Quantum computers promise to provide groundbreaking speed in solving complex problems. However, in the present-day noisy intermediate-scale quantum era, algorithms that require fewer resources are highly desired. In this work, we develop a new method called the variational rodeo eigensolver (VRE) for efficiently searching eigenstates and estimating eigenvalues with shallow circuits. We experimentally demonstrate this method on a programmable photonic chip with a single-qubit exciton transfer Hamiltonian whose eigenstates are searched with fidelities of more than 99% and their eigenvalues are estimated, reaching chemical accuracy. Furthermore, we experimentally estimate the ground energies of the simplified Hamiltonian of the hydrogen molecule with different atomic separations. To verify the scalability of VRE, we numerically search eigenstates of a tapered two-qubit Hamiltonian of hydrogen-helium ion. Our work provides a systematic and promising approach for the efficient estimation of Hamiltonian spectra.
Quantum computers promise to provide groundbreaking speed in solving complex problems. However, in the present-day noisy intermediate-scale quantum era, algorithms that require fewer resources are highly desired. In this work, we develop a new method called the variational rodeo eigensolver (VRE) for efficiently searching eigenstates and estimating eigenvalues with shallow circuits. We experimentally demonstrate this method on a programmable photonic chip with a single-qubit exciton transfer Hamiltonian whose eigenstates are searched with fidelities of more than 99% and their eigenvalues are estimated, reaching chemical accuracy. Furthermore, we experimentally estimate the ground energies of the simplified Hamiltonian of the hydrogen molecule with different atomic separations. To verify the scalability of VRE, we numerically search eigenstates of a tapered two-qubit Hamiltonian of hydrogen-helium ion. Our work provides a systematic and promising approach for the efficient estimation of Hamiltonian spectra.
The efficient calculation of Hamiltonian spectra, a problem often intractable on classical machines, can find application in many fields, from physics to chemistry. We introduce the concept of an "eigenstate witness" and, through it, provide a new quantum approach that combines variational methods and phase estimation to approximate eigenvalues for both ground and excited states. This protocol is experimentally verified on a programmable silicon quantum photonic chip, a mass-manufacturable platform, which embeds entangled state generation, arbitrary controlled unitary operations, and projective measurements. Both ground and excited states are experimentally found with fidelities >99%, and their eigenvalues are estimated with 32 bits of precision. We also investigate and discuss the scalability of the approach and study its performance through numerical simulations of more complex Hamiltonians. This result shows promising progress toward quantum chemistry on quantum computers.
Quantum computers have the potential to deliver speed-ups for solving certain important problems that are intractable for classical counterparts, making them a promising avenue for advancing modern computation. However, many quantum algorithms require deep quantum circuits, which are challenging to implement on current noisy devices. To address this limitation, variational quantum algorithms (VQAs) have been actively developed, enabling practical quantum computing in the noisy intermediate-scale quantum (NISQ) era. Among them, the variational quantum eigensolver (VQE) stands out as a leading approach for solving problems in quantum chemistry, many-body physics, and even integer factorization. The VQE algorithm can be implemented on various quantum hardware platforms, including photonic systems, quantum dots, trapped ions, neutral atoms, and superconducting circuits. In particular, photonic platforms offer several advantages: they operate at room temperature, exhibit low decoherence, and support multiple degrees of freedom, making them suitable for scalable, high-dimensional quantum computation. Here we present methodologies for realizing VQE on photonic systems, highlighting their potential for practical quantum computing. We first provide a theoretical overview of the VQE framework, focusing on the procedure for variationally estimating ground state energies. We then explore how photonic systems can implement these processes, showing that a wide variety of problems can be addressed using either multiple qubit states or a single qudit state.
The efficient calculation of Hamiltonian spectra, a problem often intractable on classical machines, can find application in many fields, from physics to chemistry. Here, we introduce the concept of an eigenstate witness and through it provide a new quantum approach which combines variational methods and phase estimation to approximate eigenvalues for both ground and excited states. This protocol is experimentally verified on a programmable silicon quantum photonic chip, a mass-manufacturable platform, which embeds entangled state generation, arbitrary controlled-unitary operations, and projective measurements. Both ground and excited states are experimentally found with fidelities >99%, and their eigenvalues are estimated with 32-bits of precision. We also investigate and discuss the scalability of the approach and study its performance through numerical simulations of more complex Hamiltonians. This result shows promising progress towards quantum chemistry on quantum computers.
