[PPT] - Quantum computational universality of hypergraph states with Pauli-X PowerPoint Presentation

SLIDE 1

Quantum computational universality of hypergraph states with Pauli-X and Z basis measurements

Yuki Takeuchi

NTT Communication Science Laboratories, NTT Corporation

Joint work with Tomoyuki Morimae (Kyoto) and Masahito Hayashi (Nagoya, SUSTech, and NUS)

QIST2019@YITP 11th June, 2019

SLIDE 2

Measurement-based quantum computation

A universal quantum computing model

proposed by R. Rasussebdorf and H. J. Briegel in 2001.

It is equivalent to the quantum circuit model.

SLIDE 3

Measurement-based quantum computation

Step 1. Prepare a universal resource state (e.g. 2D cluster state). (This step does not depend on what we want to compute.)

A universal quantum computing model

proposed by R. Rasussebdorf and H. J. Briegel in 2001.

It is equivalent to the quantum circuit model.

CZ

<latexit sha1_base64="QGuCZT/4DwDB14DhWaTi/gq7iI=">AB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GOxF49V7Ae2oWy2m3bpZhN2J0IJ/QdePCji1X/kzX/jts1Bqw8GHu/NMDMvSKQw6LpfTmFldW19o7hZ2tre2d0r7x+0TJxqxpslrHuBNRwKRvokDJO4nmNAokbwfj+sxvP3JtRKzucZJwP6JDJULBKFrprv7QL1fcqjsH+Uu8nFQgR6Nf/uwNYpZGXCGT1Jiu5yboZ1SjYJPS73U8ISyMR3yrqWKRtz42fzSKTmxyoCEsbalkMzVnxMZjYyZRIHtjCiOzLI3E/zuimGV34mVJIiV2yxKEwlwZjM3iYDoTlDObGEMi3srYSNqKYMbTglG4K3/PJf0jqreudV9/aiUrvO4yjCERzDKXhwCTW4gQY0gUEIT/ACr87YeXbenPdFa8HJZw7hF5yPb0QtjS0=</latexit>

|+i ⌘ |0i + |1i p 2

<latexit sha1_base64="HsrEpC8VraJ6Tgc8GF3Sj5Nhw=">ACHXicbZDLSsNAFIYnXmu9RV26CRZBKJSkFnRZdOygr1AE8pkOmHTibpzEmhpH0RN76KGxeKuHAjvo3TNgt/WHg4z/nzMz5/ZgzBb9baytb2xubed28rt7+weH5tFxQ0WJLROIh7Jlo8V5UzQOjDgtBVLikOf06Y/uJ3VmyMqFYvEA4xj6oW4J1jACAZtdczKpOhKLHqcunSYsJFLAolJOrEztzhxMpqmrhpKSMvTacs2CV7LmsVnAwKFOtY363YgkIRVAOFaq7dgxeCmWwIi+Oe8misaYDHCPtjUKHFLlpfPtpta5drpWEl9BFhz9/dEikOlxqGvO0MfbVcm5n/1doJBNdeykScABVk8VCQcAsiaxaV1WSEuBjDZhIpv9qkT7W8YAONK9DcJZXoVGueRcluz7SqF6k8WRQ6foDF0gB12hKrpDNVRHBD2iZ/SK3own48V4Nz4WrWtGNnOC/sj4+gGhk6OA</latexit>

SLIDE 4

Measurement-based quantum computation

A universal quantum computing model

proposed by R. Rasussebdorf and H. J. Briegel in 2001.

It is equivalent to the quantum circuit model.

Step 1. Prepare a universal resource state (e.g. 2D cluster state). (This step does not depend on what we want to compute.) Step 2. Adaptively measure each qubit in an appropriate basis. (The measurement bases decide what we compute.)

SLIDE 5

Measurement-based quantum computation

A universal quantum computing model

proposed by R. Rasussebdorf and H. J. Briegel in 2001.

It is equivalent to the quantum circuit model.

