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PLDI '21論文読み会: Quantum Abstract Interpretation
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Idein
June 08, 2022
Research
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1.4k
PLDI '21論文読み会: Quantum Abstract Interpretation
Idein
June 08, 2022
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Transcript
தଜߊҰ 2VBOUVN"CTUSBDU*OUFSQSFUBUJPO 1-%*จಡΈձBU*EFJO
ಡΜͩจ w 2VBOUVN"CTUSBDU*OUFSQSFUBUJPO w ྔࢠϓϩάϥϜͷநղऍख๏ΛఏҊ͢Δจ /FOHLVO:VBOE+FOT1BMTCFSH2VBOUVNBCTUSBDUJOUFSQSFUBUJPO *O1SPDFFEJOHTPGUIFOE"$.4*(1-"/*OUFSOBUJPOBM$POGFSFODFPO1SPHSBNNJOH -BOHVBHF%FTJHOBOE*NQMFNFOUBUJPO 1-%*
"TTPDJBUJPOGPS$PNQVUJOH.BDIJOFSZ /FX:PSL /: 64" r %0*IUUQTEPJPSH
ΞδΣϯμ w நղऍͱ w ྔࢠܭࢉͱ w ຊจͷհ
w ϓϩάϥϜͷ੩తղੳͷϑϨʔϜϫʔΫͷҰͭ w ϓϩάϥϜΛԿΒ͔ͷநྖҬ BCTUSBDUEPNBJO ͷ্Ͱ࣮ߦ w நྖҬଋ MBUUJDF ͱͯ͠දݱ͞ΕΔ
நղऍ "CTUSBDU*OQUFSQSFUBUJPO \Y Z^ [Y Z \Y Z [^ நత ۩ମత
ྔࢠϏοτ 2VBOUVN#JU 2CJU w RVCJUೋ͕ͷෳૉͭͰදݱ w ͜ͷRVCJUΛ؍ଌ͢Δͱɺ֬ Ͱঢ়ଶ ɺ֬ Ͱঢ়ଶ
͕ಘΒΕΔ |α2 | |0⟩ |β2 | |1⟩ α|0⟩ + β|1⟩ = (α, β)T (α2 + β2 = 1) |0⟩ = (1,0)T, |1⟩ = (0,1)T
༧උࣝϒϥͱέοτ w Λέοτ LFU ϕΫτϧͱݺͿɻ w ͜ΕͷਵΛϒϥ CSB ϕΫτϧͱݺͼ ͱॻ͘
w ௨ৗͷੵʹͳΔ |ψ⟩ ⟨ψ| ⟨ψ|ϕ⟩ |ψ⟩ = ( α β), ⟨ψ| = (α* β*)
ྔࢠϏοτ 2VBOUVN#JU 2CJU w RVCJUෳૉ ݸͰද͞ΕΔ n 2n α|00⟩ +
β|01⟩ + γ|10⟩ + δ|11⟩ (α2 + β2 + γ2 + δ2 = 1) ྫRCJUঢ়ଶͷॏͶ߹Θͤ
ิෳ2VCJUͷܭࢉ |ϕ, ψ⟩ > = |ϕ⟩ ⊗ |ψ⟩ = (
α β) ⊗ ( γ δ) = αγ αδ βγ βδ
ྔࢠήʔτ 2VBOUVN-PHJD(BUF w ྔࢠϏοτͷঢ়ଶΛม͑Δૢ࡞ XJLJQFEJB2VBOUVNMPHJDHBUFΑΓҾ༻ |ψ⟩ |ψ′  ⟩
ྔࢠήʔτ 2VBOUVN-PHJD(BUF w ྔࢠϏοτͷঢ়ଶΛม͑Δૢ࡞ w ྫ)BEBNBSE(BUF XJLJQFEJB2VBOUVNMPHJDHBUFΑΓҾ༻ |0⟩ 1 2
|0⟩ + 1 2 |1⟩ H
ྔࢠճ࿏ 2VBOUVN$JSDVJU w ྔࢠήʔτΛΈ߹Θͤͯɺճ࿏Λߏͨ͠ͷ w ճ࿏Λతؒతʹදݱ͢Δͷ͕ྔࢠϓϩάϥϜ w ֤ԋࢉճ࿏શମϢχλϦߦྻ 6OJUBSZ.BUSJY
