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arXiv:quant-ph/0311116 · Physical Review A

Quantum-error correction on linear-nearest-neighbor qubit arrays

Austin G. Fowler, Charles D. Hill, Lloyd C. L. Hollenberg

Published Apr 20, 2004• 59 citations
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Abstract

A quantum circuit implementing 5-qubit quantum-error correction on a linear-nearest-neighbor architecture is described. The canonical decomposition is used to construct fast and simple gates that incorporate the necessary swap operations allowing the circuit to achieve the same depth as the current least depth circuit. Simulations of the circuit's performance when subjected to discrete and continuous errors are presented. The relationship between the error rate of a physical qubit and that of a logical qubit is investigated with emphasis on determining the concatenated error correction threshold.

DOI
10.1103/physreva.69.042314

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A quantum circuit implementing 5-qubit quantum-error correction on a linear-nearest-neighbor architecture is described. The canonical decomposition is used to construct fast and simple gates that incorporate the necessary swap operations allowing the circuit to achieve the same depth as the current least depth circuit. Simulations of the circuit's performance when subjected to discrete and continuous errors are presented. The relationship between the error rate of a physical qubit and that of a logical qubit is investigated with emphasis on determining the concatenated error correction threshold.
The five-qubit quantum error correcting code encodes one logical qubit to five physical qubits, and protects the code from a single error. It was one of the first quantum codes to be invented, and various encoding circuits have been proposed for it. In this paper, we propose a systematic procedure for optimization of encoder circuits for stabilizer codes. We start with the systematic construction of an encoder for a five-qubit code, and optimize the circuit in terms of the number of quantum gates. Our method is also applicable to larger stabilizer codes. We further propose nearest neighbor compliant (NNC) circuits for the proposed encoder using a single swap gate, as compared to three swap gates in a prior design.
Real quantum computing technologies have different restrictions and constraints which need to be considered during circuit synthesis. In certain technologies, only physically adjacent qubits can interact. It is common to use a pair of SWAP gates to bring the control and target qubits of any quantum gate to adjacent lines and maintain correct circuit functionality. This paper introduces a new method to synthesize linear nearest neighbor quantum circuit on the base of the analysis of quantum state in circuit. For all 3-qubit quantum circuit, experimental results shows that our algorithm can reduce the quantum cost by approximately 30% on average compared with one recent proposed method.
Design and synthesis of quantum circuits have gained huge momentum during the last few years. With various physical constraint sone of the major issues is in making quantum circuits nearest neighbor complaint. The existing one and two dimensional architectures for the placement of qubits does not scale well for larger circuits. The situation thus demands for a faster approach to solve the problem. To make a quantum circuit nearest neighbor compliant, SWAP gates are inserted in the quantum gate circuit to bring the interacting qubits of a gate adjacent. We have proposed a faster method with a simple yet effective algorithm to minimize the number of SWAP gates for adjacent interactions. The time required for processing does not exceed one second even for larger benchmark circuits. The proposed method is found to reduce the number of SWAP gates on an average by 9.5% compared to the best results in previous works on 2D architectures, and by 45.7% compared to the best results on 1D architecture.
This thesis deals with a series of quantum computer implementation issues from the Kane 31P in 28Si architecture to Shor's integer factoring algorithm and beyond. The discussion begins with simulations of the adiabatic Kane CNOT and readout gates, followed by linear nearest neighbor implementations of 5-qubit quantum error correction with and without fast measurement. A linear nearest neighbor circuit implementing Shor's algorithm is presented, then modified to remove the need for exponentially small rotation gates. Finally, a method of constructing optimal approximations of arbitrary single-qubit fault-tolerant gates is described and applied to the specific case of the remaining rotation gates required by Shor's algorithm.
A significant hurdle towards realization of practical and scalable quantum computing is to protect the quantum states from inherent noises during the computation. In physical implementation of quantum circuits, a long-distance interaction between two qubits is undesirable since, it can be interpreted as a noise. Therefore, multiple quantum technologies and quantum error correcting codes strongly require the interacting qubits to be arranged in a nearest neighbor (NN) fashion. The current literature on converting a given quantum circuit to an NN-arranged one mainly considered chained qubit topologies or Linear Nearest Neighbor (LNN) topology. However, practical quantum circuit realizations, such as Nuclear Magnetic Resonance (NMR), may not have an LNN topology. To address this gap, we consider an arbitrary qubit topology. We present an Integer Linear Programming (ILP) formulation for achieving minimal logical depth while guaranteeing the nearest neighbor arrangement between the interacting qubits. We substantiate our claim with studies on diverse network topologies and prominent quantum circuit benchmarks.
Experimental realization of stabilizer-based quantum error correction (QEC) codes that would yield superior logical qubit performance is one of the formidable task for state-of-the-art quantum processors. A major obstacle toward realizing this goal is the large footprint of QEC codes, even those with a small distance. We propose a circuit based on the minimal distance-3 QEC code, which requires only five data qubits and five ancilla qubits, connected in a ring with iswap gates implemented between neighboring qubits. Using a density-matrix simulation, we show that, thanks to its smaller footprint, the proposed code has a lower logical error rate than surface-17 for similar physical error rates. We also estimate the performance of a neural-network-based error decoder, which can be trained to accommodate the error statistics of a specific quantum processor by training on experimental data.