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18 Pages

### w03lecture14a

Course: CPS 237, Spring 2009
School: Duke
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Word Count: 382

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Computing Quantum Lecture 14a (notes on QEC) Michele Mosca Classical Error Correcting Codes Suppose errors in our physical system for storing 0 and 1 cause each physical bit to be toggled independently with probability p We can reduce the probability of error to be in O(p2 ) by using a repetition code e.g. : encode a logical 0 with the state 000 and a logical 1 with the state 111 Reversible networks for...

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Computing Quantum Lecture 14a (notes on QEC) Michele Mosca Classical Error Correcting Codes Suppose errors in our physical system for storing 0 and 1 cause each physical bit to be toggled independently with probability p We can reduce the probability of error to be in O(p2 ) by using a repetition code e.g. : encode a logical 0 with the state 000 and a logical 1 with the state 111 Reversible networks for encoding and decoding b 0 0 b b b b b b b 0 0 Classical Error Correcting Codes After the errors occur, decode the logical bits by taking the majority answer of the three bits and correct the encoded bits So 000 ! 000 111 ! 111 001 ! 000 010 ! 000 100 ! 000 011 ! 111 101 ! 111 110 ! 111 Classical Error Correcting Codes As long as less than 2 errors occurred, we will keep the correct value of the logical bit The probability of 2 or more errors is 3p (1 " p) + p = 3p " 2p ! O(p ) 2 3 2 3 2 1 (which is less than p if p < ) 2 Reversible network for error correction b ! e3 b ! e2 b ! e1 0 0 Assume that e3 + e2 + e1 ! 1 ei ! {0,1} b b b s1 s2 syndrome If s1s2 = 00 then no error occurred Otherwise, the error occurred in bit j where j = 2s1 + s2 Equivalently b e3 ! b ! e2 b ! e1 0 0 s1 b b b s2 Stabilizer measurement?? 0 This is implementing a Z1 measurement (interpreting 0 as +1, and 1 as 1) Stabilizer measurement?? 0 This is implementing a Z1Z 2 measurement Stabilizer measurement?? 0 H H This is implementing a X 1 X 2 measurement Notation clarification For an n-qubit system Z j denotes I " 4 44 I2 3 I "L" 14 " L " I " Z " 1 24 I 4 3 j !1 n! j E.g. n=3, then Z1Z 2 = (Z ! I ! I )(I ! Z ! I ) = (Z ! Z ! I ) Perform operations on logical bits e.g. NOT gate b b b X X X b b b Perform operations on logical bits e.g. c-NOT gate Quantum Error Correcting Codes e.g. : encode a logical 0 with t...

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6.63 15.04 14.26 15.56 16.83 16.98 14.88 14.37 14.23 15.15 15.74 26.15 26.52 25.84 25.69 26.08 25.32 25.95 26.95 25.57 29.57 36.99 38.77 39.59 37.07 37.7 37.22 37.45 37.89 38.99 38.79 49.35 48.21 48.63 410.33 48.51
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APMAM APSAB APSLAKE OPBPC OPRC OPSLAKE Y Year 9.13 3.58 3.91 4.1 7.43 6.47 54235 1948 5.28 4.82 5.2 7.55 11.11 10.26 67567 1949 4.2 3.77 3.67 9.52 12.2 11.35 66161 1950 4.6 4.46 3.93 11.14 15.15 11.13 68094 1951 7.15 4.99 4.88 16.34 20.05 22.81
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BigMac Bread BusFare EngSal EngTax Service TeachSal TeachTax VacDays WorkHrs 31 9 1.27 44.3 44.1 280 21.8 28.2 31.9 1714 33 9 0.27 19.4 23.7 170 9.4 14.8 23.5 1792 98 23 0.09 15.4 20.3 100 2.2 4.3 17.4 2152 131 27 0.09 4.7 37.6 70 1.1 11.7 30.6 2
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0 49 376 726 736 990 2008 2574 2718 2857 2920 3423 3678 3739 4465 4879 5056 5217 6027
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&quot;x1&quot;&quot;x2&quot;&quot;x3&quot;&quot;x4&quot;&quot;y&quot;8410011.42418072.27418014.610712054.97418054.67718014.771314014.65416074.54714034.85110071.481014034.72410031.641018034.56712074.7101318034.84101605
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&quot;C1&quot;&quot;C2&quot;&quot;C3&quot;1.2&quot;pH 3&quot;&quot;Diseased&quot;1.4&quot;pH 3&quot;&quot;Diseased&quot;1&quot;pH 3&quot;&quot;Diseased&quot;1.2&quot;pH 3&quot;&quot;Diseased&quot;1.4&quot;pH 3&quot;&quot;Diseased&quot;.8&quot;pH 5.5&quot;&quot;Diseased&quot;.6&quot;pH 5.5&quot;&quot;Diseased&quot;.8&quot;pH 5.5&quot;&quot;Diseased&quot;1&quot;pH 5.5&quot;&quot;Diseased&quot;.8&quot;pH 5.5&quot;&quot;Diseased&quot;1&quot;pH 7&quot;&quot;Dis
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&quot;response&quot;&quot;type&quot;&quot;subject&quot;12111022733744815926837748919152114321443111411251136124711181329123113421313102483511461217828103910411212923934745101611
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&quot;Obs:&quot;&quot;x:&quot;&quot;y:&quot;1.41.022.421.213.48.884.51.985.571.526.61.837.71.58.751.89.751.7410.781.6311.84212.952.813.992.48141.032.47151.123.05161.153.18171.23.76181.253.68191.253.82201.283.21211.3
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&quot;Linoleic&quot;&quot;Kerosene&quot;&quot;Antiox&quot;&quot;Betacaro&quot;303010.7303010.63303018.41.01340405.049303010.713.183010.120405.04204015.006540205.202303010.6330301.59.04402015.132404015.15303010.73046.8210.34630
Duke - CH - 113
Duke - CH - 113
&quot;stiffness&quot;&quot;plate lengths&quot;309.24409.543114326.54316.84349.84309.74402.16347.263616404.563316348.96381.76392.48366.283518357.18409.98367.383828346.710452.910461.410433.110410.610384.210362.6104
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&quot;temp&quot;&quot;removal%&quot;7.6898.096.5198.256.4397.825.4897.826.5797.8210.2297.9315.6998.3816.7798.8917.1398.9617.6398.916.7298.6815.4598.6912.0698.5111.4498.0910.1798.259.6498.368.5598.277.57986.9498.098.3298.2510.59
Duke - CH - 113
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Duke - CH - 113
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Duke - STA - 244
STA2441/08/2003Homework 1Due 1/15/2003.Please provide concise, neatly written or typed solutions. All work should be your own and not copied from other texts or sources. Do feel free to discuss questions with me, the TA, others in class, or po
Duke - STA - 113
Students0510152025Range: 69.38% - 96.08%, 84 Students Median = 82.07, Quantiles = [76.36, 86.09] Mean = 81.4, Std Dev = 6.255060708090100Course Averages for STA113
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Duke - STA - 103
The data come from http:/www.econstats.com/eq_d1.htm. After the date and day of week they are open high low close return(%)
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