PLDI '21論文読み会: Quantum Abstract Interpretation

Transcript

1. தଜߊҰ
   2VBOUVN"CTUSBDU*OUFSQSFUBUJPO
   1-%*࿦จಡΈձBU*EFJO
   

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2. ಡΜͩ࿦จ
   w 2VBOUVN"CTUSBDU*OUFSQSFUBUJPO
   w ྔࢠϓϩάϥϜͷந৅ղऍख๏ΛఏҊ͢Δ࿦จ
   /FOHLVO:VBOE+FOT1BMTCFSH2VBOUVNBCTUSBDUJOUFSQSFUBUJPO
   *O1SPDFFEJOHTPGUIFOE"$.4*(1-"/*OUFSOBUJPOBM$POGFSFODFPO1SPHSBNNJOH
   -BOHVBHF%FTJHOBOE*NQMFNFOUBUJPO 1-%*
   
   "TTPDJBUJPOGPS$PNQVUJOH.BDIJOFSZ /FX:PSL /: 64" r
   %0*IUUQTEPJPSH
   

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3. ΞδΣϯμ
   w ந৅ղऍͱ͸
   w ྔࢠܭࢉͱ͸
   w ຊ࿦จͷ঺հ
   

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4. w ϓϩάϥϜͷ੩తղੳͷϑϨʔϜϫʔΫͷҰͭ
   w ϓϩάϥϜΛԿΒ͔ͷந৅ྖҬ BCTUSBDUEPNBJO
   ͷ্Ͱ࣮ߦ
   w ந৅ྖҬ͸ଋ MBUUJDF
   ͱͯ͠දݱ͞ΕΔ
   ந৅ղऍ "CTUSBDU*OQUFSQSFUBUJPO
   
   \Y Z^
   [YZ
   \Y Z [^
   ந৅త
   ۩ମత
   

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5. ྔࢠϏοτ 2VBOUVN#JU 2CJU
   
   w RVCJUೋ৐࿨͕ͷෳૉ਺ͭͰදݱ
   w ͜ͷRVCJUΛ؍ଌ͢Δͱɺ֬཰ Ͱঢ়ଶ ɺ֬཰ Ͱঢ়ଶ ͕ಘΒΕΔ
   |α2 | |0⟩ |β2 | |1⟩
   α|0⟩ + β|1⟩ = (α, β)T (α2 + β2 = 1)
   |0⟩ = (1,0)T, |1⟩ = (0,1)T
   

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6. ༧උ஌ࣝϒϥͱέοτ
   w Λέοτ LFU
   ϕΫτϧͱݺͿɻ
   w ͜Εͷਵ൐Λϒϥ CSB
   ϕΫτϧͱݺͼ ͱॻ͘
   w ͸௨ৗͷ಺ੵʹͳΔ
   |ψ⟩
   ⟨ψ|
   ⟨ψ|ϕ⟩
   |ψ⟩ = (
   α
   β), ⟨ψ| = (α* β*)
   

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7. ྔࢠϏοτ 2VBOUVN#JU 2CJU
   
   w RVCJU͸ෳૉ਺ ݸͰද͞ΕΔ
   n 2n
   α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩ (α2 + β2 + γ2 + δ2 = 1)
   ྫRCJU͸ঢ়ଶͷॏͶ߹Θͤ
   

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8. ิ଍ෳ਺2VCJUͷܭࢉ
   |ϕ, ψ⟩ > = |ϕ⟩ ⊗ |ψ⟩ = (
   α
   β) ⊗ (
   γ
   δ) =
   αγ
   αδ
   βγ
   βδ
   

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9. ྔࢠήʔτ 2VBOUVN-PHJD(BUF
   
   w ྔࢠϏοτͷঢ়ଶΛม͑Δૢ࡞
   XJLJQFEJB2VBOUVNMPHJDHBUFΑΓҾ༻
   |ψ⟩ |ψ′
   
   ⟩
   

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10. ྔࢠήʔτ 2VBOUVN-PHJD(BUF
    
    w ྔࢠϏοτͷঢ়ଶΛม͑Δૢ࡞
    w ྫ)BEBNBSE(BUF
    XJLJQFEJB2VBOUVNMPHJDHBUFΑΓҾ༻
    |0⟩ 1
    2
    |0⟩ +
    1
    2
    |1⟩
    H
    