Λຬͨ͢ ͱͯ͠දݱ͞ΕΔ UU† = U†U = I U |ψ′  ⟩ = U|ψ⟩
ຊจͷऔΓΉ՝ w ྔࢠϓϩάϥϜͷ੩తղੳΛ͍ͨ͠ʂ w RVCJUΛදݱ͢Δͷʹ ݸͷෳૉ͕ඞཁɻετϨʔδɾԋࢉྔڞʹେɻ n 2n ܭࢉ݁ՌͲͷΑ͏ͳ ঢ়ଶϕΫτϧ
ࢀߟຊ࣌Ͱͷ ଟ ੈք࠷େͷྔࢠίϯϐϡʔλͷن w (PPHMFͷ#SJTUMFDPOF RVCJU w ݹయܭࢉػͰγϛϡϨʔτ͢Δʹ w
ঢ়ଶ ݸͷෳૉͰදݱ w ແཧͰ͢ 4.7 × 1021
ຊจͷߩݙ w ྔࢠϓϩάϥϜʹର͢Δநղऍख๏ΛఏҊ w RVCJUʹରͯ͠ଟ߲ࣜ࣌ؒͰ࣮ߦՄೳͰ͋Δ
ཧղ͢Δ্ͰͷϙΠϯτ w நྖҬ "CTUSBDU%PNBJO ΛͲ͏ఆΊΔ͔ʁ w ෦ઢܗۭࣹؒӨߦྻͷͳ͢ଋΛݩʹBCTUSBDUEPNBJOΛߏ͢Δ
ࣹӨߦྻ QSPKFDUJPONBUSJY w Λຬͨ͢ਖ਼ํߦྻ w ϕΫτϧΛ͋Δ෦ઢܗۭؒ 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
۩ମతঢ়ଶࣹӨߦྻͱͯ͠දݱ͞ΕΔ w ࣹӨߦྻͱͯ͠ͷੑ࣭Λຬͨ͢ ρ = |ψ⟩⟨ψ|
ࣹӨߦྻͷॱং P ⊆ Q J ff SP ⊆ SQ
நྖҬͷߏ w ORVCJUͷঢ়ଶʹରͯ͠ |ψ⟩⟨ψ| (Ps1 , ⋯, Psm ) ͷڊେߦྻ
2n × 2n Nݸͷখ͞ͳࣹӨ ߦྻͰۙࣅ
நྖҬͷఆٛ w ʹରͯ͠ ͱ͢Δɻͨͩ͠ w ORVCJUͷͷ͢ΔϏοτͷू߹Λදݱ w ͷॱংҎԼͰఆΊΔ 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
'JOFS"CTUSBDU%PNBJO AbsDom({0,1}, {1,2}) AbsDom({0,1,2}, {1,2}) AbsDom(S) ⊴ AbsDom(T) J ff
∀i si ⊆ ti
۩ମྖҬ $PODSFUF%PNBJO w நྖҬͷಛผͳ߹ɻ࠷ fi OFɻ ͱͯ͠ [n] = {0,…,
n − 1} AbsDom([n]m) = {2nࣹ࣍Өߦྻmݸͷ}
நԽࣸ૾ͱ۩ମԽࣸ૾ நԽ ۩ମԽ
ΨϩΞଓ (BMPJT$POOFDUJPO
ΨϩΞଓ (BMPJT$POOFDUJPO "CTUSBDU%PNBJOͰܭࢉͯ͠ಘͨॱংͱ $PODSFUF%PNBJOͰܭࢉͯ͠ಘͨॱংҰக
நԋࢉ RVCJUͷू߹'ʹର͢ΔϢχλϦߦྻ6ͰͷநԋࢉΛߦ͏ʹɺ 'ΛؚΉ fi OFͳEPNBJOʹҠͬͯ۩ମతʹܭࢉͯ͠ɺ"CTUSBDUEPNBJOʹΔ S = (s1 , …,
sm ) ⇒ T = (s1 ∪ sF , …, sm ∪ sF )
ओఆཧ நྖҬઢܗ෦ۭؒ ͷ ͷ-BUUJDFͩͬͨͷͰܭࢉ݁Ռͷ ͷ ʹؚ·ΕΔϏοτ෦ۭؒ ͷுΔۭؒʹؚ·ΕΔ ͱ͍ͬͨBTTFSUJPO͕ࣔͤΔ si Psi
ܭࢉྔ w ͭͷࣹӨߦྻͷαΠζΛLRVCJUͱͨ͠߹ w ۭؒܭࢉྔ w ࣌ؒܭࢉྔ ϓϩάϥϜͷήʔτ
O(|S| × (2k+3 × 2k+3)) O(|p| × 8k) |p|
ϕϯνϚʔΫ w #7G Y BY CͷB CΛݟ͚ͭΔ w ();શͯɺશ͕ͯॏͳͬͨঢ়ଶΛ࡞Δ w
(SPWFSG Y ͱͳΔYΛ୳ࡧ w .BD#PPL1SP $PSFJ()[ (#
݁ w ྔࢠܭࢉͷҝͷநղऍख๏ΛఏҊͨ͠ 4DBMBCMFͰ͋ΔRVCJUنͷγϛϡϨʔγϣϯग़དྷͨ 6TFGVMͰ͋ΔͭͷॏཁͳͰBTTFSUJPODIFDL͕ग़དྷͨ 'MFYJCMFͰ͋Δ"CTUSBDU%PNBJOͷઃܭࣗ༝͕ߴ͍