Practical block encodings of matrix polynomials that can also be trivially controlled

TL;DR

This work introduces FOQCS-LCU block encodings to implement matrix polynomials non-unitarily on quantum hardware with practical resource costs. By replacing multi-controlled gates with two parallel layers of two-qubit gates and using unary-encoded polynomial coefficients, the authors achieve circuit depths that scale linearly with the polynomial degree $d$ and are independent of system size, while enabling efficient control with only a few extra gates. They develop explicit circuits for spin Hamiltonians, provide Hadamard-test-friendly constructions, and map circuits to 2D nearest-neighbor architectures, accompanied by non-asymptotic gate counts. The results significantly lower the barriers to applying block-encoding methods to quantum simulation and linear algebra tasks on current and near-term devices, trading some qubit overhead for substantial depth reductions and practical controllability. Overall, the paper presents a practical module-based framework for block encodings and their polynomials, ready for compilation, benchmarking, and hardware experimentation.

Abstract

Quantum circuits naturally implement unitary operations on input quantum states. However, non-unitary operations can also be implemented through block encodings, where additional ancilla qubits are introduced and later measured. While block encoding has a number of well-established theoretical applications, its practical implementation has been prohibitively expensive for current quantum hardware. In this paper, we present practical and explicit block encoding circuits implementing matrix polynomial transformations of a target matrix. With standard approaches, block-encoding a degree-$d$ matrix polynomial requires a circuit depth scaling as $d$ times the depth for block-encoding the original matrix alone. By leveraging the recently introduced Fast One-Qubit Controlled Select LCU (FOQCS-LCU) framework, we show that the additional circuit-depth overhead required for encoding matrix polynomials can be reduced to scale linearly in $d$ with no dependence on system size or the cost of block encoding the original matrix. Moreover, we demonstrate that the FOQCS-LCU circuits and their associated matrix polynomial transformations can be controlled with negligible overhead, enabling efficient applications such as Hadamard tests. Finally, we provide explicit circuits for representative spin models, together with detailed non-asymptotic gate counts and circuit depths.

Figures (10)

• Figure 1: FOQCS-LCU block encoding circuits for a matrix polynomial $p_d(\mathcal{H}) = a_0 I + a_1 \mathcal{H} + a_2 \mathcal{H}^2 + \cdots + a_d \mathcal{H}^d$.
• Figure 2: Simplified $\text{P}_\text{R}$ circuit for the one-dimensional XYZ Heisenberg Hamiltonian with open boundary conditions, defined in \ref{['eq:def_heisenberg xyz']}, for the case $n=4$. The dashed region corresponds to the gate $\tilde{\text{P}}_\text{R}$ in \ref{['ass:PR-PL']}.
• Figure 3: Circuit implementing the controlled FOQCS-LCU block encoding of $\mathcal{H}$.
• Figure 4: Total CNOT depth for implementing the matrix polynomial $p_d(\mathcal{H})$ of the one-dimensional XYZ Heisenberg Hamiltonian as a function of system size $n$ and polynomial degree $d$, assuming a square-grid connectivity (See \ref{['tab:resource_estimation_xyz']}).
• Figure 5: CNOT count \ref{['fig:count_hadamard_test']} and depth \ref{['fig:depth_hadamard_test']} for a circuit implementing the Hadamard test, defined in \ref{['eq:def_hadamard_test']}. The computational cost for preparing the state $\ket{\varphi}$ is excluded. We consider the one-dimensional XYZ and XXZ Heisenberg models as well as the Ising model, with computational costs detailed in \ref{['tab:resource_estimation_xyz', 'tab:resource_estimation_xxz', 'tab:resource_estimation_ising']}.

Theorems & Definitions (29)

• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1
• proof
• Theorem 1
• proof
• Corollary 2
• proof