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arXiv:1109.4821 · Physical Review A

Experimental quantum error correction with high fidelity

Jingfu Zhang, Dorian A. Gangloff, Osama Moussa, Raymond Laflamme

Published Sep 15, 2011• 21 citations
arXivHTMLPDF

Abstract

More than ten years ago a first step toward quantum error correction (QEC) was implemented [Phys. Rev. Lett. 81, 2152 (1998)]. The work showed there was sufficient control in nuclear magnetic resonance to implement QEC, and demonstrated that the error rate changed from $\ensuremath{\epsilon}$ to $\ensuremath{\sim}{\ensuremath{\epsilon}}^{2}$. In the current work we reproduce a similar experiment using control techniques that have been since developed, such as the pulses generated by gradient ascent pulse engineering algorithm. We show that the fidelity of the QEC gate sequence and the comparative advantage of QEC are appreciably improved. This advantage is maintained despite the errors introduced by the additional operations needed to protect the quantum states.

DOI
10.1103/physreva.84.034303

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arXiv:1203.6260•Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Sep 3, 2012•47 citations
More than ten years ago a first step toward quantum error correction (QEC) was implemented [Phys. Rev. Lett. 81, 2152 (1998)]. The work showed there was sufficient control in nuclear magnetic resonance to implement QEC, and demonstrated that the error rate changed from \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ to \ensuremath∼\ensuremathϵ2\ensuremath{\sim}{\ensuremath{\epsilon}}^{2}\ensuremath∼\ensuremathϵ2. In the current work we reproduce a similar experiment using control techniques that have been since developed, such as the pulses generated by gradient ascent pulse engineering algorithm. We show that the fidelity of the QEC gate sequence and the comparative advantage of QEC are appreciably improved. This advantage is maintained despite the errors introduced by the additional operations needed to protect the quantum states.
Abstract Fault-tolerant quantum computing requires many qubits with long lifetimes and accurate quantum gate operations. However, external noise limits the computing time and hampers accurate quantum gate operations. Quantum error correction (QEC) codes may extend such limits, but imperfect gate operations during QEC cause errors, which could cancel out QEC. We used density matrix simulations to examine the performance of QEC codes with five qubits. In current quantum devices, less than ten qubits are needed to conduct sufficient gate operations within their lifetime so that it is feasible to implement QEC codes. We analyzed the maximum tolerable error rate and error correction effect of individual QEC codes according to the qubit arrangement and gate accuracy. Assuming a 0.1% gate error probability, a logical ∣ 1 〉 state encoded by a five-qubit QEC code is expected to have a 0.25 higher fidelity than its physical counterpart.
We investigate how pulse-sequences and operation times of elementary quantum gates can be optimized for silicon-based donor electron spin quantum computer architecture, complementary to the original Kane's nuclear spin proposal. This gate-sequence-optimal or time-optimal quantum gate control in a quantum circuit is in addition to the more conventional concept of optimality in terms of the number of elementary gates needed in a quantum transformation. The optimal control method we use is the so-called gradient ascent pulse engineering (GRAPE) scheme. We focus on the high fidelity controlled-NOT (CNOT) gate and explicitly find the digitized control sequences by optimizing the effective, reduced donor electron spin Hamiltonian, with external controls over the hyperfine A and exchange J interactions. We first try different piecewise constant control steps and numerically calculate the fidelity (error) against the time needed to implement a CNOT gate with stopping criteria of error in the optimizer set to 〖10〗^(-9) in order to economize the simulation time. Here, the error is defined as 1-F, where F is fidelity. The error is less than 〖10〗^(-8) for times longer than 100ns, and it is found that 30 piecewise constant control steps for the CNOT gate operation will be sufficient to meet the required fidelity (error), and the performance would not be improved further with more steps. With operation time t=100ns and stopping criteria of error set to 〖10〗^(-16), we can find that the near time-optimal, high-fidelity CNOT gate control sequence has an error of 〖1.11×10〗^(-16). We then simulate the control sequences of the CNOT gate, obtained from reduced Hamiltonian simulations, with the full spin Hamiltonian. We find the error of about 〖10〗^(-6) which is below the error threshold required for fault-tolerant (〖10〗^(-4)) quantum computation. The CNOT gate operation time of 100ns is 3 times faster than the globally controlled electron spin scheme of 297ns. One of the great advantages of this near optimal-time high fidelity CNOT gate is that the exchange interaction is not required to be strong (the maximum value is J/h=20MHz compared to the typical value of 10.2GHz. This relaxes significantly the stringent distance constraint of two neighboring donor atoms of about 10nm as reported in the original Kane's proposal to be about 30nm which is within the reach of the current fabrication technology.
