SCIENCE CHINA Information Sciences, Volume 63 , Issue 10 : 202501(2020) https://doi.org/10.1007/s11432-019-2689-4

Effect on ion-trap quantum computers from the quantum nature of the driving field

Biyao YANG 1,2,3,4, Li YANG 1,3,*
• AcceptedSep 22, 2019
• PublishedMay 18, 2020
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Abstract

In this study, we evaluate the effect on ion-trap quantum computers (QCs) from the quantum nature of the driving field, and propose a theoretical limit for ion-trap QCs that may impact the design of quantum algorithms and realization of practical QCs. We obtain, for the first time, the permitted depth of logical operation for fault-tolerant ion-trap QCs. Physically, we provide an exact (full-quantum) description of the QC system, and present for the first time its time evolution after gate operations; mathematically, we solve problems such as certain summations of trigonometric series with any given precision. Comparing the actual state after CNOT gates driven by a quantized field with the expected state, we obtain the failure probability and estimate that the numberof CNOT gates on the same pair of physical qubits is notmore than $10^2$ in one error-correction period, which is a physical limit that cannot be easily overcome. The conclusion can help determine the number of CNOT operations between coding and decoding in one error-correction period and can be used as a reference for quantum algorithm design.

Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 61672517), National Cryptography Development Fund (Grant No. MMJJ20170108), National Key RD Program of China (Grant No. 2016QY03D0503), and Beijing Municipal Science Technology Commission (Grant No. Z191100007119006).

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• Figure 1

(Color online) Failure probability from field quantization for different initial states and $\bar{n}$. Here, $N$ denotes the operation number and $p_f$ denotes the failure probability. The failure probabilities under the initial state $|1\rangle_{x} |0\rangle_{y}$ and $|1\rangle_{x} |1\rangle_{y}$ are almost the same, and the corresponding curves overlap in both the cases of $\bar{n}=10^6$ and $\bar{n}=10^4$.

• Table 1

Table 1Values of $S_i~~(i=1,2,\ldots,42)$

 Sum Value Sum Value $S_1$ $0.500000098174762910198272554299$ $S_{22}$ $0.999998883150173201808064776118$ $S_2$ $0.499999979636969416064898201137$ $S_{23}$ $0.999999566574913171200982444856$ $S_3$ $0.499999901825219523136530070842$ $S_{24}$ $6.168501644358131228504850110550\times~10^{-7}$ $S_4$ $0.499999783287456585600491222428$ $S_{25}$ $5.890483292671957696772466384108\times~10^{-7}$ $S_5$ $0.499999901825237089801727445701$ $S_{26}$ $1.963495258203965451085974149167~\times~10^{-7}$ $S_6$ $0.499999783287456585600491222428$ $S_{27}$ $0.999999383149835564186877149515$ $S_7$ $0.499999848174788132734922216027$ $S_{28}$ $0.999999566574913171200982444856$ $S_8$ $0.499999705475753242352304889827$ $S_{29}$ $0.999997532605141996802084969596$ $S_9$ $0.499999586937964823618783989023$ $S_{30}$ $3.926947317638469728959289607809\times~10^{-7}$ $S_{10}$ $0.500000294524246757647695110173$ $S_{31}$ $0.999997532604525147746750277033$ $S_{11}$ $0.707106796084216571982847264135$ $S_{32}$ $3.926986075381421964264370003863\times~10^{-7}$ $S_{12}$ $0.707106518404046899137211784607$ $S_{33}$ $-0.99999876630028797819009097432$ $S_{13}$ $0.707106707695871467210027449798\times~10^{-7}$ $S_{34}$ $2.467394858003197915030403720421\times~10^{-6}$ $S_{14}$ $0.707106707695871467210027449798$ $S_{35}$ $3.926986075381421964264370003825\times~10^{-7}$ $S_{15}$ $6.168498560109049545128396829682\times~10^{-7}$ $S_{36}$ $2.46739300745180671853161538037\times~10^{-6}$ $S_{16}$ $-1.96349546006718133137558424000\times~10^{-7}$ $S_{37}$ $-3.9269909201343626627511684800\times~10^{-7}$ $S_{17}$ $6.168497017982575442178796124039\times~10^{-7}$ $S_{38}$ $0.999997532603908304779480548015$ $S_{18}$ $-1.96349546006718133137558424000\times~10^{-7}$ $S_{39}$ $3.927024833172821399411092766930\times~10^{-7}$ $S_{19}$ $-5.89048493515295390220345714779\times~10^{-7}$ $S_{40}$ $-0.999998766299671128373754299030$ $S_{20}$ $0.999999383150143989095045487160$ $S_{41}$ $2.467396091695220519451984529811\times~10^{-6}$ $S_{21}$ $0.499958694533432311848321647856$ $S_{42}$ $-3.92699092013436266275116848001\times~10^{-7}$

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