Spin Foam Vertex Amplitudes on Quantum Computer - Preliminary Results

Vertex amplitudes are elementary contributions to the transition amplitudes in the spin foam models of quantum gravity. The purpose of this article is make the first step towards computing vertex amplitudes with the use of quantum algorithms. In our studies we are focused on a vertex amplitude of 3+1 D gravity, associated with a pentagram spin-network. Furthermore, all spin labels of the spin network are assumed to be equal $j=1/2$, which is crucial for the introduction of the \emph{intertwiner qubits}. A procedure of determining modulus squares of vertex amplitudes on universal quantum computers is proposed. Utility of the approach is tested with the use IBM's \emph{ibmqx4} 5-qubit quantum computer as well as on simulator of quantum computer provided by the same company. Finally, upper bound on the value of the vertex probability is determined employing the IBM simulator with 20-qubit quantum register.


I. INTRODUCTION
The basic objective of theories of quantum gravity is to calculate transition amplitudes between configurations of the gravitational field. The most straightforward approach to the problem is provided by the Feynman's path integral where S G and S φ are the gravitational and matter actions respectively. While the formula (1) is easy to write it is not very practical for the case of continuous gravitational field, characterized by infinite number of degrees of freedom. One of the approaches to determine (1) is based on introducing discretization of the gravitational field associated with some cut-off scale. The expectation is that continuous limit of such discretized theory can be recovered at the second order phase transition [1,2]. The essential step in this challenge is to generate different discrete space-time configurations (triangulations) contributing to the path integral (1). In Causal Dynamical Triangulations (CDT) [1] which is one of the approaches to the problem, Markov chain of elementary moves is used to explore different triangulations between initial and final state. In practice, the Markov chain is implemented after performing Wick rotation in Eq. 1. In the last over twenty years, the procedure has been extensively studied running computer simulations [3]. However, in 1+1 D case analytical methods of generating allowed triangulations are also available. In particular, it has been shown that Feynman graphs of auxiliary random matrix theories generate graphs dual to the triangulations [4]. An advantage the method is that in the large N (color) limit of such theories symmetry factors associated with given triangulations can be recovered [5].
Another path to the problem of determining (1) is provided by the Loop Quantum Gravity (LQG) [6,7] approach to the Planck scale physics. Here, discreteness of space is not due to the applied by hand cut-off but is a consequence of the procedure of quantization. Accordingly, the spatial configuration of the gravitational is encoded in the so-called spin network states [8]. In consequence, the transition amplitude (1) is calculated between two spin network states. The geometric structures (2-complexes) representing the path integral are called Spin Foams [9]. The elementary processes contributing to the spin foam amplitudes are associated with vertices of the spin foams and are called vertex amplitudes [10,11]. The employed terminology of spin networks and spin foams in clarified in Fig. 1 on example of 2+1 D gravity. In Fig. 1 two boundary spin networks with 3-valent nodes are shown. Such nodes are dual to two-dimensional triangles. In this article we will focus on the 3+1 D case in which the spin network nodes (related with non-vanishing volumes) are 4-valent. The nodes are dual to tetrahedra (3-simplex). In the example presented in Fig. 1  In analogy to the random matrix theories in case of the 2D triangulations, the spin foams (2-coplexes) can also be obtained as Feynman diagrams of some auxiliary field theory. Namely, the so-called Group Field Theories (GFTs) have been introduced to generate structure of vertices and edges associated with spin foams [12][13][14]. In particular, the 3+1 D theory with 5-valent vertices requires GFT with five-order interaction terms, known as Ooguri's model [15]. There has recently been made a great progress in the field of GFTs with many interesting results (see e.g. [16,17]).
The aim of this article is to investigate a possibility of employing universal quantum computers to compute vertex amplitudes of 3+1 D spin foams. The idea has been suggested in Ref. [18], however, not investigated there. Here, we make the first attempt to materialize this concept. In our studies, we consider a special case of spin networks with spin labels corresponding to fundamental representations of the SU (2) group, for which intertwiner qubits [18][19][20] can be introduced. The qubits will be implemented on IBM's 5-qubit quantum computer (ibmqx4) as well as with the use of QASM simulator of quantum computer provided by the same company [21]. In the case of real 5-qubit quantum computer the qubits are physically realized as superconducting circuits [22] operating at millikelvin temperatures.
The studies contribute to our broader research program focused on exploring the possibility of simulating Planck scale physics with the use of quantum computers. The research is in the spirit of the original Feynman's idea [23] of performing the so-called exact simulations of quantum systems with the use of quantum information processing devices.
In our previous articles [20,24] we have preliminary explored possibility of utilizing Adiabatic Quantum Computers [25] to simulate quantum gravitational systems. Here, we are making our first steps towards the application of Universal Quantum Computers [26,27].

