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Robust determination of molecular spectra on a quantum processor
James I. Colless, Vinay V. Ramasesh, Dar Dahlen, Machiel S. Blok, Jarrod R. McClean, Jonathan Carter, Wibe A. de Jong, Irfan Siddiqi
TL;DR
The paper addresses how quantum processors with limited coherence can calculate molecular spectra and mitigate errors. It combines VQE with a Quantum Subspace Expansion using additional measurements and classical optimization. For H2, the approach produced near-chemical-accuracy ground- and excited-state estimates and reduced incoherent-error effects, while the exact error-correction conditions remain unresolved.
Problem
The paper seeks efficient molecular eigenvalue calculations on quantum processors with limited quantum-bit lifetime and imperfect operations.
Method
The authors augment the Variational Quantum Eigensolver with a Quantum Subspace Expansion based on additional tomographic measurements and classical optimization.
Results
Near-chemical-accuracy ground- and excited-state energy estimates were obtained for H2, while the complete linear-response expansion reduced energy-estimate error by almost two orders of magnitude.
Takeaways & Limitations
QSE can extract molecular excited states and mitigate incoherent errors without increasing the entanglement required in the quantum state.
Takeaways & Limitations
The exact conditions under which operator sets correct ground-state errors for a given Hamiltonian and error channel remain an open problem.
Abstract
from arXiv · showhide
Harnessing the full power of nascent quantum processors requires the efficient management of a limited number of quantum bits with finite lifetime. Hybrid algorithms leveraging classical resources have demonstrated promising initial results in the efficient calculation of Hamiltonian ground states--an important eigenvalue problem in the physical sciences that is often classically intractable. In these protocols, a Hamiltonian is parsed and evaluated term-wise with a shallow quantum circuit, and the resulting energy minimized using classical resources. This reduces the number of consecutive logical operations that must be performed on the quantum hardware before the onset of decoherence. We demonstrate a complete implementation of the Variational Quantum Eigensolver (VQE), augmented with a novel Quantum Subspace Expansion, to calculate the complete energy spectrum of the H2 molecule with near chemical accuracy. The QSE also enables the mitigation of incoherent errors, potentially allowing the implementation of larger-scale algorithms without complex quantum error correction techniques.
General Approach
The paper combines VQE with a quantum subspace expansion to estimate H2 ground and excited energies while mitigating incoherent errors. A hybrid quantum-classical workflow evaluates Hamiltonian expectations on a two-qubit processor and performs optimization and matrix diagonalization classically.
- Hamiltonian mapping: The electronic-structure Hamiltonian is projected into the STO-3G basis and mapped to a two-qubit Hamiltonian whose coefficients depend on internuclear separation.For a two-qubit state, the Hamiltonian expectation is evaluated through repeated Pauli-correlator measurements.
- VQE workflow: VQE prepares a parameterized trial state, evaluates the Hamiltonian expectation term-wise on quantum hardware, and classically minimizes it to estimate the ground state.The minimizing state |ψ(θmin)⟩ provides the approximate ground state used by the subsequent expansion.
- Quantum subspace expansion: The QSE measures additional Pauli correlators to form and classically diagonalize an expanded Hamiltonian matrix, yielding refined ground-state and low-lying excited-state energies.The linear-response subspace uses operators acting on the VQE reference state; its matrix elements are evaluated as quantum inner products.
- Quantum hardware: The experiment uses two superconducting transmon qubits, a heralded |00⟩ initialization, a parameterized circuit with single-qubit rotations and a bSWAP entangler, followed by tomography and readout.The full preparation-and-measurement sequence lasts less than approximately 1.5 µs, below the stated qubit coherence times.
- Classical optimization: A particle-swarm optimizer searches the circuit parameters, converging after approximately 12 iterations or 240 function evaluations for a 20-particle swarm at 1.55 Å.The swarm energy decreases toward the theoretical value, followed by a reduction in energy variance.
- Results and limitations: Chemical accuracy is achieved for the ground and highest excited states across a wide range of internuclear distances, while the second and third excited states are generally within an order of magnitude of that level.The QSE approximates excited states using additional local measurements and classical computation without increasing qubit-state entanglement.
- Results and limitations: The complete linear-response expansion reduces ground-state energy-estimate error by almost two orders of magnitude across the computed range, mitigating incoherent errors beyond bare VQE.An unphysical spurious state appears between approximately 1.2 Å and 1.7 Å when error channels make the prepared state sufficiently mixed.