Step 1. Prepare a universal resource state (e.g. 2D cluster state). (This step does not depend on what we want to compute.) Step 2. Adaptively measure each qubit in an appropriate basis. (The measurement bases decide what we compute.) Step 3. A desired output state is prepared:)

|ψfi

<latexit sha1_base64="8TAcbvY0GJgXmcnGkoEOfSQENE=">AB9XicdVDLSgMxFM34rPVdekmWARXQ6atY7srunFZwT6gM5ZMmlDM5khyShl7H+4caGIW/FnX9j+hBU9MCFwzn3cu89QcKZ0gh9WEvLK6tr67mN/ObW9s5uYW+/peJUEtokMY9lJ8CKciZoUzPNaSeRFEcBp+1gdDH127dUKhaLaz1OqB/hgWAhI1gb6ebeSxTrhZ7EYsBpr1BEdq2GKo4LkX2KUMmtGYLKparQsdGMxTBAo1e4d3rxySNqNCEY6W6Dkq0n2GpGeF0kvdSRNMRnhAu4YKHFHlZ7OrJ/DYKH0YxtKU0HCmfp/IcKTUOApMZ4T1UP32puJfXjfVYdXPmEhSTQWZLwpTDnUMpxHAPpOUaD42BPJzK2QDLHERJug8iaEr0/h/6RVsp2yja4qxfr5Io4cOARH4AQ4AzUwSVogCYgQIH8ASerTvr0XqxXuetS9Zi5gD8gPX2CVmgkxI=</latexit>

|ψfi

<latexit sha1_base64="0wrDLxYi8cvbV4jF/y5QwlUezgk=">AB9XicbVBNS8NAEJ34WetX1aOXxSJ4KokKeix68VjBfkAby2Y7aZduNmF3o5TY/+HFgyJe/S/e/Ddu2xy09cHA470ZuYFieDauO63s7S8srq2Xtgobm5t7+yW9vYbOk4VwzqLRaxaAdUouMS64UZgK1FIo0BgMxheT/zmAyrNY3lnRgn6Ee1LHnJGjZXunzqJ5t2wo6jsC+yWym7FnYIsEi8nZchR65a+Or2YpRFKwTVu25ifEzqgxnAsfFTqoxoWxI+9i2VNItZ9Nrx6TY6v0SBgrW9KQqfp7IqOR1qMosJ0RNQM9703E/7x2asJLP+MySQ1KNlsUpoKYmEwiID2ukBkxsoQyxe2thA2oszYoIo2BG/+5UXSOK14ZxX39rxcvcrjKMAhHMEJeHABVbiBGtSBgYJneIU359F5cd6dj1nrkpPHMAfOJ8/64iSxg=</latexit>

SLIDE 6

Measurement-based quantum computation

Step 1. Prepare a universal resource state (e.g. 2D cluster state). (This step does not depend on what we want to compute.) Step 2. Adaptively measure each qubit in an appropriate basis. (The measurement bases decide what we compute.) Step 3. A desired output state is prepared:) |ψfi

<latexit sha1_base64="0wrDLxYi8cvbV4jF/y5QwlUezgk=">AB9XicbVBNS8NAEJ34WetX1aOXxSJ4KokKeix68VjBfkAby2Y7aZduNmF3o5TY/+HFgyJe/S/e/Ddu2xy09cHA470ZuYFieDauO63s7S8srq2Xtgobm5t7+yW9vYbOk4VwzqLRaxaAdUouMS64UZgK1FIo0BgMxheT/zmAyrNY3lnRgn6Ee1LHnJGjZXunzqJ5t2wo6jsC+yWym7FnYIsEi8nZchR65a+Or2YpRFKwTVu25ifEzqgxnAsfFTqoxoWxI+9i2VNItZ9Nrx6TY6v0SBgrW9KQqfp7IqOR1qMosJ0RNQM9703E/7x2asJLP+MySQ1KNlsUpoKYmEwiID2ukBkxsoQyxe2thA2oszYoIo2BG/+5UXSOK14ZxX39rxcvcrjKMAhHMEJeHABVbiBGtSBgYJneIU359F5cd6dj1nrkpPHMAfOJ8/64iSxg=</latexit>

Step 2 is the only online operation in MBQC.

