Quantum logic gate

The current notation for quantum gates was developed by many of the founders of quantum information science including Adriano Barenco, Charles Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter, building on notation introduced by Richard Feynman in 1986. == Representation ==

states that are not entangled and lack global phase can be represented as points on the surface of the Bloch sphere, written as |\psi\rangle = \cos\left(\theta /2\right)|0 \rangle + e^{i\varphi}\sin\left(\theta /2\right) |1\rangle.Rotations about the axes of the Bloch sphere are represented by the rotation operator gates. Quantum logic gates are represented by unitary matrices. A gate that acts on n qubits (a register) is represented by a 2^n \times 2^n unitary matrix, and the set of all such gates with the group operation of matrix multiplication is the unitary group U(2n). :| \psi \rangle = v_{00} | 00 \rangle + v_{01} | 0 1 \rangle + v_{10} | 1 0 \rangle + v_{11} | 1 1 \rangle \rightarrow \begin{bmatrix} v_{00} \\ v_{01} \\ v_{10} \\ v_{11} \end{bmatrix}. The action of the gate on a specific quantum state is found by multiplying the vector |\psi_1\rangle, which represents the state by the matrix U representing the gate. The result is a new quantum state :U|\psi_1\rangle = |\psi_2\rangle. Relation to the time evolution operator The Schrödinger equation describes how quantum systems that are not observed evolve over time, and is i\hbar\frac{d}{dt}|\Psi\rangle = \hat{H}|\Psi\rangle. When the system is in a stable environment, so it has a constant Hamiltonian, the solution to this equation is U(t) = e^{-i\hat{H}t/\hbar}. If the time t is always the same it may be omitted for simplicity, and the way quantum states evolve can be described as U|\psi_1\rangle = |\psi_2\rangle, just as in the above section. That is, a quantum gate is how a quantum system that is not observed evolves over some specific time, or equivalently, a gate is the unitary time evolution operator U acting on a quantum state for a specific duration. == Notable examples ==

There exists an uncountably infinite number of gates. Some of them have been named by various authors, and below follow some of those most often used in the literature. Identity gate The identity gate is the identity matrix, usually written as I, and is defined for a single qubit as : I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} , where I is basis independent and does not modify the quantum state. The identity gate is most useful when describing mathematically the result of various gate operations or when discussing multi-qubit circuits. Pauli gates (X,Y,Z) The Pauli gates (X,Y,Z) are the three Pauli matrices (\sigma_x,\sigma_y,\sigma_z) and act on a single qubit. The Pauli X, Y and Z equate, respectively, to a rotation around the x, y and z axes of the Bloch sphere by \pi radians. The Pauli-X gate is the quantum equivalent of the NOT gate for classical computers with respect to the standard basis which distinguishes the z axis on the Bloch sphere. It is sometimes called a bit-flip as it maps |0\rangle to |1\rangle and |1\rangle to |0\rangle. Similarly, the Pauli-Y maps |0\rangle to i|1\rangle and |1\rangle to . Pauli Z leaves the basis state |0\rangle unchanged and maps |1\rangle to Due to this nature, Pauli Z is sometimes called phase-flip. These matrices are usually represented as : X = \sigma_x =\operatorname{NOT} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} , : Y = \sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, : Z = \sigma_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. The Pauli matrices are involutory, meaning that the square of a Pauli matrix is the identity matrix. :I^2 = X^2 = Y^2 = Z^2 = -iXYZ = I The Pauli matrices also anti-commute, for example ZX=iY=-XZ. The matrix exponential of a Pauli matrix \sigma_j is a rotation operator, often written as e^{-i\sigma_j\theta/2}. Controlled gates Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some operation. Hadamard gate The Hadamard or Walsh-Hadamard gate, named after Jacques Hadamard () and Joseph L. Walsh, acts on a single qubit. It maps the basis states |0\rangle \mapsto \frac \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ \end{bmatrix} The Toffoli gate is related to the classical AND (\land) and XOR (\oplus) operations as it performs the mapping |a, b, c\rangle \mapsto |a, b, c\oplus (a \land b)\rangle on states in the computational basis. The Toffoli gate can be expressed using Pauli matrices as : \mbox{Toff} = e^{i\frac{\pi}{8}(I-Z_1)(I-Z_2)(I-X_3)}= e^{-i\frac{\pi}{8}(I-Z_1)(I-Z_2)(I-X_3)}. == Universal quantum gates ==

