Quantum approximate optimization of bosonic finite-state systems

TL;DR

This work develops a Hamiltonian-based QAOA framework to tackle finite-dimensional bosonic problems on gate-based quantum hardware by designing problem-preserving mixers for three encodings: binary, symmetric, and unary. It shows that symmetric encoding, combined with the standard mixer, minimizes entangling resources and outperforms the other encodings in both initial-state preparation and measurement efficiency, while binary and unary encodings incur a $p$-fold growth in CX gate counts. The authors apply this approach to quantum approximate thermalization and to ground-state searches in the Bose-Hubbard model, demonstrating high-fidelity Gibbs-state approximation and strong-ground-state convergence in the strong-interaction regime, with more depth required for weakly interacting or highly entangled regimes. The results indicate that carefully engineered mixers can confine the variational search to feasible subspaces, offering a scalable route to simulate bosonic and other multi-level systems on near-term quantum devices, and point to future work in error mitigation, qudit-based hardware, and adaptive mixing strategies.

Abstract

There exist numerous problems in nature inherently described by finite $D$-dimensional states. Formulating these problems for execution on qubit-based quantum hardware requires mapping the qudit Hilbert space to that of multiqubit which may be exponentially larger. To exclude the infeasible subspace, one common approach relies on penalizing the objective function. However, this strategy can be inefficient as the size of the illegitimate subspace grows. Here we propose to employ the Hamiltonian-based quantum approximate optimization algorithm (QAOA) through devising appropriate mixing Hamiltonians such that the infeasible configuration space is ruled out. We investigate this idea by employing binary, symmetric, and unary mapping techniques. It is shown that the standard mixing Hamiltonian (sum of the bit-flip operations) is the optimal option for symmetric mapping, where the controlled-NOT gate count is used as a measure of implementation cost. In contrast, the other two encoding schemes witness a $p$-fold increase in this figure for a $p$-layer QAOA. We apply this framework to quantum approximate thermalization and find the ground state of the repulsive Bose-Hubbard model in the strong and weak interaction regimes.

Figures (5)

• Figure 1: (Color online). The number of CNOT gates required by a best candidate mixing Hamiltonian for $D$-state systems for binary (dark blue), symmetric (pink) and unary (light blue) encoding schemes. Note that the symmetric mapping is overall more efficient than the other two encoding techniques. For the sake of clarity, zero count of the entangling gate is set to a small number ($0.1$).
• Figure 2: (Color online). Logarithmic negativity of $\rho_{1,s}^K = |{\psi_1^K}\rangle_s\langle{\psi_1^K}|$, Eq. (\ref{['ket:map:s']}), as a function of dimension $D=K+1$. The partial transpose is calculated with respect to the first $j$ qubits with Hilbert space bipartitioning $2^j|2^{K-j}$.
• Figure 3: (Color online). Approximate thermalization of two coupled harmonic oscillators. (a) Top row is the exact thermal density matrix (real and imaginary parts) obtained via Eq. (\ref{['rho:th:exact:gen']}). (b) The middle and (c) bottom rows show the simulation results using the binary $\rho{(\Theta_b^*)}$, and symmetric $\rho{(\Theta_s^*)}$ encodings, respectively. The quantum relative entropy, Eq. (\ref{['eq:rel:ent']}), for these mappings are ${\cal S}(\rho{(\Theta_b^*)}|\rho_{\rm th}) = 0.26$ and ${\cal S}(\rho{(\Theta_s^*)}|\rho_{\rm th}) = 0.14$, and the respective state fidelities, Eq. (\ref{['mixed:fidelity']}), are ${\cal F}(\rho{(\Theta_b^*)},\rho_{\rm th}) = 0.89$, and ${\cal F}(\rho{(\Theta_s^*)},\rho_{\rm th}) = 0.93$. Here $\Theta_m^*$ are optimal circuit parameters for mapping scheme 'm' with $p=5$ QAOA layers. Consult the main text for the effect of noisy (depolarized) CNOT gate on the algorithm's efficiency. We set $\omega_1 = \omega_2 = 2$, $\lambda = 1$, and $\beta = 0.5$.
• Figure 4: (Color online). Quantifying approximate thermalization. (a) and (b) demonstrate variations of fidelity and quantum relative entropy in terms of inverse temperature. Simulations show that symmetric encoding (pink squares) is more robust to temperature changes compared to the standard mapping. For binary encoding the choice of mixing Hamiltonian is going to impact accuracy of computations. For example, the $XY$ driver Hamiltonian has the worst performance among all other options. (c) and (d) depict the same metrics as a function of the number $p$ of QAOA layers. The profiles suggest that as the circuit depth increases the classical optimization methods struggle with finding the optimal solution. Here we set $\omega_1 = \omega_2 = 2$, $\lambda = 1$; for (a) and (b) $p=5$ and for (c) and (d) $\beta = 0.5$.
• Figure 5: (Color online). Distribution of boson number per site in the (a) strong and (b) weak interaction regime for symmetric and binary encoding schemes. For the latter the system is initialized in the ground state of the respective mixing Hamiltonian, built upon Eqs. (\ref{['H:mix:d3s1']}) and (\ref{['H:mix:d3s2']}). The corresponding initial mean particle number is indicated by $p=0$. For the interaction dominated case, both mappings converge to the average occupation number $\langle \hat{n}_\ell \rangle$ obtained over the ground state $|{\Psi_{\rm gr}}\rangle$ of $\hat{H}_{\rm BH}$ (for the chosen parameters, this is the vacuum state $|{{\mathbb 0}{\mathbb 0}{\mathbb 0}{\mathbb 0}}\rangle$). The fidelity with respect to this minimum-energy eigenstate is ${\cal F}_m \ge 0.95$. In contrast, the ground state of the kinetically dominated scenario is quantum correlated. Therefore, an ansatz circuit with a higher depth, for example $p=50$, is required to represent a larger range of quantum states. This implies the classical optimizer needs more resources. With these parameters, the final state reaches ${\cal F}_{b} \approx (0.92,0.99)$ for the corresponding mixer $\hat{H}_{M,b}^{(j)}$, and ${\cal F}_s \approx 0.84$. The target mean particle number is $\langle \hat{n}_\ell \rangle \approx (0.28,0.72,0.72,0.28)$.