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Quantum Gravity
on a
Quantum Chip
Jakub Mielczarek
Jagiellonian University, Cracow
Centre de Physique Théorique, Marseille
5th POTOR Conference, Wojanów, 26.09.2018
Quantum computing industry is booming!
Superconducting qubits
The recent progress has been made mainly thanks to the
technology of manufacturing superconducting qubits.
Semba, 2008
Quantum computers are becoming accessible
You can program quantum computer this evening…
Visit www.quantumexperience.ng.bluemix.net
and get access to the IMB Q 5-qubit quantum computer.
Let’s create the Bell state…
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CNOT( ˆH ⌦ ˆI)( ˆX ⌦ ˆX)(|0i ⌦ |0i) =
1
p
2
(|0i ⌦ |1i |1i ⌦ |0i)
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Why it can better than using classical
supercomputers?
Dimension of the Hilbert space for N qubits grows exponentially with N:
dim(H ⌦ H ⌦ · · · ⌦ H
| {z }
N
) = 2N
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dim(H) = 2<latexit sha1_base64="Fi0+JyqfIo24foYquVHPkt/6gWg=">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</latexit><latexit sha1_base64="Fi0+JyqfIo24foYquVHPkt/6gWg=">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</latexit><latexit sha1_base64="Fi0+JyqfIo24foYquVHPkt/6gWg=">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</latexit><latexit sha1_base64="OkH1sCw5ED9/6tjbYldxQvpD+MU=">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</latexit>
For a sigle qubit
Quantum parallelism > reduction of computational complexity of some classical problems
(Deutsch, Grover, Shor, … ) > quantum speed-up
With the present most powerful classical supercomputers you can
simulate quantum systems with N=56 at most.
IBM and Rigetti Computing are developing quantum chips with N>100
and certain topologies of couplings between the qubits > quantum supremacy
Quantum computers will allow to perform exact
simulations of quantum systems with large N
In particular, we may think about simulating Planck scale physics…
But, can we express QG DOF with qubits?
Loop Quantum Gravity
Holonomies of Ashtekar connection along some curveA 2 su(2)<latexit sha1_base64="PH7MRzi1UbHp7jilv25eElcakx0=">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</latexit><latexit 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h[A, e] = Pe
R
e
A
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Under the gauge transformations
A ! Ag = g 1
dg + g 1
Ag<latexit sha1_base64="oidyHZBtlLWMnJmtOUtyGk1aJnc=">AAAC6XicjVHLSsNAFD3G97vq0k2wCoJYEhF0I7S6calgtVBrSdJpOjRNwmSilNIfcOdO3PoDbvVHxD/Qv/DOGEEtohOSnDn3njNz73XjgCfSsl6GjOGR0bHxicmp6ZnZufncwuJpEqXCY2UvCiJRcZ2EBTxkZcllwCqxYE7HDdiZ2z5Q8bNLJhIehSeyG7Nax/FD3uSeI4mq51ZL54L7LekIEV2Zpbpv7pn+RW/T7jf8jQ9Q8uu5vFWw9DIHgZ2BPLJ1FOWecY4GInhI0QFDCEk4gIOEnipsWIiJq6FHnCDEdZyhjynSppTFKMMhtk1fn3bVjA1przwTrfbolIBeQUoTa6SJKE8QVqeZOp5qZ8X+5t3TnupuXfq7mVeHWIkWsX/pPjP/q1O1SDSxq2vgVFOsGVWdl7mkuivq5uaXqiQ5xMQp3KC4IOxp5WefTa1JdO2qt46Ov+pMxaq9l+WmeFO3pAHbP8c5CE63CrZVsI+388X9bNQTWMYK1mmeOyjiEEcok/c1HvCIJ6Nt3Bi3xt1HqjGUaZbwbRn371GmnPI=</latexit><latexit sha1_base64="oidyHZBtlLWMnJmtOUtyGk1aJnc=">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</latexit><latexit sha1_base64="oidyHZBtlLWMnJmtOUtyGk1aJnc=">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</latexit><latexit sha1_base64="llI3DvttFW49isVKJLgurfWrurg=">AAAC6XicjVHLSsNAFD2N7/qqunQTrIIglsSNbgSrG5cKthVsLUk6jUPTJEwmihR/wJ07cesPuNUfEf9A/8I70xTUIjohyZlz7zkz9143DngiLestZ4yMjo1PTE7lp2dm5+YLC4vVJEqFxypeFETi1HUSFvCQVSSXATuNBXO6bsBqbudAxWuXTCQ8Ck/kdcwaXccPeZt7jiSqWVgt1wX3L6QjRHRllpu+uWv6571N+6blb/RB2W8WilbJ0sscBnYGisjWUVR4RR0tRPCQoguGEJJwAAcJPWewYSEmroEecYIQ13GGG+RJm1IWowyH2A59fdqdZWxIe+WZaLVHpwT0ClKaWCNNRHmCsDrN1PFUOyv2N++e9lR3u6a/m3l1iZW4IPYv3SDzvzpVi0QbO7oGTjXFmlHVeZlLqruibm5+qUqSQ0ycwi2KC8KeVg76bGpNomtXvXV0/F1nKlbtvSw3xYe6JQ3Y/jnOYVDdKtlWyT7eKu7tZ6OexDJWsE7z3MYeDnGECnnf4gnPeDE6xp1xbzz0U41cplnCt2U8fgJRBpzw</latexit>
h[A, e] ! h[Ag, e] = g(e(0))h[A, e]g(e(1)) 1
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Gauge invariant objects > Wilson loops
W[A, e] = tr (h[A, e])<latexit sha1_base64="i2+boC2DEaxp+Kwx4ymV5hu3870=">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</latexit><latexit sha1_base64="i2+boC2DEaxp+Kwx4ymV5hu3870=">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</latexit><latexit sha1_base64="i2+boC2DEaxp+Kwx4ymV5hu3870=">AAAC6HicjVHLSsNAFD2N73fVpZtgERSkJCLoRvCxcalgbaEtksRpOzp5MJkIpfgB7tyJW3/ArX6J+Af6F94ZU1CL6IQkZ84958zcGT8RPFWO81qwhoZHRsfGJyanpmdm54rzC6dpnMmAVYJYxLLmeykTPGIVxZVgtUQyL/QFq/qXB7pevWIy5XF0oroJa4ZeO+ItHniKqLNiqVrfW2fNnV5DhraS1w3BWmq1Y0i7IXm7o9ZI5ZQdM+xB4OaghHwcxcUXNHCOGAEyhGCIoAgLeEjpqcOFg4S4JnrESULc1BmuMUnejFSMFB6xl/Rt06yesxHNdWZq3AGtIuiV5LSxQp6YdJKwXs029cwka/a37J7J1Hvr0t/Ps0JiFTrE/uXrK//r070otLBteuDUU2IY3V2Qp2TmVPTO7S9dKUpIiNP4nOqScGCc/XO2jSc1veuz9Uz9zSg1q+dBrs3wrndJF+z+vM5BcLpRdp2ye7xZ2t3Pr3ocS1jGKt3nFnZxiCNUKPsGj3jCs3Vh3Vp31v2n1CrknkV8G9bDB8iqnMU=</latexit><latexit sha1_base64="1cVcIdpfEDeMX72Ejq/EqHbKv6s=">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</latexit>
The key idea behind LQG is to built a Hilbert space of the theory out of the 

Wilson loops. However, such basis is in general over-complete and the 

solution comes with the constructions of spin networks which are 

certain linear combination of products of the Wilson loops.
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Spin Networks
„Quantum of space”
| i = | , jl, ini = ⌦n|ini<latexit sha1_base64="y5l4YLNEI0Bgip8zB2xsNtpv/hQ=">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</latexit><latexit sha1_base64="y5l4YLNEI0Bgip8zB2xsNtpv/hQ=">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</latexit><latexit sha1_base64="y5l4YLNEI0Bgip8zB2xsNtpv/hQ=">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</latexit><latexit sha1_base64="HKL/4JCoixxPiMdoKMUeU92E6Lo=">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</latexit>
= {links (l), nodes (n)}<latexit sha1_base64="qJ8UdaSd9EMBrdwqw/wpv5CzQ20=">AAAC+HicjVHLSsNAFD2N7/qqunQzWAQFKYkKuhFEF7pUsLZgRCbpqKF5kUxELf0Qd+7ErT/gVr9A/AP9C++MI/hAdEKSM+fec2buvV4aBrm07eeS1dPb1z8wOFQeHhkdG69MTO7nSZH5ou4nYZI1PZ6LMIhFXQYyFM00EzzyQtHw2psq3jgTWR4k8Z68SMVhxE/i4DjwuSTqqLLkbvEo4myNuR3mSnEuO2TVzrsumwvnFwwVJy2hqXieud2jStWu2Xqxn8AxoAqzdpLKE1y0kMBHgQgCMSThEBw5PQdwYCMl7hAd4jJCgY4LdFEmbUFZgjI4sW36ntDuwLAx7ZVnrtU+nRLSm5GSYZY0CeVlhNVpTMcL7azY37w72lPd7YL+nvGKiJU4JfYv3Ufmf3WqFoljrOoaAqop1Yyqzjcuhe6Kujn7VJUkh5Q4hVsUzwj7WvnRZ6Y1ua5d9Zbr+IvOVKza+ya3wKu6JQ3Y+T7On2B/sebYNWd3ubq+YUY9iGnMYI7muYJ1bGMHdfK+wj0e8GhdWtfWjXX7nmqVjGYKX5Z19wZxe6MC</latexit><latexit sha1_base64="qJ8UdaSd9EMBrdwqw/wpv5CzQ20=">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</latexit><latexit sha1_base64="qJ8UdaSd9EMBrdwqw/wpv5CzQ20=">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</latexit><latexit sha1_base64="DqFMXO96V4GcizDChf53P9ejaIg=">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</latexit>
Graph:
Spin network state:
jl<latexit sha1_base64="VqVUo/lMd7S1rBJtIiHh0mR3yuA=">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</latexit><latexit sha1_base64="VqVUo/lMd7S1rBJtIiHh0mR3yuA=">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</latexit><latexit sha1_base64="VqVUo/lMd7S1rBJtIiHh0mR3yuA=">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</latexit><latexit sha1_base64="PuctsTK81E2eGUjtB/7OnKafvAg=">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</latexit>
