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arXiv:1412.7921 · Physical Review A

Entanglement generation by communication using phase-squeezed light with photon loss

Fumiaki Matsuoka, Akihisa Tomita, Atsushi Okamoto

Published Mar 7, 2016• 7 citations
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Abstract

To implement fault-tolerant quantum computation, entanglement generation with low error probability and high success probability is required. In a previous paper, we proposed the use of squeezed coherent light as a probe to generate entanglement between two atoms by communication and showed that the error probability is reduced well below the threshold of fault-tolerant quantum computation [F. Matsuoka et al., Phys. Rev. A 88, 022313 (2013)]. In this paper, we investigate the effect of photon loss mainly due to finite coupling efficiency to the cavity. The error probability with photon loss is calculated using a beam-splitter model for homodyne measurement of probe light. We examine the optimum conditions of the amplitude of the probe light and the degree of squeezing to minimize the error probability. We show that the phase-squeezed probe light yields lower error probability than a coherent-light probe, even with photon losses. A fault-tolerant quantum computation algorithm can be implemented under 87% transmittance by concatenating a seven-qubit error correction code for the phase-flip error.

DOI
10.1103/physreva.93.032308

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To implement fault-tolerant quantum computation, entanglement generation with low error probability and high success probability is required. In a previous paper, we proposed the use of squeezed coherent light as a probe to generate entanglement between two atoms by communication and showed that the error probability is reduced well below the threshold of fault-tolerant quantum computation [F. Matsuoka et al., Phys. Rev. A 88, 022313 (2013)]. In this paper, we investigate the effect of photon loss mainly due to finite coupling efficiency to the cavity. The error probability with photon loss is calculated using a beam-splitter model for homodyne measurement of probe light. We examine the optimum conditions of the amplitude of the probe light and the degree of squeezing to minimize the error probability. We show that the phase-squeezed probe light yields lower error probability than a coherent-light probe, even with photon losses. A fault-tolerant quantum computation algorithm can be implemented under 87% transmittance by concatenating a seven-qubit error correction code for the phase-flip error.
In order to improve the error probability of generating entanglement by communication for quantum computation, we propose the use of squeezed light. When generating entanglement between two atoms by communication, the error probability can be reduced by increasing the distance between quantum states of probe light in phase space. The phase rotation of light depends on the atom-photon coupling strength and the light amplitude, which are limited in practice. A large error probability has been expected for coherent probe light. If we assume typical values of light amplitude and phase rotation, $\ensuremath{\alpha}=100$ and $\ensuremath{\theta}=0.01$, the error probability is estimated to be Pcoh(min)=0.14{P}_{\mathrm{coh}}^{(\mathrm{min})}=0.14Pcoh(min)​=0.14 and Pcoh(hom)=0.23{P}_{\mathrm{coh}}^{(\mathrm{hom})}=0.23Pcoh(hom)​=0.23 for minimum error discrimination and homodyne measurements. The error probability can be reduced to ${P}_{\mathrm{squ}}^{(\mathrm{min})}=1.73\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$ and ${P}_{\mathrm{squ}}^{(\mathrm{hom})}=4.09\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ using squeezed coherent light, where the same values of the mean photon number and the phase rotation angle are assumed for the coherent light probe. These values satisfy the requirements for scalable quantum computation.
We analyze the requirements for fault-tolerant quantum computation with atom-atom gates based on cavity quantum electrodynamics (cQED) mediated by a photon with a finite pulse length. For short photon pulses, the distorted shape of the reflected pulses from the cQED system is a serious error source. In the previous study by Goto and Ichimura [Phys. Rev. A 82, 032311 (2010)], only the photon loss is minimized without considering the shape distortion to optimize the system parameters. Here we show an improved optimization method to minimize the infidelity due to the shape distortion and the photon losses in a well-balanced manner for the fault-tolerant scheme using probabilistic gates [Phys. Rev. A 80, 040303(R) (2009)]. Under this optimization, we discuss the fault-tolerant quantum computing requirement for short pulses. Finally, we show that reducing the cavity length is an effective way to reduce the errors of this type of gate in the case of short photon pulses.
The scalability of photonic implementations of fault-tolerant quantum computing based on Gottesman-Kitaev-Preskill (GKP) qubits is injured by the requirements of inline squeezing and reconfigurability of the linear optical network.In this work we propose a topologically error-corrected architecture that does away with these elements at no cost-in fact, at an advantage-to state preparation overheads.Our computer consists of three modules: a two-dimensional (2D) array of probabilistic sources of GKP states; a depthfour circuit of static beam splitters, phase shifters, and short delay lines; and a 2D array of homodyne detectors.The symmetry of our proposed circuit allows us to combine the effects of finite squeezing and uniform photon loss within the noise model, resulting in more comprehensive threshold estimates.These jumps over both architectural and analytical hurdles considerably expedite the construction of a photonic quantum computer.
Fault-tolerant quantum computing requires architectures that simultaneously address scalability, connectivity, and error correction under realistic noise constraints. We present a compound photonic-atomic quantum computing platform that uses cavity QED to realize near-deterministic entangling operations between flying photonic qubits and stationary atomic qubits. Photons provide long-range connectivity and scalability via measurement-based quantum computing (MBQC), while atoms supply reusable, near-deterministic resources for photon generation and entanglement, overcoming the inefficiency of purely photonic platforms. The core primitive is a symmetrized Duan-Kimble photon-atom controlled-phase (CZ) gate, robust to experimental imperfections and high-fidelity. Using single 87^{87}87Rb atoms coupled to optical cavities, we give protocols for state preparation, measurement, photon generation, and entangling gates on tens-of-nanosecond timescales, and show how large-scale cluster states with effectively unrestricted connectivity and reduced overhead can be generated through atomic reuse. We analyze fault tolerance on the Raussendorf-Harrington-Goyal (RHG) lattice with a hardware-aware noise model capturing asymmetric loss and correlated photonic-atomic errors. Logical memory simulations yield a photon-loss threshold near 2.6%2.6\%2.6% per physical gate ($\sim$15\% total per trajectory). The full Clifford set -- Hadamard, phase, CNOT -- is implementable transversally or fold-transversally at thresholds matching the identity channel, and we propose two non-Clifford resource-state routes (code teleportation and magic state cultivation) within the foliated cluster-state architecture.
In order to place an upper bound on the tolerable amount of photon loss at the detectors, we calculate the error threshold for the linear optics quantum computing proposal by Knill, Laflamme, and Milburn [Nature (London) 409, 46 (2001)] under an error model where photon detectors have efficiency <100<100%<100 but all other components---such as single photon sources, beam splitters, and phase shifters---are perfect and introduce no errors. We make use of the fact that the error model induced by the lossy hardware is that of an erasure channel, i.e., the error locations are always known. Using a method based on a Markov chain description of the error correction procedure, our calculations show that, with the 7 qubit Calderbank--Shor--Steane quantum code, the gate error threshold for fault tolerant quantum computation is bounded below by a value between 1.78% and 11.5% depending on the construction of the entangling gates.
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.