Abstract
The aim of this work is to give an introduction for a non-practical reader to the growing field of quantum machine learning, which is a recent discipline that combines the research areas of machine learning and quantum computing. This work presents the most notable scientific literature about quantum machine learning, starting from the basics of quantum logic to some specific elements and algorithms of quantum computing (such as QRAM, Grover and HHL), in order to allow a better understanding of latest quantum machine learning techniques. The main aspects of quantum machine learning are then covered, with detailed descriptions of some notable algorithms, such as quantum natural gradient and quantum support vector machines, up to the most recent quantum deep learning techniques, such as quantum neural networks.
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Notes
The C01 can be referred in literature as \(\overline {\mathrm {C}}\)NOT, and its control value should be |0〉 in order to change.
Kullback–Leibler divergence or KL divergence is a measure of how far away two probability distributions are from each other.
Fubini-Study metric, seen as the Fisher information matrix in quantum field.
See https://www.statista.com/statistics/993634/quantum-computers-by-number-of-qubits/ for a general idea of the trend.
For a better comprehension of the phenomenon, see Feynman (1965)
References
Aaronson S (2007) The learnability of quantum states. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463(2088):3089–3114. https://doi.org/10.1098/rspa.2007.0113
Acampora G, Schiattarella R (2021) Deep neural networks for quantum circuit mapping. Neural Comput Appl:1–21
Aerts D, Czachor M (2004) Quantum aspects of semantic analysis and symbolic artificial intelligence. J Phys A Math Gen 37(12):L123–L132. https://doi.org/10.1088/0305-4470/37/12/l01
Aleksandrowicz G, Alexander T, Barkoutsos P, Bello L, Ben-Haim Y, Bucher D, Cabrera-Hernández F J, Carballo-Franquis J, Chen A, Chen C-F et al (2019) Qiskit: An open-source framework for quantum computing. Accessed on: May https://doi.org/10.5281/zenodo.2562111
Allcock J, Hsieh C-Y, Kerenidis I, Zhang S (2020) Quantum algorithms for feedforward neural networks. ACM Trans Quantum Comput 1(1):1–24
Altaisky M V (2001) Quantum neural network. arXiv:quant-ph/0107012
Ameur E, Brassard G, Gambs S (2006) Machine learning in a quantum world. Adv Artif Intell Lect Notes Comput Sci:431–442. https://doi.org/10.1007/11766247_37
An D, Fang D, Lin L (2020) Time-dependent unbounded hamiltonian simulation with vector norm scaling
Anguita D, Ridella S, Rivieccio F, Zunino R (2003) Quantum optimization for training support vector machines. Neural Netw 16(5-6):763–770
Arjovsky M, Chintala S, Bottou L (2017) Wasserstein generative adversarial networks. In: International conference on machine learning. PMLR, pp 214–223
Arthur D, Vassilvitskii S (2006) k-means++: the advantages of careful seeding. Technical report, Stanford
Arute F, Arya K, Babbush R et al (2019) Quantum supremacy using a programmable superconducting processor. Nature. https://doi.org/10.1038/s41586-019-1666-5
Arute F, Arya K, Babbush R, Bacon D, Bardin J C, Barends R, Biswas R, Boixo S, Brandao F G S L, Buell D A et al (2019) Quantum supremacy using a programmable superconducting processor. Nature 574(7779):505–510. https://doi.org/10.1038/s41586-019-1666-5
Bausch J (2020) Recurrent quantum neural networks. arXiv:2006.14619
Beer K, Bondarenko D, Farrelly T, Osborne T J, Salzmann R, Scheiermann D, Wolf R (2020) Training deep quantum neural networks. Nat Commun 11(1). https://doi.org/10.1038/s41467-020-14454-2
