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Quantum Gravity on a Quantum Chip

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Jakub Mielczarek

This talk aims at emphasizing the potential of a synergy between quantum gravity and the quantum computing technologies. Such a combination would be beneficial for both understanding the Planck scale physics and the stimulation of development of the quantum technologies. This is especially important in the present early days of commercial quantum computers, when challenges originating from the basic research may catalyze the technological progress. Our attention will be focused on simulations of Planck scale physics with the use of existing universal and adiabatic quantum computers. Current possibilities, technological challenges and prospects for the future will be outlined.

Science
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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… |01i<latexit sha1_base64="avqpVuSR6xrS70ag56rVsmiIdwc=">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</latexit><latexit 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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 <latexit sha1_base64="9RzfZbuunItQuP54Lr1ii1inNFE=">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</latexit><latexit 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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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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=">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</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=">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</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=">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</latexit><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 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sha1_base64="i8Tphzq0Oz2ACKHZhfakb6RBA7g=">AAAC+3icjVHLSsNAFD3GV62vqks3wSK0oCVxoW4EHxuXFWwr1FqSOKaDaRImE0VK/8SdO3HrD7jVvfgH+hfeGVPwgeiEJPeee86ZuXPdOOCJtKyXIWN4ZHRsPDeRn5yanpktzM3XkygVHqt5URCJI9dJWMBDVpNcBuwoFszpugFruOd7qt64YCLhUXgor2LW6jp+yM+450iC2oX1TnNnhbXMY8H9jnSEiC5Ngtq+Arf8EitZ5fIHRyV2uXzSW7X77ULRqlh6mT8DOwuKyFY1KjzjGKeI4CFFFwwhJMUBHCT0NGHDQkxYCz3CBEVc1xn6yJM2JRYjhkPoOX19ypoZGlKuPBOt9miXgF5BShPLpImIJyhWu5m6nmpnhf7m3dOe6mxX9Hczry6hEh1C/9INmP/VqV4kzrCpe+DUU6wR1Z2XuaT6VtTJzU9dSXKICVPxKdUFxZ5WDu7Z1JpE967u1tH1V81UqMq9jJviTZ2SBmx/H+fPoL5Wsa2KfbBW3N7NRp3DIpZQonluYBv7qKJG3td4wCOejL5xY9wadx9UYyjTLODLMu7fAYW+ohU=</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=">AAACzHicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdKFQcONKKtiHtEWS6bQO5kUyEUrp1h9wq98l/oH+hXfGKahFdEKSM+eec2bujJ8EIpOO81qw5uYXFpeKy6WV1bX1jfLmVjOL85TxBouDOG37XsYDEfGGFDLg7STlXugHvOXfnal6656nmYijKzlKeC/0hpEYCOZJoq67AUn73qlzU644VUcPexa4BlRgRj0uv6CLPmIw5AjBEUESDuAho6cDFw4S4noYE5cSErrOMUGJvDmpOCk8Yu/oO6RZx7ARzVVmpt2MVgnoTclpY488MelSwmo1W9dznazY37LHOlPtbUR/32SFxErcEvuXb6r8r0/1IjHAse5BUE+JZlR3zKTk+lTUzu0vXUlKSIhTuE/1lDDTzuk529qT6d7V2Xq6/qaVilVzZrQ53tUu6YLdn9c5C5oHVdepupeHldqJueoidrCLfbrPI9RwjjoalB3iEU94ti4saY2tyafUKhjPNr4N6+EDDKySeQ==</latexit><latexit 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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 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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="23v61J2CbBM5sYx8aCUq5t10FqY=">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</latexit><latexit sha1_base64="s7+TyWLiTs6dlnrEKzF1zY12cd0=">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</latexit> dim Inv(H1/2 ⌦ H1/2 ⌦ H1/2 ⌦ H1/2) = 2<latexit sha1_base64="qaQ0sQuGe0zcHy67zS4I4wfO3NE=">AAADDHicjVHLSsNAFD3G97vq0s1gEeqmJkXQjSC6sO4qWBXaUpJ01MG8mEwEKf0F/8SdO3HrD7gQQff6F94ZI6hFdEKSc8+958zcuV4SiFTZ9tOANTg0PDI6Nj4xOTU9M1uYmz9M40z6vO7HQSyPPTflgYh4XQkV8ONEcjf0An7kne/o/NEFl6mIowN1mfBW6J5G4kT4riKqXdjtNmXIOiJssr3oosdK1XbXWa30mrESIU/Zv8KVzUq7ULTLtlmsHzg5KCJftbjwiCY6iOEjQwiOCIpwABcpPQ04sJEQ10KXOElImDxHDxOkzaiKU4VL7Dl9Tylq5GxEsfZMjdqnXQJ6JSkZlkkTU50krHdjJp8ZZ83+5t01nvpsl/T3cq+QWIUzYv/SfVb+V6d7UTjBhulBUE+JYXR3fu6SmVvRJ2dfulLkkBCncYfykrBvlJ/3zIwmNb3ru3VN/tVUalbHfl6b4U2fkgbs/BxnPzislB277OyvFbe281GPYRFLKNE817GFKmqok/c1HvCMF+vKurFurbuPUmsg1yzg27Lu3wGveqnz</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:
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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=">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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
  • 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="j+1GIrWtjJIvhXSuc2pONuSaHsI=">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</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 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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 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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. = 0<latexit sha1_base64="VgQTEJ9SfjSQpS8GWCz2oPqUi30=">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</latexit><latexit 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  • 9. Spin Networks „Quantum of space” | i = | , jl, ini = ⌦n|ini<latexit sha1_base64="y5l4YLNEI0Bgip8zB2xsNtpv/hQ=">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</latexit><latexit 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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="23v61J2CbBM5sYx8aCUq5t10FqY=">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</latexit><latexit sha1_base64="s7+TyWLiTs6dlnrEKzF1zY12cd0=">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</latexit> 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: