Fast Expectation Value Calculation Speedup of Quantum Approximate Optimization Algorithm: HoLCUs QAOA

TL;DR

The paper tackles the bottleneck of computing expectation values for non-unitary operators in variational quantum algorithms, particularly QAOA, by introducing HoLCUs, a deterministic Hadamard+LCU method that uses a single-qubit measurement to evaluate \(\langle \psi|A|\psi\rangle\) for \(A=\sum_k \alpha_k \xi_k U_k\) with logarithmic ancilla overhead. It then specializes the approach to QAOA and QUBO/Ising problems, demonstrating substantial runtime speedups in circuit simulators over the conventional Hadamard test, especially when LCU coefficients are degenerate. The authors provide a detailed experimental design and report speedups between \(2.5\times\) and \(22.5\times\), while noting scalability challenges due to additional ancilla qubits. The work offers a practical pathway to accelerate variational circuit training and suggests future optimizations and extensions to error-prone devices and tensor-network methods.

Abstract

In this paper, we present a new method for calculating expectation values of operators that can be expressed as a linear combination of unitary (LCU) operators. This method allows to perform this calculation in a single quantum circuit measuring a single qubit, which speeds up the computation process. This method is general for any quantum algorithm and is of particular interest in the acceleration of variational quantum algorithms, both in real devices and in simulations. We analyze its application to the parameter optimization process of the Quantum Approximate Optimization Algorithm (QAOA) and the case of having degenerate values in the matrix of the Ising problem. Finally, we apply it to several Quadratic Unconstrained Binary Optimization (QUBO) problems to analyze the speedup of the method in circuit simulators.

Figures (8)

• Figure 1: QAOA circuit with one layer and three qubits.
• Figure 2: Quantum circuit for the Hadamard Test of state $|\psi\rangle$ and a unitary operator $U$. The $S^\dagger$ gate will be on if we want the imaginary part of the expectation value and off if we want the real part.
• Figure 3: Quantum circuit for the LCU of a 2-qubits state $|\psi\rangle$ and an operator $A=\sum_{k=0}^{3} \alpha_k \xi_k U_k$, with 2 ancilla qubits.
• Figure 4: Quantum circuit for the HoLCUs of a 2-qubits state $|\psi\rangle$ and an operator $A=\sum_{k=0}^{3} \alpha_k \xi_k U_k$, with two ancilla qubits for the LCU and one for the Hadamard.
• Figure 5: Quantum circuit for the uniform initialization process for the HoLCUs with four ancilla qubits.