in<latexit sha1_base64="KxFfG48b6eMKKrKZiFkCryK07dM=">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</latexit><latexit sha1_base64="KxFfG48b6eMKKrKZiFkCryK07dM=">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</latexit><latexit sha1_base64="KxFfG48b6eMKKrKZiFkCryK07dM=">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</latexit><latexit sha1_base64="u+zXQwJ0eLXINj/wuBo+s/GlSiU=">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</latexit>
- spin labels - intertwiners
in<latexit sha1_base64="KxFfG48b6eMKKrKZiFkCryK07dM=">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</latexit><latexit sha1_base64="KxFfG48b6eMKKrKZiFkCryK07dM=">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</latexit><latexit sha1_base64="KxFfG48b6eMKKrKZiFkCryK07dM=">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</latexit><latexit sha1_base64="u+zXQwJ0eLXINj/wuBo+s/GlSiU=">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</latexit>
( ˆE1
+ ˆE2
+ ˆE3
+ ˆE4
)|ini = 0<latexit sha1_base64="LOhJaxFgrzzqOR3RrmnohFNqyZg=">AAADBnicjVHLSsNAFD3GV31XXboJFkERSlILuhF8ILhUsCpYLZM41qFpEiYTQWr3/ok7d+LWH3Br8Q/0L7wzRqmK6IQkZ86958zce704EIlynOceq7evf2AwNzQ8Mjo2PpGfnNpPolT6vOJHQSQPPZbwQIS8ooQK+GEsOWt6AT/wGps6fnDBZSKicE9dxvy4yeqhOBM+U0TV8uvz1XOmWlvtk5bbXvzEpS681IXL7YUrUQvtqmRhPeD2qu3U8gWn6Jhl/wRuBgrI1k6U76CKU0TwkaIJjhCKcACGhJ4juHAQE3eMFnGSkDBxjjaGSZtSFqcMRmyDvnXaHWVsSHvtmRi1T6cE9EpS2pgjTUR5krA+zTbx1Dhr9jfvlvHUd7ukv5d5NYlVOCf2L91H5n91uhaFM6yYGgTVFBtGV+dnLqnpir653VWVIoeYOI1PKS4J+0b50WfbaBJTu+4tM/EXk6lZvfez3BSv+pY0YPf7OH+C/VLRdYrubrmwtpGNOocZzGKe5rmMNWxjBxXyvsEjntCxrq1b6866f0+1ejLNNL4s6+ENjxSpVg==</latexit><latexit sha1_base64="LOhJaxFgrzzqOR3RrmnohFNqyZg=">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</latexit><latexit sha1_base64="LOhJaxFgrzzqOR3RrmnohFNqyZg=">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</latexit><latexit sha1_base64="46fIFt86bhMZ/IoL79Q2tAQ6mxE=">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</latexit>
Gauss constraint:
Qubit
H1/2 ⌦ H1/2 ⌦ H1/2 ⌦ H1/2 = H0 H0
| {z }
3H1 H2
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dim Inv(H1/2 ⌦ H1/2 ⌦ H1/2 ⌦ H1/2) = 2<latexit sha1_base64="qaQ0sQuGe0zcHy67zS4I4wfO3NE=">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</latexit><latexit sha1_base64="qaQ0sQuGe0zcHy67zS4I4wfO3NE=">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</latexit><latexit sha1_base64="qaQ0sQuGe0zcHy67zS4I4wfO3NE=">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</latexit><latexit sha1_base64="uuG1aQRYhPDKdOAkdL9fln4uKNw=">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</latexit>
We consider spin networks composed of 4-valent vertices and spin
labels corresponding to fundamental representations of the SU(2) group
i.e. j=1/2.The Hilbert space at each vertex is:
such that the invariant subspace is two dimensional:
We associate the two dimensional invariant space with the qubit space:
Qubit base states
|1si =
1
p
3
✓
| ""##i + | ##""i
| "#"#i + | #"#"i + | "##"i + | #""#i
2
◆
<latexit sha1_base64="n25PQt4yY8V5UOGbewpMEfl03zQ=">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</latexit><latexit sha1_base64="n25PQt4yY8V5UOGbewpMEfl03zQ=">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</latexit><latexit sha1_base64="n25PQt4yY8V5UOGbewpMEfl03zQ=">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</latexit><latexit sha1_base64="dJU2urBq+dlWPfIqV88w3XL3rXw=">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</latexit>
|0si =
1
2
(| "#"#i + | #"#"i | "##"i | #""#i)
<latexit sha1_base64="RC3kJ9pBeI5WYPFYdvtn6CY+XJk=">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</latexit><latexit sha1_base64="RC3kJ9pBeI5WYPFYdvtn6CY+XJk=">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</latexit><latexit sha1_base64="RC3kJ9pBeI5WYPFYdvtn6CY+XJk=">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</latexit><latexit sha1_base64="RGoN+akc8pQ6u4uIzJoO7U7czvU=">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</latexit>
s-channel basis:
Eigenbasis of the volume operator :
|1i =
1
p
2
(|0si i|1si)
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|0i =
1
p
2
(|0si + i|1si)
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ˆV |1i = +V0|1i<latexit sha1_base64="bg/Q8wh6nmkz59VcAjnRf0keEd4=">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</latexit><latexit sha1_base64="bg/Q8wh6nmkz59VcAjnRf0keEd4=">AAAC5HicjVHLSsNAFD2N7/qqunRhsAiCUBIRdCMU3LhUsLFgpUzSaRuaF5OJUKpLd+7ErT/gVr9F/AP9C++MqahFdEKSc8+958zcuW4S+Km0rJeCMTY+MTk1PVOcnZtfWCwtLTtpnAmP17w4iEXdZSkP/IjXpC8DXk8EZ6Eb8FO3d6DypxdcpH4cnch+ws9D1on8tu8xSVSztNboMjlwri7thmBRJ+D75pbTtD7DZqlsVSy9zFFg56CMfB3FpWc00EIMDxlCcESQhAMwpPScwYaFhLhzDIgThHyd57hCkbQZVXGqYMT26Nuh6CxnI4qVZ6rVHu0S0CtIaWKDNDHVCcJqN1PnM+2s2N+8B9pTna1Pfzf3ComV6BL7l25Y+V+d6kWijT3dg089JZpR3Xm5S6ZvRZ3c/NKVJIeEOIVblBeEPa0c3rOpNanuXd0t0/lXXalYFXt5bYY3dUoasP1znKPA2a7YVsU+3ilXq/mop7GKdWzSPHdRxSGOUCPvazzgEU9G27gxbo27j1KjkGtW8G0Z9+/yeJu4</latexit><latexit sha1_base64="bg/Q8wh6nmkz59VcAjnRf0keEd4=">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</latexit><latexit sha1_base64="H4d5Nb5e29IU85Xsab52T1i99Mc=">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</latexit>
ˆV |0i = V0|0i<latexit sha1_base64="Y1pzJJwf8veqRWsDV00gEIrwn4c=">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</latexit><latexit sha1_base64="Y1pzJJwf8veqRWsDV00gEIrwn4c=">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</latexit><latexit sha1_base64="Y1pzJJwf8veqRWsDV00gEIrwn4c=">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</latexit><latexit sha1_base64="cpUqtvkCnusEsbqbk2sJHI+BPVw=">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</latexit>
V0 =
p
3
4
l3
P l
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ˆV<latexit sha1_base64="ayP5B8FArjllzopLb5MNLKuXP10=">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</latexit><latexit sha1_base64="ayP5B8FArjllzopLb5MNLKuXP10=">AAACynicjVHLSsNAFD2Nr/quunQTLIKrkoigy4IbFy4q2Ae0RZLptB2aF5OJUEJ3/oBb/TDxD/QvvDOmoBbRCUnOnHvOnbn3+kkgUuU4ryVraXllda28vrG5tb2zW9nbb6VxJhlvsjiIZcf3Uh6IiDeVUAHvJJJ7oR/wtj+51PH2PZepiKNbNU14P/RGkRgK5imi2r2xp/LW7K5SdWqOWfYicAtQRbEaceUFPQwQgyFDCI4IinAADyk9XbhwkBDXR06cJCRMnGOGDfJmpOKk8Iid0HdEu27BRrTXOVPjZnRKQK8kp41j8sSkk4T1abaJZyazZn/LnZuc+m5T+vtFrpBYhTGxf/nmyv/6dC0KQ1yYGgTVlBhGV8eKLJnpir65/aUqRRkS4jQeUFwSZsY577NtPKmpXffWM/E3o9Ss3rNCm+Fd35IG7P4c5yJondZcp+benFXr9WLUZRziCCc0z3PUcYUGmqbKRzzh2bq2pDW18k+pVSo8B/i2rIcPHYWSJg==</latexit><latexit sha1_base64="ayP5B8FArjllzopLb5MNLKuXP10=">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</latexit><latexit sha1_base64="9GFBf+IjRJYf9w9jHdWAg8RrL0o=">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</latexit>
where
Transition amplitude
W(x, x0
) = hx0
| ˆP|xi<latexit sha1_base64="+r5gEOQat59u8jQIe295YEFhOV0=">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</latexit><latexit sha1_base64="+r5gEOQat59u8jQIe295YEFhOV0=">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</latexit><latexit sha1_base64="+r5gEOQat59u8jQIe295YEFhOV0=">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</latexit><latexit sha1_base64="zan6bvJeZ3vjSbEOaluY72PPgqE=">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</latexit>
ˆP = ( ˆC) = lim
T !1
1
2T
Z T
T
d⌧ei⌧ ˆC
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ˆC| i = 0<latexit sha1_base64="w9fbT5UfFINJYWTqLXi7h1Pb5Lg=">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</latexit><latexit sha1_base64="w9fbT5UfFINJYWTqLXi7h1Pb5Lg=">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</latexit><latexit sha1_base64="w9fbT5UfFINJYWTqLXi7h1Pb5Lg=">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</latexit><latexit sha1_base64="zPzoHE6okzNgGWWexkMvci5WKfg=">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</latexit>