II. INTERTWINER QUBIT
The basic question a skeptic can ask is why it is worth considering quantum computers to study Planck scale physics at all? Can't we just do it employing classical supercomputers as in the case of CDT approach to quantum gravity? Let me answer to this questions by giving two arguments. The first concerns the huge dimensionality of a Hilbert space for many-body quantum system. For a single spin-1/2 (qubit) Hilbert space H 1/2 = span{|0 , |1 } the dimension is equal 2. However, considering N such spins (qubits) the resulting Hilbert space is a tensor product of N copies of the qubit Hilbert space. The dimension of such space grows exponentially with N:  [29]. The second arguments concerns quantum speed-up leading to reduction of computational complexity of some classical problems. Such possibility is provided by certain quantum algorithms (e.g. Deutsch, Grover, Shor,...) thanks to the so-called quantum parallelism.
Taking the above arguments into account we are convicted that it is justified to explore the possibility of simulating quantum gravitational physics on quantum computers. The fundamental question is, however, whether gravitational degrees of freedom can be expressed with qubits, which are used in the current implementations of the quantum computers 1 ? Fortunately, it has recently been shown that at least in the Loop Quantum Gravity approach to quantum gravity notion of qubit degrees of freedom can be introduced and is associated with the intertwiner space of a certain class of spin networks (see Refs. [18][19][20]24]). 1 In general, quantum variables associated with higher dimensional Hilbert spaces may be considered.
Let us briefly explain it. Namely, nodes of the spin networks are where Hilbert spaces associated with the links meet. The gauge invariance (enforced by the Gauss constraint) implies that the total spin at the node has to be equal zero. The 4-valent nodes are of special interest since they are associated with the non-vanishing eigenvalues of the volume operator (see e.g. Ref. [7]). As already mentioned in Introduction, in the picture of discrete geometry, the 4-valent nodes are dual to tetrahedra. The class of spin networks that we are focused on here are those with links of the spin networks labelled by fundamental representations of the SU (2) group (i.e. the spin labels are equal j = 1/2) and the nodes are 4-valent. For such spin networks the Hilbert spaces at the nodes are given by the following tensor products: There Gauss constraint implies that only singlet configurations (H 0 ) are allowed. Because there are two copies of the spin-zero configurations in the tensor product (3), the so-called intertwiner Hilbert space is two-dimensional: We associate the two-dimensional invariant subspace with the intertwiner qubit |I ∈ H 0 ⊕ H 0 . The 4-valent node (at which the intertwiner qubit is defined) together with the entering links is dual to the tetrahedron in a way shown in Fig. 2.
The two basis states of the intertwiner qubit |I are basically the two singlets we can obtain for a system of four spins 1/2. The basis states can be expressed composing familiar singlets and triplet states for two spin-1/2 particles: Namely, in the s-channel (which is one of the possible superpositions) the intertwiner qubit basis states can be expressed as follows: The |0 s state is simply a tensor product of two singlets for two spin-1/2 particles, while the state |1 s does not have such simple product structure. The states |0 s and |1 s form an orthonormal basis of the intertwiner qubit. Worth stressing is that other bases being linear compositions of |0 s and |0 s might be considered. In particular, the eigenbasis of the volume operator turns out to be useful (see Ref. [24]). Here we stick to the s-channel basis {|0 s , |1 s } in which a general intertwiner state (neglecting the total phase) can be expressed as where θ ∈ [0, π] and φ ∈ [0, 2π) are angles parametrizing the Bloch sphere.
In the context of quantum computations it is crucial to define a quantum algorithm (a unitary operationÛ I acting on a input state) which will allows us to create the intertwiner state (11) from the input state |0000 , i.e.
The general construction of the operatorÛ I can be performed applying the procedure introduced in Ref. [30], and will be discussed in a sequel to this article [31]. Here, for the purpose of illustration of the method of computing vertex amplitude we will focus on the special case of the intertwiner states being the first basis state: |I = |0 s = |S ⊗ |S . The contributing two-partice singlet states can easily be generated as a sequence of elementary gates used to construct quantum circuits: Here, theX is the so-called bit-flip (NOT) operator (Pauli σ x matrix) which transforms TheĤ is the Hadamard and ⊕ is the XOR (exclusive or) logical operation, such that 0⊕0 = 0, 0⊕1 = 1, 1 ⊕ 0 = 1 and 1 ⊕ 1 = 0. In consequence, the |0 s basis state can be expressed as follows: In Fig. 3 a quantum circuit generating (and measuring) the intertwiner state |0 s has been presented. The final state can be written as a superposition of 16 basis states in the product space of four qubit Hilbert spaces: where the normalization condition implies that ijkl∈{0,1} |a ijkl | 2 = 1.
We have executed the quantum algorithm (14) with the use of both IBM simulator of quantum computer and real IBM Q 5-qubit quantum chip ibmqx4. In the case the simulator, 1000 runs were performed while in the case of the quantum chip the algorithm has been executed 1024 times. Results of the measurements of probabilities P (i) = |a i | 2 are summarized in Table I. In order to compare the results from the Table I  are density matrices of the compared states [32]. For this purpose (i.e. reconstruction of the density matrix) full tomography of the obtained quantum state has to be performed.

