100 years of many-particle quantum mechanics: from Bose and Fermi to quantum materials and black holes (with 2 spooky dreams)
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Talk online: sachdev.physics.harvard.edu ICTP-SAIFR distinguished public lecture IFT-UNESP Sao Paolo, Brazil October 31, 2025 Subir Sachdev HARVARD 100 years of many-particle quantum mechanics: from Bose and Fermi to quantum materials and black holes (with 2 spooky dreams)
Quantum mechanics to quantum materials: the first 100 years
<latexit sha1_base64="trWnrZ96RKDAXHYth1kMYXo8LJ4=">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</latexit> )1010 meters ( <latexit sha1_base64="4vbYp/uxnYaIe3iHtQFjUzxZ00Y=">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</latexit> The motion of the electron around the proton is not described by the same theory as the motion of the planets around the sun. <latexit sha1_base64="0wWwkSNUbMgwszY7POUUxbiiBBw=">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</latexit> It is described by the quantum theory of Schr¨odinger and Heisenberg (1925). Hydrogen atom
<latexit sha1_base64="AyQDesllIusaFw+1t7VnZnFsfvE=">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</latexit> •Schr¨odinger, Heisenberg (1925): Discovery of the equation obeyed by a single electron, replacing Newton’s laws of motion. These equations precisely described the light emission spectrum of a single hydrogen atom. •Sommerfeld (1927): The same equations also describe the motion of →1023 electrons in a metal. Each electron is a fermion, named after Fermi (1926), which obey exclusion—at most one fermion can occupy each quantum orbital. •Bose, Einstein (1924): Particles now known as bosons, which do not obey exclusion. Many bosons can condense into a single macroscopic quantum state, which is today understood to be the key to superfluidity and superconductivity. •Bardeen, Cooper, Schrie!er (1957): Pairs of electrons behave like bosons, and this is the explanation for superconductivity.
<latexit sha1_base64="iwcLJ7gzGLunMyUJU7xk7XpgVWs=">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</latexit> •Schr¨odinger, Heisenberg (1925): Discovery of the equation obeyed by a single electron, replacing Newton’s laws of motion. These equations precisely described the light emission spectrum of a single hydrogen atom. •Sommerfeld (1927): The same equations also describe the motion of →1023 electrons in a metal. Each electron is a fermion, named after Fermi (1926), which obey exclusion—at most one fermion can occupy each quantum orbital. •Bose, Einstein (1924): Particles now known as bosons,which do not obey exclusion. Many bosons can condense into a single macroscopic quantum state, which is today understood to be the key to superfluidity and superconductivity. •Bardeen, Cooper, Schrie!er (1957): Pairs of electrons behave like bosons, and this is the explanation for superconductivity.
<latexit sha1_base64="iwcLJ7gzGLunMyUJU7xk7XpgVWs=">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</latexit> •Schr¨odinger, Heisenberg (1925): Discovery of the equation obeyed by a single electron, replacing Newton’s laws of motion. These equations precisely described the light emission spectrum of a single hydrogen atom. •Sommerfeld (1927): The same equations also describe the motion of →1023 electrons in a metal. Each electron is a fermion, named after Fermi (1926), which obey exclusion—at most one fermion can occupy each quantum orbital. •Bose, Einstein (1924): Particles now known as bosons,which do not obey exclusion. Many bosons can condense into a single macroscopic quantum state, which is today understood to be the key to superfluidity and superconductivity. •Bardeen, Cooper, Schrie!er (1957): Pairs of electrons behave like bosons, and this is the explanation for superconductivity.
<latexit sha1_base64="iwcLJ7gzGLunMyUJU7xk7XpgVWs=">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</latexit> •Schr¨odinger, Heisenberg (1925): Discovery of the equation obeyed by a single electron, replacing Newton’s laws of motion. These equations precisely described the light emission spectrum of a single hydrogen atom. •Sommerfeld (1927): The same equations also describe the motion of →1023 electrons in a metal. Each electron is a fermion, named after Fermi (1926), which obey exclusion—at most one fermion can occupy each quantum orbital. •Bose, Einstein (1924): Particles now known as bosons,which do not obey exclusion. Many bosons can condense into a single macroscopic quantum state, which is today understood to be the key to superfluidity and superconductivity. •Bardeen, Cooper, Schrie!er (1957): Pairs of electrons behave like bosons, and this is the explanation for superconductivity.
<latexit sha1_base64="H282CCMc9hk/ig/SGf4QAuUsDaQ=">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</latexit> •Today: Many particles exhibit many emergent phenomena, related to quantum entanglement. These are crucial to understanding modern quantum materials, such as the high temperature superconductors. •Ideas on multi-particle entanglement in quantum materials have strongly influenced quantum computing, especially quantum error correction, and the theory of black holes (and vice versa).
<latexit sha1_base64="H282CCMc9hk/ig/SGf4QAuUsDaQ=">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</latexit> •Today: Many particles exhibit many emergent phenomena, related to quantum entanglement. These are crucial to understanding modern quantum materials, such as the high temperature superconductors. •Ideas on multi-particle entanglement in quantum materials have strongly influenced quantum computing, especially quantum error correction, and the theory of black holes (and vice versa).
YBa2Cu3O6+x Cu P.W. Anderson and G. Baskaran (1988): The key to high temperature superconductivity is the formation of a “resonating valence bond state” (a type of quantum spin liquid) which entangles the electrons on Cu
Quantum entanglement
The double slit experiment Interference of electrons Principles of Quantum Mechanics: 1. Quantum Superposition TWO$ SLITS$ Unlike water waves, electrons arrive one-by-one ! (so is it like a particle ?)
