Quantum random power method for ground state computation

TL;DR

QRPM addresses ground-state preparation by hybridizing quantum polynomial filtering with a classical randomized power method. The quantum part samples matrix elements of a matrix polynomial $p(H)$ using either Fejér-filtered Hamiltonian simulation or QSVT/QETU block encodings, while the classical part performs a stochastic gradient-like iteration to iteratively approach the ground state. The authors prove convergence with probability one under mild noise assumptions and show favorable iteration and quantum-depth scaling, especially when $p(H)$ is sparse, with numerical experiments on TFIM, XXZ, and Hubbard models validating performance. The work establishes a pathway to quantum-accelerated ground-state computation with constant per-iteration classical cost and noise-robust fidelity guarantees.

Abstract

We present a quantum-classical hybrid random power method that approximates a ground state of a Hamiltonian. The quantum part of our method computes a fixed number of elements of a Hamiltonian-matrix polynomial via quantum polynomial filtering techniques with either Hamiltonian simulation or block encoding. The use of the techniques provides a computational advantage that may not be achieved classically in terms of the degree of the polynomial. The classical part of our method is a randomized iterative algorithm that takes as input the matrix elements computed from the quantum part and outputs an approximation of ground state of the Hamiltonian. We prove that with probability one, our method converges to an approximation of a ground state of the Hamiltonian, requiring a constant scaling of the per-iteration classical complexity. The required quantum circuit depth is independent of the initial overlap and has no or a square-root dependence on the spectral gap. The iteration complexity scales linearly as the dimension of the Hilbert space when the quantum polynomial filtering corresponds to a sparse matrix. We numerically validate this sparsity condition for well-known model Hamiltonians. We also present a lower bound of the fidelity, which depends on the magnitude of noise occurring from quantum computation regardless of its charateristics, if it is smaller than a critical value. Several numerical experiments demonstrate that our method provides a good approximation of ground state in the presence of systematic and/or sampling noise.

Figures (7)

• Figure 1: An illustration of quantum random power method (QRPM). At every iteration, a set of indices, $M$, is sampled randomly, and then the corresponding elements of a matrix polynomial filter, $\{\widetilde{p(H)}_{ij}\}_{(i,j)\in M}$, are computed from a quantum computer. We call this quantum polynomial filtering. In practice, matrix elements obtained from the quantum polynomial filtering may be biased due to various types of noise (e.g. systematic error from the Trotter formula, sampling noise from a Hadamard test, and so on). This implies that we essentially estimate elements of a perturbation of the true matrix polynomial, $p(H)$, from a quantum computer. Next, we use the matrix elements as input for a classical randomized algorithm, which we call random power method. After a sufficient number of iterations, the method yields a classical vector, $\bm x$, close to a ground state of $H$, $\psi_{GS}$.
• Figure 2: Estimating matrix elements using the Hadamard test given an access to Hamiltonian evolution $e^{iHt}$ or a block encoding of $H$, $U_H$. For \ref{['eq: esimatemat']}, $U=e^{iHt}$ and for the QSP techniques GSLW19dong2021efficient, $U$ encodes the matrix polynomial $p(H)$. H is the Hadamard gate and $U_i$ ($U_j$) applies the tensor product of single-qubit X gates corresponding to the computational basis ket $\ket{\bm i}$ ($\ket{\bm j}$). $W$ is either $I$ or $S^{\dagger}$ where $S$ is the phase gate.
• Figure 3: An illustration of the Chebyshev filtering applied to a Hamiltonian matrix $H$. The x-axis labels the index of the eigenvalue of $H$, $\lambda$, in increasing order (e.g. the first index corresponds to the ground-state energy), and the y-axis labels the value of $p(\lambda)$ corresponding to the eigenvalue $\lambda$.
• Figure 4: Top: Comparison of the averaged fidelities obtained after 100 iterations of \ref{['alg: algorithm1']} with and without the polynomial filter, and with and without scheduling decaying step sizes for the TFIM (left) and the XXZ model (right) for different $D/J$. For each model, we used the same initial guess. For the optimization with step-size scheduling, we decrease the step size by a rate of 0.5 every 10 iteration steps. Bottom: Comparison of the growing rate of average fidelity for the cases with and without polynomial filtering.
• Figure 5: Top: comparison of the Rayleigh quotients obtained from \ref{['alg: algorithm1']} when using the polynomial filter with different shot numbers ($n_{shot}=10^5(\text{orange}),\; 4\times 10^5 (\text{blue}),\; 4\times 10^6 (\text{green})$) and without the filter (yellow) for the TFIM (left), the HM in a weak coupling regime (middle), and the HM in a strong regime (right). In all optimization results, we decrease the step size by a rate of 0.8 every 500 iteration steps. Bottom: comparison of the fidelity of the iteration with the true ground state or the subspace of true ground states. For each of the three models, we use a fixed initial guess.

Theorems & Definitions (13)

• Definition 1
• Proposition 2
• Theorem 3: Implication from \ref{['thm: convergence2']} and \ref{['lem: overlap1']}
• Theorem 4
• Lemma 5
• Proposition 6
• proof
• Lemma 7
• proof
• Theorem 8: Overall complexity