The simulation of quantum systems is crucial across diverse fields, including physics, chemistry, nanotechnology, and biology. However, classical computers face limitations due to the exponential increase in computational resources required as the number of particles increases. Quantum computers, utilizing one quantum system to simulate another, aim to overcome this hurdle. Two primary approaches exist: universal quantum computers, capable of performing arbitrary unitary operations, and quantum orreries, which are less versatile but better capture the physical characteristics of the system. This research focuses on universal quantum computers, leveraging the readily available IBM quantum computers accessed via the Quantum Experience program. While current devices have limitations in qubit number and noise levels, they provide a valuable platform to test applications and develop practical understanding of quantum computing capabilities. The study focuses on advancing hybrid variational algorithms, which combine quantum and classical computations. These algorithms rely on preparing a quantum register, applying parameterized unitary operations, measuring operators, and utilizing a classical subroutine to optimize parameters. The core element is the ansatz, the choice of which impacts efficiency. This paper concentrates on optimizing decompositions for three-qubit gates, which are building blocks for larger circuits and facilitate error reduction techniques. These decompositions allow for modular circuit construction, piecewise process tomography, and parameter optimization to mitigate systematic errors from cross-talk, and potentially incorporate error detection methods. Statistical errors inherent in quantum computation are acknowledged and may, when carefully managed, even aid in simulating the effects of noise in open quantum systems. The paper uses the Fenna-Matthews-Olsen (FMO) photosynthetic protein as a complex, real-world example to test this approach.

Significant research has explored the role of quantum effects in energy transport within photosynthetic organisms. Various models, ranging from semi-classical to fully quantum, have been employed to explain the efficient energy transfer in FMO protein complexes. The current consensus points towards FMOs as finely-tuned thermalization devices using thermodynamic steering and thermal noise to optimize energy transport. Studies suggest that observed quantum beats are largely due to vibrational modes rather than excitonic coherences. The prevailing model involves steering quasi-particles down an energy gradient, with environmental interactions playing a vital role in incoherent relaxation. Environment-assisted quantum transport (ENAQT) describes the balance between Anderson localization and the quantum Zeno effect, both potentially inhibiting transport. Delocalization and quantum interference allow exploration of multiple paths simultaneously, leading to enhanced transport efficiency, a concept analogous to the Grover algorithm. Invariant subspaces also play a role, enabling the avoidance of sites causing excitation loss. Understanding these processes in biological systems has implications for diverse engineering applications, including more efficient computer chip architectures, solar cells, and nanomedicine.

The research focuses on simulating the equations of motion of systems where the environment significantly impacts the dynamics using Noisy Intermediate-Scale Quantum (NISQ) devices. The approach centers on constructing circuits from fundamental blocks acting on a restricted number of qubits, adhering to current quantum computer architecture limitations. The authors develop software for generating arbitrary three-qubit circuits and benchmark their effectiveness in simulating unitary evolutions. An error model is introduced to study circuit behavior and its approximation to a general quantum error channel. This model demonstrates that errors primarily depend on circuit structure rather than specific parameterization, enabling the incorporation of errors in open quantum systems simulations. The core method involves unitaries of the form U(t) = e^(i∑tᵢPᵢ), where Pᵢ are Pauli strings acting on three qubits, and tᵢ are parameters optimized classically. The paper utilizes a Cartan decomposition, a mathematical technique to split the unitary matrix into a sequence of one-and two-qubit gates, which can then be implemented on a quantum computer. This decomposition guarantees correct behavior for all input states, unlike approximate methods. The algorithm is implemented in MATLAB and Python (using Qiskit), enabling integration with IBM Q Experience. Two IBM quantum computers, ibmq_santiago and ibmqx2, were used for implementation, with error rates considered. The authors classify errors into three categories: state preparation and measurement (SPAM) errors (mitigated using calibration circuits and filtering), statistical errors (minimized by increasing shot numbers), and systematic errors (addressed using a noise model). The noise model assumes a probability 'p' of jumping to a random state. The methodology uses a metric A² to quantify the accuracy of the mitigation technique by comparing simulated and quantum computer data. Simulations were conducted for a linear-ordered chain and the FMO molecule, with the latter representing a more complex system.

Simulations were performed on both a linear chain and the FMO complex. For the linear chain, the simulation successfully demonstrated an excitation propagating through and reflecting at the boundaries of the chain, consistent with classical simulations. The results showed an excitation starting at site 1 and propagating along the chain, then reflecting at the boundaries. This aligns with expectations for a system experiencing ENAQT. The FMO molecule simulation, while using the same techniques, showed a larger deviation from the expected curve, indicating that the systematic error mitigation technique is less accurate for more complex Hamiltonians. An overestimation of population peaks in sites 1 and 2 at the expense of site 6 was noted, highlighting a discrepancy between the real device behavior and the simplified noise model. Site 8, a non-participating site, was used to assess the error rate. The average probability of measuring site 8 was 0.053, with a low standard deviation, suggesting error probability independence from gate parameters. Using the noise model, an estimated fidelity of ~0.65 was obtained. Mitigation of SPAM and statistical errors improved accuracy significantly, as shown by the A² metric. The analysis revealed that certain gates in the linear chain decomposition may be redundant, minimizing noise and improving precision. For the FMO, the results indicated a need for a more sophisticated calibration procedure to better account for systematic errors. The discrepancy between the quantum computation results and the ideal simulation highlighted the potential of using inherent quantum computer noise to simulate realistic open quantum systems.

The findings demonstrate the feasibility of using near-term quantum computers to simulate complex molecular dynamics, even with current noise levels. The Cartan decomposition approach proved effective for transpiling unitary evolutions into quantum circuits, even if circuit optimality is not guaranteed. Error mitigation techniques improved accuracy, but more sophisticated methods may be needed for larger and more complex Hamiltonians. The observation that inherent quantum computer noise can reflect realistic system noise suggests a pathway for more faithful simulation of open systems. The results highlight the need for advancements in quantum error correction to improve the purity of the output states. Scaling up the algorithm to more than three qubits requires exploring advanced transpiling techniques and the integration of error correction codes. Furthermore, simulating dephasing, relaxation, and probability loss in open systems requires techniques like projective measurements on ancillary qubits, a challenge for current technology. The paper provides several suggestions to address these limitations, such as using two-state systems with a four-state environment or using statistical averages of multiple circuit runs.

This work presents a significant step towards simulating open quantum systems on NISQ devices. The methodology efficiently transpiles unitary evolutions into quantum circuits and offers a novel approach to error mitigation. The use of a noise model to account for experimental errors in quantum computations is particularly insightful. Although limitations remain, including the need for improved error mitigation strategies and scaling to larger qubit systems, the results show the potential of this approach for simulating complex molecular dynamics. Future research should focus on developing more advanced error correction and transpilation techniques, enabling more accurate simulations of a wider range of quantum systems.

The current method's accuracy is limited by the noise levels in available quantum computers. While error mitigation techniques were implemented, the approach showed limitations when applied to more complex systems like the FMO complex. Scaling the method to larger systems requires improvements in error correction and circuit optimization. Simulating open systems fully, incorporating relaxation and dephasing, remains a technological challenge.