And |R!represents the state with the electron in the right slit Let |L!represent the state with the electron in the left slit |L! |R! The double slit experiment Actual state of each electron is |Li+|Ri Principles of Quantum Mechanics 101: Quantum Superposition
.DESC RI PT ION OF PHYSI CAL REALITY of lanthanum is 7/2, hence the nuclear magnetic moment as determined by this analysis is 2.5 nuclear magnetons. This is in fair agreement with the value 2.8nuclear magnetons determined, from La III hyperfine structures by the writer and N. S. Grace. 9 'M. F. Crawford and N. S. Grace, Phys. Rev. 4'7, 536 (1935). This investigation was carried out under the supervision of Professor G. Breit, and, Iwish to thank him for the invaluable advice and assistance so freely given. Ialso take this opportunity to acknowledge the award of aFellowship by the Royal Society of Canada, and to thank the University of Wisconsin and the Department of Physics for the privilege of working here. MAY 15, 1935 PH YSI CAL REVI EW VOLUM E47 Can Quantum-Mechanical Description of Physical Reality Be Considered Complete' ? A. EINsTEIN, B. PQDoLsKY AND N. RosEN, Institute for Advanced Study, Princeton, New Jersey (Received March 25, 1935) In acomplete theory there is an element corresponding to each element of reality. AsufFicient condition for the reality of aphysical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning asystem on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by awave function is not complete. ANY serious consideration of aphysical theory must take into account the distinction between the objective reality, which is independent of any theory, and the physical concepts with which the theory operates. These concepts are intended to correspond with the objective reality, and by means of these concepts we picture this reality to ourselves. In attempting to judge the success of a physical theory, we may ask ourselves two questions: (1) "Is the theory correct?" and (2) "Is the description given by the theory complete?" It is only in the case in which positive answers may be given to both of these questions, that the concepts of the theory may be said to be satisfactory. The correctness of the theory is judged by the degree of agreement between the conclusions of the theory and human experience. This experience, which alone enables us to make inferences about reality, in physics takes the form of experiment and measurement. It is the second question that we wish to consider here, as applied to quantum mechanics. Whatever the meaning assigned to the term conzp/eEe, the following requirement for acomplete theory seems to be anecessary one: every element of the physical reality must have acounter part in the physical theory We shall ca. 11 this the condition of completeness. The second question is thus easily answered, as soon as we are able to decide what are the elements of the physical reality. The elements of the physical reality cannot be determined by apriori philosophical considerations, but must be found by an appeal to results of experiments and measurements. A comprehensive definition of reality is, however, unnecessary for our purpose. We shall be satisfied with the following criterion, which we regard as reasonable. If, without in any way disturbing a system, we can predict with certainty (i.e.,with probability equal to unity) the value of aphysical quantity, then there exists an element of physical reality corresponding lo this physical quantity. It seems to us that this criterion, while far from exhausting all possible ways of recognizing a physical reality, at least provides us with one
Quantum Entanglement _ Einstein, Podolsky, Rosen (1935)
Quantum Entanglement _ Einstein, Podolsky, Rosen (1935)
Quantum Entanglement _ Einstein, Podolsky, Rosen (1935)
_ Quantum Entanglement Einstein, Podolsky, Rosen (1935) Measurement of one electron instantaneously determines the state of the other electron very far away
Quantum Entanglement _ Einstein, Podolsky, Rosen (1935) Measurement of one electron instantaneously determines the state of the other electron very far away
Patrik Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974). <latexit sha1_base64="44EQSLd248yeLy8H80pBg3XRVUI=">AAACl3icdVHbbhMxEPUutxJuofCCeLGIqHiKnF5oi4SoaAV9bCXSVspGkdeZ3Vj1ZWXPAtGyfBIfwxt/gzfZSiXASJaOz5kznhmnhZIeGfsVxTdu3rp9Z+1u5979Bw8fdR+vn3lbOgFDYZV1Fyn3oKSBIUpUcFE44DpVcJ5eHjb6+WdwXlrzCecFjDXPjcyk4BioSffHRqIgw2/0I02czGeYOG5yBfQtTXypJ1WiOc4EV9VRXVOxcm+917iVKknS2fhb5c7ZL8GV2q/VVGpwVNjQpDR53Rjo96tYptiMKo4oBdSTbo/12Q57vcVoAIsIYH9re4/t00HL9EgbJ5Puz2RqRanBoFDc+9GAFTiuuAvlFNSdpPRQcHHJcxgFaLgGP64We63py8BMaWZdOAbpgr3uqLj2fq7TkNnM6Fe1hvyXNiox2xtX0hQlghHLh7JSUbS0+SQ6lQ4EqnkAXDgZeqVixh0XGL6yE5ZwNSn9Pzjb7A92+ux0u3fwvl3HGnlOXpBXZEB2yQE5JidkSET0NHoTHUZH8bP4XfwhPl6mxlHreUL+iPj0N2tCzWs=</latexit> |Gi=X D cD|Di D!dimer covering of lattice Spin liquid: resonating valence bonds <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974). <latexit sha1_base64="44EQSLd248yeLy8H80pBg3XRVUI=">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</latexit> |Gi=X D cD|Di D!dimer covering of lattice Spin liquid: resonating valence bonds <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974). <latexit sha1_base64="44EQSLd248yeLy8H80pBg3XRVUI=">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</latexit> |Gi=X D cD|Di D!dimer covering of lattice Spin liquid: resonating valence bonds <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">AAACknicfVFdT9swFHUyNlj3QTf2xou1ahJ7WOVUBaZJMCgvPPDApBWQmi5y3JvUwnEy+wZUhfyg/Z297d/gloAYTLuSpeNz7peP40JJi4z98fwnS0+fLa88b714+er1avvN2xObl0bAUOQqN2cxt6CkhiFKVHBWGOBZrOA0Pj+Y66cXYKzM9XecFTDOeKplIgVHR0XtXzs0TAwXVVBXof1psOrVNQ0VJLhBr2hYFtyY/JKGk/xSN9BwnSqgn+b6PfoutdFDI9MpfqT/mTCIgh/hhKcpGNduEPXubrfFV+y2X9TusG4QsD7boqzLFkGDh6BDmjiO2r/ddqLMQKNQ3NpRwAocV9ygFArqVlhaKLg45ymMHNQ8AzuuFpbW9INjJjTJjTsa6YK9X1HxzNpZFrvMjOPUPtTm5L+0UYnJ53EldVEiaHEzKCkVxZzO/4dOpAGBauYAF0a6XamYcucful9sORMePfkxOOl1g61u/1u/szdo7Fgh6+Q92SAB2SZ75JAckyER3qq36e16X/13/hd/3z+4SfW9pmaN/BX+0TVg2MeU</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974). <latexit sha1_base64="44EQSLd248yeLy8H80pBg3XRVUI=">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</latexit> |Gi=X D cD|Di D!dimer covering of lattice Spin liquid: resonating valence bonds <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974). <latexit sha1_base64="44EQSLd248yeLy8H80pBg3XRVUI=">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</latexit> |Gi=X D cD|Di D!dimer covering of lattice Spin liquid: resonating valence bonds <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974). <latexit sha1_base64="44EQSLd248yeLy8H80pBg3XRVUI=">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</latexit> |Gi=X D cD|Di D!dimer covering of lattice Spin liquid: resonating valence bonds <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i
<latexit sha1_base64="pxXYg6H6qbyZqyyyHVia4OXaH+c=">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</latexit> |!→= C1C2 C3C4C5 + + + + + … Spin liquid: resonating valence bonds <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i
<latexit sha1_base64="pxXYg6H6qbyZqyyyHVia4OXaH+c=">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</latexit> |!→= C1C2 C3C4C5 + + + + + … Spin liquid: resonating valence bonds <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i <latexit sha1_base64="4M78XEQB5kFc7NWj3D5asOmDvyY=">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</latexit> Key feature: fractionalization. Excitations are particle-like, but cannot be created by local operators. The excitations are classified under distinct anyon sectors.
<latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i Spin liquid: resonating valence bonds Anyon: a “spinon”
<latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i Spin liquid: resonating valence bonds Anyon: a “spinon”
<latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i Spin liquid: resonating valence bonds Anyon: a “spinon”
<latexit sha1_base64="pxXYg6H6qbyZqyyyHVia4OXaH+c=">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</latexit> |!→= C1C2 C3C4C5 + + + + + … Spin liquid: resonating valence bonds <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">AAACknicfVFdT9swFHUyNlj3QTf2xou1ahJ7WOVUBaZJMCgvPPDApBWQmi5y3JvUwnEy+wZUhfyg/Z297d/gloAYTLuSpeNz7peP40JJi4z98fwnS0+fLa88b714+er1avvN2xObl0bAUOQqN2cxt6CkhiFKVHBWGOBZrOA0Pj+Y66cXYKzM9XecFTDOeKplIgVHR0XtXzs0TAwXVVBXof1psOrVNQ0VJLhBr2hYFtyY/JKGk/xSN9BwnSqgn+b6PfoutdFDI9MpfqT/mTCIgh/hhKcpGNduEPXubrfFV+y2X9TusG4QsD7boqzLFkGDh6BDmjiO2r/ddqLMQKNQ3NpRwAocV9ygFArqVlhaKLg45ymMHNQ8AzuuFpbW9INjJjTJjTsa6YK9X1HxzNpZFrvMjOPUPtTm5L+0UYnJ53EldVEiaHEzKCkVxZzO/4dOpAGBauYAF0a6XamYcucful9sORMePfkxOOl1g61u/1u/szdo7Fgh6+Q92SAB2SZ75JAckyER3qq36e16X/13/hd/3z+4SfW9pmaN/BX+0TVg2MeU</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i
<latexit sha1_base64="pxXYg6H6qbyZqyyyHVia4OXaH+c=">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</latexit> |!→= C1C2 C3C4C5 + + + + + … Another anyon—involves subtle changes in sign of superposition <latexit sha1_base64="fpzmcPrOyFWnLeqKEhS2W2xQOMQ=">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</latexit> =1 p2(|"#i|#"i)= 1 p2⇣B† 1B† 2⌘|0i <latexit sha1_base64="5+XMvSV7f4a/Mj211smTrrlztKA=">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</latexit> Cn→Cn(↑1)Number of dimers across green line
<latexit sha1_base64="AoJxmobMc5vA65R9VwL8O7qO34g=">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</latexit> Read and Sachdev (1990): Determined the anyon structure of the resonating valence bond spin liquid. The anyon structure of this spin liquid is the same as Kitaev’s toric code (1997). Kitaev (1997): Place the quantum information in a superposition of anyon sectors for fault-tolerant quantum computation.
676 | Nature | Vol 614 | 23 February 2023 Article Suppressing quantum errors by scaling a surface code logical qubit Google Quantum AI* Practical quantum computing will require error rates well below those achievable with physical qubits. Quantum error correction1,2 offers a path to algorithmically relevant error rates by encoding logical qubits within many physical qubits, for which increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low for logical performance to improve with increasing code size. Here we report the measurement of logical qubit performance scaling across several code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find that our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, in terms of both logical error probability over 25 cycles and logical error per cycle ((2.914 ± 0.016)% compared to (3.028 ± 0.023)%). To investigate damaging, low-probability error sources, we run a distance-25 repetition code and observe a 1.7 % 10−6 logical error per cycle floor set by a single high-energy event (1.6 % 10−7 excluding this event). We accurately model our experiment, extracting error budgets that highlight the biggest challenges for future systems. These results mark an experimental demonstration in which quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation. Since Feynman’s proposal to compute using quantum mechanics 3 , many potential applications have emerged, including factoring4, optimization5, machine learning6, quantum simulation7 and quantum chemistry 8 . These applications often require billions of quantum operations 9–11 and state-of-the-art quantum processors typically have error rates around 10−3 per gate12–17, far too high to execute such large circuits. Fortunately, quantum error correction can exponentially suppress the operational error rates in a quantum processor, at the expense of temporal and qubit overhead18,19. Several works have reported quantum error correction on codes able to correct a single error, including the distance-3 Bacon–Shor20, colour 21 , five-qubit 22 , heavy-hexagon 23 and surface 24,25 codes, as well as continuous variable codes 26–29 . However, a crucial question remains of whether scaling up the error-correcting code size will reduce logical error rates in a real device. In theory, logical errors should be reduced if physical errors are sufficiently sparse in the quantum processor. In practice, demonstrating reduced logical error requires scaling up a device to support a code that can correct at least two errors, without sacrificing state-of-the-art performance. In this work we report a 72-qubit superconducting device supporting a 49-qubit distance-5 (d = 5) surface code that narrowly outperforms its average subset 17-qubit distance-3 surface code, demonstrating a critical step towards scalable quantum error correction. Surface codes with superconducting qubits Surface codes 30–34 are a family of quantum error-correcting codes that encode a logical qubit into the joint entangled state of a d % d square of physical qubits, referred to as data qubits. The logical qubit states are defined by a pair of anti-commuting logical observables XL and ZL. For the example shown in Fig.(1a, a Z L observable is encoded in the joint Z-basis parity of a line of qubits that traverses the lattice from top to bottom, and likewise an X L observable is encoded in the joint X-basis parity traversing left to right. This non-local encoding of information protects the logical qubit from local physical errors, provided we can detect and correct them. To d e te c t er ro r s , we p e r io d i ca l l y m e as ure X and Z parities of adjacent clusters of data qubits with the aid of d 2 − 1 measure qubits interspersed throughout the lattice. As shown in Fig.