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arXiv:2104.14949 · Physical Review A

Automatically differentiable quantum circuit for many-qubit state preparation

Peng-Fei Zhou, Rui Hong, Shi-Ju Ran

Published Oct 1, 2021• 22 citations
arXivHTMLPDF

Abstract

Constructing quantum circuits for efficient state preparation are central topics in the field of quantum information and computation. As the number of qubits grows quickly, methods to derive large-scale quantum circuits are strongly desired. In this article, we propose the automatically differentiable quantum circuit (ADQC) approach to efficiently prepare quantum many-qubit states in the form of a matrix product state (MPS). A key ingredient is to introduce latent gates whose decompositions give unitary gates that form a quantum circuit. The circuit is optimized by updating the latent gates using backpropagation to minimize the distance between the evolved and target states. Taking the ground states of one-dimensional spin lattice models and random matrix product states as examples, with the number of qubits where processing the full coefficients is unlikely, ADQC obtains high fidelities with small numbers of layers ${N}_{L}\ensuremath{\sim}O(1)$. Much higher fidelities are obtained compared with the existing state-preparation approach based on the matrix product disentangler. In addition, ADQC provides an efficient representation of many-qubit states, whose number of parameters can be compressed to about $0.1%\text{--}1%$ of that in the MPS. Our work sheds light on the ``intelligent construction'' of quantum circuits for many-qubit systems by combining them with machine learning methods.

DOI
10.1103/physreva.104.042601

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Constructing quantum circuits for efficient state preparation are central topics in the field of quantum information and computation. As the number of qubits grows quickly, methods to derive large-scale quantum circuits are strongly desired. In this article, we propose the automatically differentiable quantum circuit (ADQC) approach to efficiently prepare quantum many-qubit states in the form of a matrix product state (MPS). A key ingredient is to introduce latent gates whose decompositions give unitary gates that form a quantum circuit. The circuit is optimized by updating the latent gates using backpropagation to minimize the distance between the evolved and target states. Taking the ground states of one-dimensional spin lattice models and random matrix product states as examples, with the number of qubits where processing the full coefficients is unlikely, ADQC obtains high fidelities with small numbers of layers NL\ensuremath∼O(1){N}_{L}\ensuremath{\sim}O(1)NL​\ensuremath∼O(1). Much higher fidelities are obtained compared with the existing state-preparation approach based on the matrix product disentangler. In addition, ADQC provides an efficient representation of many-qubit states, whose number of parameters can be compressed to about 0.10.1%\text{--}1%0.1 of that in the MPS. Our work sheds light on the ``intelligent construction'' of quantum circuits for many-qubit systems by combining them with machine learning methods.
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.
Efficient quantum circuit optimization schemes are central to quantum simulation of strongly interacting quantum many body systems. Here, we present an optimization algorithm which combines machine learning techniques and tensor network methods. The said algorithm runs on classical hardware and finds shallow, accurate quantum circuits by minimizing scalar cost functions. The gradients relevant for the optimization process are computed using the reverse mode automatic differentiation technique implemented on top of the time-evolved block decimation algorithm for matrix product states. A variation of the ADAM optimizer is utilized to perform a gradient descent on the manifolds of charge conserving unitary operators to find the optimal quantum circuit. The efficacy of this approach is demonstrated by finding the ground states of spin chain Hamiltonians for the Ising, three-state Potts and the massive Schwinger models for system sizes up to L=100. The first ten excited states of these models are also obtained for system sizes L=24. All circuits achieve high state fidelities within reasonable CPU time and modest memory requirements.
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.
Effective quantum computation relies upon making good use of the exponential information capacity of a quantum machine. A large barrier to designing quantum algorithms for execution on real quantum machines is that, in general, it is intractably difficult to construct an arbitrary quantum state to high precision. Many quantum algorithms rely instead upon initializing the machine in a simple state, and evolving the state through an efficient (i.e. at most polynomial-depth) quantum algorithm. In this work, we show that there exist families of quantum states that can be prepared to high precision with circuits of linear size and depth. We focus on real-valued, smooth, differentiable functions with bounded derivatives on a domain of interest, exemplified by commonly used probability distributions. We further develop an algorithm that requires only linear classical computation time to generate accurate linear- depth circuits to prepare these states, and apply this to well-known and heavily-utilized functions including Gaussian and lognormal distributions. Our procedure rests upon the quantum state representation tool known as the matrix product state (MPS). By efficiently and scalably encoding an explicit amplitude function into an MPS, a high fidelity, linear-depth circuit can directly be generated. These results enable the execution of many quantum algorithms that, aside from initialization, are otherwise depth-efficient.
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.