誤り訂正符号を用いた量子力学的性質の保護 : 量子誤り訂正符号入門 (諸分野との協働による数理科学のフロンティア)

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1 * 1 ( : ) [1] [2] 2 $*$ ; hagiwara. hagiwara@aist.go.jp, University $\backslash j_{;}$ of Hawaii, : http: $//manau.j_{p}/b\log/sub/$ http: //staff. aist. go. jp/hagiwara. hagiwara/,

$jp, University $\backslash j_{;}$ of Hawaii, : http:$

2 8 CD( ) CD CD CD LAN $MAN$ LAN DVD Blu-ray Disc ( ) CD 1: 12: $0_{m}$

3 $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$ 9 ( ) ( ) ( ) ( ) 12:30 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 $k$ $\mathbb{c}$ $\mathbb{c}^{2\otimes k}$ $k$ $\mathbb{c}^{2\otimes k}:=\overline{\mathbb{c}^{2}\otimes \mathbb{c}^{2}\otimes\cdots\otimes \mathbb{c}^{2}}$ $\mathbb{c}^{2\otimes k}$ $2^{k}$ $\otimes$ $\mathbb{c}^{2}$ 2

$$\mathbb{c}^{2\otimes k}:=\overline{\mathbb{c}^{2}\otimes$

4 $ m\rangle$ pad 10 $\mathbb{c}^{2\otimes k}$ 1 $\mathbb{c}^{2\otimes k}$ $\langle,$ $\rangle$ $\sqrt{\langle m\rangle, m)\rangle}$ 1 $\mathbb{c}^{2\otimes k}$ 1 $k$ $k$ $q$ $k$ $n$ $n$ $n\geq k$ $f$ ( pad $kn$ : $\mathbb{c}^{2\otimes ) enc $o$ $\mathbb{c}^{2\otimes pad : $f$ k}arrow \mathbb{c}^{2\otimes n}$ enc : $\mathbb{c}^{2\otimes n}arrow \mathbb{c}^{2\otimes n}$ k}arrow \mathbb{c}^{2\otimes n}$ pad $k,n$, enc $k,$ $n$ $ m\rangle$ $k<n$ $k$ $q$ pad $k,n$ $\mathbb{c}^{2\otimes : k}arrow \mathbb{c}^{2\otimes n}$ pad $k,n( m\rangle)$ $:= m \rangle\otimes\frac{n-k}{ 0\rangle\otimes 0\rangle\otimes\cdots\otimes 0\rangle}$ $ 0\rangle:=(1,0)^{T}\in \mathbb{c}^{2}$ enc $\mathbb{c}^{2\otimes n}$ $T$ $2^{k}$ $2^{n}$ encopad $kn( m\rangle)$ $q$ $q $ $n$ $q $ $n$ $ 1\rangle$ $ c_{1}c_{2}\ldots c_{n}\rangle$ $c_{1},$ $c_{2},$ $\ldots,$ $c_{n}\in$ {0,1} $ 1\rangle;=(0,1)^{T}$ $ c_{1}c_{2}\ldots c_{n}\rangle:= c_{1}\rangle\otimes c_{2}\rangle\otimes\cdots\otimes c_{n}\rangle$ $k,n$ enc $opad^{k,n}$ $Q$ $Q:=$ Imenc $\circ$ $\mathbb{q}^{2\otimes k}$ $0$ 1 $\text{ _{}1_{\lrcorner}}$ $o_{\lrcorner}$ $ c_{1}c_{2}\ldots c_{n}\rangle$ 1 $ m\rangle,$ $ m^{l})$ ( $ m\rangle,$ $ m \rangle\rangle$ $\langle m m^{j}$ )

$n}$ pad $k,n$, enc $k,$ $n$ $ m\rangle$ $k<n$ $k$ $q$ pad $k,n$ $\mathbb{c}^{2\otimes : k}arrow \mathbb{c}^{2\otimes n}$ pad $k,n( m\rangle)$ $:= m \rangle\otimes\frac{n-k}{ 0\rangle\otimes$

5 11 $n$ $\mathbb{c}^{2\otimes n}$ dec $\oplus_{s\in\{0,1\}^{r,-k}}v_{s}$ $\mathbb{c}^{2\otimes n}$ $2^{k}$ $V_{S}$ $\{0,1\}^{n-k}arrow U_{2^{\gamma}}\cdot(\mathbb{C})$ rec : $V_{s}$ $rec(s)v_{\theta}$. $rec(s)_{v_{\wedge}}$. $;V_{S}arrow Q$ rec $(s)$ $Q$ n}$ $U_{2^{f1}}(\mathbb{C})$ $\mathbb{c}^{2\otimes $*$ : $n$ $q $ $ y\rangle$ $*$ : $k$ $ m \rangle$ $q$ (1) $n$ $q $ $s$ $ y\rangle$ $ y\grave{\}}$ 2 $s\in\{0,1\}^{n-k}$ $V_{s}$ rec $(s)$ (2) $n$ $q $ 2 $ c \rangle$ (3) $n$ $q $ $ c \rangle$ $ m \rangle$ $k$ (4) $q$ $ m \rangle$ 1 $s$ $s$ $ m\rangle$ $ m \rangle$ $ \langle m m \rangle $ 1 1-2