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11. ྔࢠճ࿏ 2VBOUVN$JSDVJU
    
    w ྔࢠήʔτΛ૊Έ߹Θͤͯɺճ࿏Λߏ੒ͨ͠΋ͷ
    w ճ࿏Λ௚઀తؒ઀తʹදݱ͢Δ΋ͷ͕ྔࢠϓϩάϥϜ
    w ֤ԋࢉ΍ճ࿏શମ͸ϢχλϦߦྻ 6OJUBSZ.BUSJY
    Λຬͨ͢
    ͱͯ͠දݱ͞ΕΔ
    UU† = U†U = I U
    |ψ′
    
    ⟩ = U|ψ⟩
    

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12. ຊ࿦จͷऔΓ૊Ή՝୊
    w ྔࢠϓϩάϥϜͷ੩తղੳΛ͍ͨ͠ʂ
    w RVCJUΛදݱ͢Δͷʹ ݸ΋ͷෳૉ਺͕ඞཁɻετϨʔδɾԋࢉྔڞʹ๲େɻ
    n 2n
    ܭࢉ݁Ռ͸ͲͷΑ͏ͳ
    ঢ়ଶϕΫτϧ
    

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13. ࢀߟຊ೔࣌఺Ͱͷ ଟ෼
    ੈք࠷େͷྔࢠίϯϐϡʔλͷن໛
    w (PPHMFͷ#SJTUMFDPOF RVCJU
    
    w ݹయܭࢉػͰγϛϡϨʔτ͢Δʹ͸
    w ঢ়ଶ͸ ݸͷෳૉ਺Ͱදݱ
    w ແཧͰ͢
    4.7 × 1021
    

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14. ຊ࿦จͷߩݙ
    w ྔࢠϓϩάϥϜʹର͢Δந৅ղऍख๏ΛఏҊ
    w RVCJU਺ʹରͯ͠ଟ߲ࣜ࣌ؒͰ࣮ߦՄೳͰ͋Δ
    

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15. ཧղ͢Δ্ͰͷϙΠϯτ
    w ந৅ྖҬ "CTUSBDU%PNBJO
    ΛͲ͏ఆΊΔ͔ʁ
    w ෦෼ઢܗۭࣹؒӨߦྻͷͳ͢ଋΛݩʹBCTUSBDUEPNBJOΛߏ੒͢Δ
    

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16. ࣹӨߦྻ QSPKFDUJPONBUSJY
    
    w Λຬͨ͢ਖ਼ํߦྻ
    w ͸ϕΫτϧΛ͋Δ෦෼ઢܗۭؒ ʹҠ͢
    w ͱ ͸ҰରҰʹରԠ
    w ྫҎԼͷ ͸ ฏ໘ͱରԠ
    P = P† = P2
    P SP
    P SP
    P xy
    P =
    1 0 0 0
    0 1 0 0
    0 0 0 0
    0 0 0 0
    

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17. ۩ମతঢ়ଶ΋ࣹӨߦྻͱͯ͠දݱ͞ΕΔ
    w ͸ࣹӨߦྻͱͯ͠ͷੑ࣭Λຬͨ͢
    ρ = |ψ⟩⟨ψ|
    

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18. ࣹӨߦྻͷॱং
    P ⊆ Q J
    ff
    SP
    ⊆ SQ
    

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19. ந৅ྖҬͷߏ੒
    w ORVCJUͷঢ়ଶʹରͯ͠
    |ψ⟩⟨ψ| (Ps1
    , ⋯, Psm
    )
    ͷڊେߦྻ
    2n × 2n Nݸͷখ͞ͳࣹӨ
    ߦྻͰۙࣅ
    

