ަੑ • શੑ Z ⇤ dV = h | i <latexit sha1_base64="6fIKy+1cVFWLe/c8/adPTYK+OuE=">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</latexit> = | i <latexit sha1_base64="VjhsDkxI/60c3vQltnMXqqFbzWY=">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</latexit> ঢ়ଶϕΫτϧ ੵ |⇠µ i = U | n i <latexit sha1_base64="0Osv/JUjYU/vCiYiwEVxT9g4ovk=">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</latexit> ঢ়ଶϕΫτϧɹɹ ʹର͠ɺมߦྻɹ Λߟ͑Δ | n i <latexit sha1_base64="ch1tAdqobveVTzEVVIL7ZGtvTG4=">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</latexit> U <latexit sha1_base64="JwiRE4gitx2dIxFZDSnmZMKIIKE=">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</latexit> มޙͷঢ়ଶɹɹ·ͨਖ਼نަجఈΛுΔΑ͏ͳɹϢχλϦߦྻͰ͋Δ U <latexit sha1_base64="JwiRE4gitx2dIxFZDSnmZMKIIKE=">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</latexit> |⇠µ i <latexit sha1_base64="pJFyuE6yLNC2SATPUxz6Jff3k/8=">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</latexit> UU† = I <latexit sha1_base64="B+qNIQOEII+kjqBDDmBbidTIzLM=">AAACvHichVHLSsNQED2N73fVjeAmWCquyq0KiqAU3eiuPqIFHyVJb2NoXiRppZb+gD/gQlwoqIif4Ubcu/ATxKWCGxdO0oCotE64uTPnzpl75o7iGLrnM/YcE9raOzq7unt6+/oHBofiwyPbnl12VS6ptmG7OUX2uKFbXPJ13+A5x+WyqRh8RymtBOc7Fe56um1t+VWH75uyZulFXZV9gvLxYUmUDmp7BVnTuFsXF8W1fDzBUiw08a+TjpwEIsva8UfsoQAbKsowwWHBJ9+ADI++XaTB4BC2jxphLnl6eM5RRy9xy5TFKUMmtER/jaLdCLUoDmp6IVulWwxaLjFFJNkTu2Vv7IHdsRf22bRWLawRaKnSrjS43MkPnYxtfvzLMmn3cfjNaqnZRxHzoVadtDshEnShNviV49O3zYWNZG2SXbJX0n/Bntk9dWBV3tWrdb5x1kKPQn3LFHOq2fzlgjwTdRpj+vfQ/jrb06n0TGp6fTaRWY4G2o1xTGCKpjaHDFaRhUTVj3COa9wIS0JBKAlmI1WIRZxR/DCh8gWfRp6v</latexit> ดͨ͡ྔࢠܥͷ࣌ؒൃలɺϢχλϦߦྻʹΑΔมͰදͤΔ͜ͱ͕ΒΕ͍ͯΔ ϢχλϦม ྔࢠܭࢉʹ͓͚ΔΞϧΰϦζϜͱɺྔࢠঢ়ଶʹର͢ΔϢχλϦมͷ܁Γฦ͠ | i = X k ak | k i <latexit sha1_base64="B4YvNakM/LBXli/WKToWttzSKSQ=">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</latexit> ઢܗ݁߹ ⇔ॏͶ߹Θͤ
containing well-defined two-level quantum systems, or qubits 2. the ability to initialize the system into a well-defined and determinate initial state 3. qubit decoherence times much longer than the gate times 4. a set of universal quantum gates which can be applied to each qubit 5. the ability to read out the qubit state with high accuracy D. P. DiVincenzo, Fortschritte der Physik 48, 771 (2000). two-level quantum systems 2 ~ 5ʹهࡌ͞ΕΔॾʑͷૢ࡞͕࣮ݱՄೳͳ ɿྔࢠྗֶతͳੑ࣭Λ࣋ͪɺ2௨Γͷঢ়ଶ͔ΒͳΔܥ ݅Λຬ͍ͨͯ͑͠͞Εҙͷཧܥ͕ྔࢠίϯϐϡʔλͱͳΓ͏Δ 0, 1 <latexit sha1_base64="IUwxB2kvN8gMgeWPqyIRENd5QyU=">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</latexit>