Bengio Y, Goodfellow I, Courville A (2017) Deep learning, vol 1. MIT press Massachusetts, USA
Bergholm V, Izaac J, Schuld M, Gogolin C, Alam M S, Ahmed S, Arrazola J M, Blank C, Delgado A, Jahangiri S, McKiernan K, Meyer J J, Niu Z, Szva A, Killoran N (2020) Pennylane: automatic differentiation of hybrid quantum-classical computations
Berry D W, Ahokas G, Cleve R, Sanders B C (2007) Efficient quantum algorithms for simulating sparse hamiltonians. Commun Math Phys 270(2):359–371
Biamonte J, Wittek P, Pancotti N, Rebentrost P, Wiebe N, Lloyd S (2017) Quantum machine learning. Nature 549(7671):195–202. https://doi.org/10.1038/nature23474
Boixo S, Isakov S V, Smelyanskiy V N, Babbush R, Ding N, Jiang Z, Bremner M J, Martinis J M, Neven H (2018) Characterizing quantum supremacy in near-term devices. Nat Phys 14(6):595–600. https://doi.org/10.1038/s41567-018-0124-x
Broughton M, Verdon G, McCourt T, Martinez A J, Yoo J H, Isakov S V, Massey P, Niu M Y, Halavati R, Peters E, Leib M, Skolik A, Streif M, Von Dollen D, McClean J R, Boixo S, Bacon D, Ho A K, Neven H, Mohseni M (2020) TensorFlow quantum: a software Framework for quantum machine learning. arXiv:2003.02989
Bruza P, Cole R (2005) Quantum logic of semantic space: an exploratory investigation of context effects in practical reasoning. In: Lamb LC, Woods J, Artemov S, Barringer A, d’Avila Garcez A (eds) We Will Show Them! Essays in Honour of Dov Gabbay. https://eprints.qut.edu.au/7179/. College Publications, United Kingdom, pp 339–362
Bshouty N, Jackson J (1999) Learning dnf over the uniform distribution using a quantum example oracle. SIAM J Comput 28:1136–1153. https://doi.org/10.1137/S0097539795293123
Cao Y, Guerreschi GG, Aspuru-Guzik A (2017) Quantum neuron: an elementary building block for machine learning on quantum computers. arXiv:1711.11240
Carleo G, Cirac I, Cranmer K, Daudet L, Schuld M, Tishby N, Vogt-Maranto L, Zdeborová L (2019) Machine learning and the physical sciences. Rev Mod Phys 91:045002. https://doi.org/10.1103/RevModPhys.91.045002
Chakraborty S, Das T, Sutradhar S, Das M, Deb S (2020) An analytical review of quantum neural network models and relevant research. In: 2020 5th International Conference on Communication and Electronics Systems (ICCES). IEEE, pp 1395–1400
Chen B-Q, Niu X-F (2020) A novel neural network based on quantum computing. Int J Theor Phys 59:2029–2043
Chen J, Wang L, Charbon E (2017) A quantum-implementable neural network model. Quantum Inf Process 16(10):1–24
Childs A M, Maslov D, Nam Y, Ross N J, Su Y (2018) Toward the first quantum simulation with quantum speedup. Proc Ntl Acad Sci 115(38):9456–9461. https://doi.org/10.1073/pnas.1801723115
Childs A M, Wiebe N (2012) Hamiltonian simulation using linear combinations of unitary operations. Quantum Inf Comput 12. https://doi.org/10.26421/qic12.11-12
Ciliberto C, Herbster M, Ialongo A D, Pontil M, Rocchetto A, Severini S, Wossnig L (2018) Quantum machine learning: a classical perspective. Proc R Soc A: Math Phys Eng Sci 474 (2209):20170551. https://doi.org/10.1098/rspa.2017.0551
Cong I, Choi S, Lukin M D (2019) Quantum convolutional neural networks. Nat Phys 15 (12):1273–1278
Cross A W, Smith G, Smolin J A (2015) Quantum learning robust against noise. Phys Rev A 92(1):012327
Dallaire-Demers P-L, Killoran N (2018) Quantum generative adversarial networks. Phys Rev A 98(1):012324
de Paula Neto F M, Ludermir T B, de Oliveira W R, da Silva A J (2019) Implementing any nonlinear quantum neuron. IEEE Trans Neural Netw Learn Syst 31(9):3741–3746
Denchev V S, Ding N, Vishwanathan SVN, Neven H (2012) Robust classification with adiabatic quantum optimization. arXiv:1205.1148
