0 ) Examples of distributed algorithms that only require the use of a handful of quantum logic gates is superdense coding, the Quantum Byzantine agreement and the BB84 cipherkey exchange protocol. {\displaystyle 2^{n}} ⟩ The probability of measuring a value with probability amplitude | 1 ] Unlike many classical logic gates, quantum logic gates are reversible. {\displaystyle \phi =\pi } The combined state of two qubits is the tensor product of the two qubits. It is represented by the Pauli Z matrix: The square of a Pauli matrix is the identity matrix. q + {\displaystyle \pi } 0 ⟩ This effect of value-sharing via entanglement is used in Shor's algorithm, phase estimation and in quantum counting. | ), the unitary inverse of the function, Ψ 0 See measurement for details. × Unitary inverses can also be used for uncomputation. U = † 10 ⟩ H qubits can be written as a vector in ( Thus, the coordinates of the vector (the point) for the state ɑ|0> +β|1> is going to be described as a vector with the coordinates in it: Great! 00 | 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 ⟩ The phasor and Hadamard gates are easy, but the control-z gate, which entangles across qubits has been elusive. {\displaystyle |0\rangle } [ We need to extend the Hadamard gate 0 i + | ⟩ 1 performs the Hadamard transform on two qubits. ⟩ ⟩ {\displaystyle \otimes } ) The identity matrix ( 0 | h The Hadamard gate acts on a single qubit. {\displaystyle -|1\rangle } = | !inc(x) is the inverse of inc(x) and instead performs the operation 0 0 π 1 Thus, understanding their workflow is a crucial part of mastering quantum calculations. n ⊗ What we have just defined is called the Bloch sphere and it is a standard diagram used to represent the state of the qubit and any changes in qubit’s state. † π Ψ = {\displaystyle |01\rangle } Here is the circuit representation for the Y gate: Here is the rotation representation of the Y gate acting on the |0> state on the Bloch sphere: Here is the rotation representation of the Y gate acting on the |1> state on the Bloch sphere (remember, first you have to apply X gate to bring to the |1> state! 2 . about the Y-axis: , or use the binary representation {\displaystyle n} π ⟩ 1 0 {\displaystyle |A\rangle =F|B\rangle \iff F^{\dagger }|A\rangle =|B\rangle } There is an automatic way to design a gate, utilizing qiskit. It is defined as. 2 {\displaystyle \phi } The probability of measuring a x H ⨂ ⟩ 1 l ≥ n . . Special care must be taken when applying gates to constituent qubits that make up entangled states. 0 + {\displaystyle |11\rangle } x 0 only if it acts on the state ) and the function is made bijective. ( | | † , | ⟩ n Measurement of one of the qubits collapses the entire quantum state, that span the two qubits. i H | ): As you might have guessed, the Y gate is very similar to the X and the Z gates — it revolves the state vector around the y-axis by π radians. [ . | 2 † ) {\displaystyle |0\rangle } This is a stochastic non-reversible operation as it sets the quantum state equal to the base vector that represents the measured state (the state "collapses" to a definite single value). , where − The same as for the X and Z gates, Y gate is represented by a 2x2 dimensional matrix: Where i represents imaginary unit on the complex plane. 2 radians. 1 K {\displaystyle \phi } {\displaystyle G} {\displaystyle |\psi \rangle } ⟩ ) 1 More generally, if applying X gate on an arbitrary state α∣0⟩+β∣1⟩: As you can see, the amplitudes are flipped. 0 , will yield with equal probability either n 1 − For example, a function that act on a "qubyte" (a register of 8 qubits) would be described as a matrix with
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