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20. ந৅ྖҬͷఆٛ
    w ʹରͯ͠ ͱ͢Δɻͨͩ͠
    w ͸ORVCJUͷ಺ͷ஫໨͢ΔϏοτͷू߹Λදݱ
    w ͷॱং͸ҎԼͰఆΊΔ
    0 < m ≤ 2n S = (s1
    , …, sm
    ) si
    ⊆ {0,…, n − 1}
    si
    P, Q ∈ AbsDom(S)
    AbsDom(S) = {(Ps1
    , …, Psm
    ) ∣ Psi
    ͸2|si
    |ࣹ࣍Өߦྻ}
    P ⊑ Q J
    ff
    ∀i, Psi
    ⊆ Qsi
    

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21. 'JOFS"CTUSBDU%PNBJO
    AbsDom({0,1}, {1,2})
    AbsDom({0,1,2}, {1,2})
    AbsDom(S) ⊴ AbsDom(T) J
    ff
    ∀i si
    ⊆ ti
    

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22. ۩ମྖҬ $PODSFUF%PNBJO
    
    w ந৅ྖҬͷಛผͳ৔߹ɻ࠷΋
    fi
    OFɻ ͱͯ͠
    [n] = {0,…, n − 1}
    AbsDom([n]m) = {2nࣹ࣍Өߦྻmݸͷ૊}
    

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23. ந৅Խࣸ૾ͱ۩ମԽࣸ૾
    ந৅Խ
    ۩ମԽ
    

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24. ΨϩΞ઀ଓ (BMPJT$POOFDUJPO
    
    

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25. ΨϩΞ઀ଓ (BMPJT$POOFDUJPO
    
    "CTUSBDU%PNBJOͰܭࢉͯ͠ಘͨॱংͱ
    $PODSFUF%PNBJOͰܭࢉͯ͠ಘͨॱং͸Ұக
    

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26. ந৅ԋࢉ
    RVCJUͷू߹'ʹର͢ΔϢχλϦߦྻ6Ͱͷந৅ԋࢉΛߦ͏ʹ͸ɺ
    'ΛؚΉ
    fi
    OFͳEPNBJOʹҠͬͯ۩ମతʹܭࢉͯ͠ɺ"CTUSBDUEPNBJOʹ໭Δ
    S = (s1
    , …, sm
    ) ⇒ T = (s1
    ∪ sF
    , …, sm
    ∪ sF
    )
    

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27. ओఆཧ
    ந৅ྖҬ͸ઢܗ෦෼ۭؒ ͷ૊
    ͷ-BUUJDFͩͬͨͷͰܭࢉ݁Ռͷ
    ಺ͷ ʹؚ·ΕΔϏοτ͸෦෼ۭؒ ͷுΔۭؒʹؚ·ΕΔ
    ͱ͍ͬͨBTTFSUJPO͕ࣔͤΔ
    si
    Psi
    

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28. ܭࢉྔ
    w ͭͷࣹӨߦྻͷαΠζΛLRVCJUͱͨ͠৔߹
    w ۭؒܭࢉྔ
    w ࣌ؒܭࢉྔ ϓϩάϥϜͷήʔτ਺
    
    O(|S| × (2k+3 × 2k+3))
    O(|p| × 8k) |p|
    

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29. ϕϯνϚʔΫ
    w #7G Y
    BYCͷB CΛݟ͚ͭΔ໰୊
    w ();શͯɺશ͕ͯॏͳͬͨঢ়ଶΛ࡞Δ
    w (SPWFSG Y
    ͱͳΔYΛ୳ࡧ
    w .BD#PPL1SP $PSFJ()[ (#
    
    

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30. ݁࿦
    w ྔࢠܭࢉͷҝͷந৅ղऍख๏ΛఏҊͨ͠
    4DBMBCMFͰ͋ΔRVCJUن໛ͷγϛϡϨʔγϣϯ΋ग़དྷͨ
    6TFGVMͰ͋Δͭͷॏཁͳ໰୊ͰBTTFSUJPODIFDL͕ग़དྷͨ
    'MFYJCMFͰ͋Δ"CTUSBDU%PNBJOͷઃܭࣗ༝౓͕ߴ͍
    

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