DiVincenzo D P (2013) Quantum information processing: Lecture notes of the 44th iff spring school 2013. Forschungszentrum
Dunjko V, Briegel H J (2018) Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Rep Prog Phys 81(7):074001. https://doi.org/10.1088/1361-6633/aab406
Dunjko V, Wittek P (2020) A non-review of quantum machine learning: trends and explorations. Quantum Views 4:32
Durr C, Hoyer P (1996) A quantum algorithm for finding the minimum. arXiv:quant-ph/9607014
Edward Farhi H N (2018) Classification with quantum neural networks on near term processors
Farhi E, Goldstone J, Gutmann S, Neven H (2017) Quantum algorithms for fixed qubit architectures
Farhi E, Goldstone J, Gutmann S, Sipser M (2000) Quantum computation by adiabatic evolution. arXiv:quant-ph/0001106
Feynman R P (1965) The feynman lectures on physics, vol 3. Narosa
Friedman J, Hastie T, Tibshirani R et al (2001) The elements of statistical learning. Springer series in statistics New York 1(10)
Gill S S, Kumar A, Singh H, Singh M, Kaur K, Usman M, Buyya R (2020) Quantum computing: A taxonomy, systematic review and future directions. arXiv:2010.15559
Giovannetti V, Lloyd S, Maccone L (2008) Quantum random access memory. Phys Rev Lett 100(16). https://doi.org/10.1103/physrevlett.100.160501
Gisin N, Bechmann-Pasquinucci H (1998) Bell inequality, bell states and maximally entangled states for n qubits. Phys Lett A 246(1-2):1–6
Goodfellow I J, Pouget-Abadie J, Mirza M, Xu B, Warde-Farley D, Ozair S, Courville A, Bengio Y (2014) Generative adversarial networks. arXiv:1406.2661
Grover L K (1998) A framework for fast quantum mechanical algorithms. In: Proceedings of the thirtieth annual ACM symposium on Theory of computing, pp 53–62
Gu J, Wang Z, Kuen J, Ma L, Shahroudy A, Shuai B, Liu T, Wang X, Wang G, Cai J et al (2018) Recent advances in convolutional neural networks. Pattern Recogn 77:354–377
Guerreschi G G (2019) Repeat-until-success circuits with fixed-point oblivious amplitude amplification. Phys Rev A 99(2):022306
Gyongyosi L, Imre S (2019) A survey on quantum computing technology. Comput. Sci. Rev. 31:51–71
Harrow A, Napp J (2019) Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms
Havlíček V, Córcoles A D, Temme K, Harrow A W, Kandala A, Chow J M, Gambetta J M (2019) Supervised learning with quantum-enhanced feature spaces. Nature 567(7747):209–212
Henderson M, Shakya S, Pradhan S, Cook T (2020) Quanvolutional neural networks: powering image recognition with quantum circuits. Quantum Mach Intell 2(1):1–9
Holevo A S (1973) Bounds for the quantity of information transmitted by a quantum communication channel. Probl Pered Inf 9(3):3–11
Huang C, Newman M, Szegedy M (2018) Explicit lower bounds on strong quantum simulation. arXiv:1804.10368
Huang H-Y, Broughton M, Mohseni M, Babbush R, Boixo S, Neven H, McClean J R (2020) Power of data in quantum machine learning
Huang H-Y, Kueng R, Preskill J (2020) Predicting many properties of a quantum system from very few measurements. Nat Phys 16(10):1050–1057. https://doi.org/10.1038/s41567-020-0932-7
Jacot A, Gabriel F, Hongler C (2020) Neural tangent kernel: convergence and generalization in neural networks
Jaeger H, Haas H (2004) Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication. Science 304(5667):78–80
James D (2019) Quantum computing algorithms for applied linear algebra. https://mysite.science.uottawa.ca/hsalmasi/report/report-JamesDickens.pdf
Joachims T (2006) Training linear svms in linear time. In: Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, pp 217–226
Kak S C (1995) Quantum neural computing. Adv Imaging Electron Phys 94:259–313