In both classical and quantum gravity physical states (physical solutions) are obtained
by solving the constraints.The physical states belong to the kernel of the constraints:
| i 2 Hphys ⇢ Hkin<latexit sha1_base64="HaPRHHItzyV2Gx+G1Y13pu8KHfk=">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</latexit><latexit sha1_base64="HaPRHHItzyV2Gx+G1Y13pu8KHfk=">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</latexit><latexit sha1_base64="HaPRHHItzyV2Gx+G1Y13pu8KHfk=">AAADBnicjVHLThsxFD1My7OFpnTJxmpUqatoBiHRRRdBbFgGqQEkBkUeYxIrnodsT6UoZM+fsOuu6pYf6BbUP2j/otdmkKCoAo9m5vjce459780qrayL419z0YuX8wuLS8srr16vrr1pvV0/sGVthOyLUpfmKONWalXIvlNOy6PKSJ5nWh5m410fP/wqjVVl8cVNKnmS82GhzpTgjqhBa+c87VnFUsOLoZYsVQVLc+5Gguvp3mwwrUYTO2OprTMrHXsYG6tixtig1Y47cVjsMUga0EazemXrBilOUUKgRg6JAo6wBoel5xgJYlTEnWBKnCGkQlxihhXS1pQlKYMTO6bvkHbHDVvQ3nvaoBZ0iqbXkJLhA2lKyjOE/WksxOvg7Nn/eU+Dp7/bhP5Z45UT6zAi9indXeZzdb4WhzN8CjUoqqkKjK9ONC516Iq/ObtXlSOHijiPTyluCIugvOszCxobave95SH+O2R61u9Fk1vjj78lDTj5d5yPwcFmJ4k7yf5Wu/u5GfUSNvAeH2me2+hiDz30yfsSP3GNm+gi+hZ9j37cpkZzjeYdHqzo6i/0tqpO</latexit><latexit sha1_base64="KJA2xnWcrFoDtuwEO39AFFMr/f4=">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</latexit>
such that
Therefore, transition amplitude between some boundary states is
where the projection operator
Rovelli &Vidotto, 2015 Keren Li, et al., arXiv: 1712.08711
Spin Foams
Vertex amplitude
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„Quantum of spacetime”
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n=1
|ini
<latexit sha1_base64="jpHcqtB5Hh3aYEuoMWK6hBRTpdY=">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</latexit><latexit sha1_base64="jpHcqtB5Hh3aYEuoMWK6hBRTpdY=">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</latexit><latexit sha1_base64="jpHcqtB5Hh3aYEuoMWK6hBRTpdY=">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</latexit><latexit sha1_base64="xHaifib8PgTWc12XOgoCW5tMiPU=">AAAC+XicjVHLSgMxFD2O73fVpZtgEVyVGVF0YUFw47KCtQWrZWaMNTgvkowgtT/izp249Qfc6g+If6B/4U2cgg9EM8zMybnnnOQmQRYJpV33ZcAZHBoeGR0bn5icmp6ZLc3NH6g0lyGvh2mUymbgKx6JhNe10BFvZpL7cRDxRnC+Y+qNCy6VSJN9fZnxo9jvJOJUhL4mql1au2rVlGh7rCX9pBNxVmWtQHRSLWKu2t2k6vWO19kVE4R7fVG7VHYrrh3sJ/AKUEYxamnpGS2cIEWIHDE4EmjCEXwoeg7hwUVG3BG6xElCwtY5epggb04qTgqf2HP6dmh2WLAJzU2msu6QVonoleRkWCZPSjpJ2KzGbD23yYb9LbtrM83eLukfFFkxsRpnxP7l6yv/6zO9aJxi0/YgqKfMMqa7sEjJ7amYnbNPXWlKyIgz+ITqknBonf1zZtajbO/mbH1bf7VKw5p5WGhzvJld0gV736/zJzhYrXhuxdtbLW9vFVc9hkUsYYXucwPb2EUNdcq+xgMe8eR0nRvn1rn7kDoDhWcBX4Zz/w7t2KP2</latexit>
| 2i =
10O
l=1
|✏li
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Spin network state:
Representation:
|✏li =
1
p
2
(| "#i | #")
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- boundary state of a single vertex
h 2| 1i = A(i1, i2, i3, i4, i5)<latexit sha1_base64="vZ5HC8JUNDvWuZUHGukpjJY7cv8=">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</latexit><latexit sha1_base64="vZ5HC8JUNDvWuZUHGukpjJY7cv8=">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</latexit><latexit sha1_base64="vZ5HC8JUNDvWuZUHGukpjJY7cv8=">AAAC+3icjVFNTxRBEH0MigiKAx69TNyQYGI2M4DKAZMlXDyuiQskLNnMDM3aoecjPT0mBPafePNmvPoHvMKd8A/kX/C6dzYBidGe1HTVq3qvu7qSUsnKhOHVlDf94OHMo9nHc/NPni488xeXdqqi1qnopYUq9F4SV0LJXPSMNErslVrEWaLEbnK8bfO7X4SuZJF/MielOMjiYS6PZBobQgP/bV/F+VCJoN+t5GD1zG1R0Ndj9H2wtcL4NVO0Ndo67c2rgd8K26FbwX0napwWmtUt/Ev0cYgCKWpkEMhh6CvEqPjtI0KIktgBTolpetLlBUaYI7dmlWBFTPSY/yGj/QbNGVvNyrFTnqJomswAy+QUrNP07WmBy9dO2aJ/0z51mvZuJ9yTRisjavCZ6L94k8r/5dleDI6w4XqQ7Kl0iO0ubVRq9yr25sGtrgwVSmLWP2Re008dc/LOgeNUrnf7trHL/3aVFrVx2tTWuLa35ICjP8d539lZbUdhO/q43upsNqOexQu8xArn+Q4dfEAXPWp/xS+c48Ibed+8796Pcak31XCe487yft4AjUKieQ==</latexit><latexit sha1_base64="jmAV6JWoX8OnpmEbgpowrG2/tLw=">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</latexit>
Vertex amplitude:
see: Keren Li, et al., arXiv: 1712.08711 But, we have to identify the states which
satisfy the constraint first…C ⇡ 0<latexit sha1_base64="GDGseDEDeS1zq5a1e9TpY/vI28k=">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</latexit><latexit sha1_base64="GDGseDEDeS1zq5a1e9TpY/vI28k=">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</latexit><latexit sha1_base64="GDGseDEDeS1zq5a1e9TpY/vI28k=">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</latexit><latexit sha1_base64="TcPeeSrZCQ6nKnO5i5NkbhPo9SQ=">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</latexit>
Can be computed with the use of quantum computer (in progress).
(Bell state)
|h 2| 1i|2
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How to compute on quantum computer?
For N nodes of the spin network we need 4N qubits (20 qubits for the vertex amplitude).
The Hilbert space is spanned by base states24N
<latexit sha1_base64="m/y0oGtD83V6vQEG/8kgyAxTCs4=">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</latexit><latexit sha1_base64="m/y0oGtD83V6vQEG/8kgyAxTCs4=">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</latexit><latexit sha1_base64="m/y0oGtD83V6vQEG/8kgyAxTCs4=">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</latexit><latexit sha1_base64="mv+IN7ZBouFq13Um/wCI14LNXAU=">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</latexit>
|ii<latexit sha1_base64="IxjyaDOb75yXFEFjwK5Kxe7Xiz4=">AAAC1nicjVHLSsNAFD3GV62vVJdugkVwVRIRdFl047KCfUBbSpJO69A0CZOJUmrdiVt/wK1+kvgH+hfeGVNQi+iEJGfOvefM3Hu9OOCJtO3XOWN+YXFpObeSX11b39g0C1u1JEqFz6p+FESi4bkJC3jIqpLLgDViwdyhF7C6NzhV8foVEwmPwgs5ill76PZD3uO+K4nqmIUba9zyehafWC3hhv2AdcyiXbL1smaBk4EislWJzBe00EUEHymGYAghCQdwkdDThAMbMXFtjIkThLiOM0yQJ21KWYwyXGIH9O3TrpmxIe2VZ6LVPp0S0CtIaWGPNBHlCcLqNEvHU+2s2N+8x9pT3W1Efy/zGhIrcUnsX7pp5n91qhaJHo51DZxqijWjqvMzl1R3Rd3c+lKVJIeYOIW7FBeEfa2c9tnSmkTXrnrr6vibzlSs2vtZbop3dUsasPNznLOgdlBy7JJzflgsn2SjzmEHu9ineR6hjDNUUCXvazziCc9Gw7g17oz7z1RjLtNs49syHj4ALwWWDA==</latexit><latexit sha1_base64="IxjyaDOb75yXFEFjwK5Kxe7Xiz4=">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</latexit><latexit sha1_base64="IxjyaDOb75yXFEFjwK5Kxe7Xiz4=">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</latexit><latexit sha1_base64="B+XLZUSY6u+6lvljNxYsWHbuDyk=">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</latexit>
where i 2 0, 24N
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The initial state for quantum algorithm:
We have to find and such that:ˆU1<latexit sha1_base64="Tg+LuVxl7yCgmthJzHRehWoypBQ=">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</latexit><latexit sha1_base64="Tg+LuVxl7yCgmthJzHRehWoypBQ=">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</latexit><latexit sha1_base64="Tg+LuVxl7yCgmthJzHRehWoypBQ=">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</latexit><latexit sha1_base64="sTmjJ2BPSdSvuYYk0O8IU/XRrx8=">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</latexit>
ˆU2<latexit sha1_base64="Kj4Ev4tb7T3tdPsyCAcz8I5Qt7c=">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</latexit><latexit sha1_base64="Kj4Ev4tb7T3tdPsyCAcz8I5Qt7c=">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</latexit><latexit sha1_base64="Kj4Ev4tb7T3tdPsyCAcz8I5Qt7c=">AAACzHicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6LLoxpVUMK3SlpKk0zY0LyYToZRu/QG3+l3iH+hfeGecglpEJyQ5c+49Z+be66VhkAnLei0YS8srq2vF9dLG5tb2Tnl3r5klOfeZ4ydhwm89N2NhEDNHBCJktylnbuSFrOWNL2S8dc94FiTxjZikrBu5wzgYBL4riLrrjFwxdWa9Wq9csaqWWuYisDWoQK9GUn5BB30k8JEjAkMMQTiEi4yeNmxYSInrYkocJxSoOMMMJdLmlMUowyV2TN8h7dqajWkvPTOl9umUkF5OShNHpEkojxOWp5kqnitnyf7mPVWe8m4T+nvaKyJWYETsX7p55n91shaBAc5UDQHVlCpGVudrl1x1Rd7c/FKVIIeUOIn7FOeEfaWc99lUmkzVLnvrqvibypSs3Ps6N8e7vCUN2P45zkXQrFVtq2pfn1Tq53rURRzgEMc0z1PUcYkGHPKO8IgnPBtXhjCmxuwz1ShozT6+LePhA8Xyksw=</latexit><latexit sha1_base64="iMjhqBHC007fgpwLjSfz5n4kzX4=">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</latexit>
