Transformation of quantum states using uniformly controlled rotations

TL;DR

The paper tackles the problem of transforming an arbitrary $n$-qubit state into another using a quantum circuit. It introduces uniformly controlled rotations to construct a two-stage state-preparation process: first equalize input phases, then adjust amplitudes to reach the target state, yielding a concrete circuit with analytic rotation angles. The authors derive a gate count of $2^{n+2}-4n-4$ CNOTs and $2^{n+2}-5$ single-qubit rotations, establishing a new upper bound and a corresponding lower bound from degrees of freedom, and compare favorably to QR-based decompositions. The work provides practical initialization tools for quantum computation and highlights avenues for further optimization and simplification in special cases.

Abstract

We consider a unitary transformation which maps any given state of an $n$-qubit quantum register into another one. This transformation has applications in the initialization of a quantum computer, and also in some quantum algorithms. Employing uniformly controlled rotations, we present a quantum circuit of $2^{n+2}-4n-4$ CNOT gates and $2^{n+2}-5$ one-qubit elementary rotations that effects the state transformation. The complexity of the circuit is noticeably lower than the previously published results. Moreover, we present an analytic expression for the rotation angles needed for the transformation.

Figures (3)

• Figure 1: Definition of the $k$-fold uniformly controlled rotation $F^k_m({\bf a},\bm{\alpha})$ of qubit $m$ about the axis ${\bf a}$. The left hand side defines the gate symbol used for the uniformly controlled rotation. The enumeration of the qubits is arbitrary with the exception that the target qubit is the $m^{\rm th}$ one. The black control bits stand for value $1$ and the white for $0$. Above, $M=2^{k}$.
• Figure 2: Efficient gate decomposition for the uniformly controlled rotation $F^3_4({\bf a},\bm{\alpha})$. The relation of the angles $\{\theta_j\}$ to the angles $\{\alpha_j\}$ is shown in Eq. (\ref{['linear']}).
• Figure 3: Gate sequence for state preparation using uniformly controlled rotations. The rotation angles $\{\alpha_{j,k}^q\}$ for the uniformly controlled rotations are given in Eqs. (\ref{['alp']}) and (\ref{['alp2']}).