Glossary

$[[n,k,d]]\!$ code: A code that uses n physical qubits to encode k logical qubits and will correct ${\frac {(d-1)}{2}}$ errors.

Abelian group: A group for which all elements commute.

Abelian subgroup: See Abelian Group, Subgroup

Adjoint (of a matrix): The transpose complex conjugate of an operator. (Also referred to as the conjugate or Hermitian conjugate.)

Ancilla: An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.

Angular momentum: A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.

Anti-commutation: Two operators anti-commute when $AB+BA=0\!$

Basis: Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.

Bath system: Describes a system that has had an unwanted interaction with an open quantum system. Environment.

Bell's theorem: "No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics." See Full Article

Bit flip error: An error which takes $\left|0\right\rangle$ to $\left|1\right\rangle$ and $\left|1\right\rangle$ to $\left|0\right\rangle$. The operator which does this is the Pauli operator $\sigma _{x}$.

Bloch sphere: Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. Image

Block diagonal matrix: A matrix which has non-zero elements only in blocks along the diagonal.

Bra-ket notation: Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, bra, and the right symbol, ket. Bra: $\left\langle \phi \right\vert$; Ket: $\left\vert \psi \right\rangle$

Centralizer: Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some $v\in S$ and all $c\in S$, $vc=cv\!$, for some group $S\!$.

Checksum: See Dot Product

Classical bit: A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)

Closed quantum system: A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.

Code: Short for Quantum Code. Used in correcting errors in quantum systems. Where $[n,k]\!$ is the way we describe a code $C\!$ with $n\!$ bits that are used to encode $k\!$ bits.

Codewords: Used to describe the set of all elements in a code $C\!$. There are $2^{n}\,\!$ $n\,\!$-bit words in the space.

Commutator: The commutator of $A\!$ and $B\!$ is denoted $[A,B]\!$, which means $AB-BA\!$. Its value may be found by implementing the operators of $A\!$ and $B\!$ on a test function. If the commutator of $A\!$ and $B\!$ is zero, they are said to commute.

Complex conjugate: Two expressions, consisting of a real number ($\alpha \!$ ) and imaginary number ($\beta i\!$), where the $\beta \!$ component is of the same magnitude, but different sign. For some $\alpha ,\beta \in \mathbb {R} ,\alpha i+\beta$ and $\alpha i-\beta \!$.

Complex number: A complex number has a real and imaginary quantity. A complex number can be represented in the form $\alpha +\beta i\!$ or, $ce^{(i\theta )}\!$, where $\alpha ,\beta ,c,\theta \in \mathbb {R}$

Controlled not (CNOT gate): Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is $\left\vert {1}\right\rangle \,\!$.

Controlled operation: An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an "If-then" or "If-then-else" statement.

Coset of a group: Used in group theory where ${\mathcal {S}}\,\!$ is a subgroup and $G\,\!$ be an element of the group ${\mathcal {G}}\,\!$. The left coset is a subset of the group $G{\mathcal {S}}=\{GS|S\in {\mathcal {S}}\}.\,\!$ One can similarly define the right coset.

CSS codes: Short for Calderbank-Shor-Steane codes. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, ${\mathcal {C}}_{1}\,\!$ and ${\mathcal {C}}_{2}\,\!$, that are of the form $[[n,k,d]]\!$. Protects against both phase- and bit-flips.

Cyclic group: A group $G\!$ with elements that can be expressed as $g^{n}\!$ where $n\in \mathbb {Z} \!$ and $g\in G$. Thus, $G=\{g^{0},g^{1},g^{2},g^{3},...\}\!$.

Dagger ($\dagger \!$): See Hermitian Conjugate, Adjoint (of a matrix)

Definite matrix: See Matrix Properties.

Degenerate: Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.

Density matrix: See Density operator.

Density operator: Language useful in describing a quantum system. In a pure state, the density operator $\rho \!$ can describe a quantum system whose state is known. For a state $|\psi \rangle \!$, it can be said $\rho =|\psi \rangle \langle \psi |$. In a mixed state, it can be described as a compilation of several pure states for $\rho \!$.

Depolarizing error: Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.

Determinant: When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms. Determinants only exist for square matrices.

Diagonalizable: A matrix $M\!$ is diagonalizable when it can be put into the form $D=S^{-1}MS\!$, where $S\!$ and $S^{-1}\!$ exist and are inverses.

Differentiable manifold: A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory.

Dirac delta function: A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.

Dirac notation: See Bra-Ket Notation.

Disjointness condition: A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.

