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arxiv: 1907.10070 · v1 · pith:V2H5SL4Cnew · submitted 2019-07-23 · 🪐 quant-ph

Phase estimation with randomized Hamiltonians

Ian D. Kivlichan , Christopher E. Granade , Nathan Wiebe This is my paper

Pith reviewed 2026-05-24 17:18 UTC · model grok-4.3

classification 🪐 quant-ph
keywords phase estimationrandomized Hamiltoniansimportance samplingquantum simulationchemical Hamiltoniansground statevariance reductioniterative phase estimation
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The pith

Iterative phase estimation can use a different Hamiltonian at each step, chosen by importance sampling on ground-state expectations, to reduce the number of terms while keeping the estimate accurate if the system is gapped.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that randomizing which Hamiltonian is used during each iteration of phase estimation allows the precision to be adjusted dynamically and exploits ground-state knowledge to minimize variance through importance sampling. This leads to fewer terms in the Hamiltonian expansion for chemical systems. A rigorous bound shows that under a gapped Hamiltonian and low sample variance in term expectations, the randomization has negligible effect on the eigenvalue estimate and success probability. Numerical tests on two chemical Hamiltonians confirm large reductions in terms and sometimes qubits required.

Core claim

By applying a different Hamiltonian in each step of the simulation circuit and selecting terms via importance sampling based on ground state expectations, the variance of the phase estimate is minimized, and if the Hamiltonian is gapped with sufficiently small sample variance, the process negligibly affects the estimate and success probability of phase estimation.

What carries the argument

Randomized Hamiltonians selected by importance sampling in iterative phase estimation, which varies the simulation Hamiltonian across repetitions to exploit ground-state term expectations.

If this is right

  • Substantial reductions in the number of Hamiltonian terms for simulating chemical systems.
  • Possible reduction in the number of qubits needed for the simulation in some cases.
  • The method applies to any simulation algorithm used within phase estimation.
  • The impact on estimate accuracy remains negligible when the gap condition and variance condition hold.
  • Precision of the Hamiltonian can be changed as phase estimation precision increases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the variance condition holds for many molecular Hamiltonians, this could lower resource requirements for quantum chemistry simulations on near-term devices.
  • Testing the approach on larger systems could reveal whether qubit reductions generalize beyond the two examples.
  • The agnosticism to the simulator suggests it could integrate with future improved simulation methods.
  • The technique might combine with other variance-reduction methods in quantum algorithms, though this is not tested in the work.

Load-bearing premise

The Hamiltonian is gapped and the sample variance in the ground state expectation values of the Hamiltonian terms is sufficiently small.

What would settle it

Running the randomized phase estimation on a gapped Hamiltonian where the sample variance of term expectations is large and observing a substantial degradation in the eigenvalue estimate accuracy or success probability compared to the non-randomized case.

Figures

Figures reproduced from arXiv: 1907.10070 by Christopher E. Granade, Ian D. Kivlichan, Nathan Wiebe.

Figure 1
Figure 1. Figure 1: FIG. 1. Quantum circuit for performing iterative phase estimation. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript generalizes iterative phase estimation to allow a different Hamiltonian at each simulation step, with importance sampling used to select terms based on ground-state expectations in order to minimize variance. It asserts a rigorous guarantee that, when the Hamiltonian is gapped and the sample variance of the ground-state term expectations is sufficiently small, the randomization has negligible effect on the phase estimate and success probability. Numerical results are reported for two chemical Hamiltonians, showing substantial reductions in the number of terms and, in one case, a reduction in the number of qubits required. The method is presented as agnostic to the underlying simulation algorithm.

Significance. If the stated conditional guarantee can be verified, the approach offers a practical route to lowering the effective cost of Hamiltonian simulation within phase estimation by sparsifying the operator via importance sampling, while preserving accuracy under explicit assumptions on the gap and variance. The numerical demonstrations of term and qubit reduction on chemical systems provide concrete evidence of potential resource savings, and the algorithm-agnostic framing increases the result's reach. Credit is due for explicitly conditioning the claim on measurable properties of the ground state rather than hiding dependencies.

major comments (1)
  1. [Abstract / proof section] Abstract and main text: the central rigorous claim—that the randomization has negligible impact on the estimate and success probability—is asserted under the gapped-Hamiltonian and low sample-variance conditions, yet the full derivation, explicit error bounds, and the precise translation from variance to success-probability degradation are not supplied; without this derivation the load-bearing guarantee cannot be assessed.
minor comments (2)
  1. [Abstract] The abstract refers to 'two chemical Hamiltonians' without naming them or citing the specific systems; this information should be added for reproducibility.
  2. [Numerical experiments] Numerical results mention 'substantial reductions' and 'one case' of qubit reduction; the precise term counts, qubit counts, and the selection criterion used for the importance-sampling probabilities should be tabulated for both Hamiltonians.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater explicitness in the central rigorous claim. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / proof section] Abstract and main text: the central rigorous claim—that the randomization has negligible impact on the estimate and success probability—is asserted under the gapped-Hamiltonian and low sample-variance conditions, yet the full derivation, explicit error bounds, and the precise translation from variance to success-probability degradation are not supplied; without this derivation the load-bearing guarantee cannot be assessed.

    Authors: We agree that the manuscript asserts the guarantee under the gapped-Hamiltonian and low-variance conditions but does not supply the full derivation, explicit error bounds, or the precise mapping from sample variance to degradation in success probability. This omission prevents independent verification of the claim. In the revised manuscript we will insert a dedicated proof section (or appendix) that derives the error bounds from first principles, shows how the variance threshold controls the deviation in the phase estimate, and quantifies the resulting effect on success probability. The added material will be self-contained and will make the conditional guarantee fully assessable. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a conditional rigorous guarantee: if the Hamiltonian is gapped and the sample variance in ground-state term expectations is sufficiently small, then randomized importance sampling has negligible effect on the phase estimate and success probability. These conditions are external domain assumptions, not quantities defined or fitted by the method. No derivation step reduces by construction to its own inputs, no self-citation is load-bearing for the central claim, and no ansatz or uniqueness theorem is smuggled in. Numerical results on chemical Hamiltonians supply independent empirical support for term (and qubit) reduction. The argument is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain assumptions about the Hamiltonian; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The Hamiltonian is gapped
    Invoked as the first condition under which the rigorous guarantee holds.
  • domain assumption Sample variance in the ground state expectation values of the Hamiltonian terms is sufficiently small
    Invoked as the second condition required for negligible impact on the phase estimate.

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Reference graph

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