$;V_{S}arrow Q$ rec $(s)$ $Q$ n}$ $U_{2^{f1}}(\mathbb{C})$ $\mathbb{c}^{2\otimes $*$ : $n$ $q $ $ y\rangle$ $*$ : $k$ $ m \rangle$ $q$ (1) $n$ $q$

6 $_{\infty\ldots 0}$ 12 $_{00\ldots 1}$ $ \bullet\bullet\bullet$ $-\bullet\bullet\bullet$ $_{11\ldots 1}$ 2: $m\in\{0,1\}^{k}$ $m^{j}\in\{0,1\}^{k}$ enc $n$ $ c )$ $ \langle c d\rangle $ $n$ 1 $n$ ( ) $\vdash$ 3 10

$_{11\ldots 1}$ 2: $m\in\{0,1\}^{k}$

7 $\overline{c}_{2}$ $I=(\begin{array}{ll}l 00 1\end{array})$, $X=(\begin{array}{ll}0 ll 0\end{array})$, $Z=(\begin{array}{ll}1 00-1\end{array})$, $Y=(\begin{array}{ll}0 i-i 0\end{array})$ $i$ $X$ $0$ 1 1 $0$ $ 1\rangle$ $X$ 1 $ 0\rangle$ $X$ $X 0\rangle= 1\rangle,$ $X 1\rangle= 0\rangle$ $ \rangle$ $I\otimes X\otimes I\otimes I\otimes I$ $ c_{1}c_{2}c_{3}c_{4}c_{5}\rangle$ $X$ $(I\otimes X\otimes I\otimes I\otimes I) ccccc\rangle= c\overline{c}ccc\rangle$ $c_{2}$ $c_{2}$ $\overline{c}_{2}$ $c_{1}c_{2}c_{3}c_{4}c_{5}$ 2 $c_{2}$ $I$ $c_{i}$ $E_{1},$ $E_{2},$ $\ldots,$ $E_{n}$ $E_{n_{o}}\neq I$ $t$ $E:=E_{1}\otimes E_{2}\otimes\cdots\otimes E_{n}$ $t$ $1\leq n_{0}\leq n$ 1 depolarizing $E_{n_{\text{ }}}$ $I,$ $X,$ $Y,$ $Z$ 1 $q,$ $q/3,$ $q/3,$ $q/3$ depolarizing

$(I\otimes X\otimes I\otimes I\otimes I) ccccc\rangle= c\overline{c}ccc\rangle$ $c_{2}$ $c_{2}$ $\overline{c}_{2}$ $c_{1}c_{2}c_{3}c_{4}c_{5}$ 2 $c_{2}$ $I$ $c_{i}$ $E_{1},$ $E_{2},$ $\ldots,$

8 14 depolarizing $t$ $E$ $q^{t}(1-q)^{n-\ell}$ depolarizing $X,$ $Z,$ $Y$ $1,$ $-1$ $I$ 1 $1,$ $-1$ $n$ $X,$ $Y,$ $Z$ ( $I$ ) $1,$ $-1$ $2^{n-1}$ $n$ $m$ $M_{1},$ $M_{2},$ $\ldots,$ $\Lambda 4_{m}$ Mm $1\leq m_{0}\leq m$ 4 $\{M_{1}, M_{2}, \ldots, M_{m}\}$ [3] $M_{m_{(J}}$ 1 $Q$ $2^{m}$ $2^{m}$ $Q $ $Q $ $ y),$ $ y \rangle$ Mm tm $M_{mo} y\rangle=t_{m_{()}} y\rangle,$ $M_{m}$ $ y \rangle=t_{m}$ $ y \rangle$, $s_{m}$ 5 : $=0$ $t_{m}$ $=-1$ $(s_{1}, s_{2}, \ldots, s_{m})$ $1,$ $-1$ $s_{m\text{ }}:=1$ $t_{m}$ $=1$ 01 $Q $ $\{M_{m\text{ }}\}_{1\leq mo\leq m}$ $(0,0, \ldots,0)$ 41. $\{M_{mo}\}_{1\leq mo\leq m}$ $Q$ $E$ $4X,$ $Y,$ $Z$ $1,$ $-1$ 1 5 $Q $

