Scalable Preparation of Matrix Product States with Sequential and Brick Wall Quantum Circuits

TL;DR

This work introduces an end-to-end MPS preparation framework that combines the strengths of both strategies within a single pipeline, and provides principled and scalable protocols for preparing MPSs as quantum circuits, supporting utility-scale applications on near-term quantum devices.

Abstract

Preparing arbitrary quantum states requires exponential resources. Matrix Product States (MPS) admit more efficient constructions, particularly when accuracy is traded for circuit complexity. Existing approaches to MPS preparation mostly rely on heuristic circuits that are deterministic but quickly saturate in accuracy, or on variational optimization methods that reach high fidelities but scale poorly. This work introduces an end-to-end MPS preparation framework that combines the strengths of both strategies within a single pipeline. Heuristic staircase-like and brick wall disentangler circuits provide warm-start initializations for variational optimization, enabling high-fidelity state preparation for large systems. Target MPSs are either specified as physical quantum states or constructed from classical datasets via amplitude encoding, using step-by-step singular value decompositions or tensor cross interpolation. The framework incorporates entanglement-based qubit reordering, reformulated as a quadratic assignment problem, and low-level optimizations that reduce depths by up to 50% and CNOT counts by 33%. We evaluate the full pipeline on datasets of varying complexity across systems of 19-50 qubits and identify trade-offs between fidelity, gate count, and circuit depth. Optimized brick wall circuits typically achieve the lowest depths, while the optimized staircase-like circuits minimize gate counts. Overall, our results provide principled and scalable protocols for preparing MPSs as quantum circuits, supporting utility-scale applications on near-term quantum devices.

Figures (20)

• Figure 1: Quantum state preparation pipeline. (1) $\mathcal{O}(2^N)$ input probability amplitudes are compressed into an MPS using truncated singular value decomposition (SVD) or tensor cross interpolation (TCI), the latter requiring only $\mathcal{O}(N\chi^2)$ values. (2) Qubits are reordered to minimize quantum mutual information between distant pairs; the optimal permutation solves a quadratic assignment problem (QAP) and defines a new MPS. (3) Heuristic circuits are built layer-by-layer from the MPS, giving either sequential (SMPD) or brick wall (BMPD) architectures. For SMPD, left-, right-, and mixed-canonical MPS forms yield different gate layouts. "Enhancements" of SMPD include isometric gate decompositions using 2 CNOTs (instead of 3 for a generic two-qubit unitary) and removal of gates not affecting the fidelity, reducing CNOT depth and count; the latter also applies to BMPD. (4) Fidelity with the target MPS is variationally optimized, initialized from the heuristic circuits, using the Evenbly-Vidal or Riemannian methods. The optimized circuit may be optionally fed back to the heuristic method to generate a new layer and continue with the optimization. The final circuit approximates the MPS.
• Figure 2: Analogy between a) the standard formulation of the quadratic assignment problem and b) qubit ordering in an MPS. Factories intensively exchanging materials between each other (large flow, marked by bold lines) such as 2 and 4 should be assigned to close cities on the right, just as correlated qubits (large quantum mutual information, marked by bold lines) should be placed close together in the MPS. Numbers enumerate facilities/qubits, capital letters enumerate locations/positions.
• Figure 3: Sequential Matrix Product Disentangler construction, example of the mixed gauge bohun_scalable_2025. Initial state $\ket{\psi_d}$ is truncated to $\ket{\psi_{d,\chi=2}}$ and onsite tensors are isometries in the direction depending on the position and gauge (left/right/mixed), with the orthonormality direction denoted by arrows. Fourth "dummy" leg of dimension $1$, corresponding to $\ket{0}$, is added to each tensor with already three legs, giving onsite tensors of dimensions $(1,2,2,2)$, which are isometries when expressed as $2\times4$ matrices. One completes these matrices to $4\times4$ (2-qubit) unitaries $U_i$ by adding two more orthogonal columns with a Gram-Schmidt or QR decomposition-based orthogonalization algorithm. Completing to unitary is not needed for $U_{6,7}$ at both ends which are already $2\times2$ unitaries. Then, the quantum circuit on the right, created by arranging the gates into a layer $\mathcal{L}[U]^{(k)}$ such that the input-output relations of tensors in the MPS are matching with those in the circuit, exactly realizes $\ket{\psi_{d,\chi=2}}$. For the next layer, one applies the inverse of the currently obtained layer to $\ket{\psi_d}$, yielding a new "disentangled" state. The operator $U_1$ comes from a diagonal tensor $\Lambda$ and can be realized with $1$ CNOT and one $R_Y$ rotation. Other two-body gates are isometries due to $\ket{0}$ input on one qubit and can be decomposed into $2$ CNOT gates and single qubit rotations. Analogous constructions for left- and right-canonical forms of MPS, which do not contain the $\Lambda$ tensor, can be obtained ran_encoding_2020.
• Figure 4: Enhancements of the SMPD. a) Isometry is decomposed into 2 CNOTs and single-qubit gates. b) If the MPS bond is already disentangled, apply only single-qubit gates. c) Using the mixed-canonical gauge decreases depth in half.
• Figure 5: a) The MPS is transformed into the $\Gamma$-$\Lambda$ gauge, also known as the Vidal's gauge. Singular values stored at each bond allow for a direct entropy calculation. b) Application of a gate $U_i$ which is supposed to decrease Rényi entanglement entropy at the $i$-th bond. Contraction of the network is followed by SVD of the resulting tensor across a dashed line. New $\Gamma_{l,r}'$ matrices are obtained as $\Gamma_l'=\Lambda_l^{-1}U_l$ and $\Gamma_r'=V_r\Lambda_r^{-1}$. The MPS stays in the $\Gamma$-$\Lambda$ canonical form and only the $\Gamma_l$, $\Gamma_r$, $\Lambda_i$ tensors are affected by the gate application. c) Layer-by-layer disentangling of an MPS. Parameters of gates in a single column are variationally optimized in parallel to minimize Rényi entropy at each bond. After optimization, the gates are contracted with the MPS, resulting in a lower bond dimension on average, and the optimization proceeds to the next column of gates. The last layer consists of single-body rotations. Hermitian conjugate of this quantum circuit prepares an approximation of the MPS from the $\ket{0}^{\otimes N}$ initial state.