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Five-qubit Error Correcting Code
The five-qubit error correcting code is the smallest quantum error correcting code that can protect a logical qubit from any arbitrary single qubit error. In this code, 5 physical qubits are used to encode the logical qubit. With X and Z being Pauli matrices and I the Identity matrix, this code's generators are \langle XZZXI, IXZZX, XIXZZ,ZXIXZ \rangle. Its logical operators are \bar = XXXXX and \bar = ZZZZZ. Once the logical qubit is encoded, errors on the physical qubits can be detected via stabilizer measurements. A lookup table that maps the results of the stabilizer measurements to the types and locations of the errors gives the control system of the quantum computer enough information to correct errors. Measurements Stabilizer measurements are parity measurements that measure the stabilizers of physical qubits. For example, to measure the first stabilizer (XZZXI), a parity measurement of X of the first qubit, Z on the second, Z on the third, X on the fourth , and I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Five Qubit Code Stabilizer Measurement Circuit
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Quantum Error Correction
Quantum error correction (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault tolerant quantum computing that can reduce the effects of noise on stored quantum information, faulty quantum gates, faulty quantum preparation, and faulty measurements. Classical error correction employs redundancy. The simplest albeit inefficient approach is the repetition code. The idea is to store the information multiple times, and—if these copies are later found to disagree—take a majority vote; e.g. suppose we copy a bit in the one state three times. Suppose further that a noisy error corrupts the three-bit state so that one of the copied bits is equal to zero but the other two are equal to one. Assuming that noisy errors are independent and occur with some sufficiently low probability ''p'', it is most likely that the error is a single-bit error and the tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical And Logical Qubits
In quantum computing, a ''qubit'' is a unit of information analogous to a bit (binary digit) in classical computing, but it is affected by quantum mechanical properties such as superposition and entanglement which allow qubits to be in some ways more powerful than classical bits for some tasks. Qubits are used in quantum circuits and quantum algorithms composed of quantum logic gates to solve computational problems, where they are used for input/output and intermediate computations. A physical qubit is a physical device that behaves as a two-state quantum system, used as a component of a computer system. A logical qubit is a physical or abstract qubit that performs as specified in a quantum algorithm or quantum circuit subject to unitary transformations, has a long enough coherence time to be usable by quantum logic gates (c.f. propagation delay for classical logic gates). , most technologies used to implement qubits face issues of stability, decoherence, fault tolerance an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pauli Matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three 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. \begin \sigma_1 = \sigma_\mathrm &= \begin 0&1\\ 1&0 \end \\ \sigma_2 = \sigma_\mathrm &= \begin 0& -i \\ i&0 \end \\ \sigma_3 = \sigma_\mathrm &= \begin 1&0\\ 0&-1 \end \\ \end These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together with the iden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context. I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end. The term unit matrix has also been widely used, but the term ''identity matrix'' is now standard. The term ''unit matrix'' is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all n\times n matrices. In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, \mathbf, or called "id" (short for identity). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stabilizer Code
The theory of quantum error correction plays a prominent role in the practical realization and engineering of quantum computing and quantum communication devices. The first quantum error-correcting codes are strikingly similar to classical block codes in their operation and performance. Quantum error-correcting codes restore a noisy, decohered quantum state to a pure quantum state. A stabilizer quantum error-correcting code appends ancilla qubits to qubits that we want to protect. A unitary encoding circuit rotates the global state into a subspace of a larger Hilbert space. This highly entangled, encoded state corrects for local noisy errors. A quantum error-correcting code makes quantum computation and quantum communication practical by providing a way for a sender and receiver to simulate a noiseless qubit channel given a noisy qubit channel whose noise conforms to a particular error model. The stabilizer theory of quantum error correction allows one to import some class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lookup Table
In computer science, a lookup table (LUT) is an array that replaces runtime computation with a simpler array indexing operation. The process is termed as "direct addressing" and LUTs differ from hash tables in a way that, to retrieve a value v with key k, a hash table would store the value v in the slot h(k) where h is a hash function i.e. k is used to compute the slot, while in the case of LUT, the value v is stored in slot k, thus directly addressable. The savings in processing time can be significant, because retrieving a value from memory is often faster than carrying out an "expensive" computation or input/output operation. The tables may be precalculated and stored in static program storage, calculated (or "pre-fetched") as part of a program's initialization phase ( memoization), or even stored in hardware in application-specific platforms. Lookup tables are also used extensively to validate input values by matching against a list of valid (or invalid) items in an array and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity Measurement
Parity measurement (also referred to as Operator measurement) is a procedure in Quantum information science used for error detection in quantum qubits. A parity measurement checks the equality of two qubits to return either a true or false answer, which can be used to determine whether a correction needs to occur.Steane, Andrew M. (2006). A tutorial on quantum error correction. ''Quantum Computers, Algorithms and Chaos'', 1-32. https://www2.physics.ox.ac.uk/sites/default/files/ErrorCorrectionSteane06.pdf Additional measurements can be made for a system greater than two qubits. Because parity measurement does not measure the state of singular bits but rather gets information about the whole state, it is considered an example of a joint measurement. Joint measurements do not have the consequence of destroying the original state of a qubit as normal quantum measurements do. Mathematically speaking, parity measurements are used to project a state into an eigenstate of an operator and to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Controlled NOT Gate
In computer science, the controlled NOT gate (also C-NOT or CNOT), 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. The CNOT can be expressed in the Pauli basis as: : \mbox = e^= e^. Being both unitary and Hermitian, CNOT has the property e^=(\cos \theta)I+(i\sin \theta) U and U =e^=e^, and is involutory. The CNOT gate can be further decomposed as products of rotation operator gates and exactly one two qubit interaction gate, for example : \mbox =e^R_(-\pi/2)R_(-\pi/2)R_(-\pi/2)R_(\pi/2)R_(\pi/2). In general, any s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Set Of Commuting Observables
In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose common eigenvectors can be used as a basis to express any quantum state. In the case of operators with discrete spectra, a CSCO is a set of commuting observables whose simultaneous eigenspaces span the Hilbert space, so that the eigenvectors are uniquely specified by the corresponding sets of eigenvalues. Since each pair of observables in the set commutes, the observables are all compatible so that the measurement of one observable has no effect on the result of measuring another observable in the set. It is therefore ''not'' necessary to specify the order in which the different observables are measured. Measurement of the complete set of observables constitutes a complete measurement, in the sense that it projects the quantum state of the system onto a unique and known vector in the basis defined by the set of operators. That is, to prepare the completely specified state, we h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Logic Gate
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum 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. Unlike many classical logic gates, quantum logic gates are reversible. It is possible to perform classical computing using only reversible gates. For example, the reversible Toffoli gate can implement all Boolean functions, often at the cost of having to use ancilla bits. The Toffoli gate has a direct quantum equivalent, showing that quantum circuits can perform all operations performed by classical circuits. Quantum gates are unitary operators, and are described as unitary matrices relative to some basis. Usually we use the ''computational basis'', which unless we compare it with something, just means that for a ''d''-level quantum system (such as a qubit, a quantum register ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
