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arXiv:1102.3143 · New Journal of Physics

Design of nanophotonic circuits for autonomous subsystem quantum error correction

Joseph Kerckhoff, Dmitri S. Pavlichin, Hamidreza Chalabi, Hideo Mabuchi

Published May 31, 2011• 28 citations
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Abstract

We reapply our approach to designing nanophotonic quantum memories in order to formulate an optical network that autonomously protects a single logical qubit against arbitrary single-qubit errors. Emulating the nine-qubit Bacon–Shor subsystem code, the network replaces the traditionally discrete syndrome measurement and correction steps by continuous, time-independent optical interactions and coherent feedback of unitarily processed optical fields.

DOI
10.1088/1367-2630/13/5/055022

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We reapply our approach to designing nanophotonic quantum memories in order to formulate an optical network that autonomously protects a single logical qubit against arbitrary single-qubit errors. Emulating the nine-qubit Bacon–Shor subsystem code, the network replaces the traditionally discrete syndrome measurement and correction steps by continuous, time-independent optical interactions and coherent feedback of unitarily processed optical fields.
Quantum error correction provides a fertile context for exploring the interplay of feedback control, microscopic physics and non-commutative probability. In this paper we deepen our understanding of this nexus through high-level analysis of a class of quantum memory models that we have previously proposed, which implement continuous-time versions of well-known stabilizer codes in autonomous nanophotonic circuits that require no external clocking or control. We demonstrate that the presence of the gauge subsystem in the nine-qubit Bacon–Shor code allows for a loss-tolerant layout of the corresponding nanophotonic circuit that substantially ameliorates the effects of optical propagation losses, argue that code separability allows for simplified restoration feedback protocols, and propose a modified fidelity metric for quantifying the performance of realistic quantum memories. Our treatment of these topics exploits the homogeneous modeling framework of autonomous nanophotonic circuits, but the key ideas translate to the traditional setting of discrete time, measurement-based quantum error correction.
We propose an approach to quantum error correction based on coding and continuous syndrome readout via scattering of coherent probe fields, in which the usual steps of measurement and discrete restoration are replaced by direct physical processing of the probe beams and coherent feedback to the register qubits. Our approach is well matched to physical implementations that feature solid-state qubits embedded in planar electromagnetic circuits, providing an autonomous and "on-chip" quantum memory design requiring no external clocking or control logic.
We study the performance of Bacon-Shor codes, quantum subsystem codes which are well suited for applications to fault-tolerant quantum memory because the error syndrome can be extracted by performing two-qubit measurements. Assuming independent noise, we find the optimal block size in terms of the bit-flip error probability pXp_XpX​ and the phase error probability pZp_ZpZ​, and determine how the probability of a logical error depends on pXp_XpX​ and pZp_ZpZ​. We show that a single Bacon-Shor code block, used by itself without concatenation, can provide very effective protection against logical errors if the noise is highly biased (pZ/pX≫1)p_Z/p_X\gg 1)pZ​/pX​≫1) and the physical error rate pZp_ZpZ​ is a few percent or below. We also derive an upper bound on the logical error rate for the case where the syndrome data is noisy.
Quantum error correction is necessary to perform large-scale quantum computations in the presence of noise and decoherence. As a result, several aspects of quantum error correction have already been explored. These have been primarily studies of quantum memory[1, 2], an important first step towards quantum computation, where the objective is to increase the lifetime of the encoded quantum information. Additionally, several works have explored the implementation of logical gates[3-5]. In this work we study a next step - fault-tolerantly implementing quantum circuits. We choose the [[4,1,2]][[4, 1, 2]][[4,1,2]] Bacon-Shor subsystem code, which has a particularly simple error-detection circuit. Through both numerics and site-counting arguments, we compute pseudo-thresholds for the Pauli error rate ppp in a depolarizing noise model, below which the encoded circuits outperform the unencoded circuits. These pseudo-threshold values are shown to be as high as p=3%p=3\%p=3% for short circuits, and p=0.6%p=0.6\%p=0.6% for circuits of moderate depth. Additionally, we see that multiple rounds of stabilizer measurements give an improvement over performing a single round at the end. This provides a concrete suggestion for a small-scale fault-tolerant demonstration of a quantum algorithm that could be accessible with existing hardware.
Quantum error correction protects fragile quantum information by encoding it into a larger quantum system. These extra degrees of freedom enable the detection and correction of errors, but also increase the operational complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while operating the logical qubit, and are essential for realizing error suppression in practice. While fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. In this work, we experimentally demonstrate fault-tolerant preparation, measurement, rotation, and stabilizer measurement of a Bacon-Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6% and a Clifford gate error of 0.3% after error correction. Additionally, we prepare magic states with fidelities exceeding the distillation threshold, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant operation. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.