(1b, each measure qubit interacts with its neighbouring data qubits to map the joint data qubit parity onto the measure qubit state, which is then measured. Each parity measurement, or stabilizer, commutes with the logical observables of the encoded qubit as well as every other stabilizer. Consequently, we can detect errors when parity measurements change unexpectedly, without disturbing the logical qubit state. A decoder uses the history of stabilizer measurement outcomes to infer likely configurations of physical errors on the device. We can then https://doi.org/10.1038/s41586-022-05434-1 Received: 13 July 2022 Accepted: 10 October 2022 Published online: 22 February 2023 Open access Check for updates *A list of authors and their affiliations appears at the end of the paper. 676 | Nature | Vol 614 | 23 February 2023 Article Suppressing quantum errors by scaling a surface code logical qubit Google Quantum AI* Practical quantum computing will require error rates well below those achievable with physical qubits. Quantum error correction1,2 offers a path to algorithmically relevant error rates by encoding logical qubits within many physical qubits, for which increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low for logical performance to improve with increasing code size. Here we report the measurement of logical qubit performance scaling across several code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find that our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, in terms of both logical error probability over 25 cycles and logical error per cycle ((2.914 ± 0.016)% compared to (3.028 ± 0.023)%). To investigate damaging, low-probability error sources, we run a distance-25 repetition code and observe a 1.7 % 10−6 logical error per cycle floor set by a single high-energy event (1.6 % 10−7 excluding this event). We accurately model our experiment, extracting error budgets that highlight the biggest challenges for future systems. These results mark an experimental demonstration in which quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation. Since Feynman’s proposal to compute using quantum mechanics 3 , many potential applications have emerged, including factoring4, optimization5, machine learning6, quantum simulation7 and quantum chemistry 8 . These applications often require billions of quantum operations 9–11 and state-of-the-art quantum processors typically have error rates around 10−3 per gate12–17, far too high to execute such large circuits. Fortunately, quantum error correction can exponentially suppress the operational error rates in a quantum processor, at the expense of temporal and qubit overhead18,19. Several works have reported quantum error correction on codes able to correct a single error, including the distance-3 Bacon–Shor20, colour 21 , five-qubit 22 , heavy-hexagon 23 and surface 24,25 codes, as well as continuous variable codes 26–29 . However, a crucial question remains of whether scaling up the error-correcting code size will reduce logical error rates in a real device. In theory, logical errors should be reduced if physical errors are sufficiently sparse in the quantum processor. In practice, demonstrating reduced logical error requires scaling up a device to support a code that can correct at least two errors, without sacrificing state-of-the-art performance. In this work we report a 72-qubit superconducting device supporting a 49-qubit distance-5 (d = 5) surface code that narrowly outperforms its average subset 17-qubit distance-3 surface code, demonstrating a critical step towards scalable quantum error correction. Surface codes with superconducting qubits Surface codes 30–34 are a family of quantum error-correcting codes that encode a logical qubit into the joint entangled state of a d % d square of physical qubits, referred to as data qubits. The logical qubit states are defined by a pair of anti-commuting logical observables XL and ZL. For the example shown in Fig.(1a, a Z L observable is encoded in the joint Z-basis parity of a line of qubits that traverses the lattice from top to bottom, and likewise an X L observable is encoded in the joint X-basis parity traversing left to right. This non-local encoding of information protects the logical qubit from local physical errors, provided we can detect and correct them. To d etec t e rrors, we p eriodically m easu re X and Z parities of adjacent clusters of data qubits with the aid of d 2 − 1 measure qubits interspersed throughout the lattice. As shown in Fig.(1b, each measure qubit interacts with its neighbouring data qubits to map the joint data qubit parity onto the measure qubit state, which is then measured. Each parity measurement, or stabilizer, commutes with the logical observables of the encoded qubit as well as every other stabilizer. Consequently, we can detect errors when parity measurements change unexpectedly, without disturbing the logical qubit state. A decoder uses the history of stabilizer measurement outcomes to infer likely configurations of physical errors on the device. We can then https://doi.org/10.1038/s41586-022-05434-1 Received: 13 July 2022 Accepted: 10 October 2022 Published online: 22 February 2023 Open access Check for updates *A list of authors and their affiliations appears at the end of the paper. 676 | Nature | Vol 614 | 23 February 2023 Article Suppressing quantum errors by scaling a surface code logical qubit Google Quantum AI* Practical quantum computing will require error rates well below those achievable with physical qubits. Quantum error correction1,2 offers a path to algorithmically relevant error rates by encoding logical qubits within many physical qubits, for which increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low for logical performance to improve with increasing code size. Here we report the measurement of logical qubit performance scaling across several code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find that our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, in terms of both logical error probability over 25 cycles and logical error per cycle ((2.914 ± 0.016)% compared to (3.028 ± 0.023)%). To investigate damaging, low-probability error sources, we run a distance-25 repetition code and observe a 1.7 % 10−6 logical error per cycle floor set by a single high-energy event (1.6 % 10−7 excluding this event). We accurately model our experiment, extracting error budgets that highlight the biggest challenges for future systems. These results mark an experimental demonstration in which quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation. Since Feynman’s proposal