We utilize the principles of quantum computing to explore and discuss the possibilities of quantu... more We utilize the principles of quantum computing to explore and discuss the possibilities of quantum supremacy: by using the QUBIT4MATLAB package, we created a quantum circuit implementation of the quantum Fourier transform alongside an implementation of the radix-2 DIT case of the Cooley-Tukey Algorithm. We also compare and contrast our simulated rendition of the QFT with the theoretical, ideal rendition of the QFT. Through complexity analysis and the use of the Master Theorem, we find that: (1.) our DFT implementation is of complexity O(N log N); (2.) our simulation of the QFT is of complexity O(N 2) and (3.) the complexity of the theoretical (computationally ideal) version of the QFT is O(log 2 N). While our theoretical QFT complexity may appear to triumph over the classical DFT, we find that these results continue to support the current ambivalence of quantum supremacy.
We utilize the principles of quantum computing to explore and discuss the possibilities of quantu... more We utilize the principles of quantum computing to explore and discuss the possibilities of quantum supremacy: by using the QUBIT4MATLAB package, we created a quantum circuit implementation of the quantum Fourier transform alongside an implementation of the radix-2 DIT case of the Cooley-Tukey Algorithm. We also compare and contrast our simulated rendition of the QFT with the theoretical, ideal rendition of the QFT. Through complexity analysis and the use of the Master Theorem, we find that: (1.) our DFT implementation is of complexity O(N log N); (2.) our simulation of the QFT is of complexity O(N 2) and (3.) the complexity of the theoretical (computationally ideal) version of the QFT is O(log 2 N). While our theoretical QFT complexity may appear to triumph over the classical DFT, we find that these results continue to support the current ambivalence of quantum supremacy. To thoroughly explain the motivations of this paper, we must first see how quantum computation poses as a fundamentally different framework than our more familiar classical computation: that is, the existence of qubits. As background, we begin with qubits, for one is not merely a binary values of 0 or 1, but instead a superposition of states. For example, if we use Dirac's bra-ket notation, we can denote: |0 = 1 0 and |1 = 0 1 (1) Moreover, we can use the Kronecker tensor product to represent product basis states. For example, we could denote the product basis states of a four-dimensional linear vector space as: |00 = |0 ⊗ |0 = 1 0 0 0 , (2) |01 = |0 ⊗ |1 = 0 1 0 0 , (3) |10 = |1 ⊗ |0 = 0 0 1 0 , and (4) |11 = |1 ⊗ |1 = 0 0 0 1 . (5) a) Developed at Pioneer Academics under Professor Western. We can use Eq.(1) to represent a qubit state as a coherent superposition of the previous basis states: |ψ = α|0 + β|1. (6) Such that α and β are probability amplitudes. (For a greater number of qubits, we can represent any n qubits as a superposition state vector in 2 n-dimensional Hilbert space.) In accordance with the Born rule, 1 a measurement finds the system is of outcome |0 in |α| 2 and of outcome |1 in |β| 2 , such that: |α| 2 + |β| 2 = 1. (7) For greater emphasis, note that such a superposition does not take a value "in between 0 and 1": instead, the qubit simply has a probability |α| 2 of the value "0" and a probability |β| 2 of the value "1." To create circuits with these bits, we form quantum logic gates (analogous to classical logic gates). For example , a rotation of π about the axis (x +ẑ)/ √ 2 can be represented by the one-qubit Hadamard matrix: H = 1 √ 2 1 1 1 −1. Another common example would be the controlled phase shift gate: a single-qubit phase shift gate will leave the basis state |0 unchanged and map |1 to exp(iφ)|1 , for a phase shift φ (such that the probability of measuring either a |0 or a |1 also remains unchanged). Quantum gates, unlike classical gates, do not delete information and are fully reversible. For this reason, as proved by Richard Feynman, 2 they do not consume energy and can be reduced in size to act on subatomic systems such as a superposition of quantum states. As is the very essence of quantum information, there are two possible outcomes wherein, by principle, there exists no way to determine to which state the superposition pertains. Before we move onto the first quantum algorithms, let us first examine the limitations of classical computation via one particular algorithm: the general number field sieve.
