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Apress eBooks

Mathematical Foundation: Time to Dust Up That Linear Algebra

Vladimir Silva

Published Dec 18, 2023• 0 citations
Publisher

Abstract

Matrices, complex numbers, and tensor products are the holy trinity of quantum computing. During my quantum learning curve on the mathematical background, I wondered as you may: Where is the quantum magic in all this? How can a complex number or a matrix tell me about superposition or entanglement? Where is the spookiness? My immature understanding of superposition was that a system existed in multiple states at the same time: Like the cat in the box, it is dead and alive at once. However that is incorrect, the cat cannot be dead and alive at the same time just like a quantum system cannot exist in two simultaneous states.

DOI
10.1007/978-1-4842-9991-3_5

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Matrices, complex numbers, and tensor products are the holy trinity of quantum computing. During my quantum learning curve on the mathematical background, I wondered as you may: Where is the quantum magic in all this? How can a complex number or a matrix tell me about superposition or entanglement? Where is the spookiness? My immature understanding of superposition was that a system existed in multiple states at the same time: Like the cat in the box, it is dead and alive at once. However that is incorrect, the cat cannot be dead and alive at the same time just like a quantum system cannot exist in two simultaneous states.
In quantum theory, a particle is allowed to be in a state of coherent superposition among a set of orthogonal states. A vivid picture of this concept is the “Schrödinger cat”: a cat is alive and dead simultaneously. This would be no surprise if one means a large number of an ensemble of cats with 50% alive and 50% dead. But, “Schrödinger cat” is a cat. People are talking about a cat being alive and dead simultaneously. In mathematics, the best word to characterize the relationship between states “alive” and “dead” is perhaps “orthogonal”. In quantum mechanics, the superposition of orthogonal states is used to describe the state of a quantum object, or a particle. The superposition principle is indeed a mystery compared to our everyday classical life experience. Beyond the superposition of orthogonal single-particle states, the superposition of orthogonal multi-particle states represents an even more troubling concept in classical theory.
The reliable and flexible foundation of linear algebra functions as the framework for the mathematical infras tructure of contemporary information science. The language of vectors, matrices, and linear operators provides descriptive framework for analyzing the coherent superpositions of quantum systems, the stochastic transitions of randomized algorithms, or the deterministic logic of conventional silicon-based processors. By converting physi cal states into formal mathematical objects, this approach empowers researchers to establish algorithms, anticipate behavior, and study the fundamental boundaries of computation [1]. The change from discrete configurations to a continuous, complex-valued state space controlled via the ideas of superposition and entanglement can be ob served in the transition from classical bit-strings to quantum kets [1].
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
It may well be that you feel a bit disappointed at this point. You probably had heard about the utter magic of quantum theory, where particles may be in different places at the same time, where cats (Schrödinger’s cat) may be dead and alive at the same time, and where particles have a telepathic knowledge of the fate of remote particles.This chapter tells more about this “big magic”. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.