to compute using quantum mechanics 3 , many potential applications have emerged, including factoring4, optimization5, machine learning6, quantum simulation7 and quantum chemistry 8 . These applications often require billions of quantum operations 9–11 and state-of-the-art quantum processors typically have error rates around 10−3 per gate12–17, far too high to execute such large circuits. Fortunately, quantum error correction can exponentially suppress the operational error rates in a quantum processor, at the expense of temporal and qubit overhead18,19. Several works have reported quantum error correction on codes able to correct a single error, including the distance-3 Bacon–Shor20, colour 21 , five-qubit 22 , heavy-hexagon 23 and surface 24,25 codes, as well as continuous variable codes 26–29 . However, a crucial question remains of whether scaling up the error-correcting code size will reduce logical error rates in a real device. In theory, logical errors should be reduced if physical errors are sufficiently sparse in the quantum processor. In practice, demonstrating reduced logical error requires scaling up a device to support a code that can correct at least two errors, without sacrificing state-of-the-art performance. In this work we report a 72-qubit superconducting device supporting a 49-qubit distance-5 (d = 5) surface code that narrowly outperforms its average subset 17-qubit distance-3 surface code, demonstrating a critical step towards scalable quantum error correction. Surface codes with superconducting qubits Surface codes 30–34 are a family of quantum error-correcting codes that encode a logical qubit into the joint entangled state of a d % d square of physical qubits, referred to as data qubits. The logical qubit states are defined by a pair of anti-commuting logical observables XL and ZL. For the example shown in Fig.(1a, a Z L observable is encoded in the joint Z-basis parity of a line of qubits that traverses the lattice from top to bottom, and likewise an X L observable is encoded in the joint X-basis parity traversing left to right. This non-local encoding of information protects the logical qubit from local physical errors, provided we can detect and correct them. To d ete c t e rro rs , we pe ri odi ca lly m ea su re X and Z parities of adjacent clusters of data qubits with the aid of d 2 − 1 measure qubits interspersed throughout the lattice. As shown in Fig.(1b, each measure qubit interacts with its neighbouring data qubits to map the joint data qubit parity onto the measure qubit state, which is then measured. Each parity measurement, or stabilizer, commutes with the logical observables of the encoded qubit as well as every other stabilizer. Consequently, we can detect errors when parity measurements change unexpectedly, without disturbing the logical qubit state. A decoder uses the history of stabilizer measurement outcomes to infer likely configurations of physical errors on the device. We can then https://doi.org/10.1038/s41586-022-05434-1 Received: 13 July 2022 Accepted: 10 October 2022 Published online: 22 February 2023 Open access Check for updates *A list of authors and their affiliations appears at the end of the paper. “Surface code” is closely related to the “toric code”
<latexit sha1_base64="EW6688bEFr48wbPrkPEDbxMdquk=">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</latexit> Although it has had a dramatic impact on theory/experiments on fault-tolerant quantum computations, the theory of spin liquids with well-defined anyons does not solve the problem of high temperature superconductivity in the cuprates
The Sachdev-Ye-Kitaev model of entanglement of mobile fermions
Philosophy And Snakes & Ladders SHARE Aditi Shah October 30th 2018 Although now popular in its modern, westernized form as a kids’ game, did you know that Snakes & Ladders traces its roots to a traditional Indian board game based on religious philosophies? In the original, it served as a lesson in morality. Playing this game wasn’t just about winning or losing, but finding out how close you were, to heaven or hell. It is believed to have been invented by Jain monks to promote the concept of liberation The history of Snakes & Ladders goes back around 1000 years to 10th century CE where it is believed to have been invented by Jain monks to promote the concept of liberation from the bondage of passions. The game was symbolic of a man’s journey in life and the design had a few similarities with the ancient Jain mandalas in which various squares were illustrated with the notions of karma and moksha. Jain mandala, 16th century CE | www.mfa.org As the monks travelled with the game, it acquired many regional names like Gyan Chaupar in northern India and Mokshapat around Maharashtra, along with Leela and Parampada Sopanapata . Meanwhile, there also developed other ‘philosophical’ variations – a Hindu and a very rare, Sufi Muslim version. The ladders represented virtues while the snakes represented vices In the game, the ladders represented virtues such as faith, generosity, humility and asceticism while the snakes represented vices such as anger, theft, lust and greed. The last square represented either a God or heaven meaning you have attained liberation. The ladders conveyed that good deeds lead you to heaven and evil to a cycle of rebirths. The number of ladders was less than the number of snakes, a reminder that the path of good is much more difficult to tread, than a path of sins. Support Live History India If you like Live History India`s work, do consider extending support to us. No contribution is too small and it will only take a minute. We thank you for pitching in. DONATE *Terms And Conditions In a nut-shell the game was meant to inspire players to introspect rather than compete with each other. Jain version, painting on cloth, 19th century Interestingly, the reason the game pivoted around pure luck was because it was in keeping with the Jain philosophical notion – emphasizing the ideas of fate and destiny. This was in contrast to other ancient games such as Chaturanga which needed skill or Pachisi, which focused on a mixture of both. The Pahari style of the game could run up to 360 squares Also, it is to be noted that unlike the 100 squares game that