Eigenstate-assisted realization of general quantum controlled unitaries with a fixed cost

TL;DR

This paper introduces a fixed-cost, eigenstate-assisted method for implementing controlled unitaries in quantum computing, reducing overhead and enabling efficient applications in algorithms and machine learning.

Contribution

The authors present a universal, low-depth circuit construction for controlled unitaries that works with any unitary and eigenstate preparation method, independent of the unitary's decomposition.

Findings

Achieves controlled-$U$ with fixed 4 CNOT and 2 Toffoli gates per qubit.

Requires $2n+1$ qubits and an eigenstate of $U$ for $n$-qubit unitaries.

Applicable to variational algorithms and quantum machine learning.

Abstract

Controlled unitary gates are a basic element in many quantum algorithms. Converting a general unitary $U$ with a known decomposition into its controlled version, controlled-$U$, can introduce a large overhead in terms of the depth of the circuit. We present a general method to take any unitary $U$ into controlled-$U$ using a fixed circuit with 4 CNOT gates and 2 Toffoli gates per qubit. For $n$-qubit unitaries and one control qubit, we require $2n+1$ qubits and a circuit that can generate an eigenstate of $U$, for which there are many cost-effective known algorithms. The method also works for any black block implementation of $U$, achieving a constant-depth realization independent of its decomposition. We illustrate its use in the Hadamard test and discuss applications to variational and quantum machine-learning algorithms.