Quantum Circuit Encoding via Matrix Product Operator Approximation
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Solution Overview
Problem
Current approaches to encoding computational operations in quantum circuits face significant complexity due to the exponential scaling of circuit complexity with the number of qubits, particularly in decomposing arbitrary multi-qubit gates and achieving full connectivity, which limits the applicability and efficiency of quantum computations.
Innovation Solution
A computer-implemented method that obtains a matrix product operator representation of the intended matrix, determines an approximation rank, and iteratively optimizes an orthogonal approximation using an optimization algorithm with an isometry constraint, encoding the result into a quantum circuit using isometric sub-tensors, thereby approximating the matrix efficiently and reducing circuit depth.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If arbitrary multi-qubit gates are decomposed into single- and two-qubit gates, then the computational operation can be implemented on quantum hardware, but the circuit complexity scales exponentially with the number of qubits
Solution Approach 1:
The patent changes the parameter of matrix representation from arbitrary form to Matrix Product Operator (MPO) form with bounded bond dimension. This parameter change allows the circuit complexity to scale polynomially rather than exponentially with the number of qubits, while still enabling implementation on quantum hardware through the structured decomposition inherent in MPO representations
Solution Approach 2:
The patent segments the arbitrary matrix into a product of local operators through MPO decomposition. By representing the global operation as a product of local operations with bounded entanglement (bond dimension), the complex multi-qubit gate is broken down into manageable segments that can be implemented efficiently on quantum hardware without exponential circuit depth
2Adaptability or versatility
If full connectivity between qubits is achieved, then arbitrary quantum operations can be performed, but the device complexity and resource requirements increase significantly
Solution Approach 1:
The patent segments the quantum operation into local interactions represented by MPO tensors. This segmentation allows the system to perform complex computations using only local qubit interactions, eliminating the need for full connectivity while maintaining the capability to implement a broad class of quantum operations through the structured product form
Solution Approach 2:
The MPO representation acts as an intermediary that translates arbitrary quantum operations into a form suitable for hardware with limited connectivity. By introducing this intermediate representation with bounded bond dimension, the patent enables implementation of complex operations on hardware that lacks full qubit connectivity
3Reliability
If variational quantum algorithms are used to solve nonlinear problems, then solutions can be obtained, but many quantum circuit evaluations are required for optimization
Solution Approach 1:
The patent performs preliminary action by directly constructing the quantum circuit from the MPO representation of the intended matrix. This preliminary construction eliminates the need for iterative variational optimization, as the circuit is prepared in its optimal form from the beginning based on the mathematical structure of the problem, thereby solving nonlinear problems without requiring multiple circuit evaluations
Data Source
AI summary
A computer-implemented method for encoding an intended matrix in a quantum circuit, the method comprising obtaining an MPO representation of the intended matrix; determining an approximation rank for the intended matrix based on the MPO representation; determining an initial guess for an orthogonal approximation of the intended matrix in the form of a tensor network with isometric sub-tensors of the approximation rank; starting with the initial guess, iteratively optimizing the orthogonal approximation of the intended matrix based on an optimization algorithm minimizing a cost function subject to an isometry constraint for the isometric sub-tensors, wherein the cost function attributes a cost to the orthogonal approximation of the intended matrix based on a quality of the orthogonal approximation with respect to the intended matrix, and encoding the orthogonal approximation into a quantum circuit based on encodings of the isometric sub-tensors into quantum gates.