SLIDE 7

Measurement-based quantum computation

Step 1. Prepare a universal resource state (e.g. 2D cluster state). (This step does not depend on what we want to compute.) Step 2. Adaptively measure each qubit in an appropriate basis. (The measurement bases decide what we compute.) Step 3. A desired output state is prepared:) |ψfi

<latexit sha1_base64="0wrDLxYi8cvbV4jF/y5QwlUezgk=">AB9XicbVBNS8NAEJ34WetX1aOXxSJ4KokKeix68VjBfkAby2Y7aZduNmF3o5TY/+HFgyJe/S/e/Ddu2xy09cHA470ZuYFieDauO63s7S8srq2Xtgobm5t7+yW9vYbOk4VwzqLRaxaAdUouMS64UZgK1FIo0BgMxheT/zmAyrNY3lnRgn6Ee1LHnJGjZXunzqJ5t2wo6jsC+yWym7FnYIsEi8nZchR65a+Or2YpRFKwTVu25ifEzqgxnAsfFTqoxoWxI+9i2VNItZ9Nrx6TY6v0SBgrW9KQqfp7IqOR1qMosJ0RNQM9703E/7x2asJLP+MySQ1KNlsUpoKYmEwiID2ukBkxsoQyxe2thA2oszYoIo2BG/+5UXSOK14ZxX39rxcvcrjKMAhHMEJeHABVbiBGtSBgYJneIU359F5cd6dj1nrkpPHMAfOJ8/64iSxg=</latexit>

Step 2 is the only online operation in MBQC. It is theoretically and experimentally interesting to reduce the necessary number of measurement bases even if the initial state becomes (slightly) complex.

SLIDE 8

Measurement-based quantum computation

Step 1. Prepare a universal resource state (e.g. 2D cluster state). (This step does not depend on what we want to compute.) Step 2. Adaptively measure each qubit in an appropriate basis. (The measurement bases decide what we compute.) Step 3. A desired output state is prepared:) |ψfi

<latexit sha1_base64="0wrDLxYi8cvbV4jF/y5QwlUezgk=">AB9XicbVBNS8NAEJ34WetX1aOXxSJ4KokKeix68VjBfkAby2Y7aZduNmF3o5TY/+HFgyJe/S/e/Ddu2xy09cHA470ZuYFieDauO63s7S8srq2Xtgobm5t7+yW9vYbOk4VwzqLRaxaAdUouMS64UZgK1FIo0BgMxheT/zmAyrNY3lnRgn6Ee1LHnJGjZXunzqJ5t2wo6jsC+yWym7FnYIsEi8nZchR65a+Or2YpRFKwTVu25ifEzqgxnAsfFTqoxoWxI+9i2VNItZ9Nrx6TY6v0SBgrW9KQqfp7IqOR1qMosJ0RNQM9703E/7x2asJLP+MySQ1KNlsUpoKYmEwiID2ukBkxsoQyxe2thA2oszYoIo2BG/+5UXSOK14ZxX39rxcvcrjKMAhHMEJeHABVbiBGtSBgYJneIU359F5cd6dj1nrkpPHMAfOJ8/64iSxg=</latexit>

Step 2 is the only online operation in MBQC. (A part of) our results: We have constructed a universal resource state that only requires adaptive Pauli X and Z-basis measurements for universal MBQC. It is theoretically and experimentally interesting to reduce the necessary number of measurement bases even if the initial state becomes (slightly) complex.

