In the article we shows how to obtain a causal geometry (metric tensor of Lorentz manifold) from the fundamental abstract objects such as quantum state, operators of identity, conjugation C, parity P and their composition CP applied to an object of discrete quantum state. The objects have no a priori link to space and time as well as operators.
We are showing, that some basic algebra operation as sum of two qubits through C-NOT (XOR logical gate) create a affine connection as bilinear form of two such gates engineered from operators identity, conjugation, parity and CP. This connection is metric connection defined for each element of state's set.
We are referring to fundamental sense of entangled states which is the result of such quantum computing.
Entanglement state define a combination of squeezed operators acting on coherence states and creating with such way the SL(2,C) left action and Lorentz group representation as well as the quantum gravity propagator. Next, we derive classical general relativity equations from squeezed operators and their commutation relations in the mean field expectation value of coherent states. We are describing that von Neumann entropy of entangled states produced as squeezed coherent states by quantum computer have a entropy with area dependency law. We point out the conjuncture between entangled two-qubits state and Hopf fibrations, which generate the structure of the standard model of elementary particles, left, right-handed doublets families tuples with band structure for allowed masses/momentum for each family.
Generally speaking, we are introducing the concept of sets with open subsets, which is equivalent to introducing a topology and whole set as topology space. For each subset (vector) we introduce a matrix/gate equivalent to a metric compatible connection resulting to Lorentz transformation on the tangent space at point. This quantum computing gate preparing entangled state. Entangled states connecting areas of Hilbert and geometric space. This implicate a mechanism to tailoring of space fabric. Entangled state is eigenstate of system and is not possible to obtain it as direct multiplication, tensor product of relevant independent subsystems. That means, it can not be decomposed in separated subsystems. Such example is Bell state.
Quantum state we can write as mathematical object, abstraction of two states of quantum objects with state one or zero using Dirac's ket notation. It's representing so called qubit, the information unit transferable on classical bit 0 or 1 after measurement and collapse of wave function. Qubit is a fundamental quantum state representing a most smallest unit of quantum information containing a bit classical information accessible with measurement. In quantum computers is defined analogue of classical electric computing circuit composed from computing/logical units. Computing/logical units are called as quantum gates, analogue of classic logical gates. It is possible to show, that exist a universal set with finite element of quantum gates, from which is possible to construct any computing circuit. One of such gate is C-NOT, summation of two 1-qubits, represented with matrix on picture
Entanglement state define a combination of squeezed operators acting on coherence states and creating with such way the SL(2,C) left action and Lorentz group representation as well as the quantum gravity propagator. Next, we derive classical general relativity equations from squeezed operators and their commutation relations in the mean field expectation value of coherent states. We are describing that von Neumann entropy of entangled states produced as squeezed coherent states by quantum computer have a entropy with area dependency law. We point out the conjuncture between entangled two-qubits state and Hopf fibrations, which generate the structure of the standard model of elementary particles, left, right-handed doublets families tuples with band structure for allowed masses/momentum for each family.
Generally speaking, we are introducing the concept of sets with open subsets, which is equivalent to introducing a topology and whole set as topology space. For each subset (vector) we introduce a matrix/gate equivalent to a metric compatible connection resulting to Lorentz transformation on the tangent space at point. This quantum computing gate preparing entangled state. Entangled states connecting areas of Hilbert and geometric space. This implicate a mechanism to tailoring of space fabric. Entangled state is eigenstate of system and is not possible to obtain it as direct multiplication, tensor product of relevant independent subsystems. That means, it can not be decomposed in separated subsystems. Such example is Bell state.
Quantum state we can write as mathematical object, abstraction of two states of quantum objects with state one or zero using Dirac's ket notation. It's representing so called qubit, the information unit transferable on classical bit 0 or 1 after measurement and collapse of wave function. Qubit is a fundamental quantum state representing a most smallest unit of quantum information containing a bit classical information accessible with measurement. In quantum computers is defined analogue of classical electric computing circuit composed from computing/logical units. Computing/logical units are called as quantum gates, analogue of classic logical gates. It is possible to show, that exist a universal set with finite element of quantum gates, from which is possible to construct any computing circuit. One of such gate is C-NOT, summation of two 1-qubits, represented with matrix on picture
Source: http://mycodehere.blogspot.com/2011/10/multiple-qubit-gates.html
CNOT gate is block matrix. Set with 1-qubit gates together with "entangling" 2-qubit gate (CNOT gate) is the universal set for n-qubit operations. This allow to us to create
the affine connection operator Gamma as block matrix, result of this quantum computing operations with connection operator Gamma gate, operator on state vectors psi_1, psi_2 is "entangled" state. As we show in article, one possible realisation of qubit's notation are creation and anihilation operators acting on coherent states, because we need some not pure vacuum state which corresponding to computer state but generating with displacement the vacuum state to have a finite excitation amplitude.
