Strategies for optimizing double-bracket quantum algorithms

TL;DR

This work develops a systematic optimization framework for double-bracket quantum algorithms (DBQAs) aimed at diagonalizing target Hamiltonians and preparing eigenstates on near-term quantum devices. It analyzes three DBI families (BHMM, GWW, and adaptive variational) and a spectrum of cost functions to quantify diagonalization progress, then details scheduling and generator-selection strategies, including analytically motivated and adaptive parametrizations. Through numerical experiments on TFIM and XXZ spin chains, adaptive DBIs consistently outperform fixed BHMM approaches, especially when combined with gradient-based parametrization and hardware-friendly compilation via group commutator formulas. The study demonstrates how a three-step workflow—scheduling, generator optimization, and compilation—can yield practical, implementable DBQA protocols with improved diagonalization performance on current or near-term quantum hardware.

Abstract

Recently double-bracket quantum algorithms have been proposed as a way to compile circuits for approximating eigenstates. Physically, they consist of appropriately composing evolutions under an input Hamiltonian together with diagonal evolutions. Here, we present strategies to optimize the choice of the double-bracket evolutions to enhance the diagonalization efficiency. This can be done by finding optimal generators and durations of the evolutions. We present numerical results regarding the preparation of double-bracket iterations, both in ideal cases where the algorithm's setup provides analytical convergence guarantees and in more heuristic cases, where we use an adaptive and variational approach to optimize the generators of the evolutions. As an example, we discuss the efficacy of these optimization strategies when considering a spin-chain Hamiltonian as the target. To propose algorithms that can be executed starting today, fully aware of the limitations of the quantum technologies at our disposal, we finally present a selection of diagonal evolution parametrizations that can be directly compiled into CNOTs and single-qubit rotation gates. We discuss the advantages and limitations of this compilation and propose a way to take advantage of this approach when used in synergy with other existing methods.

Figures (4)

• Figure 1: TFIM with $L=5$, transverse field $h=3$ and $\hat{D}_0=\Delta(\hat{H}_0)$. a) We show $f_1$ for DBI with steps optimized using $f_1, f_2$ and $f_3$. The inset shows that optimization with $f_3$ is sensitive to the reference state. b) We show the validity of the Taylor expansion for various orders $n$. For all $n$ the DBR is well approximated for short DBR durations $s<0.04$. In order to match the first local minimum of the $\sigma$-decrease curve, we find that a relatively high polynomial degree is needed.
• Figure 2: DBRs (top) and DBIs (bottom) for the XXZ model with $L=5$ qubits and $\Delta = 0.5$. In panel a) we show DBRs for the analytically motivated ansatze. The GWW parametrization $\Delta(\hat{H}_0)$ yields the largest $\sigma$-decrease. Bullets indicate the first local minima extracted numerically. In panel b) we show DBRs for parametrizations which can be easily compiled using CNOT and single-qubit rotation gates. Here the global minimum is not necessarily the first. The GWW curve is repeated because it coincides with the NN Ising strategy. In panels c) and d) we show the respective BHMM iteration. We stop, for each adopted strategy, the DBI steps when they would have provided only marginal gain at the expense of long evolutions. Due to the symmetries of XXZ the GWW and magnetic field strategies yield block-diagonalization as opposed to full diagonalization. In panel d) for short $s$ the local $\hat{D}$ parametrization strategies perform comparably well with those in panel c). These results set the benchmark when moving from analytically prescribed BHMM DBIs to adaptive DBIs.
• Figure 3: DBI applied to XXZ model with $L=5$ qubits, $\delta=0.5$ using variational strategies. Variational strategies, except for the GWW strategy (blue) surpass the best diagonalization achieved with BHMM (grey dotted line). See Fig.\ref{['fig:BHMM']}. Pauli-Z strategy achieved the most diagonalization but is difficult to implement physically. We also see that despite being more computationally exhaustive, gradient descent with computational ansatz (green) is not necessarily better than gradient descent with magnetic field (red) and Ising model ansatz (purple).
• Figure 4: Group commutator approximation to DBR. In panel a) we apply DBR to the TFIM model with $L=5$ qubits, $h=3$ and see that both GC and HOPF are good approximations to the desired evolution up to $s<0.02$ and found the global minimum. Panel b) shows that on the XXZ model with $L=5$ qubits, $\delta=0.5$, HOPF performs considerably better than GC, which is far from the true minimum.