SLIDE 9

Resource state Measurement basis Class Cluster state [15] X, Y, TXT † [16] graph state Brickwork state [17] X, Y, TXT † [17] graph state Triangular lattice state [18] X, Z, H, XHX [18] graph state Raussendorf-Harrington-Goyal (RHG) lattice [19, 20] X, Y, Z, TXT † [20, 21] graph state Decorated RHG lattice [22] X, Y, TXT † [22] graph state Affleck-Kennedy-Lieb-Tasaki (AKLT) state [23, 24] qutrit bases [24] matrix-product state Union Jack state [25] X, Y, Z [25] hypergraph state Three-uniform hypergraph state [26] X, Y, Z [26] hypergraph state Mølmer-Sørensen graph state [27] X, Z [27] weighted graph state Our state X, Z hypergraph state

Several universal resource states

Ø Hypergraph state = CCZ Ø Weighted graph state

[Rossi et al. ‘13] [Hartmann et al. ‘07]

Ø Graph state = |+〉 = CZ = =

e−iθZ⊗Z

<latexit sha1_base64="i1pRBGcbHqMdygrcpzlF0k96nIM=">ACAHicbVDJSgNBEO2JW4zbqAcPXhqD4MUwo4Ieg148RjALycTQ06kTXoWumuEMzFX/HiQRGvfoY3/8bOctDEBwWP96qoqufHUmh0nG8rt7S8srqWXy9sbG5t79i7ezUdJYpDlUcyUg2faZAihCoKlNCIFbDAl1D3hzdjv/4ISosovMdRDO2A9UPRE5yhkTr2ATykp8LDASCjTS9CEYCmzaxjF52SMwFdJO6MFMkMlY795XUjngQIpdM65brxNhOmULBJWQFL9EQMz5kfWgZGjKzp51OHsjosVG6tBcpUyHSifp7ImWB1qPAN50Bw4Ge98bif14rwd5VOxVhnCEfLqol0iKER2nQbtCAUc5MoRxJcytlA+YhxNZgUTgjv/8iKpnZXc85Jzd1EsX8/iyJNDckROiEsuSZnckgqpEk4y8kxeyZv1ZL1Y79bHtDVnzWb2yR9Ynz9PqpY3</latexit>

e−i(2θ)Z⊗Z

<latexit sha1_base64="DhMW8SU1c9GmQjQms3oWDqlB7AU=">ACA3icbVDJSgNBEO2JW4zbqDe9NAYhHgwzUdBj0IvHCGYhi6GnU0ma9Cx01whCHjxV7x4UMSrP+HNv7GzHDTxQcHjvSq6nmRFBod59tKLS2vrK6l1zMbm1vbO/buXkWHseJQ5qEMVc1jGqQIoIwCJdQiBcz3JFS9wfXYrz6A0iIM7nAYQctnvUB0BWdopLZ9APfJqcgVmtgHZCe03gxR+KBpfdS2s07emYAuEndGsmSGUtv+anZCHvsQIJdM64brRNhKmELBJYwyzVhDxPiA9aBhaMDMnlYy+WFEj43Sod1QmQqQTtTfEwnztR76nun0Gfb1vDcW/MaMXYvW4kIohgh4NF3VhSDOk4ENoRCjKoSGMK2FupbzPFONoYsuYENz5lxdJpZB3z/LO7Xm2eDWLI0OyRHJEZdckCK5ISVSJpw8kmfySt6sJ+vFerc+pq0pazazT/7A+vwBloiW2A=</latexit>

[Briegel et al. ‘01]

SLIDE 10

e−i(π/8)Z⊗Z

<latexit sha1_base64="+q4aPQi5VxWL9gsFHsFPSutNp/s=">ACAHicbVC7TsMwFHXKq5RXgIGBxaJCKgMlASQ6VrAwFok+1CZUju2Vh07sh2kKsrCr7AwgBArn8HG3+C2GaDlSFc6Oude3XtPEDGqtON8W7ml5ZXVtfx6YWNza3vH3t1rKBFLTOpYMCFbAVKEU7qmpGWpEkKAwYaQajm4nfCRSUcHv9TgifogGnPYpRtpIXfuAPCSntORF9Kxy0vaEpiFRsJ127aJTdqaAi8TNSBFkqHXtL68ncBwSrjFDSnVcJ9J+gqSmJG04MWKRAiP0IB0DOXI7PGT6QMpPDZKD/aFNMU1nKq/JxIUKjUOA9MZIj1U895E/M/rxLpf8RPKo1gTjmeL+jGDWsBJGrBHJcGajQ1BWFJzK8RDJBHWJrOCcGdf3mRNM7L7kXZubsVq+zOPLgEByBEnDBFaiCW1ADdYBCp7BK3iznqwX6936mLXmrGxmH/yB9fkDTDmVkg=</latexit>

