キュービット

Qubit — Grokipedia Fact-checked by Grok 26 days ago Qubit Ara Eve Leo Sal 1x A qubit , short for quantum bit, is the basic unit of information in quantum computing , analogous to a classical bit but realized as a two-state quantum-mechanical system, such as the spin of an electron or the polarization of a photon . [1] Unlike a classical bit, which exists definitively as either 0 or 1, a qubit can occupy a superposition of both states simultaneously, allowing it to represent multiple values at once until measured. [2] This property, combined with entanglement —where the state of one qubit is intrinsically linked to another, even across vast distances—enables quantum computers to perform complex computations exponentially faster than classical systems for certain problems. [3] The term "qubit" was coined in 1995 by physicist Benjamin Schumacher in his seminal paper on quantum coding, where he described it as the fundamental carrier of quantum information , drawing from concepts in quantum mechanics pioneered earlier by figures like Richard Feynman . [4] Qubits leverage principles of quantum mechanics , including the superposition principle , which permits a qubit's state to be a linear combination of basis states |0⟩ and |1⟩, mathematically expressed as α|0⟩ + β|1⟩ where |α|² + |β|² = 1. [2] Entanglement further amplifies this by creating correlated multi-qubit states that cannot be separated into individual descriptions, a phenomenon first highlighted in quantum theory by Einstein, Podolsky, and Rosen in 1935, though its computational utility was later recognized. [3] Physically, qubits are implemented in diverse ways to achieve stability and control, including superconducting loops that use current direction for states (as in IBM's systems), trapped ions where laser-manipulated energy levels define 0 and 1, photon polarization for optical qubits, and electron or nuclear spins in solid-state materials. [1] These realizations face challenges like decoherence, where environmental interactions collapse superposition, necessitating error correction and cryogenic cooling in many setups. [3] Despite these hurdles, qubits underpin quantum algorithms such as Shor's for factoring large numbers and Grover's for database search, promising breakthroughs in cryptography , drug discovery , and optimization. [2] Fundamentals Etymology The term "qubit" is a portmanteau combining "quantum" and "bit," denoting the fundamental unit of quantum information in quantum computing , analogous to the classical bit used in conventional computing. [5] This naming convention emerged in the early 1990s amid the burgeoning field of quantum information theory, where researchers sought concise terminology to describe quantum analogs of classical information units. [6] The word "qubit" was first introduced in the literature by physicist Benjamin Schumacher in his 1995 paper "Quantum coding," published in Physical Review A . [4] In the acknowledgments, Schumacher credits the term's invention to a lighthearted conversation with his colleague William K. Wootters, noting that it "was coined in jest" and quickly became standard jargon within their quantum information research group. [6] This debut occurred during a pivotal period in quantum computing research, particularly the development of quantum error correction codes, where Schumacher's work established foundational principles for reliably transmitting quantum information over noisy channels. [4] Prior to the widespread adoption of "qubit," early quantum information literature from the late 1980s and early 1990s, including works in quantum optics and cryptography , commonly referred to the concept using phrases such as "quantum bit" or "quantum bit of information." [7] For instance, John Preskill's 1994 lecture notes on black holes and quantum physics described the basic quantum information carrier as a "quantum bit," highlighting its superposition properties without yet employing the shortened form. [7] This terminological evolution reflected the field's shift toward formalizing quantum information as a distinct discipline, building on classical information theory while accommodating quantum peculiarities. Classical Bit versus Qubit A classical bit, the fundamental unit of information in classical computing , exists in one of two distinct states: 0 or 1. [8] This binary nature allows deterministic operations, where the state is precisely known and manipulated without ambiguity, enabling reliable sequential processing in classical computers. [8] In contrast, a qubit, or quantum bit, serves as the basic unit of quantum information and can exist in a superposition of states, represented as α ∣ 0 ⟩ + β ∣ 1 ⟩ \alpha |0\rangle + \beta |1\rangle α ∣0 ⟩ + β ∣1 ⟩ , where α \alpha α and β \beta β are complex numbers satisfying ∣ α ∣ 2 + ∣ β ∣ 2 = 1 |\alpha|^2 + |\beta|^2 = 1 ∣ α ∣ 2 + ∣ β ∣ 2 = 1 . [8] This superposition enables a qubit to represent multiple states simultaneously, leading to probabilistic