Preparation of Matrix Product States with Log-Depth Quantum Circuits

Area of Science:

• Quantum computing
• Quantum information theory
• Condensed matter physics

Background:

• Matrix product states (MPS) are crucial for simulating quantum systems.
• Efficient preparation of MPS on quantum hardware is a key challenge.
• Current methods may lack scalability for large systems.

Purpose of the Study:

• To determine the fundamental limits of preparing translation-invariant normal MPS.
• To develop an optimal quantum circuit algorithm for MPS preparation.
• To explore the role of measurements and feedback in accelerating MPS synthesis.

Main Methods:

• Proving lower bounds for circuit depth using theoretical analysis.
• Developing a renormalization-group-based algorithm for MPS preparation.
• Incorporating measurement and feedback mechanisms into the quantum circuit.

Main Results:

• Established a lower bound of Ω(log N) circuit depth for preparing N-site translation-invariant normal MPS.
• Introduced an optimal algorithm with O[log(N/ε)] depth for MPS preparation with error ε.
• Demonstrated an exponential speedup to O[loglog(N/ε)] depth using measurement and feedback.
• Showed that measurements enable preparation of arbitrary translation-invariant MPS.

Conclusions:

• The developed algorithm provides an optimal and efficient method for MPS preparation on quantum devices.
• Measurement and feedback are powerful tools for enhancing the speed and versatility of quantum state preparation.
• The algorithm's extensibility to inhomogeneous MPS broadens its applicability in quantum simulations.