Distance of a quantum error correcting code: A quantum error correcting code of distance $d=2t+1\!$ can correct $t\!$ errors. Denoted by the value $d\!$ when a code is expressed in the form $[[n,k,d]]\!$.

DiVincenzo's requirements for quantum computing: Set of criteria that are required for the physical system of a quantum computer. See Full Article

Dot product: The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed.

Dual code: Denoted ${\mathcal {C}}^{\perp }\,\!$, is the set of all vectors that have zero inner product with all $v\in {\mathcal {C}}\,\!$. In other words, it is the set of all vectors $u\,\!$ such that $u\cdot v=0\,\!$ for all $v\in {\mathcal {C}}\,\!$.

Dual matrix: See Parity Check Matrix.

Eigenfunction: Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.

Eigenvalue: A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, $H|\psi \rangle =E|\psi \rangle \!$, where $E\!$ is the eigenvalue that transforms the Hamiltonian, $H\!$.

Eigenvector: If $HY=EY\!$ where $H\!$ is a matrix, $E\!$ is a scalar and $Y\!$ is a vector, then $Y\!$ is the eigenvector and $E\!$ is the eigenvalue. If $Y\!$ is a function, it is called an eigenfunction.

Entangled state: A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.

Environment: See Bath System.

EPR paradox: Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.

Epsilon tensor: Cyclic permutation that can be expressed as $\epsilon _{ijk}={\begin{cases}+1,\;{\mbox{if }}\;ijk=123,231,\;{\mbox{or}}\;312,({\mbox{These are even permutations of }}123),\\-1,\;{\mbox{if }}\;ijk=213,132,\;{\mbox{or}}\;321({\mbox{These are odd permuations of }}123),\\\;\;\;0,\;{\mbox{otherwise}},\;({\mbox{meaning any index is repeated}})\end{cases}}\,\!$

Equivalent representation: Two representations $D^{(1)}\,\!$ and $D^{(2)}\,\!$ are equivalent if and only if there is an invertible matrix $S\,\!$ such that $D^{(1)}=SD^{(2)}S^{-1}\,\!$.

Error syndrome: Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.

Euler angle parametrization: A three-dimensional rotational transformation that can be expressed as $U_{EA}=\exp(-i\sigma _{z}\alpha /2)\exp(-i\sigma _{y}\beta /2)\exp(-i\sigma _{z}\gamma /2).\,\!$ Fully explained here.

Euler's Law: $\sin(x)+i\cos(x)=e^{ix}\!$

Expectation value: The most probable outcome of a measurement in quantum physics.

Exponentiating a matrix: See Matrix Exponential.

Faithful representation: A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group

Finite (or Galois) Field: A set of finite elements that is closed under vector addition and multiplication.

Gate: See Quantum Gate.

General linear group: The set of invertible $N\times N\,\!$ matrices with complex numbers as entries. It is denoted $GL(N,\mathbb {C} )\,\!$.

Generators of a group: Consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are the generators of the group.

Generator matrix: An $n\times k\,\!$ matrix with columns that form a basis for the $k\,\!$-dimensional coding sub-space of the $n\,\!$-dimensional binary vector space. The vectors comprising the rows form a basis that will span the code space.

Gram-Schmidt Decomposition: See Schmidt Decomposition.

Group: Set of objects that is closed under multiplication, associative, invertible and contain an identity element.

Grover's algorithm: Quantum algorithm for searching unsorted databases. See Full Article

$h\!$ bar ($\hbar$): Planck's constant divided by $2\pi$, $h$ is Planck's constant

Hadamard gate: Also known as a Hadamard transformation and refers to the rotation of a single qubit used as the first step in many quantum algorithms.

Hamming bound: Restricts the rate of the error correcting code. A code that reaches the Hamming bound has perfect efficiency.

Hamming code: A $[7,4,3]\!$ code. This code, as the notation indicates, encodes $k=4\!$ bits of information into $n=7\!$ bits. One error can be detected and corrected at a distance of up to 3.

Hamming distance: The number of places where two vectors differ.

Hamming weight: The number of non-zero components of a vector or string.

Hamiltonian: An operator that corresponds with the total energy of the system.

Heisenberg exchange interaction Hamiltonian: When this Hamiltonian is between two qubits labelled $i\,\!$ and $j\,\!$, it can be expressed as $H_{ex}^{i,j}=X_{i}X_{j}+Y_{i}Y_{j}+Z_{i}Z_{j},\,\!$ where $X_{i}\,\!$ is the Pauli x-operation on the $i\,\!$th qubit and similarly for the other operators.