is ubiquitous with the Snakes & Ladder board game today, there wasn’t any standardization then. The most common types were 84-square Jain board, 101-square Sufi board and the 72 square Hindu (predominantly Vaishnav) board, followed by their expanded variants, which in Pahari style can run up to 360-squares. Often made simply of painted cloth and sometimes on paper, few boards have survived from any earlier than the mid-18th century. The iconography on it depicts cosmological elements, with upper regions depicting divine beings and the heavens. The rest of the board was covered with pictures of animals, flowers and people. Gyan Chaupar - 19th century CE | Rajasthan Oriental Research Institute, Jodhpur The appeal of this game not only transcended religious boundaries but also geographical ones. When it was first brought to Victorian England in 1892 for instance, it was a big hit. Here it was customised to suit Christian sensibilities. The squares of fulfilment, grace and success were accessible by ladders of thrift and penitence and snakes of indulgence, disobedience and indolence caused one to end up in illness, disgrace and poverty. While the Indian version of the game had snakes outnumbering ladders, the English counterpart was more forgiving, as it contained each in the same amount. This concept of equality signifies the cultural ideal that for every sin one commits, there exists another chance at redemption. Chutes and Ladders which taught kids about good and bad deeds | www.indoindians.com In 1943, it was rebranded as Chutes and Ladders in the United States by game pioneer Milton Bradley. Over time, the game was simplified, stripped of moral lessons altogether and in its recent avatar, it came to be known as Snakes and Ladders. This game serves as a perfect example of how even a simple game can evolve over time and space. In this case, also how a profound lesson in morality, became a game children play. ABOUT LIVE HISTORY INDIA Live History India is a first of its kind digital platform aimed at helping you Rediscover the many facets and layers of India’s great history and cultural legacy. Our aim is to bring alive the many stories that make India and get our readers access to the best research and work being done on the subject. If you have any comments or suggestions or you want to reach out to us and be part of our journey across time and geography, do write to us at [email protected] MOST POPULAR Cli! 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Pick a set of random positions Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 The Sachdev-Ye-Kitaev (SYK) model
Place electrons randomly on some sites Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 The Sachdev-Ye-Kitaev (SYK) model
Entangle electrons pairwise randomly Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 <latexit sha1_base64="Utq9HnZ9cYobM/K6ortiIQYaw9k=">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</latexit> U9,18;5,15 The Sachdev-Ye-Kitaev (SYK) model
Entangle electrons pairwise randomly Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 <latexit sha1_base64="Utq9HnZ9cYobM/K6ortiIQYaw9k=">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</latexit> U9,18;5,15 The Sachdev-Ye-Kitaev (SYK) model
Entangle electrons pairwise randomly Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 <latexit sha1_base64="kyfUy4iLmDHmZ1iLj3Dez4cecBc=">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</latexit> U6,8;4,14 The Sachdev-Ye-Kitaev (SYK) model
Entangle electrons pairwise randomly Sachdev, Ye (1993); Kitaev (2015) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 <latexit sha1_base64="kyfUy4iLmDHmZ1iLj3Dez4cecBc=">AAACLHicbZDLSsNAFIYnXmu9VV26CRbBRQmJxgu4KbpxqWBVaEqZTE7bwckkzJxUS+hruNUn8GnciLj1OZzULLwdGPj5/3M4Z74wFVyj675aU9Mzs3PzlYXq4tLyymptbf1KJ5li0GKJSNRNSDUILqGFHAXcpApoHAq4Dm9Pi/x6CErzRF7iKIVOTPuS9zijaKyg1c0PGkfHfsPzx91a3XXcSdmu43v7/l4hSscrRZ2Udd5ds6pBlLAsBolMUK3bnptiJ6cKORMwrgaZhpSyW9qHtpGSxqAb0ZCneiI7+eT+sb1twsjuJco8ifbE/T6c01jrURyazpjiQP/OCvO/rJ1h76iTc5lmCJJ9LeplwsbELmDYEVfAUIyMoExxc7bNBlRRhgZZNdBgeMo+DvIA4R7veGT25L6zz+WPz+VhXNDzfrP6K652He/A8S/8evOk5Fghm2SL7BCPHJImOSPnpEUYSckDeSRP1rP1Yr1Z71+tU1Y5s0F+lPXxCYJUpx4=</latexit> U6,8;4,14 The Sachdev-Ye-Kitaev (SYK) model
A solvable model of multi-particle quantum entanglement. Yields a metal whose excitations are not particle-like i.e. no bosons, fermions, anyons…. Current is carried by an “entangled quantum soup” The Sachdev-Ye-Kitaev (SYK) model Sachdev, Ye (1993); Kitaev (2015)
Yields a quantum state whose excitations are not particle-like i.e. no bosons, fermions, anyons…. The Sachdev-Ye-Kitaev (SYK) model Sachdev, Ye (1993); Kitaev (2015)
Yields a quantum state whose excitations are not particle-like i.e. no bosons, fermions, anyons…. The Sachdev-Ye-Kitaev (SYK) model Sachdev, Ye (1993); Kitaev (2015) Current is carried by an “entangled quantum soup”
Yields a quantum state whose excitations are not particle-like i.e. no bosons, fermions, anyons…. The Sachdev-Ye-Kitaev (SYK) model Sachdev, Ye (1993); Kitaev (2015) <latexit sha1_base64="/6YvObh066WHGUc2kIGiQCPA0W4=">AAAD7nicbVPLbtNAFHUTHiW8WliyuaJBgFSspFULG6RSUMUKFSlpK9VRNB5fx6OMx2bmmtayzGewQ2z5Jb6AJb/AjGOqtDCSpTv3ec654zCXwtBg8HOl0712/cbN1Vu923fu3ru/tv7gyGSF5jjmmcz0ScgMSqFwTIIknuQaWRpKPA7nb138+DNqIzI1ojLHScpmSsSCM7Ku6frKryDEmVAVR0Wo694bmGMJPFMGPxWoOEIWAyUILDTL15xpElziCynmCHjOBTUtDQgDQaEi1A5TFQiCsRIOA5NwKJnic8EUvBPGiLwpqf0ggJHtqVGy88YFJFLchH5ArOhvwlmCCnLUVOgQI5vNCBjEegGxtHlZijPWd7NndpaCsOy1xCJhcsnKlFFS91w/eA1BrBmvgiRkuq7m030Y1XAAgcSYni0HYdH3IifQYpbQ816AKrrc2CLUaHG4qgbGgulT00hJTJFlM2oihKmlwiwX3HRUVNQIGheKN9T7B02asBLmdo7dy1/NDWlUM0rcXbh1sUVFiHSGlvTyXowPC5jtYqdrGwN/0BwY+Ns7O4PhEC48w9bY8NpzOF37HUQZL1JbziUz5nQ4yGlSte2tlIXBnPE5m+GpNRVL0Uyq5kXW8MR6IogzbT+Lv/EuV1QsNaZMQ5vp9DNXY42o/4mdFhS/mlRC5QXZzS8GxYUEysA9b4iERk6ytAbjWliswBPmhLIPsBcYtH+Ik7AKCM/pTER2TrXt7wjlFBpe1eNf42jLH+76ux+3Nvb2W61WvUfeY++ZN/Reenvee+/QG3u886FDnbrzpZt3v3a/db8vUjsrbc1D79Lp/vgDFb9Kag==</latexit> A key consequence of the absence of the particle-like excitations is Universal Planckian Dissipation. The relaxation time, ω, when perturbed at a frequency εis given by ω=⊋ kBTF!⊋ε kBT" where ⊋is Planck’s constant, Tis temperature, and the function Fis independent of the strength of interaction between the particles.