Conclusion
The work extends VQE with QSE to extract molecular excited states and mitigate incoherent errors using polynomially many additional tomographic measurements. Applied to H2 with particle-swarm optimization, the approach yields near-chemical-accuracy ground- and excited-state energy estimates.
- The QSE extension extracts molecular excited states and mitigates incoherent errors in the ground-state estimate.
- The method uses only a polynomial number of additional tomographic measurements.
- For H2, QSE, VQE, and particle-swarm minimization yield ground- and excited-state energy estimates with near-chemical accuracy.
SI: Experimental Details
The experiment used a two-qubit superconducting processor to evaluate the molecular Hamiltonian through Pauli measurements and quantum-subspace expansions. The H2 Hamiltonian was reduced to a compact qubit representation, while particle-swarm optimization minimized the energy.
- SI: Experimental Details: The device consisted of two superconducting transmon qubits on a silicon chip coupled to a three-dimensional copper cavity.
- SI: Experimental Details: Qubit states were measured heterodynely through the cavity’s dispersive frequency shift, although |01⟩ and |10⟩ could not be distinguished in single-shot measurements.
- SI: Experimental Details: Particle-swarm optimization was used for the classical optimization routine, while the QSE operator choice determined whether the low-lying excited states or the full spectrum could be resolved.
- SI: Experimental Details: The H2 Hamiltonian was projected into a particle-conserving, spin-preserving configuration-interaction manifold before being mapped to computational basis states and Pauli operators.
SI: QSE with Errors
The QSE constructs an expanded operator-defined subspace from measured matrix elements and solves a generalized eigenvalue problem, with mixed states enlarging the effective space. Incomplete operator resolution can produce spurious eigenvalues, while suitable or sufficiently complete expansions recover the Hamiltonian spectrum.
- QSE construction: QSE uses operators {Oi} to generate states and measures Hamiltonian and overlap matrix elements before solving HC = SCE.The matrices H and S yield eigenvectors C and eigenvalues E, providing approximations that improve with subspace size.
- Mixed-state formulation: The density-matrix formulation Hij = Tr[O†i Ojρ] extends the construction from pure states to mixed states with rank greater than one.Vectorization clarifies the matrices' hermiticity and dimensionality.
- Operator choice: Using only the identity and one Pauli choice produces errors in calculated excited energies, while the full linear-response expansion resolves the entire spectrum.The operator choice therefore determines which excited states can be extracted.
- Mixed-state formulation: For a maximally mixed state ρ = 1/d I, the effective dimension reaches the square of the original Hamiltonian dimension and Hamiltonian eigenvalues become d-fold degenerate.The factor 1/d is incorporated through the metric matrix S.
- Spurious states: Incomplete resolution of a mixed state can yield extra predicted eigenvalues that do not coincide with the Hamiltonian eigenvalues, termed spurious states.A complete linearly independent operator set produces the Hamiltonian spectrum with d-fold degeneracies.
- Spurious states: For the illustrated mixed-state example, the measured space has dimension 7 and includes 3 erroneous eigenvalues, whereas adding operators removes the spurious values.A smaller operator set can also produce an exact spectrum when it is capable of correcting the errors.
SI: VQE and Coherent Errors
The VQE can compensate for coherent gate errors by directly optimizing microwave pulse parameters, including amplitudes, lengths, and phases. At 1.55 Å, optimization corrected a phase discrepancy likely caused by an uncalibrated Stark shift.
- VQE directly optimizes microwave pulse amplitudes, lengths, and phases, enabling compensation for coherent gate errors such as under- or over-rotations.
- At 1.55 Å, the optimized single-qubit rotation amplitudes converged nearly to zero and the bSWAP gate length agreed with simulation.
- The optimized bSWAP phase differed significantly from the expected zero phase, likely because of an uncalibrated Stark shift.
SI: QSE beyond Linear Response
Beyond the linear-response subspace, additional QSE measurement operators can provide further error mitigation. In the reported dataset, adding σxσx restored chemical-accuracy performance across a large range after calibration drift disrupted the linear-response correction.
- Additional measurement operators can extend the two-qubit QSE beyond the linear-response subspace and provide further error mitigation.
- The dataset used separate collection runs above and below 2.6 Å, with a technical restart at that separation.
- After the restart, calibration drift likely prevented linear response from reaching chemical accuracy, whereas additional QSE operators restored it over a large separation range.
- Adding the two-qubit correlator σxσx dramatically improved the ground-state energy estimate.