Abstract The variational quantum eigensolver (VQE) is one of the most promising algorithms to find eigenstates of a given Hamiltonian on noisy intermediate-scale quantum devices (NISQ). The practical realization is limited by the complexity of quantum circuits. Here we present an approach to reduce quantum circuit complexity in VQE for electronic structure calculations. Our ClusterVQE algorithm splits the initial qubit space into clusters which are further distributed on individual (shallower) quantum circuits. The clusters are obtained based on mutual information reflecting maximal entanglement between qubits, whereas inter-cluster correlation is taken into account via a new “dressed” Hamiltonian. ClusterVQE therefore allows exact simulation of the problem by using fewer qubits and shallower circuit depth at the cost of additional classical resources, making it a potential leader for quantum chemistry simulations on NISQ devices. Proof-of-principle demonstrations are presented for several molecular systems based on quantum simulators as well as IBM quantum devices.
The state-of-the-art quantum computing hardware has entered the noisy intermediate-scale quantum (NISQ) era. Having been constrained by the limited number of qubits and shallow circuit depth, NISQ devices have nevertheless demonstrated the potential of applications on various subjects. One example is the variational quantum eigensolver (VQE) that was first introduced for computing ground states. Although VQE has now been extended to the study of excited states, the algorithms previously proposed involve a recursive optimization scheme which requires many extra operations with significantly deeper quantum circuits to ensure the orthogonality of different trial states. Here we propose a new algorithm to determine many low energy eigenstates simultaneously. By introducing ancillary qubits to purify the trial states so that they keep orthogonal to each other throughout the whole optimization process, our algorithm allows these states to be efficiently computed in one quantum circuit. Our algorithm reduces significantly the complexity of circuits and the readout errors, and enables flexible post-processing on the eigen-subspace from which the eigenpairs can be accurately determined. We demonstrate this algorithm by applying it to the transverse Ising model. By comparing the results obtained using this variational algorithm with the exact ones, we find that the eigenvalues of the Hamiltonian converge quickly with the increase of the circuit depth. The accuracies of the converged eigenvalues are of the same order, which implies that the difference between any two eigenvalues can be more accurately determined than the eigenvalues themselves.
Quantum Phase Estimation (QPE) is a foundational algorithm for molecular ground-state energy estimation, but its deep circuit requirements make direct hardware execution impractical on Noisy Intermediate-Scale Quantum (NISQ) devices. We present an analytically grounded variational surrogate framework in which a shallow Variational Quantum Circuit (VQC) is trained to reproduce the QPE measurement distribution without any quantum circuit simulation. The training target is computed entirely classically via the Dirichlet kernel, evaluated directly from the Full Configuration Interaction (FCI) ground-state energy, the ancilla qubit count, and the time evolution parameter, eliminating the exponentially scaling simulation bottleneck of prior surrogate approaches. We apply this framework to the hydrogen molecule (H2) with a symmetry-tapered Hamiltonian, conducting a four-stage experimental investigation on IBM Quantum hardware. Stage 1 compares linear and full entangler topologies for the RY-RZ-CZ ansatz, with and without XpXm Dynamical Decoupling (DD), across four distributional metrics (Hellinger distance, fidelity error, total variation distance, Jensen-Shannon divergence), identifying the linear entangler as optimal. Stage 2 varies VQC layers (p=1 to 5) for the linear-entangler ansatz, identifying single-layer depth as optimal under hardware noise. Stage 3 applies this configuration to the reduced RY-CZ ansatz, comparing ideal and noisy simulator-trained parameters. A supplementary noise analysis at p∈{8,64} characterizes the depth-dependent interplay between circuit depth and DD effectiveness. The framework enables faithful QPE mimicry using a linearly scaling VQC, recovering the ground-state energy within the chemical accuracy threshold (1 kcal/mol), constituting a scalable, hardware-efficient paradigm for QPE-based molecular energy estimation on NISQ devices.