pc YBa2Cu3O6+x Strange Metal The “strange metal” has no particle-like/anyon excitations and is described by a SYK-type theory Aavishkar A. Patel, Haoyu Guo, Ilya Esterlis, S. S., Science 381, 790 (2023)
pc Strange Metal Aavishkar A. Patel, Haoyu Guo, Ilya Esterlis, S. S., Science 381, 790 (2023) This linear dependence of the scattering rate calls for a comparison with resistivity. Hence we have also measured the temperature dependence of the resistivity of our sample under two magnetic fields H= 0 T and H= 16 T. As displayed in Fig. 2a, the resistivity has a linear T-dependence ρ=ρ 0 +AT over an extended range of temperature, with A≈0.63 μΩcm/K. This is a hallmark of cuprates in this regime of doping10,13,14,20,53. It is qualitatively consistent with the observed linear frequency dependence of the scattering rate and, as discussed later in this paper, also in good quantitative agreement with the ω→0 extrapolation of our optical data within experimental uncertainties. The optical mass enhancement m*(ω)/mis displayed in Fig. 1d. With the chosen normalization, m*/mdoes not reach the asymptotic value of one in the range ℏω< 0.4 eV, which means that intraand interband and/or mid-infrared transitions overlap above 0.4 eV. The inset of Fig. 1d shows a semi-log plot of the mass enhancement evaluated at ℏω=5k B T,wherethenoiselevelislowforT⩾40 K. Despite the larger uncertainties at low T, this plot clearly reveals a logarithmic temperature dependence of m*/m. This is a robust feature of the data, independent of the choice of ϵ ∞ and K. We note that the specific heat coefficient C/Tof LSCO at the same doping level was previously reported to display a logarithmic dependence on temperature, see Fig. 2c47,48. We will further elaborate on this important finding of a logarithmic dependence of the optical mass and discuss its relation to specific heat in the next section. Scaling analysis In this section, we consider simultaneously the frequency and temperaturedependenceoftheopticalpropertiesandinvestigatewhether ℏω/k B Tscaling holds for this sample close to the pseudogap critical point. We propose a procedure to determine the three parameters ϵ ∞ , K,andmintroduced above. Putting ω/Tscaling to the test. Quantum systems close to a quantum critical point display scale invariance. Temperature being the only relevant energy scale inthe quantum criticalregime, this leads in many cases to ω/Tscaling22 (in most of the discussion below, we set ℏ=k B =1 except when mentioned explicitly). In such a system we expect the complex optical conductivity to obey a scaling behavior 1/ σ(ω,T)∝TνF(ω/T), with ν⩽1 a critical exponent. More precisely, the scaling properties of the optical scattering rate and effective mass read: 1=τðω,TÞ=Tνfτðω=TÞð4Þ m*ðω,TÞ#m*ð0,TÞ=Tν#1fmðω=TÞð5Þ with f τ and f m two scaling functions. This behavior requires that both ℏω and k B Tare smaller than a high-energy electronic cutoff, but their ratio can be arbitrary. Furthermore, we note that when ν=1(Planckiancase) the scaling is violated by logarithmic terms, which control in particular the zero-frequency value of the optical mass m*(0, T). As shown in Theory within a simple theoretical model, ω/Tscaling nonetheless holds in this case to an excellent approximation provided that m*(0, T)is subtracted, as in Eq. (5). We also note that in a Fermi liquid, the singleparticle scattering rate ∝ω2+(πT)2does obey ω/Tscaling (with formally ν=2), but the optical conductivitydoesnot.Indeed,itinvolvesω/T2 terms violating scaling, and hence depends on two scaling variables ω/T2and ω/T,asisalreadyclearfroman(approximate)generalized Drude expression 1/σ≈−iω+τ 0 [ω2+(2πT)2]. For a detailed discussion of this point, see Ref. 54.Suchviolationsofscalingbyω/Tνterms apply more generally to the case where the scattering rate varies as Tνwith ν>1.Hence,ω/Tscaling for both the optical scattering rate and optical effective mass are a hallmark of non-Fermi liquid behavior with ν⩽1. Previous work has indeed provided evidence for ω/Tscaling in the optical properties of cuprates23,24. Here, we investigate whether our optical data obey ω/Tscaling. We find that the quality of the scaling depends sensitively on the chosen value of ϵ ∞ . Different prescriptions in the literature to fixϵ ∞ yield—independentlyof the method used—values rangingfromϵ ∞ ≈4.3 for strongly underdoped Bi2212 to ϵ ∞ ≈5.6 for strongly overdoped Bi221232,55. The parameter ϵ ∞ is commonly understood to represent the dielectricconstantofthematerialintheabsenceofthechargecarriers, and is caused by the bound charge responsible for interband transitions at energies typically above 1 eV. While this definition is unambiguous for the insulating parent compound, for the doped material one is confronted with the difficulty that the optical conductivity at these higher energies also contains contributions described by the self-energy of the conduction electrons, caused for example by their coupling to dd-excitations56. Consequently, not all of the oscillator strength in the interband region represents bound charge. Our model overcomes this hurdle by determining the low-energy spectrum below 0.4 eV, and subsuming all bound charge contributions in a single constant ϵ ∞ . Its value is expected to be bound from above by the value of the insulating phase, in other words we expect to find ϵ ∞ < 4.5 (see