| 2i = ˆU2|0i<latexit sha1_base64="vsoNT/l9aUP1Ql9U16Mj4l9Oi14=">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</latexit><latexit sha1_base64="vsoNT/l9aUP1Ql9U16Mj4l9Oi14=">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</latexit><latexit sha1_base64="vsoNT/l9aUP1Ql9U16Mj4l9Oi14=">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</latexit><latexit sha1_base64="G4CTk9rGwMXEDEi+e01i/krN0D0=">AAAC8nicjVHLSsNAFD3GV31XXboZLIIbS9KNbgTRjcsKtgpGyiRO62CahMlEKLVf4c6duPUH3OpHiH+gf+GdMYoPRCckOXPuPWfm3hukkcy06z4NOcMjo2PjpYnJqemZ2bny/EIzS3IVikaYRIk6DHgmIhmLhpY6EoepErwbROIgONsx8YNzoTKZxPu6l4rjLu/Esi1Drolqldcu/HomWzXmKx53IsE2mX/Kdb8xIO6i7wdt5g4+gq1yxa26drGfwCtABcWqJ+VH+DhBghA5uhCIoQlH4MjoOYIHFylxx+gTpwhJGxcYYJK0OWUJyuDEntG3Q7ujgo1pbzwzqw7plIheRUqGFdIklKcIm9OYjefW2bC/efetp7lbj/5B4dUlVuOU2L9075n/1ZlaNNrYsDVIqim1jKkuLFxy2xVzc/apKk0OKXEGn1BcEQ6t8r3PzGoyW7vpLbfxZ5tpWLMPi9wcL+aWNGDv+zh/gmat6rlVb69W2douRl3CEpaxSvNcxxZ2UUeDvC9xh3s8ONq5cq6dm7dUZ6jQLOLLcm5fAdV+oIc=</latexit>
| 1i = ˆU1|0i<latexit sha1_base64="3vfhocPNmtdb+Ayss/1ZB1W3bf8=">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</latexit><latexit sha1_base64="3vfhocPNmtdb+Ayss/1ZB1W3bf8=">AAAC8nicjVHLSsNAFD3GV31XXboZLIIbSyKCbgTRjcsKtgpGyiRO62CahMlEKLVf4c6duPUH3OpHiH+gf+GdMYoPRCckOXPuPWfm3hukkcy06z4NOINDwyOjpbHxicmp6Zny7FwjS3IVinqYRIk6DHgmIhmLupY6EoepErwTROIgONsx8YNzoTKZxPu6m4rjDm/HsiVDrolqllcu/Fommx7zFY/bkWCbzD/lulfvE3fR84MWc/sfwWa54lZdu9hP4BWggmLVkvIjfJwgQYgcHQjE0IQjcGT0HMGDi5S4Y/SIU4SkjQv0MU7anLIEZXBiz+jbpt1Rwca0N56ZVYd0SkSvIiXDEmkSylOEzWnMxnPrbNjfvHvW09ytS/+g8OoQq3FK7F+698z/6kwtGi1s2Bok1ZRaxlQXFi657Yq5OftUlSaHlDiDTyiuCIdW+d5nZjWZrd30ltv4s800rNmHRW6OF3NLGrD3fZw/QWO16rlVb2+tsrVdjLqEBSximea5ji3sooY6eV/iDvd4cLRz5Vw7N2+pzkChmceX5dy+AtEmoIc=</latexit><latexit sha1_base64="3vfhocPNmtdb+Ayss/1ZB1W3bf8=">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</latexit><latexit sha1_base64="OxC6pMdJs7I6Bx3ZGqLg4Yyklc0=">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</latexit>
and
Our quantum algorithm is defined by
the unitary operator (quantum circuit):
ˆU = ˆU†
2 U1<latexit sha1_base64="MXGxx9+0iVqySe/jKEy94JkjHnw=">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</latexit><latexit sha1_base64="MXGxx9+0iVqySe/jKEy94JkjHnw=">AAAC5XicjVHLSsNAFD3Gd31FXboJFsFVSUTQjVB047KCqYLVMEmnaWiahMlEkNKtO3fi1h9wq78i/oH+hXfGFHwgOiHJmXPvOTP3Xj+Lo1za9suYMT4xOTU9M1uZm19YXDKXV5p5WoiAu0Eap+LUZzmPo4S7MpIxP80EZ30/5id+70DFTy65yKM0OZZXGT/vszCJOlHAJFGeabW6TA7cobU3Qt7WxaDVZmHIxdD1HM+s2jVbL+sncEpQRbkaqfmMFtpIEaBAHxwJJOEYDDk9Z3BgIyPuHAPiBKFIxzmGqJC2oCxOGYzYHn1D2p2VbEJ75ZlrdUCnxPQKUlrYIE1KeYKwOs3S8UI7K/Y374H2VHe7or9fevWJlegS+5dulPlfnapFooNdXUNENWWaUdUFpUuhu6Jubn2qSpJDRpzCbYoLwoFWjvpsaU2ua1e9ZTr+qjMVq/ZBmVvgTd2SBux8H+dP0NyqOXbNOdqu1vfLUc9gDevYpHnuoI5DNOCS9zUe8IgnIzRujFvj7iPVGCs1q/iyjPt3WWOcRw==</latexit><latexit sha1_base64="MXGxx9+0iVqySe/jKEy94JkjHnw=">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</latexit><latexit sha1_base64="KVgMrRKEjGtSbzlHyAl+vkH/lq4=">AAAC5XicjVHLSsNAFD3GV62vqks3wSK4Kkk3uhGKblxWMFWwNUzSMQbzYjIRSunWnTtx6w+41V8R/0D/wjtjCj4QnZDkzLn3nJl7r5dFYS4t62XCmJyanpmtzFXnFxaXlmsrq508LYTPHT+NUnHisZxHYcIdGcqIn2SCs9iL+LF3ua/ix1dc5GGaHMlBxnsxC5LwPPSZJMqtmd0LJofOyNwdI7d5Nuz2WRBwMXJc263VrYall/kT2CWoo1zttPaMLvpI4aNADI4EknAEhpyeU9iwkBHXw5A4QSjUcY4RqqQtKItTBiP2kr4B7U5LNqG98sy12qdTInoFKU1skialPEFYnWbqeKGdFfub91B7qrsN6O+VXjGxEhfE/qUbZ/5Xp2qROMeOriGkmjLNqOr80qXQXVE3Nz9VJckhI07hPsUFYV8rx302tSbXtaveMh1/1ZmKVXu/zC3wpm5JA7a/j/Mn6DQbttWwD5v11l456grWsYEtmuc2WjhAGw55X+MBj3gyAuPGuDXuPlKNiVKzhi/LuH8HWMOcRQ==</latexit>
such that
…
…
ˆU = ˆU†
2 U1<latexit sha1_base64="MXGxx9+0iVqySe/jKEy94JkjHnw=">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</latexit><latexit sha1_base64="MXGxx9+0iVqySe/jKEy94JkjHnw=">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</latexit><latexit sha1_base64="MXGxx9+0iVqySe/jKEy94JkjHnw=">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</latexit><latexit sha1_base64="KVgMrRKEjGtSbzlHyAl+vkH/lq4=">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</latexit>
|0i<latexit sha1_base64="FL1ldUm5msesvnxxrFeFnlt2AH4=">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</latexit><latexit sha1_base64="FL1ldUm5msesvnxxrFeFnlt2AH4=">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</latexit><latexit sha1_base64="FL1ldUm5msesvnxxrFeFnlt2AH4=">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</latexit><latexit sha1_base64="F+4HQ6Dr1GAG3LRyn2dPdkSDqak=">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</latexit>
We measure: P(0) = |a0|2
<latexit sha1_base64="PF/3lZqyHB2yiXv5eDbMthix7tE=">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</latexit><latexit sha1_base64="PF/3lZqyHB2yiXv5eDbMthix7tE=">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</latexit><latexit sha1_base64="PF/3lZqyHB2yiXv5eDbMthix7tE=">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</latexit><latexit sha1_base64="qKDfr/FM2kawzXq3OIT1frcX0EE=">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</latexit>
a0 = h0| ˆU|0i = h 2| 1i<latexit sha1_base64="HUFSo9M45onouFj/eu+ZomUBpTc=">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</latexit><latexit sha1_base64="HUFSo9M45onouFj/eu+ZomUBpTc=">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</latexit><latexit sha1_base64="HUFSo9M45onouFj/eu+ZomUBpTc=">AAADFnicjVFPa9swHH111zVL283bjr2IhUFPwS6F7lII3WXHFJa2UJcgO0oiothGlgslyffYN9ltt7HrbmWHXffnW/QnTYFuoXQytp7e771n/aS0VLIyUfRtLVh/tPF4s/GkubW98/RZ+PzFaVXUOhO9rFCFPk95JZTMRc9Io8R5qQWfpkqcpZO3tn52JXQli/y9uS7F5ZSPcjmUGTdE9cMT3o/YEUsUz0dKsFmSDlm0mLNkzM2sR8AzLNFO0bwjTrqV7O/P3RQvBawftqJ25AZbBbEHLfjRLcIbJBigQIYaUwjkMIQVOCp6LhAjQkncJWbEaULS1QUWaJK3JpUgBSd2Qt8RrS48m9PaZlbOndFfFL2anAyvyVOQThO2f2OuXrtky96XPXOZdm/XNKc+a0qswZjYh3xL5f/6bC8GQ7xxPUjqqXSM7S7zKbU7FbtzdqcrQwklcRYPqK4JZ865PGfmPJXr3Z4td/VfTmlZu868tsZvu0u64Pjf61wFp/vtOGrHJwetzrG/6gZ28Qp7dJ+H6OAduuhR9kd8xw/8DD4En4LPwZc/0mDNe17irxF8vQVgla3V</latexit><latexit sha1_base64="6CA1xZNdkgjb370u6Q8IYxtCBtQ=">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</latexit>
where
|0i = | "i ⌦ · · · ⌦ | "i
| {z }
4N<latexit sha1_base64="MFNfR52ElxihH0KkGie5ef3lAhA=">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</latexit><latexit sha1_base64="MFNfR52ElxihH0KkGie5ef3lAhA=">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</latexit><latexit sha1_base64="MFNfR52ElxihH0KkGie5ef3lAhA=">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</latexit><latexit sha1_base64="p65XN8RTGmmghVxsqtttBKAWAcE=">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</latexit>
ˆU|0i =
24N
1X
i=0
ai|ii
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Solving the constraint with the
Adiabatic Quantum Computer
The adiabatic quantum computers are designed to solve a specific
problem of finding minimum of a Hamiltonian of a coupled system
of qubits (spins). In the process of finding the minimum of one
employs a time dependent Hamiltonian in the form:
where is the so-called base Hamiltonian characterized by a simple
and easy to prepare ground state. In practice, the base Hamiltonian is
often equal


such that the ground state corresponds to the alignment of spins in the x
direction.Then, the value of is changed adiabatically from
to , such that while the system is initially in non-degenerate ground
state it will remind in the ground state during the process.
