| Z 0 Whereas the metric itself provides a way to measure the length of (or angle between) vector fields, the inverse metric supplies a means of measuring the length of (or angle between) covector fields; that is, fields of linear functionals. There are many unitary transforms available in the libraries of Q#, QCL, Qiskit, and other quantum programming languages. , To keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields, before moving on to the definition for general tensors. n {\displaystyle n} The probability of measuring a value with probability amplitude = b cannot be determined while the displacement 00 In R^2, consider the matrix that rotates a given vector v_0 by a counterclockwise angle theta in a fixed coordinate system. where Suppose that v is a tangent vector at a point of U, say, where ei are the standard coordinate vectors in n. about the axis U 0 i b In this case, the spacetime interval is written as, The Schwarzschild metric describes the spacetime around a spherically symmetric body, such as a planet, or a black hole. U x U = 2 [4]:177185 Sometimes this is shortened to just CX, CY and CZ. = X n {\displaystyle F} A {\displaystyle \land } For this reason, the system of quantities gij[f] is said to transform covariantly with respect to changes in the frame f. A system of n real-valued functions (x1, , xn), giving a local coordinate system on an open set U in M, determines a basis of vector fields on U, The metric g has components relative to this frame given by, Relative to a new system of local coordinates, say. where Dy denotes the Jacobian matrix of the coordinate change. {\displaystyle {\hat {z}}} X See measurement below for details. ) The Pauli matrices also anti-commute, for example | ) U L and , x It is worth noting that the spinor Lie derivative is independent of the metric, and hence also of the connection. This means that negative exponents of gates are unitary inverses of their positively exponentiated counterparts: By Lagrange's identity for the cross product, the integral can be written. x 1 A ^ | There must always exist a function {\displaystyle \wedge } ( . Pauli Z leaves the basis state {\displaystyle |01\rangle } Additional properties are consistent with that of the Lie bracket. {\displaystyle \oplus } | {\displaystyle \delta ^{\ast }A} Consider T to be a differentiable multilinear map of smooth sections 1, 2, , p of the cotangent bundle TM and of sections X1, X2, , Xq of the tangent bundle TM, written T(1, 2, , X1, X2, ) into R. Define the Lie derivative of T along Y by the formula. x ) {\displaystyle |1\rangle } | 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, or qutrits and qudits[1]:2223) we have labeled the orthogonal basis vectors ( ( a X with the initial condition being , a | {\displaystyle A} The entries of the matrix G[f] are denoted by gij, where the indices i and j have been raised to indicate the transformation law (5). Thus a metric tensor is a covariant symmetric tensor. and [ . {\displaystyle |0\rangle ,|1\rangle ,|2\rangle ,\cdots ,|2^{n}-1\rangle } The most common quantum gates operate on vector spaces of one or two qubits, just like the common classical logic gates operate on one or two bits. 0 WebThe outer product and Kronecker product are closely related; in fact the same symbol is commonly used to denote both operations. n T 8 ( {\displaystyle X^{3}=X\cdot X\cdot X} we have: Hence for the scalar field U | + In quantum mechanics, the basis vectors constitute an orthonormal basis. p , | H 2 by re-initializing the value of it, or by its spontaneous decoherence) that have not been uncomputed may result in errors,[39][40] as their state may be entangled with the qubits that are still being used in computations. has that all quantum states | | only. T The Lie derivative commutes with the contraction. In contrast, when taking a Lie derivative, no additional structure on the manifold is needed, but it is impossible to talk about the Lie derivative of a tensor field with respect to a single tangent vector, since the value of the Lie derivative of a tensor field with respect to a vector field X at a point p depends on the value of X in a neighborhood of p, not just at p itself. | {\displaystyle 2^{n}} It is possible to perform classical computing using only reversible gates. The identity gate is most useful when describing mathematically the result of various gate operations or when discussing multi-qubit circuits. b for some uniquely determined smooth functions v1, , vn. Row: Index of row, where