The introduction of quantum computing has presented algorithmic solutions to computationally difficult challenges that are far more efficient than those of classical computers. These algorithms leverage the properties of quantum mechanics to manipulate the quantum properties of subatomic particles, requiring immense precision and stability. Current quantum hardware, however, is too noisy and introduces too many errors for these algorithms to be useful in practice, necessitating the use of error correction algorithms. This field survey seeks to introduce various principles of quantum mechanics relevant to quantum computing and quantum error correction (QEC), detail the implementation and motivations of a basic QEC algorithm, and provide insight into the current research direction.
Quantum error correction (QEC) is all for naught if it cannot be implemented experimentally in the laboratory. Further, even if it can be implemented in small laboratory experiments, these experiments need not lead to a technology that is scalable to large quantum computers (where large is defined as “big enough to contemporaneously outperform the best classical computer on some problem”). In this chapter we will survey some of the experiments that have been performed to implement QEC. These experiments are all a long way from demonstrating viable QEC, but demonstrate the proof-of-principle methods that will need to be implemented in the future if quantum computation is to be made viable. This chapter deals with experimental implementations whose goal is to implement QEC, and not experiments using passive or open-loop methods, which are covered in Chapter 22. Experiments in liquid-state NMR The first experiments that attempted to perform QEC were performed using room-temperature liquid-state NMR. Liquid-state NMR [CFH97, GC97] is a testbed for quantum computing ideas in which one uses the internal states of coupled nuclear spins from a molecule as the qubits in a quantum computer. In its original and experimentally implemented form, liquid-state NMR is not generally thought to be scalable: the signal used to read out the result of the quantum computation decays exponentially [W97] in the number of qubits in the system and the mixed states produced in these experiments can be shown to possess no quantum entanglement.
With the development of quantum information and quantum computation science, quantum information processor has been widely used in different areas such as quantum simulation, quantum computation and quantum metrology and so on. To make quantum computer come true, we need to increase the number of controllable qubits of the system and improve the controllability to perform complex quantum manipulation. As a good experimental testbed for quantum information processing, nuclear magnetic resonance (NMR) spin system provides rich and sophisticated quantum control methods. In recent years a lot of multi-qubit experiments have been performed on the platform and a series of experimental technologies have been developed. In this paper, we firstly explain the difficulties of multi-qubit NMR experiments. Then by focusing on the experiment of 7-qubit labelled pseudo-pure state preparation and other relevant experiments, we review the technologies in multi-qubit experiments. Using the radio frequency selective method, the inhomogeneities of the radio frequency pulses are reduced and the spectral resolution is improved. After performing 1/2 spin selective sequence, we can regard the three methyl protons in the sample of crotonic acid as a single 1/2 spin nucleus and treat the whole molecule as a 7-qubit quantum information processor. We utilize Gauss pulses, Hermite pulses, composite pulses and gradient ascent pulse engineering (GRAPE) pulses to implement basic /2 and rotation operations. The GRAPE pulses are calculated by subspace GRAPE program to speed up the computation greatly. The errors of the basic pulses caused by chemical shift and J-coupling evolution can be estimated by the program of pulse compilation. It divides the errors of the pulses into a series of post-errors and pre-errors. A program of sequence compilation is used to eliminate the accumulated error of the whole pulse sequence, reduce the number of pulses and optimize the experimental duration. A variety of methods of quantum state tomography have been proposed to improve the efficiency of reading out information about quantum state. As an experimental example, we combine the above experimental technologies and perform the experiment of 7-qubit labelled pseudo-pure state preparation by using the method of cat state preparation. The sequence of cat state preparation consists of three steps:encoding procedure, phase cycling and decoding procedure. We use 14 experiments to realize the phase cycling and acquire the final 7-qubit labelled pseudo-pure state. The total duration of experimental sequence is about 132 ms. All the readout spectra have the similar shapes to the theoretical expectations. Finally we give an outlook for further research in this direction.