A set of universal quantum gates is any set of gates to which any operation possible on a quantum computer can be reduced, that is, any other unitary operation can be expressed as a finite sequence of gates from the set. Technically, this is impossible with anything less than an uncountable set of gates since the number of possible quantum gates is uncountable, whereas the number of finite sequences from a finite set is countable. To solve this problem, we only require that any quantum operation can be approximated by a sequence of gates from this finite set. Moreover, for unitaries on a constant number of qubits, the Solovay–Kitaev theorem guarantees that this can be done efficiently. Checking if a set of quantum gates is universal can be done using group theory methods and/or relation to (approximate) unitary t-designs. The spectral gap conjecture, if true, would imply that a generically chosen set of quantum gates is efficiently universal. Some universal quantum gate sets include: • The rotation operators , , , the phase shift gate {{efn|Either the or List of quantum logic gates#Identity gate and global phase| gate can be used, as R_z(\delta)\operatorname{Ph}(\delta/2)=P(\delta)}} and CNOT are commonly used to form a universal quantum gate set. • The Clifford set {CNOT, H, S} + T gate. The Clifford set alone is not a universal quantum gate set, as it can be efficiently simulated classically according to the Gottesman–Knill theorem. • The Toffoli gate + Hadamard gate. named after physicist David Deutsch. It is a general case of CC-U, or controlled-controlled-unitary gate, and is defined as : |a, b, c\rangle \mapsto \begin{cases} i \cos(\theta) |a, b , c\rangle + \sin(\theta) |a, b, 1 - c\rangle & \text{for}\ a = b = 1, \\ |a, b, c\rangle & \text{otherwise}. \end{cases} Unfortunately, a working Deutsch gate has remained out of reach, due to lack of a protocol. There are some proposals to realize a Deutsch gate with dipole–dipole interaction in neutral atoms. A universal logic gate for reversible classical computing, the Toffoli gate, is reducible to the Deutsch gate D(\pi/2), thus showing that all reversible classical logic operations can be performed on a universal quantum computer. There also exist single two-qubit gates sufficient for universality. In 1996, Adriano Barenco showed that the Deutsch gate can be decomposed using only a single two-qubit gate (Barenco gate), but it is hard to realize experimentally. This feature is exclusive to quantum circuits, as there is no classical two-bit gate that is both reversible and universal. Universal two-qubit gates could be implemented to improve classical reversible circuits in fast low-power microprocessors. == Circuit composition ==

Serially wired gates Assume that we have two gates A and B that both act on n qubits. When B is put after A in a series circuit, then the effect of the two gates can be described as a single gate C. : C = B \cdot A where \cdot is matrix multiplication. The resulting gate C will have the same dimensions as A and B. The order in which the gates would appear in a circuit diagram is reversed when multiplying them together. For example, putting the Pauli X gate after the Pauli Y gate, both of which act on a single qubit, can be described as a single combined gate C: : C = X \cdot Y = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} = iZ The product symbol (\cdot) is often omitted. Exponents of quantum gates All real exponents of unitary matrices are also unitary matrices, and all quantum gates are unitary matrices. Positive integer exponents are equivalent to sequences of serially wired gates (e.g. and the real exponents is a generalization of the series circuit. For example, X^\pi and \sqrt{X}=X^{1/2} are both valid quantum gates. U^0=I for any unitary matrix U. The identity matrix (I) behaves like a NOP and can be represented as bare wire in quantum circuits, or not shown at all. All gates are unitary matrices, so that U^\dagger U = UU^\dagger = I and {{nowrap|U^\dagger = U^{-1},}} where \dagger is the conjugate transpose. This means that negative exponents of gates are unitary inverses of their positively exponentiated counterparts: {{nowrap|U^{-n} = (U^n)^{\dagger}.}} For example, some negative exponents of the phase shift gates are T^{-1}=T^{\dagger} and {{nowrap|T^{-2}=(T^2)^{\dagger}=S^{\dagger}.}} Note that for a Hermitian matrix H^\dagger=H, and because of unitarity, HH^\dagger=I, so H^2 = I for all Hermitian gates. They are involutory. Examples of Hermitian gates are the Pauli gates, Hadamard, CNOT, SWAP and Toffoli. Each Hermitian unitary matrix H has the property that e^{i\theta H}=(\cos \theta)I+(i\sin \theta) H where H=e^{i\frac{\pi}{2}(I-H)}=e^{-i\frac{\pi}{2}(I-H)}. The exponent of a gate is a multiple of the duration of time that the time evolution operator is applied to a quantum state. E.g., in a spin qubit quantum computer the \sqrt{\mathrm{SWAP}} gate could be realized via exchange interaction on the spin of two electrons for half the duration of a full exchange interaction. Hadamard transform The gate H_2 = H \otimes H is the Hadamard gate applied in parallel