Kamruzzaman A, Alhwaiti Y, Leider A, Tappert C C (2020) Quantum deep learning neural networks. In: Arai K, Bhatia R (eds) Advances in Information and Communication. Springer International Publishing, Cham, pp 299–311
Kerenidis I, Landman J, Luongo A, Prakash A (2019) q-means: A quantum algorithm for unsupervised machine learning. In: Advances in Neural Information Processing Systems, pp 4134–4144
Killoran N, Bromley T R, Arrazola J M, Schuld M, Quesada N, Lloyd S (2019) Continuous-variable quantum neural networks. Phys Rev Res 1(3):033063
Lahtinen V, Pachos J K (2017) A short introduction to topological quantum computation. SciPost Physics 3(3)
Landau L D, Lifsits E M (2010) Fisica teorica 1 - meccanica. Editori Riuniti
LeCun Y, Bottou L, Bengio Y, Haffner P (1998) Gradient-based learning applied to document recognition. Proc IEEE 86(11):2278–2324
LeCun Y, Cortes C, Burges CJ (2010) Mnist handwritten digit database. Florham Park, NJ
Li F, Xu G (2009) Quantum bp neural network for speech enhancement. In: 2009 Asia-Pacific Conference on Computational Intelligence and Industrial Applications (PACIIA), vol 2. IEEE, pp 389–392
Lloyd S (2010) Quantum algorithm for solving linear systems of equations. APS 2010:D4–002
Lloyd S, Mohseni M, Rebentrost P (2013) Quantum algorithms for supervised and unsupervised machine learning. arXiv:1307.0411
Lloyd S, Mohseni M, Rebentrost P (2014) Quantum principal component analysis. Nat Phys 10(9):631–633
Lloyd S, Schuld M, Ijaz A, Izaac J, Killoran N (2020) Quantum embeddings for machine learning. arXiv:2001.03622
Mandic D, Chambers J (2001) Recurrent neural networks for prediction: learning algorithms, architectures and stability. Wiley
Mari A (2019) Quanvolutional neural networks. https://pennylane.ai/qml/demos/tutorial_quanvolution.html
Marshall K, Pooser R, Siopsis G, Weedbrook C (2015) Repeat-until-success cubic phase gate for universal continuous-variable quantum computation. Phys Rev A 91(3):032321
Matsui N, Takai M, Nishimura H (2000) A network model based on qubitlike neuron corresponding to quantum circuit. Electron Commun Jpn (Part III: Fund Electron Sci) 83(10):67–73. https://doi.org/10.1002/(SICI)1520-6440(200010)83:10<67::AID-ECJC8>3.0.CO;2-H
Mcclean J R, Boixo S, Smelyanskiy V N, Babbush R, Neven H (2018) Barren plateaus in quantum neural network training landscapes. Nat Commun 9(1). https://doi.org/10.1038/s41467-018-07090-4
McInnes L, Healy J, Melville J (2018) Umap: uniform manifold approximation and projection for dimension reduction. arXiv:1802.03426
Mirza M, Osindero S (2014) Conditional generative adversarial nets. arXiv:1411.1784
Mishra N, Kapil M, Rakesh H, Anand A, Mishra N, Warke A, Sarkar S, Dutta S, Gupta S, Dash A P et al (2021) Quantum machine learning: a review and current status. Data Manag Anal Innov:101–145
Novak R, Xiao L, Hron J, Lee J, Alemi A A, Sohl-Dickstein J, Schoenholz S S (2019) Neural tangents: fast and easy infinite neural networks in python
Olivares S (2020) Lecture notes on quantum computing. https://sites.unimi.it/olivares/wp-content/uploads/2020/04/lectures_qc_Olivares_v4.0.pdf. Last Accessed on 29 Apr 2021
Paetznick A, Svore K M (2013) Repeat-until-success: non-deterministic decomposition of single-qubit unitaries. arXiv:1311.1074
Radford A, Metz L, Chintala S (2015) Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv:1511.06434
Ranzato M, Huang F J, Boureau Y-L, LeCun Y (2007) Unsupervised learning of invariant feature hierarchies with applications to object recognition. In: 2007 IEEE conference on computer vision and pattern recognition. IEEE, pp 1–8
Raussendorf R, Briegel H (2001) Computational model underlying the one-way quantum computer. arXiv:quant-ph/0108067