Hypergraph states vs. weighted graph states

{X, Z}

<latexit sha1_base64="mGkuBYNAJLANUkV8Bij5Ss9LCq0=">AB7nicdVDLSsNAFJ34rPVdelmsAguJEzS0NZd0Y3LCvaBTSiT6bQdOpmEmYlQj/CjQtF3Po97vwbJ20FT1w4XDOvdx7T5hwpjRCH9bK6tr6xmZhq7i9s7u3Xzo4bKs4lYS2SMxj2Q2xopwJ2tJMc9pNJMVRyGknFzlfueSsVicaunCQ0iPBJsyAjWRur4Wf8zp/1S2VkX9SrleFyEao5rhOTtyaV/GgY5QcZbBEs1969wcxSMqNOFYqZ6DEh1kWGpGOJ0V/VTRBJMJHtGeoQJHVAXZ/NwZPDXKA5jaUpoOFe/T2Q4UmoahaYzwnqsfnu5+JfXS/WwHmRMJKmgiwWDVMOdQz3+GASUo0nxqCiWTmVkjGWGKiTUJFE8LXp/B/0nZtp2KjG6/cuFzGUQDH4AScAQfUQANcgyZoAQIm4AE8gWcrsR6tF+t10bpiLWeOwA9Yb59Vjo+S</latexit>

Our resource state state Mølmer − Sørensen

<latexit sha1_base64="O/fIp/DWj+zQUk4xok02SveSBs=">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</latexit><latexit sha1_base64="O/fIp/DWj+zQUk4xok02SveSBs=">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</latexit><latexit sha1_base64="O/fIp/DWj+zQUk4xok02SveSBs=">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</latexit><latexit sha1_base64="O/fIp/DWj+zQUk4xok02SveSBs=">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</latexit>

[Kissinger et al. ‘17]

{X, Z}

<latexit sha1_base64="2iHi84gYjg0yKYsmRs04VwgpuZg=">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</latexit><latexit sha1_base64="2iHi84gYjg0yKYsmRs04VwgpuZg=">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</latexit><latexit sha1_base64="2iHi84gYjg0yKYsmRs04VwgpuZg=">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</latexit><latexit sha1_base64="2iHi84gYjg0yKYsmRs04VwgpuZg=">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</latexit>

Class Bases (Computation) Hypergraph state Weighted graph state Bases (Verification)

{X, Z}

<latexit sha1_base64="mGkuBYNAJLANUkV8Bij5Ss9LCq0=">AB7nicdVDLSsNAFJ34rPVdelmsAguJEzS0NZd0Y3LCvaBTSiT6bQdOpmEmYlQj/CjQtF3Po97vwbJ20FT1w4XDOvdx7T5hwpjRCH9bK6tr6xmZhq7i9s7u3Xzo4bKs4lYS2SMxj2Q2xopwJ2tJMc9pNJMVRyGknFzlfueSsVicaunCQ0iPBJsyAjWRur4Wf8zp/1S2VkX9SrleFyEao5rhOTtyaV/GgY5QcZbBEs1969wcxSMqNOFYqZ6DEh1kWGpGOJ0V/VTRBJMJHtGeoQJHVAXZ/NwZPDXKA5jaUpoOFe/T2Q4UmoahaYzwnqsfnu5+JfXS/WwHmRMJKmgiwWDVMOdQz3+GASUo0nxqCiWTmVkjGWGKiTUJFE8LXp/B/0nZtp2KjG6/cuFzGUQDH4AScAQfUQANcgyZoAQIm4AE8gWcrsR6tF+t10bpiLWeOwA9Yb59Vjo+S</latexit>