$Mm tm $M_{mo} y\rangle=t_{m_{()}} y\rangle,$ $M_{m}$ $ y \rangle=t_{m}$ $ y \rangle$, $s_{m}$ 5 : $=0$ $t_{m}$ $=-1$ $(s_{1}, s_{2}, \ldots, s_{m})$ $1,$ $-1$$

9 15 $EQ;=\{E c\rangle c\rangle\in Q\}$ $\{l\mathfrak{l}1_{m\downarrow\rangle}\}_{1\leq m_{(\downarrow}\leq m}$ Proof. $M_{m_{1I}}$ $E$ $P_{1},$ $P_{2}$ $P_{1}P_{2}=P_{2}P_{1}$ $P_{1}P_{2}=-P_{2}P_{1}$ $M_{m_{0}}$ $t_{mo}\in\{1, -1\}$ $M_{m_{1I}}E=t_{m_{(\rceil}}EM_{m_{11}}$ $EQ$ $EQ$ $ y\rangle\in EQ$ $ c\rangle\in Q$ $ y\rangle=e c\rangle$ $t_{m_{0}}$ $M_{m_{t1}} y\rangle$ $=$ $M_{m_{0}}E c\rangle$ $=$ $t_{mu}em_{m_{()}} c\rangle$ $ c\rangle$ $Q$ $M_{m_{U}}$ 1 $ M_{m_{1)}} c\rangle= c\rangle$ $M_{m_{(}}, y\rangle$ $=$ $t_{mo}e c\rangle$ $=$ $t_{m_{(}}, y\rangle$ $ y\rangle\in EQ$ $EQ$ sm $Q $ $EQ$ $Q$ $Q$ $EQ\subset Q $ $\dim EQ=\dim Q $ $EQ$ $Q $

$EQ$ $ y\rangle\in EQ$ $ c\rangle\in Q$ $ y\rangle=e c\rangle$ $t_{m_{0}}$ $M_{m_{t1}} y\rangle$ $=$ $M_{m_{0}}E c\rangle$ $=$ $t_{mu}em_{m_{()}} c\rangle$$

10 $\Lambda_{i}I_{4}$ $M_{1},M_{2},$ $M_{3}$, $M_{1}$ $:=$ $Z\otimes X\otimes I\otimes X\otimes Z$ $M_{2}$ $:=$ $Z\otimes Z\otimes X\otimes I\otimes X$ $M_{3}$ $;=$ $X\otimes Z\otimes Z\otimes X\otimes I$ $M_{4}$ $;=$ $I\otimes X\otimes Z\otimes Z\otimes X$ $Q$ $Q$ 5 $\mathcal{e}$ 1 $E\in \mathcal{e}$ Proof. $EQ$ 1 1 $E\in \mathcal{e}$ 41 $M_{m_{11}}$ $XoIoIoIoI\in \mathcal{e}$ $M_{1}E$ $=$ $-EM_{1}$ $M_{2}E$ $=$ $-EM_{2}$ $M_{3}E$ $=$ $EM_{3}$ $M_{4}E$ $=$ $E_{I}1jI_{4}$ $($ $EQ$ 1, 1, $0,0)$ $\mathcal{e}$ $2^{4}=16$ $\mathcal{e}$ $\mathcal{e}$ $\mathcal{e}$ rec $E$ $E^{2}=I$ 5 $Q$ $E$ $E c)$ $ c\rangle$ $s$ rec rec $(s)=e$ rec $(s)e c\rangle= c)$

$EQ$ 1 1 $E\in \mathcal{e}$ 41 $M_{m_{11}}$ $XoIoIoIoI\in \mathcal{e}$ $M_{1}E$ $=$ $-EM_{1}$ $M_{2}E$ $=$ $-EM_{2}$ $M_{3}E$ $=$ $EM_{3}$ $M_{4}E$ $=$ $E_{I}1jI_{4}$

11 $_{80\{\}2^{-}}$ $_{0\mathfrak{X}\Omega A}$ $_{10^{-}\emptyset l}$ $\forall_{1\lambda \text{ })1}$ 17 $\vee 0\S 10$ $\vee 01\lambda 0^{-}$ $v_{10x^{-}0}$ $V_{1110}$ $_{O}21}$ $_{\zeta\}111}$ $_{1}$ 1 $_{113L}$ 3:5 $Q$ $pad^{1,5}$ 5 16 ( ) 16 $Q$ 2 6 $\mathbb{c}^{2}\otimes 000\ranglearrow Q$ $n$ $k$ depolarizing $(1-3q)^{5}+I5q(I-3q)^{4}$ 6 16 $2^{5}=32$

$$_{\zeta\}111}$ $_{1}$ 1 $_{113L}$ 3:5 $Q$ $pad^{1,5}$ 5 16 ( ) 16 $Q$ 2 6$

12 $\mathcal{e}$ 18 $\mathcal{e}$ depolarizing $q=0.022$ $n=5$ 7 $(\begin{array}{ll}cos\theta -sin\thetasin\theta cos\theta\end{array})$ $(\begin{array}{ll}l/\sqrt{2} 1/\sqrt{2}l/\sqrt{2} -1/\sqrt{2}\end{array})$ $I$ $Q$ ( ) $Q$ $\sum_{e\in \mathcal{e}}\alpha_{e}e$ $Q$ $\alpha_{e}$ (2) 8 (2,2) $t$ $t$ 20 [5][4] 7 LDPC

$$(\begin{array}{ll}l/\sqrt{2} 1/\sqrt{2}l/\sqrt{2} -1/\sqrt{2}\end{array})$ $I$ 1 5 52.$