Rodríguez-García M A, Castillo I P, Barberis-Blostein P (2020) Efficient qubit phase estimation using adaptive measurements
Rupp M, von Lilienfeld O A, Burke K (2018) Guest editorial: Special topic on data-enabled theoretical chemistry. J Chem Phys 148(24):241401. https://doi.org/10.1063/1.5043213
Sakuma T (2020) Application of deep quantum neural networks to finance
Schuld M, Killoran N (2019) Quantum machine learning in feature hilbert spaces. Phys Rev Lett 122(4). https://doi.org/10.1103/physrevlett.122.040504
Schuld M, Petruccione F (2018) Supervised learning with quantum computers. Springer
Schuld M, Sinayskiy I, Petruccione F (2014) An introduction to quantum machine learning. Contemp Phys 56(2):172–185. https://doi.org/10.1080/00107514.2014.964942
Servedio R A, Gortler S J (2004) Equivalences and separations between quantum and classical learnability. SIAM J Comput 33(5):1067–1092. https://doi.org/10.1137/s0097539704412910
Shor P W (1996) Fault-tolerant quantum computation. In: Proceedings of 37th Conference on Foundations of Computer Science. IEEE, pp 56–65
Shor P W (1999) Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev 41(2):303–332
Smith R S, Curtis M J, Zeng W J (2016) A practical quantum instruction set architecture
Stokes J, Izaac J, Killoran N, Carleo G (2020) Quantum natural gradient. Quantum 4:269. https://doi.org/10.22331/q-2020-05-25-269
Stoudenmire E M, Schwab D J (2017) Supervised learning with quantum-inspired tensor networks
Svore K, Roetteler M, Geller A, Troyer M, Azariah J, Granade C, Heim B, Kliuchnikov V, Mykhailova M, Paz A (2018) Q#. Proceedings of the Real World Domain Specific Languages Workshop 2018 on - RWDSL2018. https://doi.org/10.1145/3183895.3183901
Tacchino F, Macchiavello C, Gerace D, Bajoni D (2019) An artificial neuron implemented on an actual quantum processor. npj Quantum Inf 5(1):1–8
Tandon P, Lam S, Shih B, Mehta T, Mitev A, Ong Z (2017) Quantum robotics: a primer on current science and future perspectives. Synthesis Lect Quantum Comput 6(1):1–149. https://doi.org/10.2200/S00746ED1V01Y201612QMC010
van Dam W, Mosca M, Vazirani U (2001) How powerful is adiabatic quantum computation?. In: Proceedings 42nd IEEE Symposium on Foundations of Computer Science, pp 279–287
Ventura D, Martinez T (1999) A quantum associative memory based on grover’s algorithm. In: ICANNGA
Wan K H, Dahlsten O, Kristjánsson H, Gardner R, Kim MS (2017) Quantum generalisation of feedforward neural networks. npj Quantum Inf 3(1):1–8
Wiebe N, Kliuchnikov V (2013) Floating point representations in quantum circuit synthesis. New J Phys 15(9):093041
Wittek P (2014) Quantum machine learning: what quantum computing means to data mining. Academic Press
Yumin D, Wu M, Zhang J (2020) Recognition of pneumonia image based on improved quantum neural network. IEEE Access 8:224500–224512. https://doi.org/10.1109/ACCESS.2020.3044697
Zhao L, Zhao Z, Rebentrost P, Fitzsimons J (2019) Compiling basic linear algebra subroutines for quantum computers
Zhao R, Wang S (2021) A review of quantum neural networks: methods, models, dilemma. arXiv:2109.01840
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Appendix
Appendix
1.1 A quantum mechanics: basic elements
In this section, some basic concepts of quantum mechanics are introduced. Olivares (2020)
Since quantum systems often behave very differently from classical systems, a novel representation is needed in their description. Understanding the meaning of this representation and its mechanisms allows the reader to comprehend the basics of quantum computation and quantum machine learning (QML).