[Hayashi-Takeuchi ‘19]

e−i(π/4)Z⊗Z

<latexit sha1_base64="C8wWq/8dpv+84pnjz5VFt4Mk2xQ=">ACfHichVHNLkNBGD29/qr+igWJTaMIkTKVBrFq2Fj6K0Jp7r0Gk96/3DtUk1fwAtYWCEigqew8QIWHkEsSWxEfL29iSD4JjNz5sx3vjkzozmG8CRjDyGlr6hsSncHGlpbWviHZ2rXp2wdV5RrcN213XVI8bwuIZKaTB1x2Xq6Zm8DUtP1fdXyty1xO2tSJLDt8y1T1L7ApdlUTloj18u5wQw1lHjKdGNrK2FCb3YhuVXDTOxpgfsZ8gGYA4gliwoxfIYgc2dBRgsOCJGxAhUdtE0kwOMRtoUycS0j4+xwVREhboCxOGSqxeRr3aLUZsBatqzU9X63TKQZ1l5QxDLJ7dsme2R27Yo/s7daZb9G1UuJZq2m5U6u47B3+fVflUmzxP6n6k/PEruY9r0K8u74TPUWek1fPDh6Xp5ZGiwPsVP2RP5P2AO7pRtYxRf9fJEvHSNCH5D8/tw/werEWJLwYiqeng2+Iow+9GOY3nsKacxjARn/3DNc4yb0rgwo0qilqEAk03voQy+QESwJKI</latexit><latexit sha1_base64="C8wWq/8dpv+84pnjz5VFt4Mk2xQ=">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</latexit><latexit sha1_base64="C8wWq/8dpv+84pnjz5VFt4Mk2xQ=">ACfHichVHNLkNBGD29/qr+igWJTaMIkTKVBrFq2Fj6K0Jp7r0Gk96/3DtUk1fwAtYWCEigqew8QIWHkEsSWxEfL29iSD4JjNz5sx3vjkzozmG8CRjDyGlr6hsSncHGlpbWviHZ2rXp2wdV5RrcN213XVI8bwuIZKaTB1x2Xq6Zm8DUtP1fdXyty1xO2tSJLDt8y1T1L7ApdlUTloj18u5wQw1lHjKdGNrK2FCb3YhuVXDTOxpgfsZ8gGYA4gliwoxfIYgc2dBRgsOCJGxAhUdtE0kwOMRtoUycS0j4+xwVREhboCxOGSqxeRr3aLUZsBatqzU9X63TKQZ1l5QxDLJ7dsme2R27Yo/s7daZb9G1UuJZq2m5U6u47B3+fVflUmzxP6n6k/PEruY9r0K8u74TPUWek1fPDh6Xp5ZGiwPsVP2RP5P2AO7pRtYxRf9fJEvHSNCH5D8/tw/werEWJLwYiqeng2+Iow+9GOY3nsKacxjARn/3DNc4yb0rgwo0qilqEAk03voQy+QESwJKI</latexit><latexit sha1_base64="C8wWq/8dpv+84pnjz5VFt4Mk2xQ=">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</latexit>

Other six measurement bases in the x-y plane of the Bloch sphere

X

<latexit sha1_base64="I/CYL8yVn02Tw3nXW3R4eAfng=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cW7Ae0oWy2k3btZhN2N0IJ/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZLGLVCahGwSU2DTcCO4lCGgUC28H4bua3n1BpHsHM0nQj+hQ8pAzaqzU6PTLFbfqzkFWiZeTCuSo98tfvUHM0gilYJq3fXcxPgZVYzgdNSL9WYUDamQ+xaKmE2s/mh07JmVUGJIyVLWnIXP09kdFI60kU2M6ImpFe9mbif143NeGNn3GZpAYlWywKU0FMTGZfkwFXyIyYWEKZ4vZWwkZUWZsNiUbgrf8ipXVS9y6rbuKrUbvM4inACp3AOHlxDe6hDk1gPAMr/DmPDovzrvzsWgtOPnMfyB8/kDtmOM3g=</latexit>

＋ [Morimae-Takeuchi-Hayashi ‘17, Miller-Sanders-Miyake ‘17, Takeuchi-Morimae ‘18, Zhu-Hayashi ‘18]

SLIDE 11

Our contribution

We have constructed, for the first time, a universal hypergraph state that achieves

Universality
Verifiability

at the same time with only Pauli X and Z-basis measurements. Our result: This property cannot be achieved using graph states due to the Gottesman-Knill theorem. In the rest of this presentation, I would like to explain (the idea of) how to construct our universal hypergraph state.

