computation. Here we introduce hybrid FBQC using small optical DV-CV hybrid resource states connected through hybrid fusions, to combine the advantages of both approaches. We develop protocols for directly generating these resource states without requiring local hybrid logic such as the Hadamard gate, together with a tailored decoding strategy that exploits the error information contained in individual hybrid-fusion outcomes. We demonstrate that hybrid FBQC achieves a photon loss threshold of 2.231% while requiring only 12 hybrid fusions per surface-code unit cell, substantially improving upon the 0.89% threshold and 60-fusion requirement of previous hybrid measurement-based scheme. These results establish optical hybrid FBQC as a high-threshold resource-efficient route to scalable fault-tolerant photonic quantum computation.
Abstract Measurement-based quantum computing (MBQC) in linear optical systems is promising for near-future quantum computing architecture. However, the nondeterministic nature of entangling operations and photon losses hinder the large-scale generation of graph states and introduce logical errors. In this work, we propose a linear optical topological MBQC protocol employing multiphoton qubits based on the parity encoding, which turns out to be highly photon-loss tolerant and resource-efficient even under the effects of nonideal entangling operations that unavoidably corrupt nearby qubits. For the realistic error analysis, we introduce a Bayesian methodology, in conjunction with the stabilizer formalism, to track errors caused by such detrimental effects. We additionally suggest a graph-theoretical optimization scheme for the process of constructing an arbitrary graph state, which greatly reduces its resource overhead. Notably, we show that our protocol is advantageous over several other existing approaches in terms of the fault-tolerance and resource overhead.
We present three schemes for constructing a (2,2)-Shor-encoded 6-ring photonic resource state for fusion-based quantum computing, each relying on a different type of photon source. We benchmark these architectures by analyzing their ability to achieve the loss tolerance threshold for fusion-based quantum computation using the target resource state. More precisely, we estimate their minimum hardware requirements for fault-tolerant quantum computation in terms of the number of photon sources to achieve on-demand generation of resource states with a desired generation period. Notably, we find that a group of 12 deterministic single-photon sources containing a single matter qubit degree of freedom can produce the target resource state near-deterministically by exploiting entangling gates that are repeated until success. The approach is fully modular, eliminates the need for lossy large-scale multiplexing, and reduces the overhead for resource-state generation by several orders of magnitude compared to architectures using heralded single-photon sources and probabilistic linear-optical entangling gates. Our work shows that the use of deterministic single-photon sources embedding a qubit substantially shortens the path toward fault-tolerant photonic quantum computation.