Heisenberg uncertainty principle: See uncertainty principle.

Hermitian: An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.

Hermitian conjugate: The transpose complex conjugate of an operator.

Hidden variable theory: See Local hidden variable theory.

Hilbert-Schmidt inner product: Product of two Hilbert-Schmidt operators.

Hilbert-Schmidt operators: Bounded, compact operators that are isometrically isomorphic to the tensor product of Hilbert spaces.

Hilbert space: Mathematical concept that makes vector operations and calculus possible in spaces that are greater than three dimensions.

Homomorphism: The composition of two functions yields the product of those functions$f(A\circ B)=f(A)\cdot f(B).\,\!$

$i\!$: Denotes the square root of negative one

Identity matrix: A matrix of zeros except for the diagonal, where each element is 1. Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.

Inner product: See Dot product.

Inverse of a matrix: The inverse of a square matrix $A\!$ is the matrix, denoted $A^{-1}\!$, such that $AA^{-1}=\mathbb {I} =A^{-1}A\!$, where $\mathbb {I} \!$ is identity matrix. (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)

Invertible matrix: A matrix for which an inverse exists.

Isolated system: See Closed system.

Isomorphism: A homomorphism that is both injective and surjective.

Isotropy group: See Stabilizer.

Isotropy subgroup: See Stabilizer.

Jacobi identity: Property of certain binary operations that are possessed by Lie groups that can be written $f_{ilm}f_{jkl}+f_{jlm}f_{kil}+f_{klm}f_{ijl}=0,\,\!$.

Ket: See Bra-Ket Notation.

Kraus representation: Kraus decomposition. See SMR representation.

Kronecker delta: The symbol $\delta _{ij}\!$ that is defined as: $\delta _{ij}={\begin{cases}1,&{\mbox{if }}i=j,\\0,&{\mbox{if }}i\neq j.\end{cases}}$

Levi-Civita symbol: See Epsilon tensor.

Lie algebra: A vector space, over some field with a binary operation, called a Lie bracket, that possesses bilinearity, is alternating on the vector space, and satisfies the Jacobi identity.

Lie group: A differentiable manifold that corresponds to a continuous set of symmetries.

Linear code: Error correcting code for which any linear combination of codewords is a codeword. Can be generated by a Generator matrix.

Linear combination: A set of vectors each multiplied by a scalar and summed.

Linear map: A very general mapping of a matrix acting on a vector that produces another vector.

Little group: See Stabilizer.

Local actions: See Local operations.

Local hidden variable theory: See hidden variable theory

Local operations: Actions on an individual particle without involving any other particle.

Logical bit: Short for Logical qubit. Qubits that are encoded with information that is to be protected from errors. Represented by $k\!$ in $[[n,k,d]]\!$ codes.

Matrix exponential: A mapping of the function of a matrix according to its Taylor expansion.

Matrix properties: Defining characteristics that allow matrices to be categorized.

Matrix transformation: Representation of a change in basis from one matrix to another.

Measurement: In quantum mechanics, a probability distribution that determines the state of the quantum object. Currently, it is impossible to measure a quantum object without affecting or destroying its current state.

Minimum distance: The smallest Hamming distance between any two non-equal vectors in a code.

Modular arithmetic: When a number is divided into another and does not go evenly, there is left a remainder. Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number. For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced "three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer. This usage of modulo has nothing to do with the physics usage of modulus.

Modulus: Complex norm of the expression $\alpha +\beta i\!$ that can be expressed by $|\alpha +\beta i|=|z|={\sqrt {\alpha ^{2}+\beta ^{2}}},\alpha ,\beta \in \mathbb {R}$.

$n,k,d\!$ code: See $[[n,k,d]]\!$ code.

No cloning theorem: There is no universal copying machine. See Section 5.2.

Noise: Errors in a system created by unwanted interactions with other systems or imperfect controls.

Non-degenerate code: Codes that have eigenvalues with multiplicity of exactly one.

Normalizer: A subgroup ${\mathcal {S}}\,\!$ of a group ${\mathcal {G}}\,\!$ that leaves the set ${\mathcal {M}}\,\!$ fixed and can be expressed by ${\mathcal {S}}=\{S\in {\mathcal {S}}|S{\mathcal {M}}={\mathcal {M}}\}.\,\!$

Normalization: A process of scaling some set of numbers or functions in order that an operation including them returns a desired value. For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.

One-to-one: A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element. Injective.

Onto: A mapping where each domain element mapped to at most one range element. Surjective.