We utilize the principles of quantum computing to explore and discuss the possibilities of quantu... more We utilize the principles of quantum computing to explore and discuss the possibilities of quantum supremacy: by using the QUBIT4MATLAB package, we created a quantum circuit implementation of the quantum Fourier transform alongside an implementation of the radix-2 DIT case of the Cooley-Tukey Algorithm. We also compare and contrast our simulated rendition of the QFT with the theoretical, ideal rendition of the QFT. Through complexity analysis and the use of the Master Theorem, we find that: (1.) our DFT implementation is of complexity O(N log N); (2.) our simulation of the QFT is of complexity O(N 2) and (3.) the complexity of the theoretical (computationally ideal) version of the QFT is O(log 2 N). While our theoretical QFT complexity may appear to triumph over the classical DFT, we find that these results continue to support the current ambivalence of quantum supremacy.
We utilize the principles of quantum computing to explore and discuss the possibilities of quantu... more We utilize the principles of quantum computing to explore and discuss the possibilities of quantum supremacy: by using the QUBIT4MATLAB package, we created a quantum circuit implementation of the quantum Fourier transform alongside an implementation of the radix-2 DIT case of the Cooley-Tukey Algorithm. We also compare and contrast our simulated rendition of the QFT with the theoretical, ideal rendition of the QFT. Through complexity analysis and the use of the Master Theorem, we find that: (1.) our DFT implementation is of complexity O(N log N); (2.) our simulation of the QFT is of complexity O(N 2) and (3.) the complexity of the theoretical (computationally ideal) version of the QFT is O(log 2 N). While our theoretical QFT complexity may appear to triumph over the classical DFT, we find that these results continue to support the current ambivalence of quantum supremacy. To thoroughly explain the motivations of this paper, we must first see how quantum computation poses as a fundamentally different framework than our more familiar classical computation: that is, the existence of qubits. As background, we begin with qubits, for one is not merely a binary values of 0 or 1, but instead a superposition of states. For example, if we use Dirac's bra-ket notation, we can denote: |0 = 1 0 and |1 = 0 1 (1) Moreover, we can use the Kronecker tensor product to represent product basis states. For example, we could denote the product basis states of a four-dimensional linear vector space as: |00 = |0 ⊗ |0 = 1 0 0 0 , (2) |01 = |0 ⊗ |1 = 0 1 0 0 , (3) |10 = |1 ⊗ |0 = 0 0 1 0 , and (4) |11 = |1 ⊗ |1 = 0 0 0 1 . (5) a) Developed at Pioneer Academics under Professor Western. We can use Eq.(1) to represent a qubit state as a coherent superposition of the previous basis states: |ψ = α|0 + β|1. (6) Such that α and β are probability amplitudes. (For a greater number of qubits, we can represent any n qubits as a superposition state vector in 2 n-dimensional Hilbert space.) In accordance with the Born rule, 1 a measurement finds the system is of outcome |0 in |α| 2 and of outcome |1 in |β| 2 , such that: |α| 2 + |β| 2 = 1. (7) For greater emphasis, note that such a superposition does not take a value "in between 0 and 1": instead, the qubit simply has a probability |α| 2 of the value "0" and a probability |β| 2 of the value "1." To create circuits with these bits, we form quantum logic gates (analogous to classical logic gates). For example , a rotation of π about the axis (x +ẑ)/ √ 2 can be represented by the one-qubit Hadamard matrix: H = 1 √ 2 1 1 1 −1. Another common example would be the controlled phase shift gate: a single-qubit phase shift gate will leave the basis state |0 unchanged and map |1 to exp(iφ)|1 , for a phase shift φ (such that the probability of measuring either a |0 or a |1 also remains unchanged). Quantum gates, unlike classical gates, do not delete information and are fully reversible. For this reason, as proved by Richard Feynman, 2 they do not consume energy and can be reduced in size to act on subatomic systems such as a superposition of quantum states. As is the very essence of quantum information, there are two possible outcomes wherein, by principle, there exists no way to determine to which state the superposition pertains. Before we move onto the first quantum algorithms, let us first examine the limitations of classical computation via one particular algorithm: the general number field sieve.
Uploads
Papers by Damian R Musk
Drafts by Damian R Musk