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Efficient Encoding of Matrix Product States into Quantum Circuits of One- and Two-Qubit Gates

Shi-Ju Ran

Published Aug 21, 2019• 2 citations
OpenAlex

Abstract

Matrix product state (MPS) belongs to the most important mathematical models in, for example, condensed matter physics and quantum information sciences. However, to realize an $N$-qubit MPS with large $N$ and large entanglement on a quantum platform is extremely challenging, since it requires high-level qudits or multi-body gates of two-level qubits to carry the entanglement. In this work, an efficient method that accurately encodes a given MPS into a quantum circuit with only one- and two-qubit gates is proposed. The idea is to construct the unitary matrix product operators that optimally disentangle the MPS to a product state. These matrix product operators form the quantum circuit that evolves a product state to the targeted MPS with a high fidelity. Our benchmark on the ground-state MPS's of the strongly-correlated spin models show that the constructed quantum circuits can encode the MPS's with much fewer qubits than the sizes of the MPS's themselves. This method paves a feasible and efficient path to realizing quantum many-body states and other MPS-based models as quantum circuits on the near-term quantum platforms.

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The matrix product state (MPS) belongs to the most important models in, for example, quantum information sciences and condensed-matter physics. However, realizing an $N$-qubit MPS with large $N$ and large entanglement on a quantum platform is extremely challenging, since it requires high-level qudits or $n$-qubit gates with $n\ensuremath{\gg}2$ to carry or produce the entanglement. In this work, an efficient method that accurately encodes a given MPS into a quantum circuit with only one- and two-qubit gates is proposed. Essentially different from the existing compiling methods, our idea is to construct the unitary matrix product operators that optimally disentangle the MPS to a product state. These matrix product operators form the quantum circuit that evolves a product state to the targeted MPS with a high fidelity. Our benchmark on the ground-state MPSs of the strongly correlated spin models show that the constructed quantum circuits can simulate the MPSs with much fewer qubits than the sizes of the MPSs themselves. This method paves a feasible and efficient path to realizing useful and/or exotic quantum states and MPS-based models as quantum circuits on near-term quantum platforms.

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Matrix product state (MPS) belongs to the most important mathematical models in, for example, condensed matter physics and quantum information sciences. However, to realize an NNN-qubit MPS with large NNN and large entanglement on a quantum platform is extremely challenging, since it requires high-level qudits or multi-body gates of two-level qubits to carry the entanglement. In this work, an efficient method that accurately encodes a given MPS into a quantum circuit with only one- and two-qubit gates is proposed. The idea is to construct the unitary matrix product operators that optimally disentangle the MPS to a product state. These matrix product operators form the quantum circuit that evolves a product state to the targeted MPS with a high fidelity. Our benchmark on the ground-state MPS's of the strongly-correlated spin models show that the constructed quantum circuits can encode the MPS's with much fewer qubits than the sizes of the MPS's themselves. This method paves a feasible and efficient path to realizing quantum many-body states and other MPS-based models as quantum circuits on the near-term quantum platforms.
The matrix product state (MPS) belongs to the most important models in, for example, quantum information sciences and condensed-matter physics. However, realizing an NNN-qubit MPS with large NNN and large entanglement on a quantum platform is extremely challenging, since it requires high-level qudits or nnn-qubit gates with $n\ensuremath{\gg}2$ to carry or produce the entanglement. In this work, an efficient method that accurately encodes a given MPS into a quantum circuit with only one- and two-qubit gates is proposed. Essentially different from the existing compiling methods, our idea is to construct the unitary matrix product operators that optimally disentangle the MPS to a product state. These matrix product operators form the quantum circuit that evolves a product state to the targeted MPS with a high fidelity. Our benchmark on the ground-state MPSs of the strongly correlated spin models show that the constructed quantum circuits can simulate the MPSs with much fewer qubits than the sizes of the MPSs themselves. This method paves a feasible and efficient path to realizing useful and/or exotic quantum states and MPS-based models as quantum circuits on near-term quantum platforms.
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.
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.
Matrix product states (MPSs) and matrix product operators (MPOs) have been proven to be a powerful tool to study quantum many-body systems but are restricted to moderately entangled states as the number of parameters scales exponentially with the entanglement entropy. While MPSs can efficiently find ground states of one-dimensional systems, their capacities are limited when simulating their dynamics, where the entanglement can increase ballistically with time. On the other hand, quantum devices appear as a natural platform to encode and perform the time evolution of correlated many-body states. However, accessing the regime of long-time dynamics is hampered by quantum noise. In this paper we use the best of worlds: the short-time dynamics is efficiently performed by MPSs, compiled into short-depth quantum circuits, and performed further in time on a quantum computer thanks to efficient MPO-optimized quantum circuits. We quantify the capacities of this hybrid classical-quantum scheme in terms of fidelities taking into account a noise model. We show that using classical knowledge in the form of tensor networks provides a way to better use limited quantum resources and lowers the noise requirements to reach a practical quantum advantage. Finally, we successfully demonstrate our approach with an experimental realization of the technique. Combined with efficient circuit transpilation we simulate a ten-qubit system on an actual quantum device over a longer time scale than low-bond-dimension MPSs and purely quantum Trotter evolution.
We study the classical compilation of quantum circuits for the preparation of matrix product states (MPS), which are quantum states of low entanglement with an efficient classical description. Our algorithm represents a near-term alternative to previous sequential approaches by reverse application of a disentangler, which can be found by minimizing bipartite entanglement measures after the application of a layer of parametrized disentangling gates. Since a successful disentangler is expected to decrease the bond dimension on average, such a layer-by-layer optimization remains classically efficient even for deep circuits. Additionally, as the Schmidt coefficients of all bonds are locally accessible through the canonical Γ − Λ form of an MPS, the optimization algorithm can be heavily parallelized. We discuss guarantees and limitations to trainability and show numerical results for ground states of one-dimensional, local Hamiltonians as well as artificially spread out entanglement among multiple qubits using error-correcting codes.
Simulating quantum error correction (QEC) circuits including non-Clifford gates at scale is important to accelerate progress toward fault-tolerant quantum computing. Here we demonstrate that matrix product state (MPS) techniques can handle many QEC circuits exactly and without restriction on gate types. Crucially, we find that MPS efficiency depends sensitively on implementation choices, and we introduce a series of targeted optimizations that reduce bond dimensions and simulation time by several orders of magnitude compared to naive approaches. We illustrate this with examples including: (a) a rotated surface code quantum memory up to distance 11, (b) logical Bell-state preparation up to distance 9, (c) a 15-to-1 magic-state distillation circuit including hundreds of QEC rounds that we optimize to be simulated with only 11 logical qubits (187 physical qubits) and a maximal bond dimension of 64 in under 40 seconds, and (d) a narrow, deep random circuit that scales linearly with the number of T gates. These results demonstrate the importance of circuit-level optimizations and position MPS as a valuable complement to near-Clifford simulators for QEC circuits.