SLIDE 12

X

<latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">ACZHichVHLSsNAFD2Nr1ofrYogCFIsiqty09bWuhLduLRqtaAiSZzW0DQJSVqoxR/QreLClYKI+Blu/AEX/oAgLhXcuPA2rYgL9Q4zc+bMPXfOzKi2obse0WNA6ujs6u4J9ob6+gcGw5Gh4Q3XqjqayGuWYTkFVXGFoZsi7+meIQq2I5SKaohNtbzU3N+sCcfVLXPdq9tip6KUTL2oa4rHVK6wG4lRXE7KydRslOLZbDadyTCgRDKdSEXlOPkRQztWrMg1trEHCxqEDAhMfYgAKX2xZkEGzmdtBgzmGk+/sChwixtspZgjMUZs8lni1WZNXjdrur5a41M7g4ro5iB7qhV7qnW3qmj19rNfwaTS91ntWVti74aOxtfd/VRWePex/q/707KGIOd+rzt5tn2neQmvpawdnr2vzq1ONabqkF/Z/QY90xzcwa2/aVU6sniPEH/D1ytHfwUYiLjPOpWILi+2vCGIck5jh985gActYQZ7PFTjGCU4DT1K/NCKNtlKlQFszgh8hTXwCUsWKQ=</latexit><latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit><latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit><latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit>

X

<latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit><latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit><latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit><latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit>

Our universal resource state

Idea of our construction Ø Fact 1: Teleportation & Decoupling

H|ψi

<latexit sha1_base64="xuyVNTvwFN13hXYzCEGxAJ+x6EY=">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</latexit><latexit sha1_base64="xuyVNTvwFN13hXYzCEGxAJ+x6EY=">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</latexit><latexit sha1_base64="xuyVNTvwFN13hXYzCEGxAJ+x6EY=">ACcHichVFNSwJBGH7cvsw+tLoEHbLEiA4yG0HRKeriMTU1SIndbTBdXfbXQWz/kB/oEOXDCKin9GlP9DBnxDdMujSod1ISqd5iZ5n3emVEtXTguY+2A1Nc/MDgUHA6NjI6NhyMTkznHrNkaz2qmbtq7quJwXRg86wpX57uWzZWqvO8Wtnq7ufr3HaEaey4DYsXq0rZECWhKS5RxeRJwXJEwVaMs73IzGWYF5EfwLZBzH4sW1GblDAUxoqKEKDgMuYR0KHGp7kMFgEVdEkzibkPD2OU4RIm2NsjhlKMRWaCzTas9nDVp3azqeWqNTdOo2KaOIs0d2yzrsgd2xJ/b+a62mV6PrpUGz2tNyaz98Np15+1dVpdnF4afqT8uSljzvArybnlM9xZaT18/Pu9k1tPx5gK7Ys/kv8Xa7J5uYNRftesUT18gRB8gf3/unyC3nJAJp1ZiG5v+VwQxg3ks0nuvYgNJbCNL5x7hHJdoBV6kaWlWmulSgFfM4UvIS19AGkjzk=</latexit><latexit sha1_base64="xuyVNTvwFN13hXYzCEGxAJ+x6EY=">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</latexit>

|ψi

<latexit sha1_base64="28XfF90KGOpdVx/Lk+dZsBxL2sE=">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</latexit><latexit sha1_base64="28XfF90KGOpdVx/Lk+dZsBxL2sE=">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</latexit><latexit sha1_base64="28XfF90KGOpdVx/Lk+dZsBxL2sE=">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</latexit><latexit sha1_base64="28XfF90KGOpdVx/Lk+dZsBxL2sE=">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</latexit>