Open system: A system that is not necessarily conserved as it is exposed to an external environment. An environment in which noise may be created.

Operator: The basis of theory in quantum mechanics. A function which acts on information obtained in an environment to describe the environment and some relevant characteristics of that data.

Operator-sum representation: See SMR representation.

Order of a group: The number of elements contained in a group.

Ordered basis: An organizational method that lists the basis of a vector in a logical order.

Orthogonal: Two vectors are orthogonal when their dot product is zero.

Outer product: Tensor product of two vectors.

P gate (Not phase gate):

Parity: The property of an equation that either remains the same (even parity) or changes (odd parity).

Parity check: See Inner product.

Parity check matrix: Matrix that can be found using the generator matrix. It has the property that it annihilates a code word.

Partial trace: The trace over one of the subsystems (particle states) of a composite system.

Partition of a group: A division of a group into disjoint, nonempty sets.

Pauli group: The three matrices that, along with the $2\times 2\,\!$ identity matrix, form a basis for the set of $2\times 2\,\!$ Hermitian matrices and can be used to describe all $2\times 2$ unitary transformations.

Pauli matrices: The $X,Y,Z\!$ gates.

Permutation: A bijective map from a set to itself.

Phase flip error: In this case a $\left\vert 1\right\rangle \,\!$ will acquire a (-1) sign change due to some noise, but a $\left\vert 0\right\rangle \,\!$ is unaffected. If a quantum phase-flip error occurs with some probability $p\,\!$, we may express the phase flip error as $\rho ^{\prime }=(1-p)\rho +p\sigma _{z}\rho \sigma _{z}\,\!$.

Phase gate: See Z gate.

Planck's constant: A constant that is a quantum of action in quantum mechanics. The value of Planck's constant is $h=6.626\times 10^{-34}J\cdot s\!$.

Polarization: The process that forces waves to only oscillate in one plane.

Positive definite matrix: Matrix whose eigenvalues are all greater than zero.

Semi-definite matrix: Matrix whose eigenvalues are nonnegative.

Probability for existing in a state: The probability that a quantum system is in the state $\psi _{n}\!$. It can be expressed as $\left\vert \psi _{n}\right\rangle =\alpha _{0}\left\vert 0\right\rangle +\alpha _{1}\left\vert 1\right\rangle ,\!$ and $|\alpha _{0}|^{2}+|\alpha _{1}|^{2}=1\,\!$, where $|\alpha _{0}|^{2}\!$ and $|\alpha _{1}|^{2}\!$ are the probabilities.

Projector: A transformation such that $P^{2}=P\!$.

Projection postulate: If we measure a system, i.e. look to see if it is Well 0 or Well 1, we will “project it into one state or the other.” In other words, suppose the system is in the state $\left|\psi \right\rangle \,\!$ above. If we look to see where the particle is and find it in Well 1, then the probability is clearly zero that it is in the other well.

Pure state: A state that cannot be expressed as a mixture of other states.

QKD: See Quantum key distribution.

Quantum bit: See Qubit.

Quantum cryptography: Use of quantum mechanics to encode messages.

Quantum dense coding: Sending two bits of information only using one qubit, using entanglement.

Quantum gate: A unitary transformation applied to one or more qubits.

Quantum hamming bound: See Hamming Bound.

Quantum key distribution: Use of quantum cryptography to generate a shared random key.

Quantum NOT gate: One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.

Qubit: A Qubit is represented by two states of a quantum mechanical system. (1.3)

Rank: Describes the number of states that make up a density operator. A rank one projection is a pure state density operator.

Rate of a code: Given by the ration of the number of logical bits to the number of bits, $k/n\,\!$.

Reduced density operator: A test for observable quantum characteristics that uses the partial trace operation.

Representation space: A vector space used to describe quantum fields.

Reversible quantum operation: Operation for which every state on which the operator can act there exists an operation which restores it to its original state.

RSA encryption: A public-key cryptography algorithm which uses prime factorization and is rendered useless by Shor's algorithm.

Schmidt decomposition: A method that takes a density operator and transforms it into a reduced density operator.

Schrodinger's Equation: Partial differential equation that describes how the quantum states of physical systems change with time.

Set: Any mathematical construct, often represented by a capital letter.

Shor's algorithm: Used to find prime factors, named after its inventor, Peter Shor.

Shor's nine-bit quantum error correcting code: The first example of a quantum error correcting code which, in principle, can correct arbitrary single-qubit errors and can be understood in terms of the simple classical error correcting code.