HI<latexit sha1_base64="JOADAC7E4cg7sft61l2F1aIQIRw=">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</latexit><latexit sha1_base64="JOADAC7E4cg7sft61l2F1aIQIRw=">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</latexit><latexit sha1_base64="JOADAC7E4cg7sft61l2F1aIQIRw=">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</latexit><latexit sha1_base64="l+v9s2xWuR6pqiPZHXkd6JHMeeY=">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</latexit>
HI<latexit sha1_base64="JOADAC7E4cg7sft61l2F1aIQIRw=">AAAC1nicjVHLSsNAFD2Nr1pfqS7dBIvgqiQi6MJFwU3dVbAPaEtJ0mkNTZMwmSgl1J249Qfc6ieJf6B/4Z0xBbWITkhy5tx7zsy914l8Lxam+ZrTFhaXllfyq4W19Y3NLb243YjDhLus7oZ+yFuOHTPfC1hdeMJnrYgze+z4rOmMzmS8ec147IXBpZhErDu2h4E38FxbENXTix3lkTp+wqbVXno+7ekls2yqZcwDKwMlZKsW6i/ooI8QLhKMwRBAEPZhI6anDQsmIuK6SInjhDwVZ5iiQNqEshhl2MSO6DukXTtjA9pLz1ipXTrFp5eT0sA+aULK44TlaYaKJ8pZsr95p8pT3m1CfyfzGhMrcEXsX7pZ5n91shaBAU5UDR7VFClGVudmLonqiry58aUqQQ4RcRL3Kc4Ju0o567OhNLGqXfbWVvE3lSlZuXez3ATv8pY0YOvnOOdB47BsmWXr4qhUOc1Gnccu9nBA8zxGBVXUUCfvGzziCc9aS7vV7rT7z1Qtl2l28G1pDx8cZZbT</latexit><latexit sha1_base64="JOADAC7E4cg7sft61l2F1aIQIRw=">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</latexit><latexit sha1_base64="JOADAC7E4cg7sft61l2F1aIQIRw=">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</latexit><latexit sha1_base64="l+v9s2xWuR6pqiPZHXkd6JHMeeY=">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</latexit>
H( ) = (1 )HB + HI<latexit sha1_base64="Plrabj4frmfH/84PRTEohXloxmc=">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</latexit><latexit sha1_base64="Plrabj4frmfH/84PRTEohXloxmc=">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</latexit><latexit sha1_base64="Plrabj4frmfH/84PRTEohXloxmc=">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</latexit><latexit sha1_base64="RCidrVARvYSZyRf514E6RwvDcM0=">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</latexit>
HB<latexit sha1_base64="+wjAH0fvdAZFAkC9kw4sRz9ml5c=">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</latexit><latexit sha1_base64="+wjAH0fvdAZFAkC9kw4sRz9ml5c=">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</latexit><latexit sha1_base64="+wjAH0fvdAZFAkC9kw4sRz9ml5c=">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</latexit><latexit sha1_base64="LgDgx+CNgJeB5cos/fUWh1g7wCQ=">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</latexit>
HB =
X
I
x
i
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sha1_base64="VgQTEJ9SfjSQpS8GWCz2oPqUi30=">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</latexit><latexit sha1_base64="UbfqJIp68SVVU1XZpMCBSR9danw=">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</latexit>
= 1<latexit sha1_base64="V5Cbyg6cStIzk2HNOaoIu6j5oLc=">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</latexit><latexit sha1_base64="V5Cbyg6cStIzk2HNOaoIu6j5oLc=">AAACznicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdKFQcOOygn1ALZJMpzV08mAyKZRS3PoDbvWzxD/Qv/DOmIJaRCckOXPuOXfm3usnIkiV47wWrIXFpeWV4mppbX1jc6u8vdNM40wy3mCxiGXb91Iugog3VKAEbyeSe6EveMsfXuh4a8RlGsTRtRonvBt6gyjoB8xTRHVuBEl7nn1uu7flilN1zLLngZuDCvJVj8svuEEPMRgyhOCIoAgLeEjp6cCFg4S4LibESUKBiXNMUSJvRipOCo/YIX0HtOvkbER7nTM1bkanCHolOW0ckCcmnSSsT7NNPDOZNftb7onJqe82pr+f5wqJVbgj9i/fTPlfn65FoY9TU0NANSWG0dWxPEtmuqJvbn+pSlGGhDiNexSXhJlxzvpsG09qate99Uz8zSg1q/cs12Z417ekAbs/xzkPmkdV16m6V8eV2lk+6iL2sI9DmucJarhEHQ3T8Uc84dmqWyNrat1/Sq1C7tnFt2U9fAD45ZLO</latexit><latexit sha1_base64="V5Cbyg6cStIzk2HNOaoIu6j5oLc=">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</latexit><latexit sha1_base64="xv7r5dPoOe4tm4kmzl/F3r9xO0o=">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</latexit>
time
Let us observe that, the problem finding solution to the constraint:
C ⇡ 0<latexit sha1_base64="4eimVysgZAR0RugxiUZPTXu89So=">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</latexit><latexit sha1_base64="4eimVysgZAR0RugxiUZPTXu89So=">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</latexit><latexit sha1_base64="4eimVysgZAR0RugxiUZPTXu89So=">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</latexit><latexit sha1_base64="+rEngd3d1/4lfJ+n6fQRVhRVBdo=">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</latexit>
can be mapped into the problem of finding minimum of the Hamiltonian:
HI / C2
<latexit sha1_base64="BkV5VfYwV49vM+VoiGkpFX3CAzA=">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</latexit><latexit sha1_base64="BkV5VfYwV49vM+VoiGkpFX3CAzA=">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</latexit><latexit sha1_base64="BkV5VfYwV49vM+VoiGkpFX3CAzA=">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</latexit><latexit sha1_base64="SsjxnlL39rEFzk7aO/X59vOm7M4=">AAAC1HicjVHLTsJAFD3UF+ID1KWbRmLiirRsdElkgztM5JEAkrYM2FA6zXRqQtCVcesPuNVvMv6B/oV3xpKoxOg0bc+ce86dufe6UeDH0rJeM8bS8srqWnY9t7G5tZ0v7Ow2Y54IjzU8HnDRdp2YBX7IGtKXAWtHgjkTN2Atd1xV8dY1E7HPwws5jVhv4oxCf+h7jiSqX8jX+mdmNxI8ktysXpb7haJVsvQyF4GdgiLSVeeFF3QxAIeHBBMwhJCEAziI6enAhoWIuB5mxAlCvo4z3CJH3oRUjBQOsWP6jmjXSdmQ9ipnrN0enRLQK8hp4pA8nHSCsDrN1PFEZ1bsb7lnOqe625T+bpprQqzEFbF/+ebK//pULRJDnOgafKop0oyqzkuzJLor6ubml6okZYiIU3hAcUHY0855n03tiXXtqreOjr9ppWLV3ku1Cd7VLWnA9s9xLoJmuWRbJfu8XKycpqPOYh8HOKJ5HqOCGupo6Jk/4gnPRtO4Me6M+0+pkUk9e/i2jIcP/IKUww==</latexit>
In such case the Hamiltonian is bounded from below and at the ground state
the constrain is satisfied.
In physical implementations the most considered form of the base
Hamiltonian corresponds to the Ising problem:
HI =
X
<i,j>
bij
z
i
z
i +
X
i
hi
z
i
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bij<latexit sha1_base64="ctCFCUoqMf209U+Dx8tKnPxoq6c=">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</latexit><latexit sha1_base64="ctCFCUoqMf209U+Dx8tKnPxoq6c=">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</latexit><latexit sha1_base64="ctCFCUoqMf209U+Dx8tKnPxoq6c=">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</latexit><latexit sha1_base64="CjV0i77ZaWbxevD7KwO4c2P8MhM=">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</latexit>
hi<latexit sha1_base64="cbaYPS+CO5ffE6TuiEM5YvRsCFY=">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</latexit><latexit sha1_base64="cbaYPS+CO5ffE6TuiEM5YvRsCFY=">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</latexit><latexit sha1_base64="cbaYPS+CO5ffE6TuiEM5YvRsCFY=">AAAC1XicjVHLSsNAFD2Nr1pfUZdugkVwVRIRdFl047KCfYAtJUmn7dA0CZNJoZTsxK0/4FZ/SfwD/QvvjCmoRXRCkjPn3nNm7r1eHPBE2vZrwVhaXlldK66XNja3tnfM3b1GEqXCZ3U/CiLR8tyEBTxkdcllwFqxYO7YC1jTG12qeHPCRMKj8EZOY9YZu4OQ97nvSqK6pjlra5OZYL1s2OVZ1yzbFVsvaxE4OSgjX7XIfEEbPUTwkWIMhhCScAAXCT23cGAjJq6DGXGCENdxhgwl0qaUxSjDJXZE3wHtbnM2pL3yTLTap1MCegUpLRyRJqI8QVidZul4qp0V+5v3THuqu03p7+VeY2IlhsT+pZtn/lenapHo41zXwKmmWDOqOj93SXVX1M2tL1VJcoiJU7hHcUHY18p5ny2tSXTtqreujr/pTMWqvZ/npnhXt6QBOz/HuQgaJxXHrjjXp+XqRT7qIg5wiGOa5xmquEINdfKe4BFPeDaaRmbcGfefqUYh1+zj2zIePgCIA5ai</latexit><latexit sha1_base64="K2G0tg/399isa0+DVNwhLngTII8=">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</latexit>
and are couplers. NP-hard problem in general.
=
Quantum processor (16x16 blocks)
Cryogenic system Adiabatic quantum computer
A single block (8 qubits) Chimera graph
Spin networks
=
Spin networks on adiabatic quantum computer, JM, arXiv:1801.06017
Chimera graph
Elementary 8-qubit block of the quantum chip.
D-Wave 2000 - a matrix 16x16=256 of 8-qubit blocks (2048 qubits in total).
Regular lattice spin network
Pentagram spin network
C =
NX
i=1
si c ⇡ 0
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H =
C2
N c2
2
=
NX
<i,j>
sisj + h
NX
i=1
si
<latexit sha1_base64="TV3pXvS24gxh9GY0vf4pvwIBhLY=">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</latexit><latexit sha1_base64="TV3pXvS24gxh9GY0vf4pvwIBhLY=">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</latexit><latexit sha1_base64="TV3pXvS24gxh9GY0vf4pvwIBhLY=">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</latexit><latexit sha1_base64="F/yT1LmxO3KtI88Gy3wal7+VdCQ=">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</latexit>
h := c<latexit sha1_base64="NavKG9++/KXU5GVAKFJEBI3vlW4=">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</latexit><latexit sha1_base64="NavKG9++/KXU5GVAKFJEBI3vlW4=">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</latexit><latexit sha1_base64="NavKG9++/KXU5GVAKFJEBI3vlW4=">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</latexit><latexit sha1_base64="IjyTuZOKGUT1S29NF/tP8XoT8Dg=">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</latexit>
Physical implementation of the spin network:
Prototype constraint:
where
N = 5<latexit sha1_base64="0tV2ku6vAGpk3/dHwGVAiWBHXeI=">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</latexit><latexit sha1_base64="0tV2ku6vAGpk3/dHwGVAiWBHXeI=">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</latexit><latexit sha1_base64="0tV2ku6vAGpk3/dHwGVAiWBHXeI=">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</latexit><latexit sha1_base64="8ApYU1hHFCB5q2/oBjnz5KvIEBM=">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</latexit>
si 2 { 1, 1}<latexit sha1_base64="YweI2I3TY7M8tzyxwrFBAEJQr7A=">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</latexit><latexit sha1_base64="YweI2I3TY7M8tzyxwrFBAEJQr7A=">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</latexit><latexit sha1_base64="YweI2I3TY7M8tzyxwrFBAEJQr7A=">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</latexit><latexit sha1_base64="NEXXZY7tNQKomp6nGeXEZiqTOnE=">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</latexit>