non-zero element is located; for all Hermitian gates. Indeed, changing basis to fA gives. I unitary matrix, and the set of all such gates with the group operation of matrix multiplication[a] is the symmetry group U(2n). The gate The algebraic definition for the Lie derivative of a tensor field follows from the following four axioms: If these axioms hold, then applying the Lie derivative The prefix ! H 1 F and has the same valence as There is thus a natural one-to-one correspondence between symmetric bilinear forms on TpM and symmetric linear isomorphisms of TpM to the dual TpM. {\displaystyle \oplus } are the three Pauli matrices 11 H 2 0 | {\displaystyle e^{-\sigma _{i}\theta /2}.}. {\displaystyle e^{i\theta H}=(\cos \theta )I+(i\sin \theta )H} {\displaystyle 2^{n}} This is called entanglement. is defined at point , With coordinates. + x i 2 {\displaystyle x+h} | a 2 the volume density of weight 1), then its Lie derivative is a tensor density of the same type and weight. I ) {\displaystyle |0\rangle } ( 00 Upon changing the basis f by a nonsingular matrix A, the coefficients vi change in such a way that equation (7) remains true. By the universal property of the tensor product, any bilinear mapping (10) gives rise naturally to a section g of the dual of the tensor product bundle of TM with itself, The section g is defined on simple elements of TM TM by, and is defined on arbitrary elements of TM TM by extending linearly to linear combinations of simple elements. For more information see the Bell test experiments. The restriction of the Lie derivative to the space of differential forms is closely related to the exterior derivative. ) | 01 I , ( B = For a pair and of covector fields, define the inverse metric applied to these two covectors by, The resulting definition, although it involves the choice of basis f, does not actually depend on f in an essential way. There are some proposals to realize a Deutsch gate with dipole-dipole interaction in neutral atoms.[20]. [ a L The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. | 2 Functions that are not invertible can be made invertible by adding ancilla qubits to the input or the output, or both. {\displaystyle X=\sin(x)\partial _{y}-y^{2}\partial _{x}} {\displaystyle X\,} so that g is regarded also as a section of the bundle T*M T*M of the cotangent bundle T*M with itself. x {\displaystyle T_{P(t,p)}M} {\displaystyle |0\rangle } {\displaystyle |0\rangle ,|1\rangle ,\dots ,|d-1\rangle } A The group U(2q) is the symmetry group for the gates that act on U Quantum algorithms often contain both a classical and a quantum part. X U p [10] Functions can be conditioned on superposition states. A Even though the quantum logic gates belong to continuous symmetry groups, real hardware is inexact and thus limited in precision. {\displaystyle p\in M} {\displaystyle 2^{n}} = 1 {\displaystyle F(a,b)=a+b{\pmod {2^{n}}}} 0 such that . WebIn mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. L If the surface M is parameterized by the function r(u, v) over the domain D in the uv-plane, then the surface area of M is given by the integral, where denotes the cross product, and the absolute value denotes the length of a vector in Euclidean space. x M Moreover, for unitaries on a constant number of qubits, the SolovayKitaev theorem guarantees that this can be done efficiently. | | For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. M n R The Cartan formula can be used as a definition of the Lie derivative of a differential form. | n | {\displaystyle T} 1 = {\displaystyle \nabla _{a}X_{b}=\nabla _{[a}X_{b]}} a ( K ) U The Clifford set alone is not universal and can be efficiently simulated classically by the GottesmanKnill theorem. One natural such invariant quantity is the length of a curve drawn along the surface. I X {\displaystyle |11\rangle } This is the covariant Lie derivative. 0 Y The Lie derivative of a tensor is another tensor of the same type, i.e., even though the individual terms in the expression depend on the choice of coordinate system, the expression as a whole results in a tensor. X The Lie derivative acting on a differential form is the anticommutator of the interior product with the exterior derivative. For a scalar field = | WebNote: =1 =i =j =k Most off-diagonal elements of the table are antisymmetric, making it almost a skew-symmetric matrix except for the elements on the main diagonal, as well as the row and column for which e 0 is an operand.. = Since length If we, as in the picture, combine the Pauli-Y gate with the Pauli-X gate in parallel, then this can be written as: Both the Pauli-X and the Pauli-Y gate act on a single qubit. {\displaystyle (|0\rangle +|1\rangle )/{\sqrt {2}}} = ) | The contravariance of the components of v[f] is notationally designated by placing the indices of vi[f] in the upper position. is a unitary matrix, then ( , thus showing that all reversible classical logic operations can be performed on a universal quantum computer. 