1.2 A.1 System, states, measures
With the term physical system, we refer to any portion of the physical universe that can be described and analyzed. Classical systems are described by the laws of classical physics, an object falling from a cliff, an electric charge in space, etc.
Quantum systems, on the other hand, are described by the laws of quantum physics. Ensemble of atoms or particles are typical example of quantum systems. Actually, any event happening at atomic or subatomic scale can be described in term of a quantum system. Now, it is known that classical physical systems have a deterministic nature: given an initial state, the system will produce always the same output. For example, the behaviour of a moving object can be predicted given a certain set of initial parameters. More precisely, the result of any measurement on that object (position at a certain time, velocity, rotational speed) can be predicted in advance. On the other hand, quantum systems have an intrinsic stochastic behaviour. One cannot predict with certainty the result of a measure, but only its probability. In other words, a certain quantum measurement on a given system can have multiple outcomes, and each of these outcomes will manifest itself with a given probability, which can be predicted in advance. To represent this situation, the concept of quantum state is introduced. A state is an object that allows for the calculation of events probabilities, and contains all the information about a given system. To refer to a state, the Dirac’s notation is used: a state of the system is associated with an object called ket, through the notation |x〉.
The simplest possible quantum system is called qubit, that has the following representation:
where a+ and a− are complex numbers. When a measurement is realized on this system, it can only have one of two possible outcomes, which are represented by |+〉 and |−〉. Each of these outcomes has a probability of respectively |a+|2 and |a−|2. It is said that the system is a superposition of two states. The two states “coexists” inside the system, and one or the other can be revealed with a measurement. Mathematically, the state |ϕ〉 is a vector in Hilbert Space \({\mathscr{H}}\), of which |+〉 and |−〉 are a basis. A practical example of this situation might be the following: the state |ϕ〉 represent the position of a single, static particle. This particle can either be in two (fixed) position. As said, a quantum state (a ket) represent a physical system. In the case reported in Eq. 77, the state |+〉 represents the particle in the first position, while the state |−〉 represents the particle in the second position. The state |ϕ〉, finally, represents the particle whose position is unknown. The particle is both in the first and second position, with given probabilities.
It is important to remark that the state |ϕ〉 does not simply mean since the position is unknown, it can be one or the other, but physically represent a particle that is both in position one and two Footnote 5.
By implicitly assuming that the only possible outcomes of the measure are |+〉 and |−〉, it is possible to write:
In other words, the state is represented as a linear combination of the basis vectors of the Hilbert space \({\mathscr{H}}\).
where:
Equivalently, another representation for states is defined: the bra.
where the symbol (∗) denotes the adjointness operation. Given |ϕ〉 in Eq. 77:
It is then clear that the product between bra and ket is equivalent to the inner product between vectors:
While the ket-bra product yields a matrix:
1.3 A.2 Operators
An operator is an object that, when applied to a quantum state, yields another quantum state:
For a two-level system, it’s possible to write:
So, in this case, an operator can be thought as a square matrix. In quantum mechanics, every observable (speed, momentum, position, etc.) is an operator. It it possible to obtain the mean value of an observable O on a given state|ϕ〉 with the following expression:
It is important to remark that any information on a given quantum state must be obtained through a measurement. For this reason, in quantum computing and quantum machine learning the concepts of observable and measurement are very important, since they represent the only way to interact with a quantum system.