Source: http://wikipedia.com
Displacement of base state can be result of spontaneous breaking of symmetry and non-zero vacuum mean value. Coherent states represent a states of displaced linear harmonic oscillator and have a link with nilpotent groups.
Biquaternions have nilpotent algebra structure also. Unit biquaternion is representing S7 sphere which play important role in standard model of elementary particles schema as we will describe in section 4 of article.
Exponential map of biquaternion algebra create representation od SL(2,C) group which is isomorphic with Lorentz group SO(1,3).
We define a affine connection operator Gamma $ for each point/element of algebra producing as bilinear form the entanglement state, which in notation of squeezing operators has a shape of two qubit Bell state.
Such structure create from coherent states a squeezed coherent states. These states representing a displaced (shifted) linear harmonic oscillator in finite region.
Mentioned structure in such form define a left action(=representation) SL(2,C) group, projection to Lorentz group/manifold. The spacetime.
We obtained a spacetime as derived emerging system from quantum computing on base of qubits.In formalism of operators we can recognize a unitary transformation similar to Bogoljubov transformation, which create from vacuum state a basic state of superconductor. In our case transformation create the universe as exotic weak superconductor (weak bosons acquire non zero mass). In other section we introduce curvature as emergent geometry attribute from squeezing operators and Einstein's general relativity equations as mean value of squeezed coherent states commutations relation. We are showing with analogy of condensed matter example the universal attractive force like gravity in environment with some periodicity or lattice like structure. If we will consider squeezed coherence states as base structure of vacuum we get similar structure. Here we can derive a value of classical Newton gravitation constant as well as Lambda comological constant.
After we are interpreting the coherent state as displaced oscillator with non zero vacuum value (Higgs field) where we have Yukawa interaction of this "quaternions fields" with Higgs field and mass definition for them. Quaternions in doublet form generated by two two-qubits phi_1 , phi_2 system entangled with affine connection Gamma represents biquaternion, which is Grassmann element of Clifford algebra with behaviour of fermions. More details about masses leptons and quarks from such schema and their gauge interactions (S15 Hopf fibration) produced from quantum computer with C-NOT gateway is described in other my work. One interesting conjecture is a band structure of allowed masses for particle families doublets and possible constrain for their SuSy partners. Standard solution of band structure as outcome from equation which is a solution of particle movement in periodic environment of squeezed coherent states fabric of spacetime. It is in principle same as for electrons in solid state matter. But in this picture band of allowed masses/momentum give a possible explanation of mass hierarchy values character for each family of quarks doublet as well as their superpartners.
Operator CP is represented as part of C-ROT quantum gate. This gate in quantum computing creating controlled Rabi rotation, cycle. This is effect of oscillation of two level quantum system in presence of oscillatory driving field. For such system with defined Rabi frequency and time value like Pi/2 is system in sharp state |1> or |2>. Meantime is in superposition of both states. This give us possibility to define a quantum time event as sharp qubit state. Presence of C-ROT gate as part of Gamma connexion create creating permanent Rabi oscillation and than ticks of quantum timer. Classical time is than a expectation mean value in squeezed coherent states. Squeezed states works like oscillatory driving field on displaced oscillator vacuum state. Existing Rabi vacuum oscillation of two level quantum states like qubits or mentioned neutral particles oscillation creating a emergent category of time. We reminding that qubit is the information unit transferable on classical bit 0 or 1 after measurement and collapse of wave function. That mean we can represent time as set of qubits manipulated by C-ROT quantum gate to ordered set. Because time/qubit is a quantum state transferable to classical bit we can observe time as classical quantity and we observing in time a sharp values (Schrodinger cat alive, or not, particle decayed, or not decayed) of state, not their superposition in experiment.
