**6. 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 shown how one of the two basis states of the intertwiner qubit can be implemented with the use available IBM 5-qubit quantum computer. To the best of our knowledge it was the first time ever a quantum gravitational quantity has been simulated on superconducting quantum chip.

Thereafter, a quantum algorithm allowing to determine modulus square of spin foam vertex amplitude (| *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, probabilities of the associated boundary states have been determined with the use of IBM and QX quantum computer simulators. Finally, the algorithm has been applied to the case of pentagram spin network, representing boundary of the

spin foam vertex. Value of the modulus square of the amplitude in a certain quantum state has been measured with the use of 20-qubit register of the IBM and QX quantum simulators. While the QX results were close to the analytically predicted value, the outcomes of the IBM simulator failed to reproduce theoretical predictions.

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 the possibility of studying collective behavior of the Planck scale systems composed of huge number of elementary constituents ("atoms of space/spacetime"). Exploration of the many-body Planck scale quantum systems [55] may allow to extract continuous and semi-classical limits from the dynamics of the "fundamental" degrees of freedom. This is crucial from the perspective of making contact between Planck scale physics and empirical sciences.

Worth stressing is that the results presented in this article are rather preliminary and only set up the stage for further, more detailed studies. In particular, the following points have to be addressed:


Some of the tasks will be subject of a sequel to this article [34].

**Funding:** This research was funded by the Sonata Bis Grant DEC-2017/26/E/ST2/00763 of the National Science Centre Poland and the Mobilno´s´c Plus Grant 1641/MON/V/2017/0 of the Polish Ministry of Science and Higher Education.

**Conflicts of Interest:** The author declare no conflict of interest.

### **Appendix A. Quantum Computing Technologies**

The domain of quantum computing is currently experiencing an unprecedented speedup. The recent progress is mainly due to advances in development of the superconducting qubits [23]. In particular, utilizing the superconducting circuits, the IBM company has developed 5 and 20 qubit (noisy) quantum computers, which are accessible in cloud [22]. Furthermore, a prototype of 50 qubit quantum computer by this company has been built and is currently in the phase of tests. The company has recently also unveiled its first commercial 20-qubit quantum computer IBM Q System One. This is intended to be the first ever commercial universal quantum computer, and the second commercially available quantum computer after the adiabatic quantum computer (quantum annealer [57]) provided by D-Wave Systems [58]. The latests D-Wave 2000Q annealer uses a quantum chip with 2048 superconducting qubits, connected in the form of the so-called *chimera* graph. Another important player in the quantum race is Intel, which recently developed its 49-qubit superconducting quantum chip named Tangle Lake [59]. However, outside of the superconducting qubits, the company in collaboration with QuTech [60] advanced centre for Quantum Computing and Quantum Internet is also developing an approach to quantum computing based on electron's spin based qubits, stored in quantum dots. Further advances in the area of superconducting quantum circuits come from Google [61] and Rigetti Computing [62]. The first company has recently announced their 72-qubit quantum chip, while the second one is currently developing its 128-qubit universal quantum chip. On the other hand, the world's leading software company—Microsoft has focused its approach to quantum computing on topological qubits through Majorana fermions [63]. The alternative to superconducting qubits is also developed by IonQ Inc. startup, which is developing a trapped ion quantum computer based on ytterbium atoms [64]. The most recent quantum computer by this company allows to operate on 79 qubits, which is the current world record. The above are only the most sound examples of the advancement which has been made in the recent years in the area of hardware dedicated to quantum computing. There are still many challenges to be addressed, including reduction of the gate errors and increase of fidelity of the quantum states. However, even in pessimistic scenarios, the current momentum of the quantum computing technologies will undoubtedly lead to emergence of reliable and advantageous quantum machines (which cannot be emulated on classical supercomputers) in the coming decade. There are no fundamental physical reasons identified, which could stop the progress. However, the rate of the progress will depend on whether commercial applications of quantum computing technologies will emerge in the coming five years, stimulating further funding of research and development. See e.g., Ref. [65] for more detailed discussion of this issue.

Major players in the field, with large financial resources, such as IBM or governments may sustain the progress independently on short-term returns (which is not the case for start-ups). This may allow for a stable long term progress. In particular, IBM has recently announced a possibility of doubling a measure called *Quantum Volume* [66] every year [67]. The Quantum Volume *VQ* is basically a maximal size of a certain random circuit, with equal width and depth, which can be successfully implemented on a given quantum computer. The current (2019) IBM's value of *VQ* is 16 and corresponds to the IBM Q System One quantum computer mentioned above. This means that any quantum algorithm employing 4 qubits and four layers (time steps) of quantum circuit can be successfully implemented on the computer. If the trend will follow the hypothesized geometric trajectory (a sort of a new Moore's law [68] for integrated circuits), then one could expect the quantum volume *VQ* to be of the order of 10<sup>3</sup> in 2025 and 10<sup>5</sup> in 2030. This means that in 2025, algorithms employing roughly log2 10<sup>3</sup> ≈ 10 qubits and the same number of time steps will be possible to execute. This number will increase to approximately 16 qubits until the end of the coming decade. While this may not sound very optimistic, the prediction is very conservative and does not rule out that much bigger (non-random) circuits (especially well-fitted to the hardware) will be possible to execute at the same time.

### **Appendix B. Basics of Quantum Computing**

The aim of the appendix is to provide a basic introduction of the concepts in quantum computing used in this article. This appendix will allow quantum gravity researchers who are not familiar with quantum computing, to grasp the relevant concepts.