Teleportation Decoupling

Z

|ψi

<latexit sha1_base64="28XfF90KGOpdVx/Lk+dZsBxL2sE=">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</latexit><latexit sha1_base64="28XfF90KGOpdVx/Lk+dZsBxL2sE=">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</latexit><latexit sha1_base64="28XfF90KGOpdVx/Lk+dZsBxL2sE=">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</latexit><latexit sha1_base64="28XfF90KGOpdVx/Lk+dZsBxL2sE=">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</latexit>

Ø Fact 2: Universal gate set

{H, CCZ}

<latexit sha1_base64="3ZF7ic4Acs5+cI6GQHXY+oapknA=">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</latexit><latexit sha1_base64="3ZF7ic4Acs5+cI6GQHXY+oapknA=">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</latexit><latexit sha1_base64="3ZF7ic4Acs5+cI6GQHXY+oapknA=">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</latexit><latexit sha1_base64="3ZF7ic4Acs5+cI6GQHXY+oapknA=">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</latexit>

[Shi ‘02]

X

<latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit><latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit><latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit><latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit>

X

<latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit><latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit><latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">ACZHichVHLSsNAFD2Nr1ofrYogCFIsiqty09bWuhLduLRqtaAiSZzW0DQJSVqoxR/QreLClYKI+Blu/AEX/oAgLhXcuPA2rYgL9Q4zc+bMPXfOzKi2obse0WNA6ujs6u4J9ob6+gcGw5Gh4Q3XqjqayGuWYTkFVXGFoZsi7+meIQq2I5SKaohNtbzU3N+sCcfVLXPdq9tip6KUTL2oa4rHVK6wG4lRXE7KydRslOLZbDadyTCgRDKdSEXlOPkRQztWrMg1trEHCxqEDAhMfYgAKX2xZkEGzmdtBgzmGk+/sChwixtspZgjMUZs8lni1WZNXjdrur5a41M7g4ro5iB7qhV7qnW3qmj19rNfwaTS91ntWVti74aOxtfd/VRWePex/q/707KGIOd+rzt5tn2neQmvpawdnr2vzq1ONabqkF/Z/QY90xzcwa2/aVU6sniPEH/D1ytHfwUYiLjPOpWILi+2vCGIck5jh985gActYQZ7PFTjGCU4DT1K/NCKNtlKlQFszgh8hTXwCUsWKQ=</latexit><latexit sha1_base64="s4dOPo437QJ9/uQynLr3okt9cg=">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</latexit>

SLIDE 15

Our universal resource state

Ø Identity gate Idea of our construction

SLIDE 16

Our universal resource state

Ø Identity gate

Ø CCZ gate Idea of our construction

SLIDE 20

Our universal resource state

Ø CCZ gate

SLIDE 24

Our universal resource state

Ø One-depth quantum computing on three input qubits Three output qubits (up to nonlocal byproducts) Three input qubits

Local Pauli byproduct

The case where CZ1,2 does not occur Ø One-depth quantum computing on three input qubits Idea of our construction

SLIDE 28

Our universal resource state

Three output qubits Three input qubits The graph state for the correction of nonlocal byproducts. Ø One-depth quantum computing on three input qubits Idea of our construction

SLIDE 29

Conclusion

Resource state Measurement basis Class Cluster state [15] X, Y, TXT † [16] graph state Brickwork state [17] X, Y, TXT † [17] graph state Triangular lattice state [18] X, Z, H, XHX [18] graph state Raussendorf-Harrington-Goyal (RHG) lattice [19, 20] X, Y, Z, TXT † [20, 21] graph state Decorated RHG lattice [22] X, Y, TXT † [22] graph state Affleck-Kennedy-Lieb-Tasaki (AKLT) state [23, 24] qutrit bases [24] matrix-product state Union Jack state [25] X, Y, Z [25] hypergraph state Three-uniform hypergraph state [26] X, Y, Z [26] hypergraph state Mølmer-Sørensen graph state [27] X, Z [27] weighted graph state Our state X, Z hypergraph state

arXiv:1809.07552 We have constructed, for the first time, a universal hypergraph state that achieves

Universality
Verifiability