Energy landscape
dim kin = 25
= 32<latexit sha1_base64="8ttnudbIJ8BaufolYI9IMaTL6KY=">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</latexit><latexit sha1_base64="8ttnudbIJ8BaufolYI9IMaTL6KY=">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</latexit><latexit 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sha1_base64="NIoLN9ei+nBlj8A64UWkFUqXka8=">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</latexit><latexit 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sha1_base64="HEEpUH5cuAcdQu8J2VPJN/snL5A=">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</latexit><latexit 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sha1_base64="8ttnudbIJ8BaufolYI9IMaTL6KY=">AAAC6XicjVHLLgRBFD3a+z1Y2lQMidWkexA2kwgLliQGiWFS3VOoTL9SXS2RyfyAnZ3Y+gFbfkT8AX/hVmmJR4TqdPepc+85VfdePw1lpl33ucfp7esfGBwaHhkdG5+YLE1N72dJrgJRD5IwUYc+z0QoY1HXUofiMFWCR34oDvz2pokfXAiVySTe05epOI74WSxPZcA1Uc3SfKehItaSUbfBGls8injTMm0Zd1mNVU9WakvVZqnsVly72E/gFaCMYu0kpSc00EKCADkiCMTQhENwZPQcwYOLlLhjdIhThKSNC3QxQtqcsgRlcGLb9D2j3VHBxrQ3nplVB3RKSK8iJcMCaRLKU4TNaczGc+ts2N+8O9bT3O2S/n7hFRGrcU7sX7qPzP/qTC0ap1izNUiqKbWMqS4oXHLbFXNz9qkqTQ4pcQa3KK4IB1b50WdmNZmt3fSW2/iLzTSs2QdFbo5Xc0sasPd9nD/BfrXiuRVvd7m8vlGMegizmMMizXMV69jGDurkfYV7PODRaTvXzo1z+57q9BSaGXxZzt0bxYyctA==</latexit><latexit 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sha1_base64="8ttnudbIJ8BaufolYI9IMaTL6KY=">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</latexit><latexit sha1_base64="WvYk0OvPrUqNx/eb1y40sD1Cwvw=">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</latexit>
Dimension of the kinematical space:
Dimension of the physical space: dim phys =
✓
5
5 1
2
◆
= 10
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phys ⇢ kin<latexit sha1_base64="fwoh0Eso/wUkQLhjWUD4Nrh3qSA=">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</latexit><latexit sha1_base64="fwoh0Eso/wUkQLhjWUD4Nrh3qSA=">AAAC7nicjVHLSsNAFD2N73fVpZtgEXRTEhF0KbrQZQVbBStlEkcdmhczE6GUfoM7d+LWH3CrnyH+gf6Fd8YUqkV0QpIz595zZu69QRYJpT3vreSMjI6NT0xOTc/Mzs0vlBeXGirNZcjrYRql8jRgikci4XUtdMRPM8lZHET8JGjvm/jJDZdKpMmx7mT8PGZXibgUIdNEtcobzQMWx6zVbcrYza47qtdUeaC4dgcDbZH0WuWKV/XscoeBX4AKilVLy69o4gIpQuSIwZFAE47AoOg5gw8PGXHn6BInCQkb5+hhmrQ5ZXHKYMS26XtFu7OCTWhvPJVVh3RKRK8kpYs10qSUJwmb01wbz62zYX/z7lpPc7cO/YPCKyZW45rYv3T9zP/qTC0al9ixNQiqKbOMqS4sXHLbFXNzd6AqTQ4ZcQZfUFwSDq2y32fXapSt3fSW2fi7zTSs2YdFbo4Pc0sasP9znMOgsVn1vap/tFXZ3StGPYkVrGKd5rmNXRyihjp53+IJz3hxMufOuXcevlKdUqFZxrflPH4CAq6gwA==</latexit><latexit sha1_base64="fwoh0Eso/wUkQLhjWUD4Nrh3qSA=">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</latexit><latexit sha1_base64="oZlG1IOa13w7SKbTcqL9NtvrdZA=">AAAC7nicjVHLSsNAFD2N7/qqunQTLIJuSuJGl0UXulSwVbBSJnHUoXkxMxFK6Te4cydu/QG3+hniH+hfeGdMoVpEJyQ5c+49Z+beG2SRUNrz3krO2PjE5NT0THl2bn5hsbK03FRpLkPeCNMolacBUzwSCW9ooSN+mknO4iDiJ0Fnz8RPbrhUIk2OdTfj5zG7SsSlCJkmql3ZbO2zOGbtXkvGbnbdVf2WygPFtTsc6Iik365UvZpnlzsK/AJUUazDtPKKFi6QIkSOGBwJNOEIDIqeM/jwkBF3jh5xkpCwcY4+yqTNKYtTBiO2Q98r2p0VbEJ746msOqRTInolKV2skyalPEnYnObaeG6dDfubd896mrt16R8UXjGxGtfE/qUbZP5XZ2rRuMSOrUFQTZllTHVh4ZLbrpibu0NVaXLIiDP4guKScGiVgz67VqNs7aa3zMbfbaZhzT4scnN8mFvSgP2f4xwFza2a79X8o61qfbcY9TRWsYYNmuc26jjAIRrkfYsnPOPFyZw75955+Ep1SoVmBd+W8/gJAg6gvg==</latexit>
The ground states of are the physical states of the theory withH<latexit sha1_base64="Lr9C6B0PGzIdBQ4tgj+aFHqNeCI=">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</latexit><latexit sha1_base64="Lr9C6B0PGzIdBQ4tgj+aFHqNeCI=">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</latexit><latexit sha1_base64="Lr9C6B0PGzIdBQ4tgj+aFHqNeCI=">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</latexit><latexit sha1_base64="CzwSaKETwgL6oIa7fGtJsouXMhY=">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</latexit>
C ⇡ 0<latexit sha1_base64="4eimVysgZAR0RugxiUZPTXu89So=">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</latexit><latexit sha1_base64="4eimVysgZAR0RugxiUZPTXu89So=">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</latexit><latexit sha1_base64="4eimVysgZAR0RugxiUZPTXu89So=">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</latexit><latexit sha1_base64="+rEngd3d1/4lfJ+n6fQRVhRVBdo=">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</latexit>
Outlook
128-qubit
New architectures to be explored.
Simulations of mesoscopic Planck scale systems
on adiabatic quantum computers.
Evaluation of the vertex amplitudes
utilizing quantum algorithms.
Extraction of physical states adopting
Quantum Phase Estimation Algorithm
to the projection operator.
Simulating gravity using gauge/gravity duality.
Application to spin foam cosmology.
Analysis of quantum states of BH horizons.
A lot can be done using available
simulators of quantum computers.
More general spin networks.

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Quantum Gravity on a Quantum Chip

  • 1. Quantum Gravity on a Quantum Chip Jakub Mielczarek Jagiellonian University, Cracow Centre de Physique Théorique, Marseille 5th POTOR Conference, Wojanów, 26.09.2018
  • 3. Superconducting qubits The recent progress has been made mainly thanks to the technology of manufacturing superconducting qubits. Semba, 2008
  • 4. Quantum computers are becoming accessible You can program quantum computer this evening… Visit www.quantumexperience.ng.bluemix.net and get access to the IMB Q 5-qubit quantum computer.
  • 5. 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  • 6. Why it can better than using classical supercomputers? Dimension of the Hilbert space for N qubits grows exponentially with N: dim(H ⌦ H ⌦ · · · ⌦ H | {z } N ) = 2N <latexit sha1_base64="9RzfZbuunItQuP54Lr1ii1inNFE=">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</latexit><latexit 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sha1_base64="OkH1sCw5ED9/6tjbYldxQvpD+MU=">AAAC4nicjVHLSsNAFD2N7/qquhQhWATdlKQb3Qiimy4r2FawUibTUQfzIpmIpXTlzp249Qfc6seIf6B/4Z0xBbWITkhy5tx7zsy914t9mSrHeS1YY+MTk1PTM8XZufmFxdLScjONsoSLBo/8KDn2WCp8GYqGksoXx3EiWOD5ouVdHuh460okqYzCI9WLxWnAzkN5JjlTRHVKa20lrlW/K4OBvdkOmLrgzO/XBlv2rl0tdkplp+KYZY8CNwdl5KselV7QRhcRODIEEAihCPtgSOk5gQsHMXGn6BOXEJImLjBAkbQZZQnKYMRe0vecdic5G9Jee6ZGzekUn96ElDY2SBNRXkJYn2abeGacNfubd9946rv16O/lXgGxChfE/qUbZv5Xp2tROMOOqUFSTbFhdHU8d8lMV/TN7S9VKXKIidO4S/GEMDfKYZ9to0lN7bq3zMTfTKZm9Z7nuRne9S1pwO7PcY6CZrXiOhX3sFre289HPY1VrGOT5rmNPdRQR4O8b/CIJzxbXevWurPuP1OtQq5ZwbdlPXwARZqaPw==</latexit> For a sigle qubit Quantum parallelism > reduction of computational complexity of some classical problems (Deutsch, Grover, Shor, … ) > quantum speed-up With the present most powerful classical supercomputers you can simulate quantum systems with N=56 at most. IBM and Rigetti Computing are developing quantum chips with N>100 and certain topologies of couplings between the qubits > quantum supremacy
  • 7. Quantum computers will allow to perform exact simulations of quantum systems with large N In particular, we may think about simulating Planck scale physics… But, can we express QG DOF with qubits?
  • 8. Loop Quantum Gravity Holonomies of Ashtekar connection along some curveA 2 su(2)<latexit sha1_base64="PH7MRzi1UbHp7jilv25eElcakx0=">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</latexit><latexit 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sha1_base64="mJ8GqMi7I82JglfZ7jvtceK19vA=">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</latexit><latexit sha1_base64="mJ8GqMi7I82JglfZ7jvtceK19vA=">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</latexit><latexit sha1_base64="mJ8GqMi7I82JglfZ7jvtceK19vA=">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</latexit><latexit sha1_base64="EjaGcGlklO+6et+cqRRe9RZHRPE=">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</latexit> h[A, e] = Pe R e A <latexit sha1_base64="j+1GIrWtjJIvhXSuc2pONuSaHsI=">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</latexit><latexit sha1_base64="j+1GIrWtjJIvhXSuc2pONuSaHsI=">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</latexit><latexit sha1_base64="j+1GIrWtjJIvhXSuc2pONuSaHsI=">AAAC5nicjVHLSsNAFD3G97vq0s1oEVxISUTQjeBj47KCVaGtZTKOdjAvkolQStfu3Ilbf8Ctfor4B/oX3hkj+EB0QpIz595zZu69fhKoTLvuc5/TPzA4NDwyOjY+MTk1XZqZPcziPBWyJuIgTo99nslARbKmlQ7kcZJKHvqBPPIvdk386FKmmYqjA91JZDPk55E6U4JrolqlhXZ9e0U22SZrhFy3BQ+61R6TJ92GinRLsu1eq1R2K65d7CfwClBGsapx6QkNnCKGQI4QEhE04QAcGT11eHCRENdEl7iUkLJxiR7GSJtTlqQMTuwFfc9pVy/YiPbGM7NqQacE9KakZFgiTUx5KWFzGrPx3Dob9jfvrvU0d+vQ3y+8QmI12sT+pfvI/K/O1KJxhg1bg6KaEsuY6kThktuumJuzT1VpckiIM/iU4ilhYZUffWZWk9naTW+5jb/YTMOavShyc7yaW9KAve/j/AkOVyueW/H218pbO8WoRzCPRSzTPNexhT1UUSPvK9zjAY9O27l2bpzb91Snr9DM4cty7t4A0Z+cCg==</latexit><latexit sha1_base64="Dv/026TQiqhmCCh6y9pd/+6l3Mg=">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</latexit> Under the gauge transformations A ! Ag = g 1 dg + g 1 Ag<latexit sha1_base64="oidyHZBtlLWMnJmtOUtyGk1aJnc=">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</latexit><latexit sha1_base64="oidyHZBtlLWMnJmtOUtyGk1aJnc=">AAAC6XicjVHLSsNAFD3G97vq0k2wCoJYEhF0I7S6calgtVBrSdJpOjRNwmSilNIfcOdO3PoDbvVHxD/Qv/DOGEEtohOSnDn3njNz73XjgCfSsl6GjOGR0bHxicmp6ZnZufncwuJpEqXCY2UvCiJRcZ2EBTxkZcllwCqxYE7HDdiZ2z5Q8bNLJhIehSeyG7Nax/FD3uSeI4mq51ZL54L7LekIEV2Zpbpv7pn+RW/T7jf8jQ9Q8uu5vFWw9DIHgZ2BPLJ1FOWecY4GInhI0QFDCEk4gIOEnipsWIiJq6FHnCDEdZyhjynSppTFKMMhtk1fn3bVjA1przwTrfbolIBeQUoTa6SJKE8QVqeZOp5qZ8X+5t3TnupuXfq7mVeHWIkWsX/pPjP/q1O1SDSxq2vgVFOsGVWdl7mkuivq5uaXqiQ5xMQp3KC4IOxp5WefTa1JdO2qt46Ov+pMxaq9l+WmeFO3pAHbP8c5CE63CrZVsI+388X9bNQTWMYK1mmeOyjiEEcok/c1HvCIJ6Nt3Bi3xt1HqjGUaZbwbRn371GmnPI=</latexit><latexit