III. VERTEX AMPLITUDE
Gravity is a theory of constraints. Specifically, in LQG three types of constraints are present. The first is the mentioned Gauss constraint, which has already been imposed at the stage of constructing spin networks states. The second is the spatial diffeomorphism constraint which is satisfied by introducing equivalence relation between all spinnetworks characterized by the same topology. The third is the so-called scalar or Hamil-tonian constraint, which encodes temporal dynamics and is the most difficult to satisfy. In quantum theory, this constraint takes a form of an operator. Let us denote this operator aŝ C. Following the Dirac procedure for constrained quantum systems, the physical states are those belonging to the kernel of the constraints, i.e.Ĉ|Ψ = 0. Due to the complicated form of the gravitational scalar constraint (see e.g. [6]), finding the physical states is in general a difficult task. However, for certain simplified scalar constraints, such as for the symmetry reduced cosmological models, the physical states are possible to extract. Furthermore, it has recently been proposed in Ref. [20] that the problem of solving simple constraints can be implemented on Adiabatic Quantum Computers.
Another approach to the problem of constraints is to consider a projection operator which projects kinematical states onto physical subspace. In particular, the formula (16) is valid forĈ characterized by discrete spectrum of eigenvalues. Specifically, the projection operator (16) can be used to evaluate transition amplitude between any two kinematical states |x and |x : The state |x might correspond to the initial and |x to the final boundary spin network states (confront with Fig. 1). While the notion of the boundary initial and final hypersurfaces is well defined in the case with preferred time foliation, the general relativistic case deserves generalization of the transition amplitude to the form being independent of the background time variable. This leads to the concept of boundary formulation [33] of transition amplitudes in which the transition amplitude is a function of boundary state only.
Taking the particular boundary physical spin network state |Ψ the transition amplitude can be, therefore, written as where the state |Ψ corresponds to representation in which the amplitude is evaluated.
The object of our interest in this article, namely the vertex amplitude is the amplitude (18) of boundary enclosing a single vertex. As we have already explained in Introduction, the spin network enclosing the single vertex has pentagram structure and can be written as: The associated spin network is shown in Fig. 4. The pentagram spin network state is a tensor product of the five intertwiner qubits: Since in the vertex amplitude (18) physical states have to be considered the intertwiner qubits |I n have to be selected such that the state is annihilated by the scalar constraint: Due to the difficulty of the issue for general form of the scalar constraint operator, we de not address the problem of selecting |Ψ states here. As we have mentioned, for either symmetry reduced or simplified scalar constraints the physical states can be identified with the use of existing methods.
Another issue is the choice of the state |W . Usually the representation of holonomies associated with the links of the spin networks are considered. Here, following Ref. [18] we will evaluate the boundary spin network state in the state: where are Bell states associated with the links. Such choice is interesting since the Bell states introduce entanglement between faces of the adjacent tetrahedra. Such a way of "gluing" tetrahedra by quantum entanglement has been recently studied in Ref. [34]. Since the spin network state |Ψ is disentangled one can also interpret W |Ψ as an amplitude of transition between disentangled and strongly entangled piece of quantum geometry.
Going further, possibly the quantum entanglement is the key ingredient which merge the chunks of space associated the nodes of spin networks into a geometric structure. This reasoning is consistent with the recent advances in the domain of entanglement/gravity duality, example of which is provided by the AdS/CFT correspondence [35], ER=EPR conjecture [36] and considerations of holographic entanglement entropy [37][38][39]. Interestingly, it has been recently argued that indeed the spin networks may represent structure of quantum entanglement [40], indicating for relation between spin networks and tensor networks [41]. This is actually not such surprising since the the holonomies associated with links of the spin-networks can be perceived as "mediators" of entanglement.
The holonomies are parallel transport maps between two vector (Hilbert) spaces at the In the case considered in this article, the Hilbert spaces are related with the elementary qubits H 1/2 "living" at the ends of the links of the spin-networks (keep in mind that these are not the intertwiner qubits but the elementary qubits our of which the intertwiner qubits are built). As an example of the holonomy of the Ashtekar connection A considered in LQG (i.e. h e := P exp e A, see e.g. Ref. [6]) let us consider where α is an angle variable and σ x is the Pauli matrix. The holonomies as the one given by Eq. 24 are associated with homogeneous models and are consider in Loop Quantum Cosmology [42] and Spinfoam Cosmology [43,44]. The special case is when α = π/2 for which h x (π/2) = iσ x , which written as an operator 1/2 at the point b 2 . This naturally introduces relation between the quantum states at distant points a and b, which possibly can be associated with entanglement. In order to illustrate the "entanglement" let us consider a superposition |Ψ a = 1 1/2 . If the two states at a and b would be disentangled then performing measurement on the quantum state at a would not influence the quantum state at b. However, once the measurement is performed at the state |Ψ a reducing the state to for instance |0 , the state at b has to be consequently reduced to i|1 . The same works in the reverse direction. Worth mentioning is that the exemplary correlation via holonomies is consistent with the entanglement resulting from the Bell state |E l (22), which we associated with the links. This gives further support to to the choice of the representation state |W given by Eq. 21. However, the issue of relation between holonomies and entanglement requires further more detailed studies, also in the spirit of the recent proposal of Entanglement holonomies [45]. In particular, it has to be confirmed that the correlation introduced via holonomies is the true quantum entanglement violating Bell inequalities.