1 00 The phase shift gate is represented by the matrix: where x e mod 0 and = {\displaystyle n} | {\displaystyle |\psi \rangle } n {\displaystyle T_{\Gamma _{X}^{t}(p)}M} . T t . This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. Z and the vector field For input matrices A and B, the result X is such that A*X == B when A is square. is the unique solution of the system, of first-order autonomous (i.e. ) behaves like a NOP[22][23] and can be represented as bare wire in quantum circuits, or not shown at all. quantum logic gates) applied to it are rotations on the sphere. , = One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending the surface without stretching it), or a change in the particular parametric form of the same geometrical surface. is an alternating multilinear map from {\displaystyle a+b} = The components ai transform when the basis f is replaced by fA in such a way that equation (8) continues to hold. i The above system of differential equations is more explicitly written as a system. {\displaystyle T^{\dagger }} | | 0 X . + 0 The matrix that represents a quantum function acting on H {\displaystyle (\sigma _{x},\sigma _{y},\sigma _{z})} The signature of g is the pair of integers (p, n p), signifying that there are p positive signs and n p negative signs in any such expression. For example, say c , where [3] For example, the controlled NOT gate (or CNOT or CX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is series and parallel circuits) of the available primitive gates. H The main difference between the Lie derivative and a derivative with respect to a connection is that the latter derivative of a tensor field with respect to a tangent vector is well-defined even if it is not specified how to extend that tangent vector to a vector field. {\displaystyle 2^{n}\times 2^{n}} T , that is, a real- or complex-valued function on M, one has. ( 0 The Lie derivative constitutes an infinite-dimensional Lie algebra representation of this Lie algebra, due to the identity. A n and {\displaystyle X} [1]:1517[27][28][29] This is known as the Born rule and appears[d] as a stochastic non-reversible operation as it probabilistically sets the quantum state equal to the basis vector that represents the measured state. or All products and tensor products (i.e. when is a Hilbert space) the concept of orthogonality can be used. While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita, who first codified the notion of a tensor. | {\displaystyle \partial _{a}={\frac {\partial }{\partial x^{a}}}} , and is involutory. denotes the product of f and X. which is independent of any coordinate system and of the same type as {\displaystyle |11\rangle } 0 1 is the wedge product on differential forms. {\displaystyle U^{-n}=(U^{n})^{\dagger }} }, The matrix exponential of a Pauli matrix This is called the induced metric. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in , .Inner products allow formal definitions of intuitive geometric notions, such as If T is a tensor field and X is a vector field, then the Lie derivative of T with respect to X is denoted x i This often leads to simpler formulas by avoiding the need for the square-root. [f] where a {\displaystyle |+\rangle } n t When has an inner product and is complete (i.e. This section has the same smoothness as g: it is continuous, differentiable, smooth, or real-analytic according as g. The mapping Sg, which associates to every vector field on M a covector field on M gives an abstract formulation of "lowering the index" on a vector field. along a vector field , . f are Dirac matrices. | If not measured, it is a quantum state with equal probability amplitude can be constructed: Because | ( A 'nave' attempt to define the derivative of a tensor field with respect to a vector field would be to take the components of the tensor field and take the directional derivative of each component with respect to the vector field. The resulting matrix, known as the matrix product, has the number of rows of the first and the Now, the metric tensor gives a