Classical shadows help quantum computer analyze a complex spin model

Laurin E. Fischer of IBM Quantum and École Polytechnique Fédérale de Lausanne and colleagues have demonstrated a large-scale implementation of quantum subspace expansion, using informationally complete measurements known as classical shadows. The work evaluated O(10^(14)) Pauli traces, representing one of the most significant experimental realizations of classical shadows to date, to probe a quantum phase transition in a spin model featuring complex three-body interactions across systems of up to 80 qubits.

By reformulating quantum subspace expansion as a constrained optimization problem, the researchers achieved accurate ground state energy recovery and mitigated local order parameters. This approach addresses the large measurement overhead that has limited the scalability of previous quantum algorithms for spectral calculations.

Quantum Subspace Expansion with Classical Shadows for Spectral Calculations

Over 3×${10}^4$ measurement basis randomizations were performed per circuit in a recent experiment, establishing a new scale for implementations of classical shadows and demanding substantial computational resources. The team’s approach successfully recovered accurate ground state energies and mitigated local order parameters across these varying system sizes, demonstrating the method’s potential for analyzing detailed quantum systems. These Pauli traces were essential for reconstructing the quantum state from the measurement data, a computationally intensive task that required substantial high-performance computing resources.

Informationally complete positive operator-valued measurements, or IC POVMs, formed the basis of the experimental setup, using uniformly randomized single-qubit X, Y, and Z measurements. Each system size and parameter value was subjected to 32,768 measurement bases, with eight shots taken per basis, allowing for efficient estimation of observables with low Pauli weight.

The researchers explain that this approach allows for the efficient estimation of observables of low Pauli weight, with statistical errors increasing exponentially with weight, providing a balance between accuracy and computational cost. This careful selection of measurements was critical to managing the exponential growth of computational complexity as the system size increased. The team’s implementation of quantum subspace expansion used the high sampling rates of leading superconducting quantum processors, collecting measurements in over 32,000 randomized bases per input state.

This data was then processed to evaluate over one hundred trillion individual Pauli traces, a feat that represents an advancement in the integration of quantum and classical computing. “This not only represents the realization of classical shadow measurements at an unprecedented scale, but also offers a blueprint for future workflows combining quantum and high-performance classical computing,” the researchers state.

The success of this large-scale implementation suggests a viable path toward tackling increasingly complex quantum simulations. The method relies on expanding a space of states around a root state, denoted ρ₀, using L Hermitian expansion operators σ_i. This expansion, represented as ρ_SE = /, allows access to states beyond the immediate capabilities of the quantum processor and is a form of noise-agnostic error mitigation.

The expectation value of an observable O over a state in this subspace is then calculated, requiring the estimation of matrix entries O_ij and S_ij, the computational cost of which scales with the dimension L of the subspace. The researchers’ ability to efficiently estimate these matrix entries, despite the large number of Pauli traces involved, is a key achievement of this work.

This work presents the first large-scale implementation of quantum subspace expansion powered by classical shadows, demonstrating a pathway for future quantum simulations. The combination of advanced quantum hardware and high-performance classical computing has enabled the analysis of a complex spin model with accurate results and scale.

Three-Body Spin Model Probed with Up to 80 Qubits

The ability to accurately model quantum phase transitions in systems with complex interactions has been advanced through a new implementation of quantum subspace expansion. Researchers demonstrated the technique on a spin model incorporating three-body interactions, a step beyond commonly studied two-body systems, and successfully recovered accurate ground state energies using up to 80 qubits. This work distinguishes itself through the scale of its implementation, using informationally complete measurements to overcome limitations in previous approaches.

The method’s success is not simply about scale, but also about the ability to refine the expansion operators to minimize energy while respecting statistical precision. The experimental setup involved probing a one-dimensional spin chain governed by a Hamiltonian with parameters designed to traverse a quantum phase transition. Error bars representing the estimated statistical error were included in the analysis, providing a measure of confidence in the results.

For a system of 16 qubits, the team validated their approach by comparing results to theoretical values obtained through exact diagonalization, demonstrating a close match between the experimental data and established theory. The researchers’ approach also offers flexibility in analyzing the obtained energy-optimized state. By constructing an estimator, they can calculate additional observables without requiring additional measurements, streamlining the analysis process. This is achieved by replacing the Hamiltonian with the observable of interest in a key equation, allowing for efficient computation of various system properties.

The implementation used the 156-qubit device, ibm_fez, and the physical qubits chosen for the experiment were documented in the supporting data. A significant aspect of the method involves a trade-off between circuit depth and fidelity when preparing the initial approximate ground state.

Further refinement was achieved by incorporating the Hamiltonian itself as an expansion operator, though this introduced a computationally intensive term with over 2. The success of this implementation demonstrates the potential of combining advanced quantum hardware with high-performance classical computing to tackle increasingly complex quantum simulations.

Constrained Optimization Formulates QSE for Error Estimation

The implementation of quantum subspace expansion (QSE) benefited from a reformulation as a constrained optimization problem, enabling the calculation of rigorous statistical error estimates and circumventing potential numerical instability. Previous statistical analyses of similar methods relied on worst-case upper bounds, lacking the reliable error bars now achieved through this new approach, and until now, no experimental demonstration of this level of statistical control existed. This advancement addresses a key limitation in scaling quantum algorithms, particularly those requiring spectral calculations.