on 2 qubits. It can be written as: :H_2 = H \otimes H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \otimes \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end{bmatrix} This "two-qubit parallel Hadamard gate" will, when applied to, for example, the two-qubit zero-vector create a quantum state that has equal probability of being observed in any of its four possible outcomes; and We can write this operation as: :H_2 |00\rangle = \frac{1}{2} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} = \frac{1}{2} |00\rangle + \frac{1}{2} |01\rangle +\frac{1}{2} |10\rangle +\frac{1}{2} |11\rangle = \frac{|00\rangle + |01\rangle + |10\rangle + |11\rangle}{2} register Here the amplitude for each measurable state is . The probability to observe any state is the square of the absolute value of the measurable states amplitude, which in the above example means that there is one in four that we observe any one of the individual four cases. See measurement for details. H_2 performs the Hadamard transform on two qubits. Similarly the gate \underbrace{ H \otimes H \otimes \dots \otimes H }_{n\text{ times}} = \bigotimes_{i=0}^{n-1} H = H^{\otimes n} = H_n performs a Hadamard transform on a register of n qubits. When applied to a register of n qubits all initialized to the Hadamard transform puts the quantum register into a superposition with equal probability of being measured in any of its 2^n possible states: :\bigotimes_{i=0}^{n-1}(H|0\rangle) = \frac{1}{\sqrt{2^n}} \begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix} = \frac{1}{\sqrt{2^n}} \Big( |0\rangle + |1\rangle + \dots + |2^n-1\rangle \Big)= \frac{1}{\sqrt{2^n}}\sum_{i=0}^{2^{n}-1}|i\rangle This state is a uniform superposition and it is generated as the first step in some search algorithms, for example in amplitude amplification and phase estimation. Measuring this state results in a random number between |0\rangle and How random the number is depends on the fidelity of the logic gates. If not measured, it is a quantum state with equal probability amplitude \frac{1}{\sqrt{2^n}} for each of its possible states. The Hadamard transform acts on a register |\psi\rangle with n qubits such that |\psi\rangle = \bigotimes_{i=0}^{n-1} |\psi_i\rangle as follows: :\bigotimes_{i=0}^{n-1}H|\psi\rangle = \bigotimes_{i=0}^{n-1}\frac{|0\rangle + (-1)^{\psi_i}|1\rangle}{\sqrt{2}} = \frac{1}{\sqrt{2^n}}\bigotimes_{i=0}^{n-1}\Big(|0\rangle + (-1)^{\psi_i}|1\rangle\Big) = H|\psi_0\rangle \otimes H|\psi_1\rangle \otimes \cdots \otimes H|\psi_{n-1}\rangle Application on entangled states If two or more qubits are viewed as a single quantum state, this combined state is equal to the tensor product of the constituent qubits. Any state that can be written as a tensor product from the constituent subsystems are called separable states. On the other hand, an entangled state is any state that cannot be tensor-factorized, or in other words: An entangled state can not be written as a tensor product of its constituent qubits states. Special care must be taken when applying gates to constituent qubits that make up entangled states. If we have a set of N qubits that are entangled and wish to apply a quantum gate on M H only act on 1 qubit, but |\psi\rangle is an entangled quantum state that spans 2 qubits. In our example, |\psi\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}. For example, the Hadamard gate acts on a single qubit, but if we feed it the first of the two qubits that constitute the entangled Bell state {{nowrap|\frac{|00\rangle + |11\rangle}{\sqrt{2}},}} we cannot write that operation easily. We need to extend the Hadamard gate H with the identity gate I so that we can act on quantum states that span two qubits: :K = H \otimes I = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \otimes \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1\end{bmatrix} The gate K can now be applied to any two-qubit state, entangled or otherwise. The gate K will leave the second qubit untouched and apply the Hadamard transform to the first qubit. If applied to the Bell state in our example, we may write that as: :K \frac{|00\rangle + |11\rangle}{\sqrt{2}} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1\end{bmatrix} \frac{1}{\sqrt{2}} \begin{bmatrix}1 \\ 0 \\ 0 \\ 1\end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1 \\ 1 \\ 1 \\ -1 \end{bmatrix} = \frac{|00\rangle + |01\rangle + |10\rangle - |11\rangle}{2} Computational complexity and the tensor product The time complexity for multiplying two n \times n-matrices is at least if using a classical machine. Because the size of a gate that operates on q qubits is 2^q \times 2^q it means that the time for simulating a step in a quantum circuit (by means of multiplying the gates) that operates on generic entangled states is {{nowrap|\Omega({2^q}^2 \log({2^q})).