The quantum computing is basically processing of quantum information. While the elementary portion of classical information is a *bit* {0, <sup>1</sup>}, its quantum counterpart is what we call a *qubit*. A single qubit is a state |Ψ in two-dimensional Hilbert space, which we denote as H = span{|<sup>0</sup> , |1 }. The space is spanned by two orthonormal basis states |0 and |1 , so that 1|0 = 0 and 0|0 = 1 = 1|1 . A general qubit is a superposition of the two basis states:

$$
\langle \Psi \rangle = a \vert 0 \rangle + \beta \vert 1 \rangle,\tag{A1}
$$

where, *α*, *β* ∈ C (complex numbers), and the normalization condition Ψ|Ψ = 1 implies that |*α*| 2 + |*β*| 2 = 1.

There are different unitary quantum operations which may be performed on the quantum state |Ψ . The elementary quantum operations are called *gates*, in analogy to electric circuits implementing Boolean logic. For instance, the so-called bit-flip operator *X*ˆ which transforms |0 into |1 and |1 into |0 (*X*ˆ |0 = |1 and *X*ˆ |1 = |0 ) can be introduced. The *X*ˆ operator introduces the NOT operation on a single qubit, and has representation in the form of the Pauli *x* matrix. Similarly, one can introduce *Y*ˆ and *Z*ˆ operators corresponding to the other two Pauli matrices. The *computational basis* {|0 , |1 } is usually introduced such that the basis states are eigenvectors of the *Z*ˆ operator: *Z*ˆ |0 = |0 and *Z*ˆ |1 = −|1 .

Another important operator (which does not have its classical counterpart) is the Hadamard operator *H* ˆ which is defined by the following action on the qubit basis states:

$$
\hat{H}|0\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle + |1\rangle\right) \text{ and } \hat{H}|1\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle - |1\rangle\right). \tag{A2}
$$

The above are examples of operators acting on a single qubit. However, quantum information processing usually concerns a multiple qubit system called *quantum register*. The quantum state of the register of *N* qubits belongs to a tensor product of *N* copies of single qubit Hilbert spaces: /*Ni*=<sup>1</sup> H*i*. The dimension of the product Hilbert space is dim /*Ni*=<sup>1</sup> H*i* = 2*N*. This exponential dependence of the dimensionality on *N* is the main obstacle behind simulating quantum systems on classical computers.

A *quantum algorithm* is a unitary operator *U*ˆ acting on the initial state of the quantum registes |**0** := <sup>⊗</sup>*Ni*=<sup>1</sup>|<sup>0</sup> ∈ /*Ni*=<sup>1</sup> H*i*, together with a sequence of measurements. The outcome of the quantum algorithm is obtained by performing measurements on the final sate: *U* ˆ |**0** . Because of the probabilistic nature of quantum mechanics, the procedure has to be performed repeatedly in order to reconstruct the final state. In general, full reconstruction of the final state *U* ˆ |**0** requires the so-called *quantum tomography* to be applied. In the procedure, states of the qubits are measured in different bases (not only in the computational basis). The quantum state tomography, which reconstructs the density matrix *ρ* ˆ = *U* ˆ |**0 0**|*U*<sup>ˆ</sup> †, is however not always required. In most of the considered quantum algorithms, only probabilities (not complex amplitudes) of the basis states are necessary to measure, which is much simpler and faster than the quantum state tomography.

As already mentioned, the unitary operator *U* ˆ can be decomposed into elementary operations called quantum gates. The already introduced *X* ˆ and *H* ˆ operators are examples of single-qubit gates. However, the gates may also act on two or more qubits. An example of 2-qubit gate relevant for the purpose of this article is the so-called controlled-NOT (CNOT) gate, which we denote as *C* ˆ . The operator is acting on 2-qubit state |*ab* ≡|*a* ⊗|*b* , where |*a* and |*b* are single quibit states. Action of the CNOT operator on the basis states can be expressed as follows: *C* <sup>ˆ</sup>(|*a* ⊗|*b* ) = |*a* ⊗|*a* ⊕ *b* , where *a*, *b* ∈ {0, <sup>1</sup>}. The ⊕ is the XOR (exclusive or) logical operation (equivalent to addition modulo 2), defined as 0 ⊕ *b* = *b*, and 1 ⊕ *b* = <sup>¬</sup>*b*, where by ¬*b* we denote negation (NOT) of *b*. This explains why the gate is called the controlled-NOT (CNOT). The first qubit (|*a* ) is a control qubit, while the second (|*b* ) is a target qubit. The first qubits acts as a switch, which turns on negation of the second quibit if *a* = 1 and remain the second qubit unchanged if *a* = 0.

The diagrammatic representation of the of the unitary operator *U* ˆ composed of elementary quantum gates is called a *quantum circuit*, examples of which can be found through this article. Each computational qubit is associated with a horizontal line, which arranges the order at which the operations are performed (direction of time). The operations are executed from the left to the right. Then, the symbols representing gates can be place on either a single-qubit line (e.g., X,Y,Z,H gates) or by joining two or more lines (e.g., CNOT, Toffoli gates).

One of the advantages of quantum algorithms is the possibility of implementing the so-called *quantum parallelism*, which allows to reduce computational complexity of certain problems. The most known example is the Shor algorithm [69] which allows to reduce classical NP complexity of the factorization problem into the BQP complexity class (see e.g., Ref. [70] for definitions of complexity classes). Another seminal example is the Grover algorithm [71] which, statistically, reduces the number of steps needed to find an element in the random database containing *N* elements from classical *N*/2 to <sup>O</sup>(√*N*). Even if we do not make use of quantum parallelism and the resulting reduction of computational complexity in this paper, the methods may also find application in the context of simulations of spin networks. This especially concerns the Quantum Phase Estimation Algorithm [56] which may possibly be applied to effectively measure phases of the spin foam amplitudes.
