Quantum Multiplexer Simplification for State Preparation

TL;DR

An algorithm is proposed that detects whether a given quantum state can be factored into substates, increasing the efficiency of compiling the QSP circuit when the authors initialize states with some level of unentanglement, and is competitive with the methods in the literature.

Abstract

The initialization of quantum states or Quantum State Preparation (QSP) is a basic subroutine in quantum algorithms. In the worst case, general QSP algorithms are expensive due to the application of multi-controlled gates required to build the quantum state. Here, we propose an algorithm that detects whether a given quantum state can be factored into substates, increasing the efficiency of compiling the QSP circuit when we initialize states with some level of unentanglement. The simplification is done by eliminating controls of quantum multiplexers, significantly reducing circuit depth and the number of CNOT gates with a better execution and compilation time than the previous QSP algorithms. Considering efficiency in terms of depth and number of CNOT gates, our method is competitive with the methods in the literature. However, when it comes to run-time and compilation efficiency, our result is significantly better, and the experiments show that by increasing the number of qubits, the gap between the temporal efficiency of the methods increases.

Figures (7)

• Figure 1: Quantum multiplexer with three control qubits. The multiplexer can be represented by a list with the multicontrolled gates $U_j$. The $k$th control of $U_j$ is open (closed) when the $k$th bit in the binary representation of $j$ is equal to zero (one).
• Figure 2: Decomposition of a quantum multiplexer
• Figure 3: (a) Abstract tree representing a state preparation with quantum multiplexers. (b) Quantum state preparation circuit produced with the abstract tree, where $U_k = R_z(\beta_k)R_y(2\arcsin(\gamma_k))$.
• Figure 4: One controlled multiplexer with several targets.
• Figure 5: (left) Tree of operators to initialize a quantum state with multiplexers. Each level of the tree defines a multiplexer that is stored in an array. Dotted (solid) arrows correspond to open (closed) controls of the gate in the multiplexer. (right) Circuit corresponding to the state preparation circuit if $[U_7, U_8]=[U_9, U_{10}]$ and $[U_{11}, U_{12}]=[U_{13}, U_{14}]$.