This method’s core innovation lies in its ability to explore only specific regions within the computational subspace, those where statistical error remains well-controlled. The researchers achieved this by developing an estimator, inserting sample means and standard deviations into an equation to quantify statistical error, and then defining a constrained optimization problem to minimize energy while adhering to a pre-defined maximum tolerable error.

This process effectively trades potential bias for variance, providing a tunable balance between accuracy and reliability in the results. The team’s approach allows for parallel evaluation of matrix entries within the subspace expansion, further enhancing efficiency. The framework also offers a significant advantage in its flexibility. Once the optimal energy state is determined, additional observables can be estimated without requiring further measurements. This is accomplished by substituting the observable of interest for the Hamiltonian in a key equation, using the previously calculated subspace coefficients.

This reuse of data streamlines the process and reduces the overall computational burden, a critical factor when dealing with complex quantum systems. “Thus, the same measurements can be reused to iteratively optimize the chosen expansion operators to minimize the energy while the constrained optimization guarantees that the variational principle is respected up to the chosen statistical precision,” the researchers write.

This represents a substantial increase in the complexity of classical shadows implementations, demanding significant computational resources and demonstrating the feasibility of performing such calculations on current hardware. The constrained optimization process is visually represented by a three-dimensional plot showing estimated energy alongside statistical error, where minimizing energy alone can lead to statistically unstable regions.

The researchers demonstrate that by imposing a maximum allowable error, the optimization process avoids these unstable areas, ensuring the validity of the variational principle. The team’s approach also allows for the identification of regions where the statistical error is minimized, providing a valuable tool for assessing the reliability of the results. This combination of advanced hardware and refined analytical techniques represents a step toward realizing the full potential of quantum simulation and error mitigation.

The ability to efficiently estimate additional observables without requiring additional measurements further enhances the versatility of this approach. The researchers’ work demonstrates a pathway to overcome the limitations of traditional methods, offering a robust and scalable solution for performing accurate spectral calculations on quantum processors.

Informationally Complete Measurements and Classical Shadow Estimation

The experimental setup involved collecting measurements in 32,768 randomized bases for each system configuration, a figure that establishes a new benchmark for the scale of classical shadows implementations and demanded substantial computational resources for subsequent data processing. This approach avoids the numerical ill-conditioning that can plague traditional methods, ensuring the reliability of the results even as system sizes increase. By framing the problem in this way, the selection of expansion operators, the building blocks of the quantum state approximation, could occur a posteriori, after the measurements were completed, offering flexibility and efficiency.

This contrasts with methods requiring pre-defined expansion operators, which can limit the accuracy and scalability of the simulation. Each measurement basis was sampled 8 times, generating a dataset that could then be used to estimate observables of low Pauli weight.

The statistical variance of the estimator is inversely proportional to the number of samples, highlighting the importance of the large-scale data acquisition. While the method excels at estimating low-weight observables, the computational cost of extending it to higher weights quickly becomes prohibitive, as evidenced by the example of a term requiring the evaluation of 35 x ${10}^9$ Pauli terms with a weight of 15. The ability to choose expansion operators after measurements is a significant advantage, as it allows for optimization based on the specific characteristics of the measured data.

Accurate ground state energy recovery and mitigation of local order parameters were observed across system sizes of up to 80 qubits. The combination of advanced hardware, capable of high sampling rates and refined analytical techniques, provides a robust framework for exploring complex quantum systems.

Statistical Variance and Unbiased Estimators for Matrix Elements

Estimating the ground state energy of a quantum system requires calculating the trace of the Hamiltonian multiplied by the ground state density matrix, a task complicated by the exponential growth of the Hilbert space with qubit number. This approach yields statistical error estimates that are not simply worst-case bounds, but reflect the actual uncertainty in the calculated energy. A key challenge in this process lies in the potential for numerical instability when inverting the overlap matrix, a component of the calculation susceptible to ill-conditioning from experimental noise and imperfections.

This allows for a tunable balance between accuracy and computational cost, a significant refinement over previous methods. The team constructed estimators for the Hamiltonian and its denominator, using sample means and standard deviations to approximate the variance of the final energy estimate.

The researchers’ method addresses this by incorporating a covariance term into the error calculation, capturing contributions from reusing the same measurement samples across multiple matrix entries. This is achieved through the definition of random variables $x_k$ and $y_k$, distributed according to a probability $p_k$, and then approximating the variance of the ratio x/y using a second-order Taylor expansion around the sample means. The resulting estimators, H and ε, are based on informationally complete samples and provide a pathway to more reliable and scalable quantum simulations.

The team’s approach differs from standard error estimation by directly calculating the statistical uncertainty, rather than relying on approximations or regularization. “We obtain the estimated error ε by inserting the sample means x for ${\mu }_x$, the sample standard deviations sum_(i = 1)^S|$x_{\text{k^(i)}}–x$|^2/(S-1) for Var[$x_k$] (similarly for y_k) and the sample covariance sum_(i = 1)^S(x_(k^(i))-x)(y_(k^(i))-y)/(S-1) for Cov[$x_k,y_k$] into Eq. (5),” the paper details, outlining the specific calculations used to quantify the uncertainty in the estimated ground state energy.

This level of precision is essential for advancing the field of quantum simulation and exploring increasingly complex quantum phenomena. By providing accurate statistical error estimates and a tunable trade-off between bias and variance, this work offers a robust and scalable solution for performing accurate spectral calculations on quantum processors.

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