Controlled Gate Networks: Theory and Application to Eigenvalue Estimation

TL;DR

The paper introduces controlled gate networks, a framework that minimizes two-qubit gate counts by routing between unitary operators through simple transformation gates controlled by ancillas, enabling efficient linear combinations of unitaries. It establishes general theory with a lower bound on controlled-unitary gate costs and demonstrates substantial resource reductions across three quantum-information tasks: variational subspace calculations, rodeo-based eigenvalue estimation with controlled reversal gates, and controlled time evolution for nuclear lattice simulations. Empirical results on IBM Perth and Quantinuum H1-2 show order-of-magnitude reductions in CNOTs and robust eigenvalue resolution despite device noise, highlighting practical potential for quantum simulations of quantum many-body systems. The work posits controlled gate networks as a versatile, broadly applicable circuit-design paradigm that complements existing transpilation and paves the way for more scalable quantum simulations in nuclear physics and related domains.

Abstract

We introduce a new scheme for quantum circuit design called controlled gate networks. Rather than trying to reduce the complexity of individual unitary operations, the new strategy is to toggle between all of the unitary operations needed with the fewest number of gates. We present the general theory of controlled gate networks and show that, under quite general conditions, it can significantly reduce the number of two-qubit gates needed to produce linear combinations of unitary operators. The first example we consider is a variational subspace calculation for a two-qubit system. The second example is estimating the eigenvalues of a two-qubit Hamiltonian via the rodeo algorithm using operators that we call controlled reversal gates. We use the Quantinuum H1-2 and IBM Perth devices to realize the quantum circuits. The third example is the application of controlled gate networks to the controlled time evolution of a free nucleon on a three-dimensional lattice. For all of the examples, we show very substantial reductions in the number of two-qubit gates required. Our work demonstrates that controlled gate networks are a useful tool for reducing gate complexity in quantum algorithms for quantum many-body problems such as those relevant to nuclear physics.

Figures (20)

• Figure 1: Example of a controlled gate network. The left side shows a circuit using two ancilla qubits to toggle between four possible unitary operators, $U_1, U_2, U_3, U_4$. The filled (open) dot means that the controlled operator is actuated when the attached ancilla is in the $\ket{1}$$(\ket{0})$ state. The right side achieves the same results using a controlled gate network. Each of the transformation gates $G_i$ act on a small number of qubits. $U_1$ corresponds to no $G_i$ actuated, $U_2$ corresponds to actuating $\{G_1,G_3,G_4\}$, $U_3$ corresponds to $\{G_2,G_5\}$, and $U_4$ corresponds to $\{G_1,G_2,G_3,G_4,G_5\}$.
• Figure 2: Standard approach for producing linear combinations of $U$ and $V$ with one ancilla qubit by controlling $U_1, U_2, \cdots, U_K$ with the $0$ state and controlling $V_1, V_2, \cdots, V_K$ with the $1$ state.
• Figure 3: Controlled gate network approach for producing linear combinations of $U$ and $V$ with one ancilla qubit. We control each $A_k$ and $B_k$ with the 1 state.
• Figure 4: Standard approach for producing a linear combination of eight unitary operators using three ancilla qubits.
• Figure 5: Controlled gate network approach for producing a linear combination of eight unitary operators using three ancilla qubits.