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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Dependence of quantum error correction criteria on concentration of qubits in an ion-trap quantum computer

Naoki Fukuda, Shota Sakaguchi, Koji Nakamae

Published May 28, 2013• 0 citations
Publisher

Abstract

In order to carry out different quantum algorithms on the same quantum circuit, we have to start the next algorithm from the qubit arrangement at the end of the previous one. The arrangement of trapped ions after some algorithm execution is generally not optimal for the following algorithm in an ion-trap quantum computer. We simulated the sequences in the Grover's quantum circuits for searching an unsorted database in order to obtain sequence error rates under a random initial arrangement of qubits. When the sequence error rate for the circuit with QEC is less than that for the circuit without QEC, it is worth introducing the QEC circuit. From the condition, we obtained the quantum error correction criteria that depends on two parameters: concentrations flip of qubits and error rates. If we can know the flip error rate in an ion-trap quantum computer and determine the concentration, the criteria help us to realize a quantum circuit execution with less error rates in an ion-trap quantum computer.

DOI
10.1117/12.2015643

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In order to carry out different quantum algorithms on the same quantum circuit, we have to start the next algorithm from the qubit arrangement at the end of the previous one. The arrangement of trapped ions after some algorithm execution is generally not optimal for the following algorithm in an ion-trap quantum computer. We simulated the sequences in the Grover's quantum circuits for searching an unsorted database in order to obtain sequence error rates under a random initial arrangement of qubits. When the sequence error rate for the circuit with QEC is less than that for the circuit without QEC, it is worth introducing the QEC circuit. From the condition, we obtained the quantum error correction criteria that depends on two parameters: concentrations flip of qubits and error rates. If we can know the flip error rate in an ion-trap quantum computer and determine the concentration, the criteria help us to realize a quantum circuit execution with less error rates in an ion-trap quantum computer.
A Quantum Computer is a new type of computer which can solve problems such as factoring and database search very efficiently. The usefulness of a quantum computer is limited by the effect of two different types of errors, decoherence and inaccuracies. In this paper we show the results of simulations of a quantum computer which consider both decoherence and inaccuracies. We simulate circuits which factor the numbers 15, 21, 35, and 57 as well as circuits which use database search to solve the circuit satisfaction problem. Our simulations show that the error rate per gate is on the order of 10^-6 for a trapped ion quantum computer whose noise is kept below pi/4096 per gate and with a decoherence rate of 10^-6. This is an important bound because previous studies have shown that a quantum computer can factor more efficiently than a classical computer if the error rate is of order 10^-6.
In the presence of physical noise of all platforms for quantum computation, quantum error correction (QEC) becomes a critical way to realize quantum algorithms with large quantum volumes. In order to understand the influence of quantum noise on QEC codes and further improve the performance of logical circuits, the noises should be accurately analyzed with proper models. Here we focus on the trapped-ion system. Fundamentally, we start from the laser pulses of the quantum gates in the circuits and extract the noise components from the complete evolution of the quantum states, beyond the standard depolarizing model and other simplified models. Our simulations indicate that the logical performance under real noises is significantly better than that predicted by previous models. Meanwhile, the advantage of QEC is shown in the levels of one, two, and more logical qubits. Moreover, we can increase the logical fidelity by the method of ion mapping, which is based on knowledge of the specific noise distribution of different ions. Some powerful evidence from the numerical results demonstrates the possibility to access fault-tolerant quantum computation with the trapped-ion system.
We propose a model for quantum computing with long chains of trapped ions and we design quantum error correction schemes for this model. The main components of a quantum error correction scheme are the quantum code and a quantum circuit called the syndrome extraction circuit, which is executed to perform error correction with this code. In this work, we design syndrome extraction circuits tailored to our ion chain model, a syndrome extraction tuning protocol to optimize these circuits, and we construct new quantum codes that outperform the state-of-the-art for chains of about 50 qubits. To establish a baseline under the ion chain model, we simulate the performance of surface codes and bivariate bicycle (BB) codes equipped with our optimized syndrome extraction circuits. Then, we propose a new variant of BB codes defined by weight-five measurements, that we refer to as BB5 codes and we identify BB5 codes that achieve a better minimum distance than any BB codes with the same number of logical qubits and data qubits, such as a [ [ 48 , 4 , 7 ] ] BB5 code. For a physical error rate of 10 − 3 , the [ [ 48 , 4 , 7 ] ] BB5 code achieves a logical error rate per logical qubit of 5 ⋅ 10 − 5 , which is four times smaller than the best BB code in our baseline family. It also achieves the same logical error rate per logical qubit as the distance-7 surface code but using four times fewer physical qubits per logical qubit.
The computational potential of a quantum processor can only be unleashed if errors during a quantum computation can be controlled and corrected for. Quantum error correction works if imperfections of quantum gate operations and measurements are below a certain threshold and corrections can be applied repeatedly. We implement multiple quantum error correction cycles for phase-flip errors on qubits encoded with trapped ions. Errors are corrected by a quantum-feedback algorithm using high-fidelity gate operations and a reset technique for the auxiliary qubits. Up to three consecutive correction cycles are realized, and the behavior of the algorithm for different noise environments is analyzed.
We study the problem of compilation of quantum algorithms into optimized physical-level circuits executable in a quantum information processing (QIP) experiment based on trapped atomic ions. We report a complete strategy: starting with an algorithm in the form of a quantum computer program, we compile it into a high-level logical circuit that goes through multiple stages of decomposition into progressively lower-level circuits until we reach the physical execution-level specification. We skip the fault-tolerance layer, as it is not within the scope of this work. The different stages are structured so as to best assist with the overall optimization while taking into account numerous optimization criteria, including minimizing the number of expensive two-qubit gates, minimizing the number of less expensive single-qubit gates, optimizing the runtime, minimizing the overall circuit error, and optimizing classical control sequences. Our approach allows a trade-off between circuit runtime and quantum error, as well as to accommodate future changes in the optimization criteria that may likely arise as a result of the anticipated improvements in the physical-level control of the experiment.
Quantum Error Correction (QEC) is an important technique for protecting quantum information against decoherence and errors. This involves the design and implementation of algorithms and techniques to minimize error rates and increase the stability of quantum circuits. One of the key parameters in QEC is the distance of the error- correcting code, which determines the number of errors that can be corrected. Another important parameter is the error probability, which quantifies the likelihood of errors occurring in the quantum system. In this context, the goal of a simulation sweeps like the one performed in the code is to study the performance of the QEC code for different values of the distance and error probability, and to optimize the code for maximum accuracy. By varying these parameters and observing the performance of the code, researchers can gain insights into how to design better codes and improve the reliability of quantum computing systems. We also discuss the challenges that need to be addressed for quantum computing to realize its potential in solving practical Error-correction problems.