Variational Quantum Eigensolver Models of Molecular Quantum Dot Cellular Automata

TL;DR

This work demonstrates that the variational quantum eigensolver (VQE) can be used to model the ground states of molecular quantum-dot cellular automata (QCA) circuits by mapping each QCA cell to a qubit in an Ising-like Hamiltonian. The authors formulate a Hamiltonian with nearest-neighbor kink energy $E_k$ and next-nearest-neighbor energy $E'_k$, implement VQE with shallow, reduced-parameter ansätze, and validate against exact diagonalization for small networks while testing on IBM QPUs and simulators for binary wires, inverters, and a majority gate. Results show good agreement in noise-free or low-noise simulations; on real hardware, decoherence and gate noise degrade accuracy, with performance strongly improving when reducing the number of variational parameters and increasing shot counts (e.g., ~16k shots). The study highlights the feasibility and limitations of VQE for QCA modeling on NISQ devices, and suggests that fault-tolerant quantum computing or more robust hardware could enable scalable QCA circuit modeling and future quantum-phase-estimation-based approaches.

Abstract

Molecular quantum-dot Cellular Automata (QCA) may provide low-power, high-speed computational hardware for processing classical information. Simulation and modeling play an important role in the design of QCA circuits because fully-coherent models of QCA scale exponentially with the number of devices, and such models are severely limited in size. For larger circuits, approximations become necessary. In the era of fault-tolerant quantum computation, however, it may become possible to model large QCA circuits without such limitations. Presently, this work explores the use of the noisy-intermediate scale quantum (NISQ) variational quantum eigensolver (VQE) method for estimating the ground state of QCA circuits. This is relevant because the computational result of a QCA calculation is encoded in the circuit's ground state. In this study, VQE is used to model logic circuits, including binary wires, inverters, and majority gates. VQE models are performed ideal simulators, noisy simulators, and actual quantum hardware. This study demonstrates that VQE may indeed be used to model molecular QCA circuits. It is observed that using modern NISQ hardware, results are still quite sensitive to noise, so measures should be taken to minimize noise. These include simplifying the ansatz circuit whenever possible, and using low-noise hardware.

Figures (26)

• Figure 1: A single bit of information is encoded by two different localized electronic charge configurations of electrons (solid red spheres) on a system of four quantum dots (translucent grey spheres). Device switching occurs through charge tunneling, via paths indicated as black bars. Bit zero "0" [subfigure (a)] and bit "1" [subfigure (b)] are assigned the polarizations $P = \mp 1$, respectively, and are respectively assigned quantum states $\ket{0}$ and $\ket{1}$.
• Figure 2: A logically complete set of circuits is possible in QCA. (a) The binary wire provides a path for data transmission. (b) The inverter and (c) majority gate support boolean logic operations. The inverter flips the input bit using diagonal coupling, and the majority gate functions as a programmable two-input AND/OR gate.
• Figure 3: Molecules may provide systems of coupled quantum dots. (a) Two iron centers (atoms colored purple) provide a coupled pair of molecular quantum dots.quardokus2013adsorption (b) Two iron centers and a carborane cage provide a three-dot QCA system.christie2015synthesis (c) Four iron centers provide corner dots, and a central Co atom provides a fifth dot that provides an ET path between the corner dots.jiao2005properties In this paper we model molecules with direct tunneling between the corner dots and have no fifth central dot (see Figure \ref{['fig:1']}).
• Figure 4: While nearest-neighbor interactions are straight-forward, we use a simplified description of next-nearest-neighbor interactions. (a) Aligned states are favored for wo cells coupling horizontally as nearest-neighbors. Aligned interactions are degenerate and lie $E_k$ below anti-aligned interaction energies, which also are degenerate. These same relationships hold for vertically-arranged nearest-neighbors (not shown here). (b) Diagonal (next-nearest-neighbor coupling) favors anti-aligned states, "10" or "01," over the aligned states "00" or "11." In a "forward-slash" configuration, the "11" state has a higher interaction energy than does the "00" state. (c) In a "back-slash" configuration, diagonal coupling again favors the anti-aligned states, but the "00" state is more energetic than the "11" state. (d) In this work, we use a simplified description of next-nearest-neighbor interactions, in which both the anti-aligned interactions and the aligned interactions are degenerate, with a new diagonal kink energy, $E_{k}^{\prime}$.
• Figure 5: The VQE workflow begins with the construction of a parameterized quantum circuit (ansatz), which prepares a trial quantum state. This state is used to estimate the expectation value of the Hamiltonian via quantum measurements. The resulting energy estimate is then fed into a classical optimizer, which updates the circuit parameters to iteratively minimize the energy. This hybrid quantum-classical feedback loop continues until convergence to the approximate ground-state energy is achieved.