Similarity transformation: A matrix transformation that leave both the trace and the determinant of a matrix unchanged.

Singular values: The entries of the diagonal matrix, $D\!$, that are produced from some complex matrix $M\!$.

Singular value decomposition: A form in which a complex matrix $M\,\!$ has a diagonal matrix $D\,\!$ such that $M=UDV,\,\!$ where $U\,\!$ and $V\,\!$ are unitary matrices.

SMR representation: Method for representing open system evolution. Full description here.

Special unitary matrix: The set of special unitary matrices is denoted $SU(n)\,\!$ and has determinant of one.

Spin: A two-state system being either up or down when measured.

Spooky action at a distance: Used to describe entangled states at great distances. For example, an entangled electron-positron pair are at opposite ends of the galaxy when one of them is measured. The conservation of angular momentum says that the other particle all of the way on the other side of the galaxy must instantly be the opposite spin as the measured particle. An idea explored in the EPR paradox.

Stabilizers of a group: The subgroup ${\mathcal {S}}\,\!$ of a group ${\mathcal {G}}\,\!$ that leaves the element $m\,\!$ fixed: ${\mathcal {S}}=\{S\in {\mathcal {S}}|sm=m\}\,\!$.

Stabilizer code: Family of quantum error correcting codes which are describable by using the stabilizer of a state (really a set of states) in the Hilbert space, conveniently described by their operators rather than their states.

Standard deviation: The amount of variation from the mean.

Stationary subgroup: See Stabilizer.

Stirling's formula: A formula used to approximate $n!\!$, denoted by $n!~={\sqrt {2\pi n}}({\frac {n}{e}})^{n}.$

Subgroup: A subgroup ${\mathcal {S}}\,\!$ of a group ${\mathcal {G}}\,\!$ is a subset of the group elements that satisfies all the properties in the definition of a group under the inherited multiplication rule.

Superposition: A qubit state in superposition, where $\phi$ may be written as $|\phi >=\alpha |0>+\beta |1>$ where $\alpha$ and $\beta$ are complex numbers.

Syndrome measurement: Measurement to find the error syndrome that gives information about the error and not the actual code word.

Taylor expansion: can be used to exponentiate a matrix by letting the matrix replace $x\!$ in the equation, $e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.\!$ Teleportation: process by which quantum information can be transmitted from one location to another using entanglement.

Tensor product: Used extensively in quantum mechanics. It is commonly denoted with a $\otimes \,\!$ symbol, although this is often left out.

Trace: The sum of the diagonal elements of a matrix.

Transpose: The same set of elements, but now the first row becomes the first column, the second row becomes the second column, and so on. Thus the rows and columns are interchanged.

Trivial representation: Map of a group where all of the elements of the group act as identity operators of the map.

Turing machine: Machine used to emulate classic computational algorithms.

Uncertainty principle: Inequalities that describe the limitations of the measurements of quantum objects that can be made simultaneously.

Unitary matrix: One whose inverse is also its Hermitian conjugate, $U^{\dagger }=U^{-1}\,\!$, so that $U^{\dagger }U=UU^{\dagger }=\mathbb {I} .\!$

Unitary transformation: A transformation which leaves the magnitude of any object it transforms the same.

Universal quantum computing: The ability to implement a special class of two-qubit gates between any two qubits, plus the ability to implement all single-qubit unitary transformations.

Universal set of gates: Universality, or the ability to perform any arbitrary computation with quantum computing.

Variance: How far a set of numbers is spread out from the mean.

Vector: In physics: A quantity with both magnitude and direction. More generally, it is an array of numbers that is written in a single column or single row. It is a special kind of matrix.

Vector space: Collection of vectors, added together and multiplied by scalars, used to describe a space that is the focus of studies in linear algebra.

Weight of a vector: See Hamming weight.

Weight of an operator: The number of non-identity elements in the tensor product.

Wigner-Clebsch-Gordon Coefficients: Coefficients that allow a new basis to be expressed in terms of the old basis. They also block-diagonalize the tensor products of the operators.

X gate: A bit-flip gate, represented by the matrix $X=\left({\begin{array}{cc}0&1\\1&0\end{array}}\right).$.

Y gate: Acts on states with both $X\!$ and $Z\!$ gates, represented by the matrix $Y=\left({\begin{array}{cc}0&-i\\i&0\end{array}}\right).$

Z gate: Phase-flip gate, represented by the matrix $Z=\left({\begin{array}{cc}1&0\\0&-1\end{array}}\right).$