sha1_base64="oidyHZBtlLWMnJmtOUtyGk1aJnc=">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</latexit><latexit sha1_base64="llI3DvttFW49isVKJLgurfWrurg=">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</latexit> h[A, e] ! h[Ag, e] = g(e(0))h[A, e]g(e(1)) 1 <latexit sha1_base64="ysTXmwGG5FhlFh+5qGMJVy2kDV8=">AAAC+3icjVHLSsNAFD2Nr/quunQTLEILWhIRdSP42LhUsFWotSRxTIemSZhMlFL6J+7ciVt/wK3uxT/Qv/DOGMEHohOS3HvuOWfmznXjgCfSsp5zxsDg0PBIfnRsfGJyarowM1tLolR4rOpFQSSOXSdhAQ9ZVXIZsONYMKfjBuzIbe+q+tEFEwmPwkPZjVmj4/ghP+eeIwlqFtZa9e0l1jBPBPdb0hEiujQJavoK3PRLrGSVy+8cldjl8mlv2e43C0WrYull/gzsLCgiW/tR4QknOEMEDyk6YAghKQ7gIKGnDhsWYsIa6BEmKOK6ztDHGGlTYjFiOIS26etTVs/QkHLlmWi1R7sE9ApSmlgkTUQ8QbHazdT1VDsr9DfvnvZUZ+vS3828OoRKtAj9S/fB/K9O9SJxjg3dA6eeYo2o7rzMJdW3ok5ufupKkkNMmIrPqC4o9rTy455NrUl07+puHV1/0UyFqtzLuCle1SlpwPb3cf4MaisV26rYB6vFrZ1s1HnMYwElmuc6trCHfVTJ+wr3eMCj0TeujRvj9p1q5DLNHL4s4+4Nhl6iFw==</latexit><latexit sha1_base64="ysTXmwGG5FhlFh+5qGMJVy2kDV8=">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</latexit><latexit 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sha1_base64="i8Tphzq0Oz2ACKHZhfakb6RBA7g=">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</latexit> Gauge invariant objects > Wilson loops W[A, e] = tr (h[A, e])<latexit 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sha1_base64="1cVcIdpfEDeMX72Ejq/EqHbKv6s=">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</latexit> The key idea behind LQG is to built a Hilbert space of the theory out of the Wilson loops. However, such basis is in general over-complete and the solution comes with the constructions of spin networks which are certain linear combination of products of the Wilson loops. = 0<latexit sha1_base64="VgQTEJ9SfjSQpS8GWCz2oPqUi30=">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</latexit><latexit 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sha1_base64="xv7r5dPoOe4tm4kmzl/F3r9xO0o=">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</latexit>
  • 9. Spin Networks „Quantum of space” | i = | , jl, ini = ⌦n|ini<latexit sha1_base64="y5l4YLNEI0Bgip8zB2xsNtpv/hQ=">AAADCHicjVHLTtwwFD0TSkuhLVO6ZGN1hNQFGiWoUtlUGpUFXU4lZkCaoMgJZnBxHrKdSojpD/RPumOH2PYHugT+oP2LXrsZ8RICR0mOz73n2PfetFLS2DC8bAUzT2afPpt7Pr/w4uWrxfbrpaEpa52JQVaqUu+k3AglCzGw0iqxU2nB81SJ7fRww8W3vwltZFls2aNK7OZ8XMh9mXFLVNLemMR9I2PNi7ES7CObxJs8z/kq+5qoVSaT4ioUl1bmwiQFmxDPpgHGknYn7IZ+sbsgakAHzeqX7XPE2EOJDDVyCBSwhBU4DD0jRAhREbeLY+I0IenjAt8xT9qasgRlcGIP6Tum3ahhC9o7T+PVGZ2i6NWkZFghTUl5mrA7jfl47Z0de5/3sfd0dzuif9p45cRaHBD7kG6a+Vidq8ViH+u+Bkk1VZ5x1WWNS+274m7OrlVlyaEizuE9imvCmVdO+8y8xvjaXW+5j//xmY51+6zJrfHX3ZIGHN0e510wXOtGYTf68r7T+9SMeg7LeIt3NM8P6OEz+hiQ90/8xgUugx/BSXAanP1PDVqN5g1urODXPw/ZqRE=</latexit><latexit 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sha1_base64="u+zXQwJ0eLXINj/wuBo+s/GlSiU=">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</latexit> ( ˆE1 + ˆE2 + ˆE3 + ˆE4 )|ini = 0<latexit 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sha1_base64="LOhJaxFgrzzqOR3RrmnohFNqyZg=">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</latexit><latexit sha1_base64="LOhJaxFgrzzqOR3RrmnohFNqyZg=">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</latexit><latexit sha1_base64="46fIFt86bhMZ/IoL79Q2tAQ6mxE=">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</latexit> Gauss constraint:
  • 10. Qubit H1/2 ⌦ H1/2 ⌦ H1/2 ⌦ H1/2 = H0 H0 | {z } 3H1 H2 <latexit sha1_base64="23v61J2CbBM5sYx8aCUq5t10FqY=">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</latexit><latexit 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sha1_base64="s7+TyWLiTs6dlnrEKzF1zY12cd0=">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</latexit> dim Inv(H1/2 ⌦ H1/2 ⌦ H1/2 ⌦ H1/2) = 2<latexit 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  • 11. Qubit base states |1si = 1 p 3 ✓ | ""##i + | ##""i | "#"#i + | #"#"i + | "##"i + | #""#i 2 ◆ <latexit sha1_base64="n25PQt4yY8V5UOGbewpMEfl03zQ=">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</latexit><latexit 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sha1_base64="Y1pzJJwf8veqRWsDV00gEIrwn4c=">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</latexit><latexit 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sha1_base64="kaY6Wc9/VKDhKe6TQDSdxmOYP1E=">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</latexit><latexit sha1_base64="kaY6Wc9/VKDhKe6TQDSdxmOYP1E=">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</latexit><latexit sha1_base64="kaY6Wc9/VKDhKe6TQDSdxmOYP1E=">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</latexit><latexit sha1_base64="hqAV5kSeUEJp7zTK3EuGeiWL0G0=">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</latexit> ˆV<latexit sha1_base64="ayP5B8FArjllzopLb5MNLKuXP10=">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</latexit><latexit sha1_base64="ayP5B8FArjllzopLb5MNLKuXP10=">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</latexit><latexit sha1_base64="ayP5B8FArjllzopLb5MNLKuXP10=">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</latexit><latexit sha1_base64="9GFBf+IjRJYf9w9jHdWAg8RrL0o=">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</latexit> where
  • 12. Transition amplitude W(x, x0 ) = hx0 | ˆP|xi<latexit sha1_base64="+r5gEOQat59u8jQIe295YEFhOV0=">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</latexit><latexit sha1_base64="+r5gEOQat59u8jQIe295YEFhOV0=">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</latexit><latexit sha1_base64="+r5gEOQat59u8jQIe295YEFhOV0=">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</latexit><latexit sha1_base64="zan6bvJeZ3vjSbEOaluY72PPgqE=">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</latexit> ˆP = ( ˆC) = lim T !1 1 2T Z T T d⌧ei⌧ ˆC <latexit sha1_base64="ApHh7qiOH85abXsGJu/SvPz8aMY=">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</latexit><latexit sha1_base64="ApHh7qiOH85abXsGJu/SvPz8aMY=">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</latexit><latexit 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  • 13. Rovelli &Vidotto, 2015 Keren Li, et al., arXiv: 1712.08711 Spin Foams
  • 14. Vertex amplitude i1<latexit sha1_base64="lfiqI4AHE7DzmKNQsiY5BJteS5U=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl002VF+4BaSjKd1qF5MZkopQj+gFv9NPEP9C+8M6agFtEJSc6ce8+Zuff6SSBS5TivBWthcWl5pbhaWlvf2Nwqb++00jiTjDdZHMSy43spD0TEm0qogHcSyb3QD3jbH5/rePuWy1TE0ZWaJLwXeqNIDAXzFFGXou/2yxWn6phlzwM3BxXkqxGXX3CNAWIwZAjBEUERDuAhpacLFw4S4nqYEicJCRPnuEeJtBllccrwiB3Td0S7bs5GtNeeqVEzOiWgV5LSxgFpYsqThPVptolnxlmzv3lPjae+24T+fu4VEqtwQ+xfulnmf3W6FoUhTk0NgmpKDKOrY7lLZrqib25/qUqRQ0KcxgOKS8LMKGd9to0mNbXr3nom/mYyNav3LM/N8K5vSQN2f45zHrSOqq5TdS+OK7WzfNRF7GEfhzTPE9RQRwNN8h7hEU94tupWZGXW3WeqVcg1u/i2rIcP6j6QEg==</latexit><latexit 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sha1_base64="vZ5HC8JUNDvWuZUHGukpjJY7cv8=">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</latexit><latexit sha1_base64="jmAV6JWoX8OnpmEbgpowrG2/tLw=">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</latexit> Vertex amplitude: see: Keren Li, et al., arXiv: 1712.08711 But, we have to identify the states which satisfy the constraint first…C ⇡ 0<latexit 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sha1_base64="GDGseDEDeS1zq5a1e9TpY/vI28k=">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</latexit><latexit sha1_base64="TcPeeSrZCQ6nKnO5i5NkbhPo9SQ=">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</latexit> Can be computed with the use of quantum computer (in progress). (Bell state)
  • 15. |h 2| 1i|2 <latexit sha1_base64="fsUpRP4I2TfLPP+cB/9SsliJGkc=">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</latexit><latexit sha1_base64="fsUpRP4I2TfLPP+cB/9SsliJGkc=">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</latexit><latexit sha1_base64="fsUpRP4I2TfLPP+cB/9SsliJGkc=">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</latexit><latexit sha1_base64="s1ZiAGqreqmAmvMd9nzttCp9/U4=">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</latexit> How to compute on quantum computer? For N nodes of the spin network we need 4N qubits (20 qubits for the vertex amplitude). The Hilbert space is spanned by base states24N <latexit sha1_base64="m/y0oGtD83V6vQEG/8kgyAxTCs4=">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</latexit><latexit 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sha1_base64="G4CTk9rGwMXEDEi+e01i/krN0D0=">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</latexit> | 1i = ˆU1|0i<latexit 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sha1_base64="3vfhocPNmtdb+Ayss/1ZB1W3bf8=">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</latexit><latexit sha1_base64="OxC6pMdJs7I6Bx3ZGqLg4Yyklc0=">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</latexit> and Our quantum algorithm is defined by the unitary operator (quantum circuit): ˆU = ˆU† 2 U1<latexit sha1_base64="MXGxx9+0iVqySe/jKEy94JkjHnw=">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</latexit><latexit sha1_base64="MXGxx9+0iVqySe/jKEy94JkjHnw=">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</latexit><latexit