PLDI '21論文読み会: Quantum Abstract Interpretation

Transcript

1. தଜߊҰ
    2VBOUVN
   "CTUSBDU
   *OUFSQSFUBUJPO 1-%*࿦จಡΈձ
   BU
   *EFJO
2. ಡΜͩ࿦จ w 2VBOUVN
   "CTUSBDU
   *OUFSQSFUBUJPO
   w ྔࢠϓϩάϥϜͷந৅ղऍख๏ΛఏҊ͢Δ࿦จ /FOHLVO
   :V
   BOE
   +FOT
   1BMTCFSH
   
   2VBOUVN
   BCTUSBDU
   JOUFSQSFUBUJPO
   *O
   1SPDFFEJOHT
   PG
   UIF
   OE
   "$.
   4*(1-"/
   *OUFSOBUJPOBM
   $POGFSFODF
   PO
   1SPHSBNNJOH
   -BOHVBHF
   %FTJHO
   BOE
   *NQMFNFOUBUJPO
   1-%*
    
   "TTPDJBUJPO
   GPS
   $PNQVUJOH
   .BDIJOFSZ
   /FX
   :PSL
   /:
   64"
   r
   %0*
   IUUQTEPJPSH
   
3. ΞδΣϯμ w ந৅ղऍͱ͸
   w ྔࢠܭࢉͱ͸
   w ຊ࿦จͷ঺հ
4. w ϓϩάϥϜͷ੩తղੳͷϑϨʔϜϫʔΫͷҰͭ
   w ϓϩάϥϜΛԿΒ͔ͷந৅ྖҬ BCTUSBDU
   EPNBJO ͷ্Ͱ࣮ߦ
   w ந৅ྖҬ͸ଋ MBUUJDF ͱͯ͠දݱ͞ΕΔ ந৅ղऍ "CTUSBDU
   *OQUFSQSFUBUJPO \Y
   