}} For this reason it is believed to be intractable to simulate large entangled quantum systems using classical computers. Subsets of the gates, such as the Clifford gates, or the trivial case of circuits that only implement classical Boolean functions (e.g. combinations of X, CNOT, Toffoli), can however be efficiently simulated on classical computers. The state vector of a quantum register with n qubits is 2^n complex entries. Storing the probability amplitudes as a list of floating point values is not tractable for large n. Unitary inversion of gates Because all quantum logical gates are reversible, any composition of multiple gates is also reversible. All products and tensor products (i.e. series and parallel combinations) of unitary matrices are also unitary matrices. This means that it is possible to construct an inverse of all algorithms and functions, as long as they contain only gates. Initialization, measurement, I/O and spontaneous decoherence are side effects in quantum computers. Gates however are purely functional and bijective. If U is a unitary matrix, then U^\dagger U = UU^\dagger = I and {{nowrap|U^\dagger = U^{-1}.}} The dagger (\dagger) denotes the conjugate transpose. It is also called the Hermitian adjoint. If a function F is a product of m gates, the unitary inverse of the function F^\dagger can be constructed: Because (UV)^\dagger = V^\dagger U^\dagger we have, after repeated application on itself :F^\dagger = \left(\prod_{i=1}^{m} A_i\right)^\dagger = \prod_{i=m}^{1} A^\dagger_{i} = A_m^\dagger \cdot \dots \cdot A_2^\dagger \cdot A_1^\dagger Similarly if the function G consists of two gates A and B in parallel, then G=A\otimes B and Gates that are their own unitary inverses are called Hermitian or self-adjoint operators. Some elementary gates such as the Hadamard (H) and the Pauli gates (I, X, Y, Z) are Hermitian operators, while others like the phase shift (S, T, P, CPhase) gates generally are not. For example, an algorithm for addition can be used for subtraction, if it is being "run in reverse", as its unitary inverse. The inverse quantum Fourier transform is the unitary inverse. Unitary inverses can also be used for uncomputation. Programming languages for quantum computers, such as Microsoft's Q#, Bernhard Ömer's QCL, and IBM's Qiskit, contain function inversion as programming concepts. == Measurement ==

Measurement (sometimes called observation) is irreversible and therefore not a quantum gate, because it assigns the observed quantum state to a single value. Measurement takes a quantum state and projects it to one of the basis vectors, with a likelihood equal to the square of the vector's length (in the 2-norm) along that basis vector. This is known as the Born rule and appears as a stochastic non-reversible operation as it probabilistically sets the quantum state equal to the basis vector that represents the measured state. At the instant of measurement, the state is said to "collapse" to the definite single value that was measured. Why and how, or even if the quantum state collapses at measurement, is called the measurement problem. The probability of measuring a value with probability amplitude \phi is where |\cdot| is the modulus. Measuring a single qubit, whose quantum state is represented by the vector {{nowrap|a|0\rangle + b|1\rangle = \begin{bmatrix} a \\ b \end{bmatrix},}} will result in |0\rangle with probability and in For example, measuring a qubit with the quantum state \frac{|0\rangle -i|1\rangle }{\sqrt{2}} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ -i \end{bmatrix} will yield with equal probability either |0\rangle or |a|^2 = \cos^2 \theta and Note: |a|^2 is the probability of measuring |0\rangle and |b|^2 is the probability of measuring A quantum state |\Psi\rangle that spans qubits can be written as a vector in 2^n complex dimensions: {{nowrap||\Psi\rangle \in \mathbb C^{2^n}.}} This is because the tensor product of qubits is a vector in 2^n dimensions. This way, a register of qubits can be measured to 2^n distinct states, similar to how a register of classical bits can hold 2^n distinct states. Unlike with the bits of classical computers, quantum states can have non-zero probability amplitudes in multiple measurable values simultaneously. This is called superposition. The sum of all probabilities for all outcomes must always be equal to . Another way to say this is that the Pythagorean theorem generalized to \mathbb C^{2^n} has that all quantum states |\Psi\rangle with qubits must satisfy 1 = \sum_{x=0}^{2^n-1}|a_x|^2, where a_x is the probability amplitude for measurable state A geometric interpretation of this is that the possible value-space of a quantum state |\Psi\rangle with qubits is the surface of the unit sphere in \mathbb C^{2^n} and that the unitary transforms (i.e. quantum logic gates) applied to it are rotations on the sphere. The rotations that the gates perform form the symmetry group