sha1_base64="MXGxx9+0iVqySe/jKEy94JkjHnw=">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</latexit><latexit sha1_base64="KVgMrRKEjGtSbzlHyAl+vkH/lq4=">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</latexit> such that … … ˆU = ˆU† 2 U1<latexit 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sha1_base64="KVgMrRKEjGtSbzlHyAl+vkH/lq4=">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</latexit> |0i<latexit 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sha1_base64="FL1ldUm5msesvnxxrFeFnlt2AH4=">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</latexit><latexit sha1_base64="FL1ldUm5msesvnxxrFeFnlt2AH4=">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</latexit><latexit sha1_base64="F+4HQ6Dr1GAG3LRyn2dPdkSDqak=">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</latexit> We measure: P(0) = |a0|2 <latexit sha1_base64="PF/3lZqyHB2yiXv5eDbMthix7tE=">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</latexit><latexit sha1_base64="PF/3lZqyHB2yiXv5eDbMthix7tE=">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</latexit><latexit sha1_base64="PF/3lZqyHB2yiXv5eDbMthix7tE=">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</latexit><latexit sha1_base64="qKDfr/FM2kawzXq3OIT1frcX0EE=">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</latexit> a0 = h0| ˆU|0i = h 2| 1i<latexit sha1_base64="HUFSo9M45onouFj/eu+ZomUBpTc=">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</latexit><latexit sha1_base64="HUFSo9M45onouFj/eu+ZomUBpTc=">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</latexit><latexit sha1_base64="HUFSo9M45onouFj/eu+ZomUBpTc=">AAADFnicjVFPa9swHH111zVL283bjr2IhUFPwS6F7lII3WXHFJa2UJcgO0oiothGlgslyffYN9ltt7HrbmWHXffnW/QnTYFuoXQytp7e771n/aS0VLIyUfRtLVh/tPF4s/GkubW98/RZ+PzFaVXUOhO9rFCFPk95JZTMRc9Io8R5qQWfpkqcpZO3tn52JXQli/y9uS7F5ZSPcjmUGTdE9cMT3o/YEUsUz0dKsFmSDlm0mLNkzM2sR8AzLNFO0bwjTrqV7O/P3RQvBawftqJ25AZbBbEHLfjRLcIbJBigQIYaUwjkMIQVOCp6LhAjQkncJWbEaULS1QUWaJK3JpUgBSd2Qt8RrS48m9PaZlbOndFfFL2anAyvyVOQThO2f2OuXrtky96XPXOZdm/XNKc+a0qswZjYh3xL5f/6bC8GQ7xxPUjqqXSM7S7zKbU7FbtzdqcrQwklcRYPqK4JZ865PGfmPJXr3Z4td/VfTmlZu868tsZvu0u64Pjf61wFp/vtOGrHJwetzrG/6gZ28Qp7dJ+H6OAduuhR9kd8xw/8DD4En4LPwZc/0mDNe17irxF8vQVgla3V</latexit><latexit sha1_base64="6CA1xZNdkgjb370u6Q8IYxtCBtQ=">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</latexit> where |0i = | "i ⌦ · · · ⌦ | "i | {z } 4N<latexit sha1_base64="MFNfR52ElxihH0KkGie5ef3lAhA=">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</latexit><latexit sha1_base64="MFNfR52ElxihH0KkGie5ef3lAhA=">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</latexit><latexit sha1_base64="MFNfR52ElxihH0KkGie5ef3lAhA=">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</latexit><latexit sha1_base64="p65XN8RTGmmghVxsqtttBKAWAcE=">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</latexit> ˆU|0i = 24N 1X i=0 ai|ii <latexit 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  • 16. Solving the constraint with the Adiabatic Quantum Computer The adiabatic quantum computers are designed to solve a specific problem of finding minimum of a Hamiltonian of a coupled system of qubits (spins). In the process of finding the minimum of one employs a time dependent Hamiltonian in the form: where is the so-called base Hamiltonian characterized by a simple and easy to prepare ground state. In practice, the base Hamiltonian is often equal such that the ground state corresponds to the alignment of spins in the x direction.Then, the value of is changed adiabatically from to , such that while the system is initially in non-degenerate ground state it will remind in the ground state during the process. 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sha1_base64="V5Cbyg6cStIzk2HNOaoIu6j5oLc=">AAACznicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdKFQcOOygn1ALZJMpzV08mAyKZRS3PoDbvWzxD/Qv/DOmIJaRCckOXPuOXfm3usnIkiV47wWrIXFpeWV4mppbX1jc6u8vdNM40wy3mCxiGXb91Iugog3VKAEbyeSe6EveMsfXuh4a8RlGsTRtRonvBt6gyjoB8xTRHVuBEl7nn1uu7flilN1zLLngZuDCvJVj8svuEEPMRgyhOCIoAgLeEjp6cCFg4S4LibESUKBiXNMUSJvRipOCo/YIX0HtOvkbER7nTM1bkanCHolOW0ckCcmnSSsT7NNPDOZNftb7onJqe82pr+f5wqJVbgj9i/fTPlfn65FoY9TU0NANSWG0dWxPEtmuqJvbn+pSlGGhDiNexSXhJlxzvpsG09qate99Uz8zSg1q/cs12Z417ekAbs/xzkPmkdV16m6V8eV2lk+6iL2sI9DmucJarhEHQ3T8Uc84dmqWyNrat1/Sq1C7tnFt2U9fAD45ZLO</latexit><latexit sha1_base64="xv7r5dPoOe4tm4kmzl/F3r9xO0o=">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</latexit> time
  • 17. Let us observe that, the problem finding solution to the constraint: C ⇡ 0<latexit sha1_base64="4eimVysgZAR0RugxiUZPTXu89So=">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</latexit><latexit sha1_base64="4eimVysgZAR0RugxiUZPTXu89So=">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</latexit><latexit sha1_base64="4eimVysgZAR0RugxiUZPTXu89So=">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</latexit><latexit sha1_base64="+rEngd3d1/4lfJ+n6fQRVhRVBdo=">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</latexit> can be mapped into the problem of finding minimum of the Hamiltonian: HI / C2 <latexit sha1_base64="BkV5VfYwV49vM+VoiGkpFX3CAzA=">AAAC1HicjVHLSsNAFD3GV62PVl26CRbBVUmKoMtiN3VXwT6grSVJpzU0zYTJRCjVlbj1B9zqN4l/oH/hnTEFtYhOSHLm3HPuzL3XjQI/lpb1umAsLi2vrGbWsusbm1u5/PZOI+aJ8Fjd4wEXLdeJWeCHrC59GbBWJJgzdgPWdEcVFW9eMxH7PLyQk4h1x84w9Ae+50iievlctXdmdiLBI8nNymWply9YRUsvcx7YKSggXTWef0EHfXB4SDAGQwhJOICDmJ42bFiIiOtiSpwg5Os4wy2y5E1IxUjhEDui75B27ZQNaa9yxtrt0SkBvYKcJg7Iw0knCKvTTB1PdGbF/pZ7qnOqu03o76a5xsRKXBH7l2+m/K9P1SIxwImuwaeaIs2o6rw0S6K7om5ufqlKUoaIOIX7FBeEPe2c9dnUnljXrnrr6PibVipW7b1Um+Bd3ZIGbP8c5zxolIq2VbTPjwrl03TUGexhH4c0z2OUUUUNdT3zRzzh2WgYN8adcf8pNRZSzy6+LePhA/0ilMU=</latexit><latexit sha1_base64="BkV5VfYwV49vM+VoiGkpFX3CAzA=">AAAC1HicjVHLSsNAFD3GV62PVl26CRbBVUmKoMtiN3VXwT6grSVJpzU0zYTJRCjVlbj1B9zqN4l/oH/hnTEFtYhOSHLm3HPuzL3XjQI/lpb1umAsLi2vrGbWsusbm1u5/PZOI+aJ8Fjd4wEXLdeJWeCHrC59GbBWJJgzdgPWdEcVFW9eMxH7PLyQk4h1x84w9Ae+50iievlctXdmdiLBI8nNymWply9YRUsvcx7YKSggXTWef0EHfXB4SDAGQwhJOICDmJ42bFiIiOtiSpwg5Os4wy2y5E1IxUjhEDui75B27ZQNaa9yxtrt0SkBvYKcJg7Iw0knCKvTTB1PdGbF/pZ7qnOqu03o76a5xsRKXBH7l2+m/K9P1SIxwImuwaeaIs2o6rw0S6K7om5ufqlKUoaIOIX7FBeEPe2c9dnUnljXrnrr6PibVipW7b1Um+Bd3ZIGbP8c5zxolIq2VbTPjwrl03TUGexhH4c0z2OUUUUNdT3zRzzh2WgYN8adcf8pNRZSzy6+LePhA/0ilMU=</latexit><latexit sha1_base64="BkV5VfYwV49vM+VoiGkpFX3CAzA=">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</latexit><latexit sha1_base64="SsjxnlL39rEFzk7aO/X59vOm7M4=">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</latexit> In such case the Hamiltonian is bounded from below and at the ground state the constrain is satisfied. In physical implementations the most considered form of the base Hamiltonian corresponds to the Ising problem: HI = X <i,j> bij z i z i + X i hi z i <latexit sha1_base64="2lepV7AwxL4sz9iMXUxcpogqNio=">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</latexit><latexit sha1_base64="2lepV7AwxL4sz9iMXUxcpogqNio=">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</latexit><latexit sha1_base64="2lepV7AwxL4sz9iMXUxcpogqNio=">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</latexit><latexit sha1_base64="yvhIb7fEZOeQpnfbVHIGA/9AXP4=">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</latexit> bij<latexit sha1_base64="ctCFCUoqMf209U+Dx8tKnPxoq6c=">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</latexit><latexit sha1_base64="ctCFCUoqMf209U+Dx8tKnPxoq6c=">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</latexit><latexit sha1_base64="ctCFCUoqMf209U+Dx8tKnPxoq6c=">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</latexit><latexit 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sha1_base64="K2G0tg/399isa0+DVNwhLngTII8=">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</latexit> and are couplers. NP-hard problem in general.
  • 18. = Quantum processor (16x16 blocks) Cryogenic system Adiabatic quantum computer A single block (8 qubits) Chimera graph Spin networks = Spin networks on adiabatic quantum computer, JM, arXiv:1801.06017
  • 19. Chimera graph Elementary 8-qubit block of the quantum chip. D-Wave 2000 - a matrix 16x16=256 of 8-qubit blocks (2048 qubits in total).
  • 21. Pentagram spin network C = NX i=1 si c ⇡ 0 <latexit sha1_base64="u20N1jeAMJeJbxF9ufkzXrMaVus=">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</latexit><latexit 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sha1_base64="NavKG9++/KXU5GVAKFJEBI3vlW4=">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</latexit><latexit 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  • 22. Energy landscape dim kin = 25 = 32<latexit sha1_base64="8ttnudbIJ8BaufolYI9IMaTL6KY=">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</latexit><latexit 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  • 23. Outlook 128-qubit New architectures to be explored. Simulations of mesoscopic Planck scale systems on adiabatic quantum computers. Evaluation of the vertex amplitudes utilizing quantum algorithms. Extraction of physical states adopting Quantum Phase Estimation Algorithm to the projection operator. Simulating gravity using gauge/gravity duality. Application to spin foam cosmology. Analysis of quantum states of BH horizons. A lot can be done using available simulators of quantum computers. More general spin networks.