means to identify vectors and covectors as follows. If a function H 1 WebAlgebraic properties. {\displaystyle \otimes } That is, put, This is a symmetric function in a and b, meaning that. 0 M A projection on a Hilbert space is called an orthogonal projection if it satisfies , = , for all ,.A projection on a Hilbert space that is not orthogonal is called an oblique projection. The Lie derivative of a function = Setting can be replaced with the covariant derivative which means replacing X (2) This is {\displaystyle M,} A 0 {\displaystyle a_{x}} }, For a coordinate chart | {\displaystyle F^{\dagger }} X p 1 has the property f | The upshot is that the first fundamental form (1) is invariant under changes in the coordinate system, and that this follows exclusively from the transformation properties of E, F, and G. Indeed, by the chain rule, Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of tangent vectors to the surface, as well as the angle between two tangent vectors. However a connection requires the choice of an additional geometric structure (e.g. Cartan's formula shows in particular that, The Lie derivative also satisfies the relation, In local coordinate notation, for a type (r, s) tensor field can now be applied to any two-qubit state, entangled or otherwise. = WebIn mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars.Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field.The operations of vector addition and scalar multiplication ( 10 2 | The two states It is also called the Hermitian adjoint. to For every {\textstyle \varphi ={\frac {\pi }{4}}} Such is not the case: the quantities on the right-hand side of Kosmann's local expression combine so as to make all metric and connection dependent terms cancel. F ) 0 | and D , ) d The phase shift gates are related to each other as follows: Note that the phase gate Originally it was solved by giving up the assumption of local realism, but other interpretations have also emerged. ( {\displaystyle U} e X [6] This isomorphism is obtained by setting, for each tangent vector Xp TpM. ) If we have a principal bundle over the manifold M with G as the structure group, and we pick X to be a covariant vector field as section of the tangent space of the principal bundle (i.e. -qubit registers a and b, {\displaystyle HH^{\dagger }=I,} {\displaystyle \cdot } {\displaystyle |\Psi \rangle } f We generally only define the length of a curve when the quantity under the square root is always of one sign or the other. F is used to emphasize the fact that the product of a function times a vector field is being taken over the entire manifold. is R-linear, and, for ) | basis vectors of an n-qubit register In components, (9) is. | b Defining the derivative of a function H M for any unitary matrix b [1]:3035[4]:22,8485,185188[32] See change of basis and Von Neumann entropy for details. That is, | [ Positive integer exponents are equivalent to sequences of serially wired gates (e.g. n X Here, In these terms, a metric tensor is a function, from the fiber product of the tangent bundle of M with itself to R such that the restriction of g to each fiber is a nondegenerate bilinear mapping. b | ( and pairwise CNOT its qubits with the qubits in register A, such that for each p the qubits {\displaystyle n} 8 0 | a | H The table can be summarized as follows: = {, =, = +, where ij is the Kronecker delta (equal to 1 if and only if i = j), and ijk is a completely The Hadamard transform acts on a register , B ) {\displaystyle \otimes } Various generalizations of the Lie derivative play an important role in differential geometry. | If two quantum states (i.e. ) 1 where the {\displaystyle U=e^{i{\frac {\pi }{2}}(I-U)}=e^{-i{\frac {\pi }{2}}(I-U)}} and , then the other qubit must also be given the output ( is the number of qubits that constitutes the register x {\displaystyle \delta ^{\ast }A} = If Kk(M, TM) and is a differential p-form, then it is possible to define the interior product iK of K and . Explicitly, let T be a tensor field of type (p, q). In the geometrical and physical settings, it is sometimes possible to associate, in a natural way, a length or magnitude and a direction to vectors. [4]:66[5]:56,65 The basis vectors (sometimes called eigenstates) are the possible outcomes if measured, and a quantum state is a linear combination of these outcomes. (See metric (vector bundle).). Certain metric signatures which arise frequently in applications are: Let f = (X1, , Xn) be a basis of vector fields, and as above let G[f] be the matrix of coefficients, One can consider the inverse matrix G[f]1, which is identified with the inverse metric (or conjugate or dual metric). for some p between 1 and n. Any two such expressions of q (at the same point m of M) will have the same number p of positive signs. , F . ) G i where | 1 ) X 1 , where ) = One can easily prove that his {\displaystyle H} ( i F sin {\displaystyle T_{p}M.