   Z
   ^
   [
   
   Y
   
   Z
   \Y
   
   Z
   
   [
   ^ ந৅త ۩ମత
5. ྔࢠϏοτ 2VBOUVN
   #JU
   2CJU w RVCJU
   
   ೋ৐࿨͕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
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*)
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
8. ิ଍
   ෳ਺2VCJUͷܭࢉ |ϕ, ψ⟩ > = |ϕ⟩ ⊗ |ψ⟩ = ( α β) ⊗ ( γ δ) = αγ αδ βγ βδ
9. ྔࢠήʔτ 2VBOUVN
   -PHJD
   (BUF w ྔࢠϏοτͷঢ়ଶΛม͑Δૢ࡞ XJLJQFEJB
   2VBOUVN
   MPHJD
   HBUFΑΓҾ༻ |ψ⟩ |ψ′  ⟩
10. ྔࢠήʔτ 2VBOUVN
    -PHJD
    (BUF w ྔࢠϏοτͷঢ়ଶΛม͑Δૢ࡞
    w ྫ
    )BEBNBSE
    (BUF XJLJQFEJB
    2VBOUVN
    MPHJD
    HBUFΑΓҾ༻ |0⟩ 1 2 |0⟩ + 1 2 |1⟩ H
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⟩
12. ຊ࿦จͷऔΓ૊Ή՝୊ w ྔࢠϓϩάϥϜͷ੩తղੳΛ͍ͨ͠ʂ
    w
    RVCJUΛදݱ͢Δͷʹ ݸ΋ͷෳૉ਺͕ඞཁɻετϨʔδɾԋࢉྔڞʹ๲େɻ n 2n ܭࢉ݁Ռ͸ͲͷΑ͏ͳ
    ঢ়ଶϕΫτϧ
13. ࢀߟ
    ຊ೔࣌఺Ͱͷ ଟ෼ ੈք࠷େͷྔࢠίϯϐϡʔλͷن໛ w (PPHMFͷ#SJTUMFDPOF 
    RVCJU
    w ݹయܭࢉػͰγϛϡϨʔτ͢Δʹ͸
    w ঢ়ଶ͸ ݸͷෳૉ਺Ͱදݱ
    w ແཧͰ͢ 4.7 × 1021
14. ຊ࿦จͷߩݙ w ྔࢠϓϩάϥϜʹର͢Δந৅ղऍख๏ΛఏҊ
    w RVCJU਺ʹରͯ͠ଟ߲ࣜ࣌ؒͰ࣮ߦՄೳͰ͋Δ
15. ཧղ͢Δ্ͰͷϙΠϯτ w ந৅ྖҬ "CTUSBDU
    %PNBJO ΛͲ͏ఆΊΔ͔ʁ
    w ෦෼ઢܗۭࣹؒӨߦྻͷͳ͢ଋΛݩʹBCTUSBDU
    EPNBJOΛߏ੒͢Δ
16. ࣹӨߦྻ QSPKFDUJPO
    NBUSJY 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
17. ۩ମతঢ়ଶ΋ࣹӨߦྻͱͯ͠දݱ͞ΕΔ w ͸ࣹӨߦྻͱͯ͠ͷੑ࣭Λຬͨ͢ ρ = |ψ⟩⟨ψ|
18. ࣹӨߦྻͷॱং P ⊆ Q J ff SP ⊆ SQ
19. ந৅ྖҬͷߏ੒ w ORVCJUͷঢ়ଶʹରͯ͠ |ψ⟩⟨ψ| (Ps1 , ⋯, Psm ) ͷڊେߦྻ 2n × 2n Nݸͷখ͞ͳࣹӨ ߦྻͰۙࣅ
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
21. 'JOFS
    "CTUSBDU
    %PNBJO AbsDom({0,1}, {1,2}) AbsDom({0,1,2}, {1,2}) AbsDom(S) ⊴ AbsDom(T) J ff ∀i si ⊆ ti
22. ۩ମྖҬ $PODSFUF
    %PNBJO w ந৅ྖҬͷಛผͳ৔߹ɻ࠷΋ fi OFɻ
    ͱͯ͠ [n] = {0,…, n − 1} AbsDom([n]m) = {2nࣹ࣍Өߦྻmݸͷ૊}
23. ந৅Խࣸ૾ͱ۩ମԽࣸ૾ ந৅Խ ۩ମԽ
24. ΨϩΞ઀ଓ (BMPJT
    $POOFDUJPO
25. ΨϩΞ઀ଓ (BMPJT
    $POOFDUJPO "CTUSBDU
    %PNBJOͰܭࢉͯ͠ಘͨॱংͱ
    $PODSFUF
    %PNBJOͰܭࢉͯ͠ಘͨॱং͸Ұக
26. ந৅ԋࢉ RVCJUͷू߹'ʹର͢ΔϢχλϦߦྻ6Ͱͷந৅ԋࢉΛߦ͏ʹ͸ɺ
    'ΛؚΉ fi OFͳEPNBJOʹҠͬͯ۩ମతʹܭࢉͯ͠ɺ"CTUSBDU
    EPNBJOʹ໭Δ S = (s1 , …, sm ) ⇒ T = (s1 ∪ sF , …, sm ∪ sF )
27. ओఆཧ ந৅ྖҬ͸ઢܗ෦෼ۭؒ ͷ૊ ͷ-BUUJDFͩͬͨͷͰܭࢉ݁Ռͷ ಺ͷ
    ʹؚ·ΕΔϏοτ͸෦෼ۭؒ
    ͷுΔۭؒʹؚ·ΕΔ
    ͱ͍ͬͨBTTFSUJPO͕ࣔͤΔ
    si Psi
28. ܭࢉྔ w ͭͷࣹӨߦྻͷαΠζΛLRVCJUͱͨ͠৔߹
    w ۭؒܭࢉྔ
    
    w ࣌ؒܭࢉྔ
    
    
    ϓϩάϥϜͷήʔτ਺
    O(|S| × (2k+3 × 2k+3)) O(|p| × 8k) |p|
29. ϕϯνϚʔΫ w #7
    G Y BY CͷB CΛݟ͚ͭΔ໰୊
    w ();
    શͯɺશ͕ͯॏͳͬͨঢ়ଶΛ࡞Δ
    w (SPWFS
    G Y ͱͳΔYΛ୳ࡧ
    w .BD#PPL
    1SP
    $PSF
    J
    ()[ (#
30. ݁࿦ w ྔࢠܭࢉͷҝͷந৅ղऍख๏ΛఏҊͨ͠
     4DBMBCMFͰ͋Δ
    RVCJUن໛ͷγϛϡϨʔγϣϯ΋ग़དྷͨ
     6TFGVMͰ͋Δ
    ͭͷॏཁͳ໰୊ͰBTTFSUJPO
    DIFDL͕ग़དྷͨ
     'MFYJCMFͰ͋Δ
    "CTUSBDU
    %PNBJOͷઃܭࣗ༝౓͕ߴ͍