Supplementary Information Sec. A). Rather than setting an a priori value for ϵ ∞ , we follow here a different route and we choose the value that yields the best scaling collapse for a given value of the exponent ν. This program is straightforwardly implemented for 1/τand indicates that the best scaling collapse is achieved with ν≈1 and ϵ ∞ ≈3, see Fig. 2b as well as Supplementary Information Sec. B and Supplementary Fig. 2. Turning to m*, we found that subtracting the dc value m*(ω=0,T) is crucial when attempting to collapse the data. Extrapolating optical data to zero frequency is hampered by noise. Hence, Fig. 2 | Scaling of scattering rate and mass enhancement. a Temperaturedependent resistivity measured in zero field (black) and at 16 teslas (red). The inset emphasizesthe linearity of the 16 T dataatlow temperature. The dashedline shows ρ 0 +AT with ρ 0 = 12.2 μΩcm and A= 0.63 μΩcm/K. bScattering rate divided by temperature plotted versus ω/T; the collapse of the curves indicates a behavior 1/ τ~Tf τ (ω/T). cEffective quasiparticle mass (in units of the indicated band mass m) deduced from the low-temperature electronic specific heat47 [m* Cp =ð3=πÞð_2dc=k2 BÞðC=TÞ] and zero-frequency optical mass enhancement; the dashed lines indicate lnTbehavior. dOptical mass minus the zero-frequency mass shown in cplotted versus ω/T; the collapse of the curves indicates a behavior m*(ω)−m*(0) ~ f m (ω/T).The data between 0.22 and 0.4 eV areshown asdotted lines. ϵ ∞ = 2.76 was used here as in Fig. 1. Article https://doi.org/10.1038/s41467-023-38762-5 Nature Communications | (2023)14:3033 3 B. Michon, C. Berthod, C. W. Rischau, A. Ataei, L. Chen, S. Komiya, S. Ono, L. Taillefer, D. van der Marel, A. Georges, Nature Comm. 14, 3033 (2023) <latexit sha1_base64="1a3o9wTml9LzDTuD4zP2XJz9u70=">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</latexit> Planckian dynamics ! of large Fermi surface Electron scattering time ω from optical conductivity ω(ε)= ⊋ kBTF!⊋ε kBT"
Quantum Entanglement across a black hole horizon Black hole horizon Quantum entanglement on the surface <latexit sha1_base64="LDW5e9z5x1j6CBP5PSvQIbBkx2A=">AAADJnicbVJLb9NAELbNq5hXCkcOjGiROCArCaXtBaktB3osgj6kbIjW60m8ynrX2l2nGCsc+S0cuMLP4IYQN/4Bf4HJQxUtjLSr0cw3r28mLZV0vt3+GUaXLl+5em3lenzj5q3bd1qrd4+cqazAQ2GUsScpd6ikxkMvvcKT0iIvUoXH6fjFzH88Qeuk0W98XWK/4CMth1JwT6bBaviAaSN1htrHezUIU5SVn/scsIrsdpa5YdKDqbyTGU7B5wip4mIMuVH4hDHY56djqUdgUs8JnjEWnwcB5bemrGOW4kjqJpOuVLwuuM+n8Wt4DmxouWh2Qbx9Om024CWwPOV2GjPU2Xn0aY4WYX13HaQDTrOCGV5oiT4r3xudUGsf6O0qBYYgFlztPBYOcj456wlK+o2dzcwVeDNLJi1MjKoKTAattXbSngu0k+72s+7WNpxZOktlLVjKwaD1m2VGUKj2QnHnep126fsNpwJCIU1UOSypUz7CHqmaF+j6zXyTU3hElgyGxtLTHubWvyMaXjhXFykhZ2y4i745Rf/x9So/3O43UtN2UYtFoWG1GJfOAjJpUXhVk8KFldQriJzTTjwdT8wc0mXpkc8b5vGdP5UZ1Wk6yYbUU2Koc5GPf5WjbtLZTDZfddd29pZcrQT3g4fB46ATbAU7wX5wEBwGIvwYfg6/hF+jT9G36Hv0YwGNwmXMveCcRL/+AM3RA8Q=</latexit> By computations outside the black hole, Hawking obtained the black hole entropy S=Ac3 4G⊋ where Ais area of the black hole horizon. All other systems have entropy proportional to their volume.
Black hole horizon Maxwell’s electromagnetism and Einstein’s general relativity allow black hole solutions with a net charge ⇣ The quantum versions of Maxwell’s and Einstein’s equations in space and time are also the equations describing electron entanglement in the SYK model! ~x ⇣
Quantum Entanglement across a black hole horizon Sakkmesterke/Science Photo Library RF/Getty Images <latexit sha1_base64="5KALZMcPDOSFA6EqTTC+joJN/3A=">AAACrHicdVFLbxMxEPYurxIeTeHIZSAFcYBot7QBblW59MAhSElaKY6iWe9sYsX2rmxvq2gVfht/ggP/BudRiSIYydL4++b5TVYp6XyS/IriO3fv3X+w97D16PGTp/vtg2cjV9ZW0FCUqrSXGTpS0tDQS6/osrKEOlN0kS2+rPmLK7JOlmbglxVNNM6MLKRAH6Bp+wfPaCZNI8h4sqvW7ptLVylcavTzgHmspw23Gqw0s/d5eW1WwJ3UwAuLouHzDO2qWUzPYBCiyeS38/sKjVhINJAvDWop3EvOv3PeOhwcgnTg5wTneL0IxSGgnnRFFn1tCcqC8w2fKRQLmJeKtg1u5m1N252k20s+JMefIekmGwtOepIcn6SQ7pAO21l/2v7J81LUOhQQCp0bp0nlJw1aL4WiMH/tqArNcEbj4IaByU2ajdIreB2QHIrShmc8bNA/MxrUzi11FiLXu7u/uTX4LtP/ose1Lz5NGmmq2pMR215FrcCXsL4c5NKS8GoZHBRWhnFBzDEcIAjh1jrcLAv/d0ZH3bTX7X076pye7RTZYy/YK/aWpewjO2XnrM+GTERvoq/RMBrF3XgQj+PJNjSOdjnP2S2Li9+A3tN6</latexit> ωring→down → ⊋ kBT Planckian dynamics! Tis the Hawking temperature of the black hole
The SYK model describes multi-particle quantum entanglement resulting in the ! loss of identity of the particles! ! In one set of variables, it describes ! the strange electrical properties of YBCO! ! ! In a dual set of variables it describes charged black holes The Sachdev-Ye-Kitaev (SYK) model Sachdev (2010), Kitaev (2015), Maldacena Stanford (2015) Sachdev, Ye (1993)
p Strange Metal The SYK model describes multi-particle quantum entanglement resulting in the ! loss of identity of the particles! ! In one set of variables, it helps describe ! the strange electrical properties of YBCO! ! ! In a dual set of variables it describes charged black holes The Sachdev-Ye-Kitaev (SYK) model Sachdev (2010), Kitaev (2015), Maldacena Stanford (2015) Sachdev, Ye (1993)
The SYK model describes multi-particle quantum entanglement resulting in the ! loss of identity of the particles! ! In one set of variables, it helps describe ! the strange electrical properties of YBCO! ! ! In a dual set of variables it describes! the interior of charged black holes The Sachdev-Ye-Kitaev (SYK) model Sachdev (2010), Kitaev (2015), Maldacena Stanford (2015) Sachdev, Ye (1993)