e and Combining What We Know to Build a Circuit. {\displaystyle |0_{10}\rangle =|0\dots 00_{2}\rangle ,|1_{10}\rangle =|0\dots 01_{2}\rangle ,|2_{10}\rangle =|0\dots 10_{2}\rangle ,\cdots ,|2^{n}-1\rangle =|111\dots 1_{2}\rangle } with So by now we’ve understood that quantum circuits consist of a sequence of gates , where each is an 8-by-8 matrix that operates “locally” on some choice of three (or fewer) qubits. ⟩ ⟩ But in order to understand quantum computing, it’s not enough just to know about quantum states. {\displaystyle U} 0 01 n For example, a function that act on a "qubyte" (a register of 8 qubits) would be described as a matrix with ⏟ | is the phase shift. 0 The y-axis is considered to be an imaginary part, if otherwise wasn’t specified. Also, the last single qubit has which equals either or which is the Hadamard gate/transform. ⋅ ) ⟺ | {\displaystyle |1\rangle } gate; is a 3-bit gate, which is universal for classical computation but not for quantum computation. i ⊗ = 2 0 a {\displaystyle |0\rangle } with probability 0 H ∑ Because all quantum logical gates are reversible, any composition of multiple gates is also reversible. n ⊕ ⟩ ⟩ ⟩ Moreover, for unitaries on a constant number of qubits, the Solovay–Kitaev theorem guarantees that this can be done efficiently. , where ⟩ | ⟩ The Hadamard gate acts on a single qubit. ) 1 Logically irreversible operations, for example addition modulo 0 ⟩ . − performs a Hadamard transform on a register of + ⟩ | ( | {\displaystyle |\psi \rangle } 2 ( gate performs half-way of a two-qubit swap. 0 ( {\displaystyle {\frac {|00\rangle +|11\rangle }{\sqrt {2}}}} | F {\displaystyle n} | {\displaystyle |10\rangle } 1 1 m o 1 ), A single-gate set of universal quantum gates can also be formulated using the three-qubit Deutsch gate 0 | h ⟩ Measurement takes a quantum state and projects it to one of the base vectors, with a likelihood equal to the square of the vector's depth (the norm or modulus) along that base vector. 1 ⟩ This is a family of single-qubit gates that map the basis states A ⟩ Measurement (sometimes called observation) is irreversible and therefore not a quantum gate, because it assigns the observed variable to a single value. H 01

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