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Dimensional Expressivity Analysis for Quantum Circuits

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"Besides tensor networks, quantum computations (QC) as well use a Hamiltonian formulation to solve physical problems. Although QC are presently very limited, since only small number of qubits are available, they have the principal advantage that they straightforwardly scale to higher dimensions. A standard tool in the QC approach are Variational Quantum Simulations (VQS) which form a class of hybrid quantum-classical algorithms for solving optimization problems. For example, the objective may be to find the ground state of a Hamiltonian by minimizing the energy. As such, VQS use parametric quantum circuit designs to generate a family of quantum states (e.g., states obeying physical symmetries) and efficiently evaluate a cost function for the given set of variational parameters (e.g., energy of the current quantum state) on a quantum device. The optimization is then performed using a classical feedback loop based on the measurement outcomes of the quantum device.

In the case of energy minimization, the optimal parameter set therefore encodes the ground state corresponding to the given Hamiltonian provided that the parametric quantum circuit is able to encode the ground state. Hence, the design of parametric quantum circuits is subject to two competing drivers. On one hand, the set of states, that can be generated by the parametric quantum circuit, has to be large enough to contain the ground state. On the other hand, the circuit should contain as few quantum gates as possible to minimize noise from the quantum device. In other words, when designing a parametric quantum circuit we want to ensure that there are no redundant parameters.

In this talk, I will consider the parametric quantum circuit as a map from parameter space to the state space of the quantum device. Using this point of view, the set of generated states forms a manifold. If the quantum circuit is free from redundant parameters, then the number of parameters is precisely the dimension of the manifold of states. This leads us to the notion of dimensional expressivity analysis. I will discuss means of analyzing a given parametric design in order to remove redundant parameters as well as any unwanted symmetries (e.g., a gate whose only effect is a change in global phase). Time permitting, I may discuss the manifold of physical states as well since this will allow us to decide whether or not a parametric quantum circuit can express all physical states (thereby ensuring that the ground state can be expressed as well)."