In article we showed, that space time, Lorentz manifold and other physical entities as momentum, location we can obtain as counting qubits operations on quantum computer. We describe a possibility to get universal attractive force comparable with gravity as feature of manifold squeezed coherent states, creating generic periodic structure similar as polarized crystal in solid states physics and UV finiteness eliminating a spacetime singularity with repulsion effect on quantum level. Base of coherent squeezed states create a area law of entropy of black hole. We pointed, that metric connections, we can represent as quantum gates and their output as spacetime. Squeezing operators representing a metric connection together with their derivations, displacement operators of coherent states creating the Lorentz group element representation in structure of Einstein tensor with cosmological constant term as mean value of commutators mentioned operators. Squeezing operators can be viewed also as mentioned Bogoljubov transformation operator acting on basic state of displaced oscillator state (basic state of electroweak superconductor as our universe look like) and creating a excited states on this basic state. And similarly standard model of elementary particles we can get from normalization condition for qubits states represented by Hopf fibration from Bloch's hyperspheres. This return us to title of this article and put a question: Is our universe a product of quantum computer?
We obtained a spacetime as derived emerging system from quantum computing on base of qubits.In formalism of operators we can recognize a unitary transformation similar to Bogoljubov transformation, which create from vacuum state a basic state of superconductor. In our case transformation create the universe as exotic weak superconductor (weak bosons acquire non zero mass). In other section we introduce curvature as emergent geometry attribute from squeezing operators and Einstein's general relativity equations as mean value of squeezed coherent states commutations relation. We are showing with analogy of condensed matter example the universal attractive force like gravity in environment with some periodicity or lattice like structure. If we will consider squeezed coherence states as base structure of vacuum we get similar structure. Here we can derive a value of classical Newton gravitation constant as well as Lambda comological constant.
After we are interpreting the coherent state as displaced oscillator with non zero vacuum value (Higgs field) where we have Yukawa interaction of this "quaternions fields" with Higgs field and mass definition for them. Quaternions in doublet form generated by two two-qubits phi_1 , phi_2 system entangled with affine connection Gamma represents biquaternion, which is Grassmann element of Clifford algebra with behaviour of fermions. More details about masses leptons and quarks from such schema and their gauge interactions (S15 Hopf fibration) produced from quantum computer with C-NOT gateway is described in other my work. One interesting conjecture is a band structure of allowed masses for particle families doublets and possible constrain for their SuSy partners. Standard solution of band structure as outcome from equation which is a solution of particle movement in periodic environment of squeezed coherent states fabric of spacetime. It is in principle same as for electrons in solid state matter. But in this picture band of allowed masses/momentum give a possible explanation of mass hierarchy values character for each family of quarks doublet as well as their superpartners.
Operator CP is represented as part of C-ROT quantum gate. This gate in quantum computing creating controlled Rabi rotation, cycle. This is effect of oscillation of two level quantum system in presence of oscillatory driving field. For such system with defined Rabi frequency and time value like Pi/2 is system in sharp state |1> or |2>. Meantime is in superposition of both states. This give us possibility to define a quantum time event as sharp qubit state. Presence of C-ROT gate as part of Gamma connexion create creating permanent Rabi oscillation and than ticks of quantum timer. Classical time is than a expectation mean value in squeezed coherent states. Squeezed states works like oscillatory driving field on displaced oscillator vacuum state. Existing Rabi vacuum oscillation of two level quantum states like qubits or mentioned neutral particles oscillation creating a emergent category of time. We reminding that qubit is the information unit transferable on classical bit 0 or 1 after measurement and collapse of wave function. That mean we can represent time as set of qubits manipulated by C-ROT quantum gate to ordered set. Because time/qubit is a quantum state transferable to classical bit we can observe time as classical quantity and we observing in time a sharp values (Schrodinger cat alive, or not, particle decayed, or not decayed) of state, not their superposition in experiment.
In article we showed, that space time, Lorentz manifold and other physical entities as momentum, location we can obtain as counting qubits operations on quantum computer. We describe a possibility to get universal attractive force comparable with gravity as feature of manifold squeezed coherent states, creating generic periodic structure similar as polarized crystal in solid states physics and UV finiteness eliminating a spacetime singularity with repulsion effect on quantum level. Base of coherent squeezed states create a area law of entropy of black hole. We pointed, that metric connections, we can represent as quantum gates and their output as spacetime. Squeezing operators representing a metric connection together with their derivations, displacement operators of coherent states creating the Lorentz group element representation in structure of Einstein tensor with cosmological constant term as mean value of commutators mentioned operators. Squeezing operators can be viewed also as mentioned Bogoljubov transformation operator acting on basic state of displaced oscillator state (basic state of electroweak superconductor as our universe look like) and creating a excited states on this basic state. And similarly standard model of elementary particles we can get from normalization condition for qubits states represented by Hopf fibration from Bloch's hyperspheres. This return us to title of this article and put a question: Is our universe a product of quantum computer?
Source:https://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/CNOT_Hadamard_Basis.svg/640px-CNOT_Hadamard_Basis.svg.png
