Quantum Subspace Expansion with Classical Shadows Achieves Accurate Ground State Recovery across 80 Qubits

Quantum subspace expansion offers a powerful approach to calculating spectral properties of complex systems, but practical implementation demands substantial measurement resources. Laurin E. Fischer from Ecole Polytechnique Fédérale de Lausanne and IBM Quantum, Daniel Bultrini from Heidelberg University, Ivano Tavernelli and Francesco Tacchino from IBM Quantum, and their colleagues demonstrate a significant advance by successfully implementing this technique on a large scale, utilising informationally complete measurements known as classical shadows. The team probes a spin model with intricate three-body interactions, accurately recovering ground state energies and effectively mitigating local order parameters across systems containing up to 80 qubits. This achievement, involving numerous measurement basis randomizations and Pauli trace evaluations, represents one of the most substantial experimental demonstrations of classical shadows to date, paving the way for more efficient and scalable quantum simulations.

Quantum state estimation (QSE) offers promising avenues to perform spectral calculations on quantum processors, but typically requires a large number of measurements. Scientists recently overcame this limitation by employing informationally complete (IC) measurements, such as classical shadows, and have now demonstrated the first large-scale implementation of QSE using this approach. The team probed a spin model with three-body interactions across system sizes of up to 80 qubits, accurately recovering ground state energy and mitigating local order parameters. This was achieved by reformulating QSE as a constrained optimization problem, allowing for rigorous statistical error estimates and avoiding numerical instability.

Krylov Expansion For Ill-Conditioned Hamiltonians

Researchers developed a sophisticated numerical method to determine the ground state energy of a quantum system, a crucial step in understanding its properties. The method builds upon the Krylov subspace technique, enhancing it with a “Krylov+” expansion to address challenges posed by “ill-conditioning”. Ill-conditioning arises when small changes in the input data can lead to significant changes in the calculated energy, a common issue in quantum simulations. To combat this, the method employs regularization, effectively filtering out problematic directions within the calculation. The core of the method involves constructing a Krylov subspace, a reduced space within which to search for the ground state.

Singular value decomposition (SVD) is then used to analyze the overlap matrix, a key component representing the relationships between states within the subspace. Regularization involves discarding the smallest singular values, stabilizing the calculation and preventing inaccurate energy results. The method solves a generalized eigenvalue problem to determine the ground state energy, providing quantifiable uncertainty through statistical error bars. Researchers carefully balanced the trade-off between bias and variance in the calculation, optimizing the method for accuracy and reliability. Experiments demonstrated that discarding a small number of singular values significantly improved the accuracy and stability of the calculation.

The need for regularization confirmed the presence of ill-conditioning in the problem. Increasing the allowed statistical error reduced bias but decreased the signal-to-noise ratio, highlighting the inherent trade-offs in the calculation. The team successfully applied this constrained optimization procedure to systems of varying sizes, including a 48-qubit system, and utilized non-linear scales to effectively visualize the data. This work represents a significant advancement in numerical methods for quantum simulation.

Large Scale Quantum Subspace Expansion Demonstrated

Scientists achieved a significant breakthrough in quantum computation by implementing quantum subspace expansion (QSE) with informationally complete (IC) measurements on a large scale. The work successfully probed a spin model with three-body interactions across system sizes of up to 80 qubits, accurately recovering ground state energy and mitigating local order parameters. This was accomplished by reformulating QSE as a constrained optimization problem, allowing for rigorous statistical error estimates and avoiding numerical instability. Experiments involved collecting measurements in over 32,768 randomized bases per circuit and evaluating Pauli traces, representing one of the most substantial experimental realizations of classical shadows to date.

The total number of single-qubit Pauli traces ranged from 4. 3×10 11 for 16 qubits to 5. 6×10 13 for 80 qubits. Results demonstrate that the “Krylov+” subspace accurately recovers true ground state expectation values while respecting the variational principle within statistical uncertainties. For a 16-qubit system, the team performed exact diagonalization, revealing that the “Krylov+” values lie below the first excited state, indicating significant overlap with the true ground state.

Researchers also examined phase transitions using order parameters, finding improvements in the energy-optimized subspace state for high values of a key parameter. Specifically, the team measured SX, which improved over unmitigated values, and SZY, a more complex observable, though statistical uncertainties dominated at larger scales. The study confirms that while energy mitigation remains robust across all system sizes, accurately reflecting complex observables requires further refinement of the subspace state. This work establishes a viable path to overcoming the measurement overhead of QSE and provides a complete characterization of its statistical error bars.

Subtle Error Characterization Enables Large Scale Simulation

This work demonstrates a large-scale implementation of quantum subspace expansion, significantly advancing the field of quantum computation and simulation. Researchers successfully combined this technique with informationally complete measurements, known as classical shadows, to probe a complex spin model with up to 80 qubits. The results reveal accurate recovery of ground state energy and effective mitigation of local order parameters across varying system sizes, representing a substantial step towards overcoming the measurement challenges inherent in quantum subspace expansion. A key achievement lies in the rigorous statistical error characterization developed alongside the method, accounting for data covariances arising from the use of informationally complete measurements.

This approach offers a viable path to noise-agnostic error mitigation, differing from other techniques that rely on precise device noise modelling. While the method effectively recovers ground state energy, the study acknowledges limitations in accurately representing complex observables, specifically the weight-five observable SZY, at larger scales. Future research directions include incorporating symmetry verification, exploring methods inspired by virtual distillation, and employing alternative estimators to further reduce sampling costs. Additionally, the team suggests broadening the scope of the method by utilizing measurement schemes with less local operators, potentially enabling the study of a wider range of quantum systems.