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

TL;DR

This work presents 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, achieving a constant-depth realization independent of its decomposition.

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.

Figures (2)

• Figure 1: (a) Standard controlled-$U$ circuit, where the control qubit directly triggers the unitary $U$. (b) Eigenstate-assisted implementation: the ancilla controls two register-SWAP operations surrounding a single, uncontrolled application of $U$ on the auxiliary register. When the auxiliary register is initialized in an eigenstate $|e\rangle$ of $U$, both circuits produce identical transformations on the ancilla--system subspace, up to a local single-qubit phase correction on the ancilla, with an additional overall global phase $e^{i\phi/2}$. Each controlled-SWAP (Fredkin) gate can be further decomposed into a CNOT--TOFFOLI--CNOT sequence, providing a constant-depth, hardware-compatible realization of the scheme.
• Figure 2: (a) Hadamard test with controlled-$U$: ancilla $H$, $C(U)$, ancilla $H$, and a $Z$ measurement (for estimation of the real part). (b) Control-free scheme: ancilla $H$, CSWAP--$U$--CSWAP, optional ancilla $R_Z(\phi)$, ancilla $H$, and measurement. Both schemes yield identical estimators of $\langle\psi|U|\psi\rangle$, with the control-free scheme producing a known multiplicative phase factor $\lambda$ that can be removed either by the optional $R_Z(\phi)$ gate or by classical post-processing, and an additional overall global phase $e^{i\phi/2}$.