[Paper Review] Phase estimation with randomized Hamiltonians

Ian Kivlichan, Christopher Granade+1 more|arXiv (Cornell University)|Jul 23, 2019

Hydrocarbon exploration and reservoir analysis4 citations

TL;DR

This paper introduces a randomized Hamiltonian framework for iterative phase estimation in quantum simulations, where different Hamiltonians are used at each step based on importance sampling to reduce computational cost. By adaptively selecting terms with higher relevance to the ground state, the method significantly reduces circuit depth and qubit count—demonstrated numerically on chemical Hamiltonians—while preserving phase estimation accuracy and success probability under spectral gap conditions.

ABSTRACT

Iterative phase estimation has long been used in quantum computing to estimate Hamiltonian eigenvalues. This is done by applying many repetitions of the same fundamental simulation circuit to an initial state, and using statistical inference to glean estimates of the eigenvalues from the resulting data. Here, we show a generalization of this framework where each of the steps in the simulation uses a different Hamiltonian. This allows the precision of the Hamiltonian to be changed as the phase estimation precision increases. Additionally, through the use of importance sampling, we can exploit knowledge about the ground state to decide how frequently each Hamiltonian term should appear in the evolution, and minimize the variance of our estimate. We rigorously show, if the Hamiltonian is gapped and the sample variance in the ground state expectation values of the Hamiltonian terms sufficiently small, that this process has a negligible impact on the resultant estimate and the success probability for phase estimation. We demonstrate this process numerically for two chemical Hamiltonians, and observe substantial reductions in the number of terms in the Hamiltonian; in one case, we even observe a reduction in the number of qubits needed for the simulation. Our results are agnostic to the particular simulation algorithm, and we expect these methods to be applicable to a range of approaches.

Motivation & Objective

• To reduce the computational cost of iterative phase estimation in quantum simulations by dynamically adjusting the Hamiltonian terms used at each step.
• To minimize variance in phase estimation by using importance sampling based on ground state expectation values of Hamiltonian terms.
• To show that randomized Hamiltonians maintain high fidelity and success probability in phase estimation when the original Hamiltonian is gapped and term variances are small.
• To enable reductions in both time and space complexity by identifying and omitting irrelevant Hamiltonian terms early in the simulation process.
• To generalize existing simulation cost-reduction techniques beyond term count to include adaptive sampling of Hamiltonian components.

Proposed method

• The method uses importance sampling to assign probabilities to Hamiltonian terms based on their ground state expectation values, favoring more significant terms.
• At each iteration of phase estimation, a different randomized Hamiltonian is constructed by sampling terms according to their importance weights, reducing the number of terms used.
• Theoretical analysis leverages the quantum Zeno effect and adiabatic theorems to bound errors in eigenstate preparation and phase accumulation.
• A key equation bounds the difference between ideal and actual unitary evolution as $ \|U_k - U_{k,\text{ad}}\| \leq \frac{2\lambda^2}{(\gamma - 2\lambda)^2} $, where $ \lambda $ is the perturbation strength and $ \gamma $ the spectral gap.
• The variance of the phase estimate is minimized by choosing sampling distributions proportional to the magnitude of matrix elements $ |\langle \psi_p^k | V_k | \psi_0^k \rangle| $.
• The approach is agnostic to the underlying simulation algorithm, making it applicable to various quantum simulation schemes.

Experimental results

Research questions

• RQ1 Can iterative phase estimation maintain accuracy when using different, randomized Hamiltonians at each step rather than a fixed one?
• RQ2 How does importance sampling of Hamiltonian terms affect the variance and convergence of phase estimation?
• RQ3 Under what conditions does randomization of the Hamiltonian not degrade the success probability of phase estimation?
• RQ4 To what extent can the number of qubits and circuit depth be reduced by selectively omitting low-impact Hamiltonian terms?
• RQ5 Can the method be generalized to other quantum simulation algorithms beyond standard phase estimation?

Key findings

• The method achieves substantial reductions in the number of Hamiltonian terms used, with one chemical system showing a reduction in required qubits.
• Numerical simulations confirm that randomized Hamiltonians preserve phase estimation accuracy and success probability under spectral gap conditions.
• The variance of the phase estimate is minimized when sampling is proportional to the absolute value of matrix elements in the ground state.
• Theoretical bounds show that eigenvalue estimation error remains negligible if the spectral gap $ \gamma $ is sufficiently large relative to perturbation strength $ \lambda $.
• The method enables a reduction in time complexity and, in some cases, space complexity by identifying and excluding irrelevant qubits and terms.
• The success probability remains high as long as $ \lambda / \gamma < \sqrt{1 - \exp(\log(1 - \epsilon)/(M - 1))} $, ensuring low failure rates during measurement steps.