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Quantum information |
A qutrit (or quantum trit) is a unit of quantum information that is realized by a 3-level quantum system, that may be in a superposition of three mutually orthogonal quantum states. [1]
The qutrit is analogous to the classical radix-3 trit, just as the qubit, a quantum system described by a superposition of two orthogonal states, is analogous to the classical radix-2 bit.
There is ongoing work to develop quantum computers using qutrits and qudits with multiple states. [2]
A qutrit has three orthonormal basis states or vectors, often denoted , , and in Dirac or bra–ket notation. These are used to describe the qutrit as a superposition state vector in the form of a linear combination of the three orthonormal basis states:
where the coefficients are complex probability amplitudes, such that the sum of their squares is unity (normalization):
The qubit's orthonormal basis states span the two-dimensional complex Hilbert space , corresponding to spin-up and spin-down of a spin-1/2 particle. Qutrits require a Hilbert space of higher dimension, namely the three-dimensional spanned by the qutrit's basis , [3] which can be realized by a three-level quantum system.
An n-qutrit register can represent 3n different states simultaneously, i.e., a superposition state vector in 3n-dimensional complex Hilbert space. [4]
Qutrits have several peculiar features when used for storing quantum information. For example, they are more robust to decoherence under certain environmental interactions. [5] In reality, manipulating qutrits directly might be tricky, and one way to do that is by using an entanglement with a qubit. [6]
The quantum logic gates operating on single qutrits are unitary matrices and gates that act on registers of qutrits are unitary matrices (the elements of the unitary groups U(3) and U(3n) respectively). [7]
The rotation operator gates [lower-alpha 1] for SU(3) are , where is the a'th Gell-Mann matrix, and is a real value (with period ). The Lie algebra of the matrix exponential is provided here. The same rotation operators are used for gluon interactions, where the three basis states are the three colors () of the strong interaction. [8] [9] [lower-alpha 2]
The global phase shift gate for the qutrit [lower-alpha 3] is where the phase factor is called the global phase.
This phase gate performs the mapping and together with the 8 rotation operators is capable of expressing any single-qutrit gate in U(3), as a series circuit of at most 9 gates.
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.
In quantum computing, a qubit or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously, a property that is fundamental to quantum mechanics and quantum computing.
The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of the SU(3) group in the defining representation.
In physics, the Rabi cycle is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of quantum computing, condensed matter, atomic and molecular physics, and nuclear and particle physics can be conveniently studied in terms of two-level quantum mechanical systems, and exhibit Rabi flopping when coupled to an optical driving field. The effect is important in quantum optics, magnetic resonance and quantum computing, and is named after Isidor Isaac Rabi.
In quantum information theory, a quantum circuit is a model for quantum computation, similar to classical circuits, in which a computation is a sequence of quantum gates, measurements, initializations of qubits to known values, and possibly other actions. The minimum set of actions that a circuit needs to be able to perform on the qubits to enable quantum computation is known as DiVincenzo's criteria.
In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.
In computer science, the controlled NOT gate, controlled-X gate, controlled-bit-flip gate, Feynman gate or controlled Pauli-X is a quantum logic gate that is an essential component in the construction of a gate-based quantum computer. It can be used to entangle and disentangle Bell states. Any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single qubit rotations. The gate is sometimes named after Richard Feynman who developed an early notation for quantum gate diagrams in 1986.
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
The one-way or measurement-based quantum computer (MBQC) is a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is "one-way" because the resource state is destroyed by the measurements.
Amplitude amplification is a technique in quantum computing which generalizes the idea behind Grover's search algorithm, and gives rise to a family of quantum algorithms. It was discovered by Gilles Brassard and Peter Høyer in 1997, and independently rediscovered by Lov Grover in 1998.
In quantum optics, an optical phase space is a phase space in which all quantum states of an optical system are described. Each point in the optical phase space corresponds to a unique state of an optical system. For any such system, a plot of the quadratures against each other, possibly as functions of time, is called a phase diagram. If the quadratures are functions of time then the optical phase diagram can show the evolution of a quantum optical system with time.
In quantum computing, the quantum phase estimation algorithm is a quantum algorithm to estimate the phase corresponding to an eigenvalue of a given unitary operator. Because the eigenvalues of a unitary operator always have unit modulus, they are characterized by their phase, and therefore the algorithm can be equivalently described as retrieving either the phase or the eigenvalue itself. The algorithm was initially introduced by Alexei Kitaev in 1995.
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems.
Linear optical quantum computing or linear optics quantum computation (LOQC) is a paradigm of quantum computation, allowing universal quantum computation. LOQC uses photons as information carriers, mainly uses linear optical elements, or optical instruments to process quantum information, and uses photon detectors and quantum memories to detect and store quantum information.
The KLM scheme or KLM protocol is an implementation of linear optical quantum computing (LOQC), developed in 2000 by Emanuel Knill, Raymond Laflamme and Gerard J. Milburn. This protocol makes it possible to create universal quantum computers solely with linear optical tools. The KLM protocol uses linear optical elements, single-photon sources and photon detectors as resources to construct a quantum computation scheme involving only ancilla resources, quantum teleportations and error corrections.
Quantum counting algorithm is a quantum algorithm for efficiently counting the number of solutions for a given search problem. The algorithm is based on the quantum phase estimation algorithm and on Grover's search algorithm.
The quantum Cramér–Rao bound is the quantum analogue of the classical Cramér–Rao bound. It bounds the achievable precision in parameter estimation with a quantum system:
In the context of quantum computing, the quantum walk search is a quantum algorithm for finding a marked node in a graph.