(kx, ky, kz) <latexit sha1_base64="5aBJDpaL7qj4U+pDxiPB886tp54=">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</latexit> ɿϕΫτϧ kx <latexit sha1_base64="i+GRJ5nsQW4JxQhO69J0p+TQ0QU=">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</latexit> ɿxํʹ͓͚Δɺ͞2"͋ͨΓʹؚ·ΕΔͷ L L L पظతڥք݅ΛՃ͑Δ (x, y, z) = (x + L, y, z) = (x, y + L, z) = (x, y, z + L) <latexit sha1_base64="y6sCPqA4z1f4FW0I+qde1F2+JgY=">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</latexit> →͞Lʹରͯ͠ݸͷΛͭΑ͏ͳಈؔͷΈڐ͞ΕΔ kx = 2⇡ L nx, ky = 2⇡ L ny, kz = 2⇡ L nz <latexit sha1_base64="di83r//Kkk8EN+6l1RxIKK/iI4E=">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</latexit> ɿ nx, ny, nz <latexit sha1_base64="GfkP6tulIP7hXEkbKavKhlUtEU4=">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</latexit> k = 1 p L3 ei(kxx+kyy+kzz) <latexit sha1_base64="47LMQXHzkmdTYNnitjG3c9cCbmk=">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</latexit> ަੑ Λຬͨ͢ Z V =L3 ⇤ k (r) k0 (r)dV = kk0 <latexit sha1_base64="MGwXOR6+JG7dGnfmsqC3f8PGSA4=">AAADAHichVE9T9tQFD24X4GWEmBBYrEa8dEO0QtUAlVCQrB0YOCjCUgYLNt5gaf4S/ZLpGB56dg/wNCpSJVADKhrO3apundg7AiMILEwcO04oi2CXst+5513z/W575q+LULJ2HGX8uDho8dPct09T5/1Pu/L9w9UQq8RWLxsebYXrJlGyG3h8rIU0uZrfsANx7T5qlmfT85XmzwIhee+ky2fbzjGlitqwjIkUXq+rAlX6lFlZmFzMlY1PxR6pJlOVI/jzVfjCQril3/QY3HcYasVdUbVqtyWRkfTSdHzBVZkaai3QSkDBWSx6OV/QkMVHiw04IDDhSRsw0BIzzpKYPCJ20BEXEBIpOccMXpI26AsThkGsXX6btFuPWNd2ic1w1Rt0V9segNSqhhhv9gBO2c/2CE7YVd31orSGomXFq1mW8t9ve/D0Mrlf1UOrRLbN6p7PUvUMJ16FeTdT5mkC6utb+7snq+8WR6JRtkeOyP/n9gx+04duM0L6/MSX/54jx+T+jZoz6nm3TeX5DlIxlj6d2i3QWWiWJosTiy9LszOZQPNYRgvME5Tm8Is3mIRZar+Db9xijPlvbKvHClf2qlKV6YZxF+hfL0Ggtq8BQ==</latexit> ަੑͷ֬ೝ ิॿεϥΠυ
= 1 L3 Z L/2 L/2 ei(kx k0 x )xdx Z L/2 L/2 ei(ky k0 y )ydy Z L/2 L/2 ei(kz k0 z )zdz <latexit sha1_base64="AiwUv4HtHQ+GgHGHFStnL1Rl5zs=">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</latexit> = 1 L3 Z L/2 L/2 ei 2⇡ L (nx n0 x )xdx Z L/2 L/2 ei 2⇡ L (ny n0 y )ydy Z L/2 L/2 ei 2⇡ L (nz n0 z )zdz <latexit sha1_base64="NtvFOOnJ3TH++15ot6AIUA/eSQ4=">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</latexit> lim x!0 sin x x = 1 <latexit sha1_base64="uJoZaoO1NkH141jpaYDKRg8IDiw=">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</latexit> = sin [⇡(nx n0 x )] ⇡(nx n0 x ) sin [⇡(ny n0 y )] ⇡(ny n0 y ) sin [⇡(nz n0 z )] ⇡(nz n0 z ) <latexit sha1_base64="arj0UW5/YTRFEQSJNmReXIl+f00=">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</latexit> = nxn0 x nyn0 y nzn0 z <latexit sha1_base64="8tIFaCCEvVlMZ7SMeT1LGnRnBUU=">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</latexit> = kk0 <latexit sha1_base64="fuuNIWkIFhl+bUNW8iAnKiXXDM8=">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</latexit> Ҏ্ΑΓɺपظతڥք݅ʹ͓͚Δ k = 1 p L3 ei(kxx+kyy+kzz) <latexit sha1_base64="47LMQXHzkmdTYNnitjG3c9cCbmk=">AAAC7XichVFLa9RQFP4aX219dGo3BTcXh0pFGG5awSIIpd24cNHXtIWmDUm8017yNLkzdOaSP9B96aIiKIiKP8ONdOFOtD9BXFZw48KTTEC0tJ6QnHO+e76T79zjJoHMFOfHA8aFi5cuXxkcGr567fqNkdrozdUsbqeeaHpxEKfrrpOJQEaiqaQKxHqSCid0A7Hm+vPF+VpHpJmMoxXVTcRm6GxHsiU9RxFk1+asJJO2ttxQ+3nOHjGrlTqeNnNtZc9SpZ9sTRMstpiWk769y3bZPebbXdYtfY/17uZ2rc4bvDR2OjCroI7KFuLaESw8RQwPbYQQiKAoDuAgo2cDJjgSwjahCUspkuW5QI5h4rapSlCFQ6hP323KNio0orzomZVsj/4S0JsSk2GCf+bv+An/yN/zb/zXmb102aPQ0iXv9rkisUf2xpd//pcVklfY+cM6V7NCCzOlVknakxIppvD6/E7v4GT54dKEvsNf8e+k/yU/5h9ogqjzw3u9KJYOz9Hj0twO5YJ6nn1zRV2IYo3mv0s7HaxONczpxtTi/frsXLXQQdzCbUzS1h5gFo+xgCZ1f4tP+IKvRmzsG8+NF/1SY6DijOEvM978Bs1csio=</latexit> ͷަੑ͕֬ೝ͞Εͨ ަੑͷ֬ೝ2 ิॿεϥΠυ
