PLDI '21論文読み会: Quantum Abstract Interpretation

by Idein

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

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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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ΞδΣϯμ w ந৅ղऍͱ͸ w ྔࢠܭࢉͱ͸ w ຊ࿦จͷ঺հ

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

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

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

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

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

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

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

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

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ࣹӨߦྻͷॱং P ⊆ Q J ff SP ⊆ SQ

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

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

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ந৅Խࣸ૾ͱ۩ମԽࣸ૾ ந৅Խ ۩ମԽ

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ΨϩΞ઀ଓ (BMPJT$POOFDUJPO

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

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

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

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

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

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