} This expression is useful for "selecting" any one of the matrices numerically by substituting values of j = 1, 2, 3, in turn and This equation has nonzero solutions that are nonsingular on [1, 1] only if and m are integers with 0 m , or with trivially equivalent negative values.When in addition m is even, the function is a polynomial. 1 T For example, the integers | series and parallel combinations) of unitary matrices are also unitary matrices. Subsets of the gates, such as the Clifford gates, or the trivial case of circuits that only implement classical boolean functions (e.g. A definition for Lie derivatives of spinors along generic spacetime vector fields, not necessarily Killing ones, on a general (pseudo) Riemannian manifold was already proposed in 1971 by Yvette Kosmann. [ i Similarly the gate ( b that have equal probability of when measured to be in any of its !inc(x) is the inverse of inc(x) and instead performs the operation {\displaystyle ({\hat {x}}+{\hat {z}})/{\sqrt {2}}} {\displaystyle |0\rangle } In the usual (x, y) coordinates, we can write. + 0 , 0 ( , For a quite long time, physicists had been using Lie derivatives, without reference to the work of mathematicians. 0 ) In our example, this can be done by passing on one of the input registers to the output: {\displaystyle T} I q b 10 2 The Hadamard-CNOT combination acts on the zero-state as follows: This resulting state is the Bell state Y b n X 0 10 | ) ) {\displaystyle X} , . is If X and Y are both vector fields, then the Lie derivative of Y with respect to X is also known as the Lie bracket of X and Y, and is sometimes denoted Measuring this state results in a random number between A system and therefore the Lie derivative to the identity z leaves the basis state { \displaystyle |01\rangle Additional! 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And Kronecker product are closely related ; in fact the same symbol is commonly used to both... Is located ; for all Hermitian gates is coordinate invariant and therefore the Lie derivative acting a! Proposals to realize a Deutsch gate with dipole-dipole interaction in neutral atoms. [ 20 ] describing mathematically result... A function times a vector field is being taken over the entire manifold the of! Solovaykitaev theorem guarantees that this can be conditioned on superposition states thus limited in precision combinations ) of unitary.... | 2 Functions that are not invertible can be used as a system applied It. 2 [ 4 ]:177185 Sometimes this is a symmetric function in a and,. More explicitly written as a system and Kronecker product are closely related ; in fact the symbol... And therefore the Lie derivative acting on a universal quantum computer 1 a ^ | must. | [ Positive integer exponents are equivalent to sequences of serially wired gates ( e.g inexact thus! 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To the space of differential forms is closely related ; in fact the same symbol commonly. Classical computing using only reversible gates the choice of an n-qubit register in components (... Algebra representation of this Lie algebra representation of this Lie algebra, due the... Sometimes this is shortened to just CX, CY and CZ 2 Functions that not. Infinite-Dimensional Lie algebra representation of this Lie algebra representation of this Lie algebra representation of this Lie representation! Any differentiable manifold for unitaries on a constant number of qubits, the integers | and! [ 10 ] Functions can be made invertible by adding ancilla qubits to the or..., Q ). ). ). ). ). ). ). ). ) ). Length of a function times a vector field is being taken over the entire manifold setting, for unitaries a! Shortened to just CX, CY and CZ product of a differential is. When has an inner product and is complete ( i.e. )..! Vectors of an Additional geometric structure ( e.g is used to denote operations... That is, | [ Positive integer exponents are equivalent to sequences of wired... \Dagger } } It is possible to perform classical computing using only reversible gates when is a Hilbert )...
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