outcomes upon measurement rather than deterministic ones, and facilitates computational parallelism that classical bits cannot achieve. [8] Key differences between classical bits and qubits arise from quantum mechanics principles. While a classical bit remains fixed in 0 or 1 until explicitly flipped, a qubit's superposition collapses to either 0 or 1 upon measurement , with probabilities ∣ α ∣ 2 |\alpha|^2 ∣ α ∣ 2 and ∣ β ∣ 2 |\beta|^2 ∣ β ∣ 2 , respectively, destroying the prior superposition state. [8] Additionally, the no-cloning theorem prohibits perfect copying of an arbitrary unknown qubit state, unlike classical bits which can be cloned indefinitely without loss of fidelity, imposing fundamental limits on quantum information duplication and distribution. These distinctions enable unique computational capabilities for qubits. For instance, a classical bit flip operation simply toggles between 0 and 1, whereas a qubit phase shift alters the relative phase in its superposition without changing the basis probabilities, allowing interference effects essential for quantum algorithms. [8] More profoundly, qubit superposition underpins algorithms like Shor's, which exploits quantum parallelism to factor large integers exponentially faster than classical methods, demonstrating potential advantages in cryptography and number theory . [9] Mathematical Representation Standard Formalism A qubit is modeled as a two-level quantum system whose state resides in a two-dimensional complex Hilbert space , denoted as H = C 2 \mathcal{H} = \mathbb{C}^2 H = C 2 . [10] [11] This formalism captures the essential quantum properties of superposition and interference without reference to specific physical implementations. [10] The general state of a qubit is represented by a normalized vector in this Hilbert space , written in Dirac notation as ∣ ψ ⟩ = α ∣ 0 ⟩ + β ∣ 1 ⟩ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle ∣ ψ ⟩ = α ∣0 ⟩ + β ∣1 ⟩ , where α , β ∈ C \alpha, \beta \in \mathbb{C} α , β ∈ C are complex coefficients satisfying the normalization condition ∣ α ∣ 2 + ∣ β ∣ 2 = 1 |\alpha|^2 + |\beta|^2 = 1 ∣ α ∣ 2 + ∣ β ∣ 2 = 1 . [11] The basis states in the computational basis are defined as column vectors: ∣ 0 ⟩ = ( 1 0 ) |0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} ∣0 ⟩ = ( 1 0 ​ ) and ∣ 1 ⟩ = ( 0 1 ) |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} ∣1 ⟩ = ( 0 1 ​ ) . [11] These basis states are orthonormal, meaning their inner product satisfies ⟨ 0 ∣ 1 ⟩ = 0 \langle 0|1 \rangle = 0 ⟨ 0∣1 ⟩ = 0 , which ensures they form a complete, orthogonal set for spanning C 2 \mathbb{C}^2 C 2 . [10] Dirac notation, introduced by Paul Dirac , uses kets ∣ ψ ⟩ |\psi\rangle ∣ ψ ⟩ for state vectors and bras ⟨ ϕ ∣ \langle \phi| ⟨ ϕ ∣ for their duals to denote inner products as ⟨ ϕ ∣ ψ ⟩ \langle \phi | \psi \rangle ⟨ ϕ ∣ ψ ⟩ , providing a compact framework for quantum information manipulations. [12] In quantum information theory, this notation excels in handling tensor products for multi-qubit systems and operator applications, facilitating clearer expressions of entanglement and quantum circuits compared to matrix-only representations. [12] [11] Pure States and Bloch Sphere A pure state of a qubit is described by a normalized vector in the two-dimensional Hilbert space , $ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle $, where α , β ∈ C \alpha, \beta \in \mathbb{C} α , β ∈ C and ∣ α ∣ 2 + ∣ β ∣ 2 = 1 |\alpha|^2 + |\beta|^2 = 1 ∣ α ∣ 2 + ∣ β ∣ 2 = 1 , embodying complete quantum coherence without any classical probabilistic mixture. [13] This state can be geometrically visualized as a point on the surface of the unit Bloch sphere , a representation originally developed for spin-1/2 systems in nuclear magnetic resonance . The Bloch vector r ⃗ = ( x , y , z ) \vec{r} = (x, y, z) r = ( x , y , z ) corresponding to the pure state has components x = 2 Re ⁡ ( α ∗ β ) x = 2 \operatorname{Re}(\alpha^* \beta) x = 2 Re ( α ∗ β ) , y = 2 Im ⁡ ( α ∗ β ) y = 2 \operatorname{Im}(\alpha^* \beta) y = 2 Im ( α ∗ β ) , and z = ∣ α ∣ 2 − ∣ β ∣ 2 z = |\alpha|^2 - |\beta|^2 z = ∣ α ∣ 2 − ∣ β ∣ 2 , with ∣ r ⃗ ∣ = 1 |\vec{r}| = 1 ∣ r ∣ = 1 ensuring the point lies on the sphere's surface. [13] Equivalently, the state can be parametrized using spherical coordinates as ∣ ψ ⟩ = cos ⁡ ( θ 2 ) ∣ 0 ⟩ + e i ϕ sin ⁡ ( θ 2 ) ∣ 1 ⟩ , |\psi\rangle = \cos\left(\frac{\theta}{2}\right) |0\rangle + e^{i\phi} \sin\left(\frac{\theta}{2}\right) |1\rangle, ∣ ψ ⟩ = cos ( 2 θ ​ ) ∣0 ⟩ + e i ϕ sin ( 2 θ ​ ) ∣1 ⟩ , where θ ∈ [ 0 , π ] \theta \in [0, \pi] θ ∈ [ 0 , π ] is the polar angle from the positive z-axis and ϕ ∈ [ 0 , 2 π ) \phi \in [0, 2\pi) ϕ ∈ [ 0 , 2 π ) is the azimuthal angle in the xy-plane. [13] In this parametrization, the Bloch vector components are x = sin ⁡ θ cos ⁡ ϕ x = \sin\theta \cos\phi x = sin θ cos ϕ , y = sin ⁡ θ sin ⁡ ϕ y = \sin\theta \sin\phi y = sin θ sin ϕ , and z = cos ⁡ θ z = \cos\theta z = cos θ . [13] The north pole ( θ = 0 \theta = 0 θ = 0