U(2n). Measurement is then a probabilistic projection of the points at the surface of this complex sphere onto the basis vectors that span the space (and labels the outcomes). In many cases the space is represented as a Hilbert space \mathcal{H} rather than some specific complex space. The number of dimensions (defined by the basis vectors, and thus also the possible outcomes from measurement) is then often implied by the operands, for example as the required state space for solving a problem. In Grover's algorithm, Grover named this generic basis vector set "the database". The selection of basis vectors against which to measure a quantum state will influence the outcome of the measurement. See change of basis and Von Neumann entropy for details. In this article, we always use the computational basis, which means that we have labeled the 2^n basis vectors of an -qubit register or use the binary representation {{nowrap||0_{10}\rangle = |0\dots 00_{2}\rangle, |1_{10}\rangle = |0\dots01_{2}\rangle, |2_{10}\rangle = |0\dots10_{2}\rangle, \cdots, |2^n-1\rangle = |111\dots1_{2}\rangle.}} In quantum mechanics, the basis vectors constitute an orthonormal basis. An example of usage of an alternative measurement basis is in the BB84 cipher. The effect of measurement on entangled states -CNOT gate, which when given the input |00\rangle produces a Bell state If two quantum states (i.e. qubits, or registers) are entangled (meaning that their combined state cannot be expressed as a tensor product), measurement of one register affects or reveals the state of the other register by partially or entirely collapsing its state too. This effect can be used for computation, and is used in many algorithms. The Hadamard-CNOT combination acts on the zero-state as follows: :\operatorname{CNOT}(H \otimes I)|00\rangle = \left( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} \left( \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \otimes \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \right) \right) \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix} = \frac{|00\rangle + |11\rangle}{\sqrt{2}} spanned by the basis vectors |00\rangle and as in the picture. The unit sphere {{nowrap|(in \mathbb C^{4})}} that represent the possible value-space of the 2-qubit system intersects the plane and |\Psi\rangle lies on the unit spheres surface. Because there is equal probability of measuring this state to |00\rangle or and because b=c=0 there is zero probability of measuring it to |01\rangle or This resulting state is the Bell state {{nowrap|\frac{|00\rangle + |11\rangle}{\sqrt{2}} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 1\end{bmatrix}.}} It cannot be described as a tensor product of two qubits. There is no solution for :\begin{bmatrix} x \\ y \end{bmatrix} \otimes \begin{bmatrix} w \\ z \end{bmatrix} = \begin{bmatrix} xw \\ xz \\ yw \\ yz \end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix}, because for example needs to be both non-zero and zero in the case of and . The quantum state spans the two qubits. This is called entanglement. Measuring one of the two qubits that make up this Bell state will result in that the other qubit logically must have the same value, both must be the same: Either it will be found in the state or in the state If we measure one of the qubits to be for example then the other qubit must also be because their combined state became Measurement of one of the qubits collapses the entire quantum state, that span the two qubits. The GHZ state is a similar entangled quantum state that spans three or more qubits. This type of value-assignment occurs instantaneously over any distance and this has as of 2018 been experimentally verified by QUESS for distances of up to 1200 kilometers. That the phenomena appears to happen instantaneously as opposed to the time it would take to traverse the distance separating the qubits at the speed of light is called the EPR paradox, and it is an open question in physics how to resolve this. Originally it was solved by giving up the assumption of local realism, but other interpretations have also emerged. For more information see the Bell test experiments. The no-communication theorem proves that this phenomenon cannot be used for faster-than-light communication of classical information. Measurement on registers with pairwise entangled qubits Take a register A with qubits all initialized to and feed it through a parallel Hadamard gate {{nowrap|H^{\otimes n}.}} Register A will then enter the state \frac{1}{\sqrt{2^n}} \sum_{k=0}^{2^{n}-1} |k\rangle that have equal probability of when measured to be in any of its 2^n possible states; |0\rangle to Take a second register B, also with qubits initialized to |0\rangle and pairwise CNOT its qubits with the qubits in register A, such that for each the qubits A_{p} and B_{p} forms the state {{nowrap||A_{p}B_{p}\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}.