Quantum State Preparation via Schmidt Spectrum Optimisation

TL;DR

An efficient algorithm for the systematic design of shallow-depth quantum circuits capable of preparing many-body quantum states represented as Matrix Product States, and provides numerical evidence that SSO mitigates the adverse time-complexity scaling observed in previous disentangling-based approaches.

Abstract

We introduce an efficient algorithm for the systematic design of shallow-depth quantum circuits capable of preparing many-body quantum states represented as Matrix Product States (MPS). The proposed method leverages Schmidt spectrum optimization (SSO) to minimize circuit depth while preserving the entanglement structure inherent to MPS representations, thereby enabling scalable state preparation on near-term quantum hardware. The core idea is to \textit{disentangle} the target MPS using a sequence of optimised local unitaries, and then reverse this process to obtain a state preparation circuit. Specifically, we define a loss function directly on the Schmidt spectra of intermediate states and use automatic differentiation to optimise each circuit layer so as to systematically reduce entanglement entropy. Once a disentangling sequence has been learned, we take the adjoints of the optimised unitaries to obtain a shallow-depth circuit that approximately reconstructs the target MPS from the computational all-zero state. We benchmark SSO across a range of MPS approximations to the ground states of local Hamiltonians and demonstrate state-of-the-art shallow-depth performance, improving accuracy by up to an order of magnitude over existing methods. Finally, we provide numerical evidence that SSO mitigates the adverse time-complexity scaling observed in previous disentangling-based approaches.

Figures (5)

• Figure 1: Computing the Schmidt decomposition for a specified bipartition of the MPS with virtual dimension $|\alpha_j|$ generates a vector of $|\alpha_j|$ non-increasing, non-negative Schmidt coefficients, which are used to define the loss function in the SSO algorithm. Here we visualise the Schmidt spectrum corresponding to the central bipartition (across $\alpha_3$) for a random real-valued MPS with $n=6$, $\chi=6$.
• Figure 2: The Schmidt Spectrum Optimisation (SSO) algorithm. The parameterized circuit layer $U_k(\theta_k)$ is contracted with the intermediate MPS $\ket{\psi^{(k)}}$ producing an updated MPS $\ket{\psi^{(k+1)}}$. The Schmidt decomposition is computed across all bipartitions of $\ket{\psi^{(k+1)}}$, and the circuit parameters $\theta_k$ are updated via gradient descent to minimise the specified objective function. The specific objective function we introduce in Eq. \ref{['eq:new_cost_fn']} corresponds to $f(\vec{\lambda}_j)=\vec{\lambda}_{1,j}^2+\vec{\lambda}_{2,j}^2$. By repeating this process, the MPS is iteratively disentangled.
• Figure 3: The normalised Schmidt probabilities ($\lambda_i^2$) across each bipartition of a 6-site random MPS and its disentangled forms, computed via the SSO algorithm and the loss function defined in Eq. \ref{['eq:new_cost_fn']}. In this example, the target MPS has $F_{\chi=2}=0.8356$ and the disentangled state after $L=10$ layers has $F_{\chi=2}=0.9998$.
• Figure 4: The quantum circuit generated by the SSO algorithm for $L$ optimised layers and 6 qubits, showing how each staircase-like layer can be parallelised. The output of the circuit is an approximation of the target MPS.
• Figure 5: The performance of the SSO algorithm and its post-processed version (SSO+All) with the MPD and its optimsied variants. We test across four MPS approximations of the ground-states of lattice Hamiltonians: (i) The 1D quantum Ising model near-criticality with $n=48$ qubits, (ii) A disordered spin model in the many-body localised regime with $n=16$ qubits, (iii) The spinless Hubbard Chain with $n=16$ qubits, and (iv) The 2D Heisenberg with nearest-neighbour coupling on a $4\times4$ grid of qubits. Above plot: the state preparation error $\epsilon_S=1-F_S$ across layers $L$, where $F_S$ is the fidelity between the simulated circuit output and the target MPS. Below plot: the scaling of $\chi_{\text{max}}$ during disentangling using $\lambda_{\text{thresh}}=10^{-7}$. Best of 2 runs reported.