ަੑ • શੑ Z ⇤ dV = h | i <latexit sha1_base64="6fIKy+1cVFWLe/c8/adPTYK+OuE=">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</latexit> = | i <latexit sha1_base64="VjhsDkxI/60c3vQltnMXqqFbzWY=">AAACvHichVHLSsNAFD2Nr/quuhHcFEvFVZlWQRGUohuXWq0W2lKSOOrQvEimlVr6A/6AC3GhoCJ+hhtx78JPEJcKblx4kwZERb0hmTNn7rk5d67mGMKTjD1GlI7Oru6eaG9f/8Dg0HBsZHTLs2uuzvO6bdhuQVM9bgiL56WQBi84LldNzeDbWnXFP9+uc9cTtrUpGw4vm+qeJXaFrkqiKrGRkuOJ+GK8VOWy6eNWJZZgKRZE/CdIhyCBMNbs2D1K2IENHTWY4LAgCRtQ4dFTRBoMDnFlNIlzCYngnKOFPtLWKItThkpslb57tCuGrEV7v6YXqHX6i0GvS8o4kuyBXbMXdsdu2BN7/7VWM6jhe2nQqrW13KkMH41vvP2rMmmV2P9U/elZYhfzgVdB3p2A8bvQ2/r64fHLxkIu2Zxi5+yZ/J+xR3ZLHVj1V/1inedO/vCjUd8q7TnV/P3m/DwT/hjT34f2E2xlUumZVGZ9NpFdDgcaxQQmMU1Tm0MWq1hDnqof4BSXuFKWlB2lqpjtVCUSasbwJZT6B0SZn2g=</latexit> hri = Z r ⇤(r) (r)dV = Z ⇤(r)r (r)dV <latexit sha1_base64="ky8MGdnuSNXlWQGlG+/YGUlXG/k=">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</latexit> Ґஔɹͷظ hˆ Li = h |ˆ L| i = Z ˆ L ⇤ dV = Z ⇤ ˆ L dV <latexit sha1_base64="1Jf/j1q8A8B/981ADayPZMtfgj4=">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</latexit> r <latexit sha1_base64="XBpPG3XJFE9IumniF9au7Mf+r8A=">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</latexit> | |2 = ⇤ <latexit sha1_base64="06ZVnLkchaQkLyBEMVonxR00JgI=">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</latexit> ݩʑ ཻࢠͷҐஔͷ֬ͩͬͨͷͰɺ Ұൠͷཧྔɹʹ͍ͭͯظΛදݱͰ͖Δ L <latexit sha1_base64="RAwY3yTen7qVIIK5BSlyv9sc4RQ=">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</latexit> ཧྔྔࢠྗֶͷੈքͰঢ়ଶϕΫτϧʹ࡞༻͢Δߦྻ ɿԋࢉࢠ ঢ়ଶϕΫτϧ ੵ ԋࢉࢠʹ͍ͭͯݴٴ൛
• Robust trap life time • Long internal-state coherence • Strong ion-ion interaction • Controlled-NOT (CNOT) gates entangling one ion’s in- ternal state with its motional state Laser or microwave drives applied to the ions allow arbitrary and high-fidelity single-qubit rotations to be per- formed achieved gate fidelities are sufficient to be compatible with error-correction schemes such as the surface code ion coherence times are much longer than gate times, Pairs of states employed for qubit levels can come from any com- bination of long-lived levels in the ground or metastable manifolds of ions The ground-state hyperfine levels are some of the most long-lived states available, with spontaneous- emission-limited lifetimes approaching the age of the uni- verse. Ion trapใ ൈਮ
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