}} If we now measure the qubits in register A, then register B will be found to contain the same value as A. If we however instead apply a quantum logic gate on A and then measure, then where F^\dagger is the unitary inverse of . Because of how unitary inverses of gates act, {{nowrap|F^\dagger |A\rangle = F^{-1}(|A\rangle) = |B\rangle.}} For example, say F(x)=x+3 \pmod{2^n}, then {{nowrap||B\rangle = |A - 3 \pmod{2^n}\rangle.}} The equality will hold no matter in which order measurement is performed (on the registers A or B), assuming that has run to completion. Measurement can even be randomly and concurrently interleaved qubit by qubit, since the measurements assignment of one qubit will limit the possible value-space from the other entangled qubits. Even though the equalities holds, the probabilities for measuring the possible outcomes may change as a result of applying , as may be the intent in a quantum search algorithm. This effect of value-sharing via entanglement is used in Shor's algorithm, phase estimation and in quantum counting. Using the Fourier transform to amplify the probability amplitudes of the solution states for some problem is a generic method known as "Fourier fishing". == Logic function synthesis ==

, given by Feynman in 1986. This puts a limit on how large functions can be brute-force factorized into primitive quantum gates. Typically quantum programs are instead built using relatively small and simple quantum functions, similar to normal classical programming. Because of the gates unitary nature, all functions must be reversible and always be bijective mappings of input to output. There must always exist a function F^{-1} such that {{nowrap|F^{-1}(F(|\psi\rangle)) = |\psi\rangle.}} Functions that are not invertible can be made invertible by adding ancilla qubits to the input or the output, or both. After the function has run to completion, the ancilla qubits can then either be uncomputed or left untouched. Measuring or otherwise collapsing the quantum state of an ancilla qubit (e.g. by re-initializing the value of it, or by its spontaneous decoherence) that have not been uncomputed may result in errors, as their state may be entangled with the qubits that are still being used in computations. Logically irreversible operations, for example addition modulo 2^n of two n-qubit registers a and b, {{nowrap|F(a, b) = a+b \pmod{2^n},}} can be made logically reversible by adding information to the output, so that the input can be computed from the output (i.e. there exists a function {{nowrap|F^{-1}).}} In our example, this can be done by passing on one of the input registers to the output: {{nowrap|F(|a\rangle \otimes |b\rangle) = |a+b \pmod{2^n}\rangle \otimes |a\rangle.}} The output can then be used to compute the input (i.e. given the output a+b and we can easily find the input; a is given and and the function is made bijective. All Boolean algebraic expressions can be encoded as unitary transforms (quantum logic gates), for example by using combinations of the Pauli-X, CNOT and Toffoli gates. These gates are functionally complete in the Boolean logic domain. There are many unitary transforms available in the libraries of Q#, QCL, Qiskit, and other quantum programming languages. It also appears in the literature. For example, \mathrm{inc}(|x\rangle) = |x + 1 \pmod{2^{x_\text{length}}}\rangle, where x_\text{length} is the number of qubits that constitutes the register is implemented as the following in QCL: cond qufunct inc(qureg x) { // increment register int i; for i = #x-1 to 0 step -1 { CNot(x[i], x[0::i]); // apply controlled-not from } // MSB to LSB } , AND and NOT respectively, and comes from the Boolean representation of Pauli-X with zero or more control qubits when applied to states that are in the computational basis. In QCL, decrement is done by "undoing" increment. The prefix ! is used to instead run the unitary inverse of the function. !inc(x) is the inverse of inc(x) and instead performs the operation {{nowrap|\mathrm{inc}^\dagger |x\rangle = \mathrm{inc}^{-1}(|x\rangle) = |x - 1 \pmod{2^{x_\text{length}}}\rangle.}} The cond keyword means that the function can be conditional. Measurement of quantum registers results in binary values that the classical computer can use in its computations. Quantum algorithms often contain both a classical and a quantum part. Unmeasured I/O (sending qubits to remote computers without collapsing their quantum states) can be used to create networks of quantum computers. Entanglement swapping can then be used to realize distributed algorithms with quantum computers that are not directly connected. Examples of distributed algorithms that only require the use of a handful of quantum logic gates are superdense coding, the quantum Byzantine agreement and the BB84 cipherkey exchange protocol. == See also ==