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The three controlled-NOT gates have directions $1\to 2$, $2\to 1$, and $1\to 2$. On computational basis bits they act asBy linearity, the whole quantum circuit is the SWAP gate. Thereforeso $|\alpha ⟩=|\varphi ⟩$ and $|\beta ⟩=|\psi ⟩$.

Use the conventionChanging the summation variable from $j$ to $j-1$ givesThus the Fourier-basis states are eigenvectors of the cyclic shift, with respective eigenvalues $e^{-2\pi ik/N}$, and the cyclic shift diagonalization by the quantum Fourier transform is

For $N=4$, write $k=2x+y$ with $x,y\in \{0,1\}$. ThensoHence the required quantum circuit applies, from input to output,The first phase gate acts on the most significant qubit $x$ and the second on the least significant qubit $y$.

The operators $X_i$ act on different qubits and therefore commute, soSince $X=HZH$,The scalar $e^{-it}$ is a global phase. Thus each factor uses two Hadamard gates and one phase gate, and applying the factors in parallel or sequentially gives an exact quantum circuit of $\boxed{3n}$ elementary gates.

QMA is the class of promise problems with a polynomial-size quantum witness checked by a uniform polynomial-size quantum circuit. YES instances have an accepting witness with probability at least $2/3$; NO instances have acceptance at most $1/3$ for every witness, including mixed states. The verifier's acceptance is linear in the witness density operator. Inverse-polynomial or exponentially small error can be obtained through QMA error reduction.

For two quantum circuits with corresponding unitary gates, their difference is a sum of terms containing one gate difference and exact or perturbed products on either side. Unitary invariance and the triangle inequality for the operator norm therefore bound the circuit difference by the sum of individual gate errors. Tensoring a gate with an identity leaves that norm unchanged. On a normalized input, the same bound controls the output vector distance with gate phases retained. This is a worst-case coherent bound; cancellation can make a particular circuit less sensitive, but cannot be assumed without further information.

Quantum complexity theory studies resources needed by quantum circuits and quantum algorithms, including time, workspace, and quantum query complexity. Its models specify which input oracles and gate implementations are available; counting oracle calls is different from counting all elementary gates.

A unitary operation on a fixed number of qubits used as a component of a quantum circuit. Examples include the Hadamard gate and Controlled-Z gate. Measurements are described separately by quantum measurement; a gate is a reversible operation on its full input state.

The quantum query complexity of a task counts uses of an input oracle by a quantum circuit, for a specified success probability. Known gates and workspace operations are not counted as oracle queries, although they contribute to the circuit's full runtime. An efficient query bound therefore need not, by itself, be an efficient gate bound.

A quantum register is a named tensor product factor of a quantum circuit's Hilbert space, usually a collection of qubits. Registers separate data, control, output and workspace roles; uncomputation returns workspace to a fixed quantum state without destroying coherence in the data.

A qudit is a quantum subsystem with $d$ distinguishable basis states and Hilbert space ${\mathbb{C}}^d$. A qubit is the special case $d=2$. Tensor products of qudits give a natural setting for local Hamiltonians and finite-dimensional quantum circuits.

A stoquastic Hamiltonian is a real Hermitian Hamiltonian operator with nonpositive off-diagonal entries in a specified basis, usually the computational basis. The property depends on that basis. Permutation matrices in a quantum circuit's propagation terms and $|-⟩⟨-|$ input penalties give a useful family of such operators.