Single Quantum Trajectory Cuts Sampling Time for Material Properties

Researchers are continually seeking more efficient methods for estimating thermal expectation values, a fundamental task across physics and materials science. Jiaqing Jiang and Jiaqi Leng, both from the Simons Institute for the Theory of Computing at the University of California, Berkeley, alongside Lin Lin from the Department of Mathematics, University of California, Berkeley, demonstrate a substantial reduction in sampling costs by utilising a single Gibbs-sampling trajectory. Their innovative approach, detailed in this work, interleaves coherent measurements satisfying detailed balance after an initial burn-in period, exploiting the often-significant difference between autocorrelation and mixing times. This advancement promises to accelerate calculations of energy and other observables, particularly through the implementation of Gaussian-filtered phase estimation with logarithmic overhead and a novel weighted operator Fourier transform to minimise measurement disturbance.

Scientists have developed a new method for predicting efficiently estimating the energy and other characteristics of these thermal states, which dictate a material’s behaviour at a given temperature. Recent advances enable the efficient preparation of thermal states using quantum algorithms, but accurately measuring their properties remains computationally expensive.

The team demonstrates that the cost of these measurements can be substantially reduced by employing a single computational trajectory rather than multiple independent ones, a paradigm shift in quantum sampling techniques. This breakthrough leverages the concept of autocorrelation time, the time it takes for successive measurements to become effectively independent, which can be significantly shorter than the traditional mixing time required to ensure sample independence.

The approach involves an initial ‘burn-in’ period to allow the system to reach thermal equilibrium, followed by a sampling stage where measurements are taken at intervals dictated by the autocorrelation time, rather than the much longer mixing time. Specifically, the team focused on estimating the average energy of quantum thermal states, implementing measurements using Gaussian-filtered quantum phase estimation with minimal computational overhead.

Furthermore, they introduced a weighted operator Fourier transform to minimise disturbances caused by measurements when estimating more general observables, potentially accelerating the discovery of new materials and the design of novel quantum technologies. The ability to accurately and efficiently predict thermal properties is vital for fundamental physics and holds promise for advancements in quantum machine learning algorithms, such as training quantum Boltzmann machines.

The research demonstrates a reduction in sampling cost for estimating thermal expectation values of observables by utilising a single Gibbs-sampling trajectory. After an initial burn-in period, interleaved coherent measurements satisfying detailed balance are performed with respect to the target Gibbs state. This approach leverages the fact that autocorrelation times can be significantly shorter than traditional mixing times, improving efficiency.

For energy estimation, and more generally for observables commuting with the Hamiltonian, measurements are implemented using Gaussian-filtered phase estimation with only logarithmic overhead. The core of this advancement lies in the discretization of an ideal target state, Ψ∗ E, through numerical integration, involving a summation over 2m qubits, where N = 22m and h = 1/2m represents the step size.

The resulting numerical summation, S[N]h, is not a quantum state with a unit 2-norm, exhibiting a convergence factor, C[N]h, which increases exponentially fast to 1 as N increases, due to the rapid decay of |bgγ(ξ)|2. This allows for a more efficient representation of the quantum state. A shifted quantum Fourier transform, Fs, is applied to implement the inverse Fourier transform, effectively performing a summation over quadrature points with a measure element of √h.

This operation, combined with a controlled unitary, W, defined as a summation over exponential terms, forms the basis for the Gaussian filtered quantum phase estimation (GQPE) measurement. The circuit implementation prepares a Gaussian resource state, bG E, and applies the controlled unitary and Fs to obtain the target state, bΨ E. Analysis of the measurement statistics reveals that the expectation of the readout, Z, differs from the expected value of the observable, O, by less than ε, given appropriate parameter choices.

The variance of Z is bounded by 3/κ2, ensuring the final output, W = κZ, maintains a precision of ε and a variance no greater than 3. The number of ancilla qubits required is m = log2(1/h) = O log(κ) + log log(ε−1), and the maximal Hamiltonian simulation time is tmax = Nh 2κ = O log(κ) + log(ε−1). This demonstrates a scalable and efficient method for estimating thermal expectation values.

A 72-qubit superconducting processor forms the foundation of the methodology for estimating thermal expectation values of observables. The team employs a single Gibbs-sampling trajectory, departing from conventional methods that rely on multiple independent trajectories separated by a full mixing time to ensure sample independence. This single trajectory begins with a burn-in period, allowing the system to reach thermal equilibrium before coherent measurements are interleaved with the Gibbs sampling dynamics.

The core innovation lies in performing these measurements in a manner that satisfies detailed balance, a condition ensuring the preservation of the target Gibbs state and avoiding the need for additional burn-in stages. To achieve detailed balance, the researchers implemented Gaussian-filtered phase estimation, a technique used to measure energy (and observables that commute with the Hamiltonian) with only logarithmic overhead.

This method allows for precise energy estimation without significantly disturbing the quantum state. For more general observables, where measurement-induced disturbance is a concern, they introduced a weighted operator Fourier transform, mitigating the impact of measurements and enabling more accurate estimation of expectation values. The rationale behind using a single trajectory stems from the concept of autocorrelation time, which can be significantly shorter than the mixing time, mirroring classical Monte Carlo methods that routinely leverage this principle.

The persistent challenge of accurately simulating complex systems at the quantum level has long been hampered by the computational cost of determining their ground state energies and thermal properties. This work offers a refinement to existing algorithms, demonstrating that fewer measurements are needed to estimate thermal expectation values than previously thought.

By intelligently interleaving measurements within a single computational trajectory, researchers have effectively decoupled the efficiency of sampling from the time it takes for the system to ‘forget’ its initial state. This is a shift in perspective from focusing on achieving full ‘mixing’ to exploiting a shorter ‘autocorrelation time’, allowing for a logarithmic overhead in measurement complexity, a substantial improvement.

However, the weighted operator Fourier transform adds complexity, and its effectiveness will depend on the specific system studied. Furthermore, the assumption of a shorter autocorrelation time requires careful validation.