IV. A QUANTUM ALGORITHM
Having the vertex amplitude (19) defined we may proceed to the task of determining | W |Ψ | 2 with the use of quantum computers. Here, we will show how to obtain modulus the amplitude modulus square (the probability) while extraction of the phase factor will be a subject of our further investigations.
Let us begin with preparation of a suitable quantum register. Because each of the intertwiner qubits is a superposition of four elementary qubits, evaluation of the spin network with N nodes requires 4N qubits in the quantum register 3 . The corresponding Hilbert space is spanned by 2 4N basis states |i , where i ∈ 0, . . . , 2 4N − 1 . The initial state for the quantum algorithm is: Now, we have to find unitary operatorsÛ Ψ andÛ W defined such that where |0 is given by Eq. 26. Utilizing the operatorsÛ Ψ andÛ W we introduce an oper-atorÛ :=Û † WÛ Ψ . Action of this operator on the initial state (26) can be expressed as a superposition of the basis states with some amplitudes a i ∈ C: It is now easy to show that the a 0 coefficient in this superposition is the transition amplitude we are looking for. Namely: By performing measurements on the final state we find the probabilities P (i) = |a i | 2 . The first of these probabilities is the modulus square of the vertex amplitude.
Before we will proceed to the discussion of the pentagram spin network associated with the vertex amplitude let us first demonstrate the algorithm on two simpler examples of spin networks with one and two nodes.

A. Example 1 -single tetrahedron
As a first example let us consider the case of a single-node spin network presented in elementary qubits according to Eq. 14. The representation state |W is a tensor product of two Bell states (22). There are basically two different choices of pairing faces of the tetrahedron. The first choice is according to the pairing of qubits entering to the twoqubit singlets |S out of which the |0 s state is built. The second choice is by linking qubits contributing to the two different singlets. The first choice is trivial since in that caseÛ W =Û Ψ and in consequence the amplitude W |Ψ = 0|Û † WÛ Ψ |0 = 0|0 = 1. Therefore, we will consider the second case for which the quantum circuit associated with theÛ =Û † WÛ Ψ operator is presented in Fig. 6. As a result of simulation (1000 shots) we obtain | W |Ψ | 2 = |a 0 | 2 ≈ 0.275, which is consistent with the theoretically expected value |a 0 | 2 = 0.25. The algorithm cannot be directly executed using the IBM Q 5-qubit quantum chip due to topological constraints of the structure of coupling between qubits. Ancilla qubits have to be introduced for this purpose.

