Quantum criticality, a phenomenon near zero-temperature phase transitions driven by quantum fluctuations, governs many collective phenomena in condensed matter systems. Characterized by diverging correlation lengths and strong finite-size effects, these systems present significant challenges for both classical and quantum simulations. Classical methods like dynamical mean field theory and singlet Monte Carlo struggle with these complexities. Tensor network techniques, particularly matrix product states (MPS), offer an advantage by working directly in the thermodynamic limit, mitigating finite-size issues. This work leverages the power of translationally invariant MPS (iMPS), adapting them into efficient quantum circuits for execution on a superconducting quantum computer. The choice of the quantum Ising model, with its well-understood quantum phase transition, serves as a robust testbed to demonstrate the capabilities and accuracy of the proposed method. The simulation of this model, even near its critical point, holds significant importance for advancing quantum computing technologies and highlighting potential quantum advantages over classical simulations, paving the way for future applications in material science and chemistry.

The literature extensively covers theoretical and numerical methods for studying quantum criticality. Classical numerical approaches, such as dynamical mean field theory (DMFT) and singlet Monte Carlo, have been instrumental in understanding various strongly correlated phenomena. However, these methods often struggle with the diverging correlation lengths near quantum critical points. Tensor network methods, particularly matrix product states (MPS), have emerged as powerful tools for tackling such challenges by directly addressing the thermodynamic limit. Previous works have explored the translation of tensor network methods into quantum circuits, aiming to exploit the potential for quantum advantage. This paper builds upon these advancements by utilizing iMPS-inspired circuits, focusing on the quantum Ising model as a benchmark system for assessing their capabilities.

The researchers implemented and optimized translationally invariant MPS (iMPS) quantum circuits on Google's Rainbow superconducting quantum device to simulate the quantum Ising model. The model's Hamiltonian is given by H = Σᵢ[J ZᵢZᵢ₊₁ + gXᵢ], where Z and X are Pauli operators, J is the exchange coupling, and g is the transverse field. The ground state optimization involved a one-dimensional sequential quantum circuit inspired by MPS, using a bond order D=2. Higher bond orders are achievable with efficient shallow circuit representations. Local observables were measured on a finite-depth, finite-width circuit with an additional unitary V=V(U) accounting for the influence of distant parts of the system. This additional unitary was determined by solving auxillary equations, optimized simultaneously with the Hamiltonian terms. Crucially, error mitigation techniques were implemented. These include careful qubit selection to minimize noise, correction for measurement errors using a confusion matrix, and correction for depolarisation using a Loschmidt echo. The simultaneous perturbation stochastic approximation (SPSA) algorithm was employed for optimization, and a quasi-adiabatic method was used to approach the critical point. For the quantum dynamics simulation, the researchers used the time-evolution operator, expanded through Trotterization. A first-order Trotterization was chosen to minimize circuit depth, and a technique was developed to work with only even bonds in the Hamiltonian. The power method was used to approximate the principal eigenvalue of the transfer matrix. The Loschmidt echo was used for error mitigation. Time evolution was conducted along a linear interpolation in the circuit parameters through the exact parameters to be determined.

The research successfully demonstrated the simulation of both the ground state and dynamics of the quantum Ising model on a noisy intermediate-scale quantum (NISQ) device. Ground state energy optimization yielded results within 2.2% of the analytically exact values even at the quantum critical point (g/J=1). Deviations arose mainly from a reduced parametrization of the two-qubit unitaries. The overlap calculation showed exponential convergence with the order (n) of the power method, allowing for accurate determination of the principal eigenvalue of the transfer matrix. The researchers demonstrated an optimization of the overlap by varying the circuit parameters, showing convergence to the expected value. Time evolution circuits accurately captured the dynamics of the quantum Ising model, including the periodic partial revivals and dynamical quantum phase transitions. The time-evolution cost function, significantly simplified compared to previous proposals, faithfully tracked the dynamics. The key to success was a careful balance between theoretical accuracy and the mitigation of errors due to the NISQ device, employing techniques such as qubit selection, confusion matrix error correction, and the Loschmidt echo.

The findings demonstrate the feasibility of simulating quantum critical systems on current NISQ devices by using tensor network methods. The ability to accurately simulate both ground states and dynamics, even near the quantum critical point, is a significant step toward harnessing quantum computers for complex many-body problems. The efficient quantum circuits and effective error mitigation strategies developed in this research offer a pathway for more sophisticated simulations in the future. The thermodynamic-limit approach avoids the limitations imposed by finite-size effects in traditional quantum simulations. This work significantly contributes to the field by showcasing a practical approach to simulating quantum critical phenomena, opening up possibilities for further explorations of complex quantum systems and materials.

This paper presents a significant advancement in using NISQ devices to simulate quantum critical systems. The researchers successfully simulated ground state and dynamical quantum phase transitions of the quantum Ising model on Google's Rainbow device. Efficient quantum circuits, inspired by iMPS, and sophisticated error mitigation techniques were instrumental to this achievement. Future research should focus on stochastic optimization of the cost function, exploring various optimization schemes to balance accuracy and sampling costs. The integration of machine learning techniques to simultaneously mitigate errors and improve simulation accuracy holds great promise. Extensions to higher bond orders and higher-dimensional systems are also important avenues for future work.

The main limitation of this study is the use of a NISQ device, which introduces noise and limits the size and complexity of the simulations. The reduced parametrization of the two-qubit unitaries may have contributed to the deviations from analytically exact results. The scalability of the confusion matrix error correction method to larger systems and higher bond dimensions needs further investigation. While error mitigation techniques were effective, further improvements in device fidelity and error correction methods would enhance the accuracy and efficiency of future simulations.