13 19 6 $M_{1},$ $M_{2},$ $M_{3},$ $l\ovalbox{\tt\small REJECT} 1_{4},$ $lti_{5},1\mathcal{v}1_{6}$ $M_{1}$ $=$ $I\otimes I\otimes I\otimes Z\otimes Z\otimes Z\otimes Z$ $M_{2}$ $=$ $I\otimes Z\otimes Z\otimes I\otimes I\otimes Z\otimes Z$ $M_{3}$ $=$ $Z\otimes I\otimes Z\otimes I\otimes Z\otimes I\otimes Z$ $\Lambda C_{4}$ $=$ $I\otimes I\otimes I\otimes X\otimes X\otimes X\otimes X$ $M_{5}$ $=$ $I\otimes X\otimes X\otimes I\otimes I\otimes X\otimes X$ $M_{6}$ $=$ $X\otimes I\otimes X\otimes I\otimes X\otimes I\otimes X$ $P$ 1 $I$ $m_{0}$ $1\leq m_{0}\leq 7$ $P=X$ $M_{4},$ $M_{5},$ $M_{6}$ $s_{4}s_{5}s_{6}$ $0$ $M_{1},$ $M_{2},$ $M_{3}$ $s_{1}s_{2}s_{3}$ 2 $m_{0}$ $X$ $P=Z$ $M_{1},$ $M_{2},$ $M_{3}$ $0$ $M_{4},$ $M_{5},$ $M_{6}$ 2 $P=Y$ $M_{1},$ $M_{2},$ $M_{3}$ $M_{4},$ $M_{5},$ $M_{6}$ $A=(\begin{array}{lllllll} l 10 I l l l 0 1\end{array})$ $H$

$I\otimes X\otimes X\otimes I\otimes I\otimes X\otimes X$ $M_{6}$ $=$ $X\otimes I\otimes X\otimes I\otimes X\otimes I\otimes X$ 7 1 5 1 $P$ 1 $I$ $m_{0}$ $1\leq m_{0}\leq 7$ $P=X$ $M_{4},$ $M_{5},$

14 20 7 LDPC LDPC CSS sum-product LDPC LDPC LDPC $\sim$ LDPC LDPC (sumproduct ) 2007 LDPC [8] LDPC [9] LDPC [10] 2 8 LDPC $LDPC$ 2 1

15 21 [1] Modern Coding Theory, T.Richardson and R.Urbanke, Cambridge University Press, $\sim$ $\sim$ [2] ( ), 2011 [3] Stabilizer Codes and Quantum Error Correction, D. Gottesman, Ph.D. thesis, Calif. Inst. Technol., Pasadena, [4] A Public-Key Cryptosystem Based On Algebraic Coding Theory R. J. McEliece, DSN Progress Report 42-44: 114, [5] Safeguarding cryptographic keys, G. R. Blakley, AFIPS 1979 Nat. Computer Conf., vol. 48, pp , [6] Good quantum error-correcting codes exist, A.R.Calderbank and P.W.Shor, Phys. Rev. A 54, pp , [7] Error Correcting Codes in Quantum Theory, A.M.Steane, Phys. Rev. Lett. 77, pp , [8] Quantum Error Correction beyond the Bounded Distance Decoding Limit, K.Kasai, M.Hagiwara, H.Imai, K.Sakaniwa, arxiv: , [9] Quantum quasi-cyclic LDPC codes, M. Hagiwara and H. Imai, in Proceedings of ISIT Nice, pp , [10] 2 QC-LDPC GF(2)

Computer Conf., vol. 48, pp. 313-317, 1979. [6] Good quantum error-correcting codes exist, A.R.Calderbank and P.W.Shor, Phys. Rev. A 54, pp. 1098-1105, 1996.

ベクトルの近似直交化を用いた高階線型常微分方程式の整数型解法

ベクトルの近似直交化を用いた高階線型常微分方程式の整数型解法 1848 2013 132-146 132 Fuminori Sakaguchi Graduate School of Engineering, University of Fukui ; Masahito Hayashi Graduate School of Mathematics, Nagoya University; Centre for Quantum Technologies, National