B. Example 2 -two tetrahedra
The second example concerns a bit more complex situation with two-node spin network presented in Fig. 7. Here, the representation state |W similarly to the previous ex- Running the algorithm with the use of IBM quantum simulator (1000 shots) we obtained | W |Ψ | 2 = |a 0 | 2 ≈ 0.070.

V. EVALUATION OF VERTEX AMPLITUDE
We are now ready to address the task of determining the vertex amplitude (19) associated with the boundary spin network state: The other possible choices of the spin network state will discussed in our further work [31].
The |W is given by Eq. 21, representing entanglement between faces of tetrahedra being connected by the links of the spin network. Due to anti-symmetricity of the Bell states (22) for the 10 links under consideration we have in general 2 10 = 1024 ways to order the states between the nodes of the spin network. Here, in order to not distinguish any of the nodes, the configuration in which every node is entangled with two other nodes by the state |E l = 1 √ 2 (|01 − |10 ) and another two nodes by the state e iπ |E l = 1 √ 2 (|10 − |01 ) is considered. The resulting quantum circuit corresponding to the operatorÛ =Û † WÛ Ψ , together with the measurements necessary to find |a 0 | 2 = | W |Ψ | 2 is shown in Fig. 9. The quantum circuit employs 20-qubit quantum register with the initial state: The algorithm introduced in Sec. IV requires finding amplitude of the initial state (32) in the final state. One has to keep in mind that the Hilbert space of the 20-qubit system is spanned by over million basis states: 2 20 = 1048576. Therefore, selecting amplitude of one of the basis states (i.e. |0 ) is not an easy task. Sufficiently high number of measurement has to be performed in order to extract the final state. For this purpose we ran the algorithm on the IBM simulator of quantum computer, performing maximal allowed number of shots equal 8192. The simulator has maximal precision of determining probability amplitude of a basis state equal 0.001. Contribution from the (32) state in the final state has been classified as smaller than the error of measurement. Therefore, the only conclusion we can draw from the measurements is that the value of the vertex amplitude is: Taking into account the fact that for homogeneous distribution of probabilities between the basis states we have 1/2 20 ≈ 10 −6 the constraint (33) is rather weak. In order to actually have chance to measure value of the probability amplitude, the precision of measurement has to be increased at least 10 3 times to reach the level of 10 −6 .

VI. SUMMARY
The purpose of this article was to explore the possibility of computing vertex amplitudes in the spin foam models of quantum gravity with the use of quantum algorithms.
The notion of intertwiner qubit being crucial to implement the vertex amplitudes on quantum computers has been pedagogically introduced. It has been show how one of the basis states of the intertwiner qubit can be implemented with the use available IBM 5qubit quantum computer. Thereafter, a quantum algorithm allowing to determine vertex amplitude modulus square (| W |Ψ | 2 ) has been introduced. Utility of the algorithm has been demonstrated on examples of single-node and two-node spin networks. For the two cases, associated boundary states probabilities have been determined with the use of simulator of universal quantum computer provided by IBM company. Finally, the algorithm has been applied to the case of pentagram spin network representing boundary of the spin foam vertex. Upper limit on the vertex amplitude in a certain quantum state has been imposed with the use of 20-qubit register of the IBM quantum simulator.
The presented results are the first step towards simulating spin foam models (associated with Loop Quantum Gravity and Group Field Theories) with the use of universal quantum computers. In particular, the vertex amplitudes can be applied as elementary building blocks in construction of more complex transition amplitudes. The aim of the developed direction is to achieve possibility of studying collective behavior of the Planck scale systems composed of huge number elementary constituents ("atoms of space/spacetime"). Exploration the many-body Planck scale quantum systems [46] may allow to extract continuous and semi-classical limits from the dynamics of the "fundamental" degrees of freedom. This is crucial to make contact between Planck scale physics and empirical sciences.
Worth stressing is that the results presented in this article are preliminary steps which set up the stage for further more detailed studies. In particular, the following points have to be addressed: • Introduction of a quantum circuit for the general intertwiner qubit |I (Eq. 11).
• Determination of the phase of vertex amplitude with the use of quantum algorithms (e.g. Quantum Phase Estimation Algorithm [47]).
• Investigation of different types of the state |W .
• Application of simulators of quantum computers, characterized by higher precision that the IBM simulator employed here.
• Solving the quantum constraints with Universal Quantum Computers.
• Investigation of the architectures of forthcoming quantum processors (with the N > 100 number of qubits) in terms of application in determining spin foam transition amplitudes.
Some of the tasks will be subject of a sequel [31] to this article.