Another important concept in quantum mechanics is the Wave Function collapse: basically, any form of measurements on a quantum state affects it irreversibly, causing the collapse of the function. For example, supposing that a measurement on a two-level quantum state as in Eq. 77 is made. The result of the measurement is |+〉. After the measurement, the state of the system will always be |+〉, as if the other component |−〉 ceased to exist. As stated by the Copenhagen interpretation of quantum mechanics: when a measurement is done on the system, its state changes, collapsing in the state that resulted by that measure. Any measurement, in conclusion, affects the system irreversibly.
1.4 A.3 Density matrix
An alternative way to represent a quantum state is through its density matrix, defined as:
This form of representation is particularly useful, since, given an observable O, its mean value on a state |ϕ〉 can be written as:
1.5 A.4 Pauli matrices
For a two level system, both operators and density matrix are 2 × 2 matrices. It is common to represent any 2 × 2 matrix as a linear combination of the following matrices:
where σn are called Pauli matrices. In other words, on a two level system, every observable and every density matrix (and so every state) can be represented as a linear combination of the Pauli Matrices and the identity. These operators, in other words, constitute a basis of the space of observables and states for two level systems, and will be very useful in the next sections.
1.6 A.5 Hamiltonians and temporal evolution
Until now, when describing a quantum system, time dependency has not been accounted for. In other words, only static system have been considered. In quantum mechanics, quantum systems are usually subject to the action of time: they evolve. It is possible, then, to write:
where the parameter t makes the time dependency explicit. As mentioned earlier, in order to modify a quantum state, an operator is needed. This holds also for time dependency. The temporal evolution operator is defined as:
It is also possible to show that this operator can be written, for infinitesimal time variations as:
In other words, the operator S(t0 + ε,t0) shifts the state from time t0 to time t0 + ε.
The operator \(\hat {H}(t)\) is the hamiltonian of the system. While the utility of such expression might not be clear at first sight, it is important to remark that hamiltonians are central concepts of mechanical physics in general (Landau and Lifsits 2010). Also, they are very important in quantum computing, inside the so-called adiabatic model (Olivares 2020). It is possible to think of the hamiltonian as that operator which completely determines the temporal evolution of the system.
1.7 A.6 Entanglement of two qubit-states
It is possible to define a two qubit state from two single qubits, namely:
Then, the two qubit state can be defined as:
This is a first example of a separable state, i.e. a state that can be written as the tensor product between two different states. This is not always the case, for example:
This state is not separable: the two subsystems are entangled.
Quantum entanglement is at the heart of quantum mechanics, and gives rise to some counter-intuitive phenomena that can be exploited in the fields of quantum information and computation. Supposing, for example, that A and B represent two different particles in the entangled state, Eq. 95; one could imagine to perform a measurement on particle A, which gives result |−A〉. From that moment on, any measurement on particle B will yield the result |−B〉, because the state has “collapsed” in the first term of Eq. 95. The two subsystems are entangled in the sense that a measurement on one of the two affects both irreversibly. In other words, measurement on the two subsystems are perfectly correlated. The utilization of this phenomenon in quantum computation is a very active area of research and development.
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Alchieri, L., Badalotti, D., Bonardi, P. et al. An introduction to quantum machine learning: from quantum logic to quantum deep learning. Quantum Mach. Intell. 3, 28 (2021). https://doi.org/10.1007/s42484-021-00056-8
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DOI: https://doi.org/10.1007/s42484-021-00056-8