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Hamiltonian dynamics simulation using linear combination of unitaries on an ion trap quantum computer

T0 review · 2 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A reduced, multiplexor-compiled LCU scheme computes squared time-evolved overlaps on a trapped-ion quantum computer with hundreds of two-qubit gates rather than thousands.

desk verdict A credible first standalone LCU dynamics run on an ion trap, with a useful symmetry-based resource reduction; the quantitative overlap numbers are weakened by using the noise-biased measured postselection probability in Eq. (18), but the demonstration itself holds up. read the letter →

arxiv 2501.18515 v2 pith:HGX63X56 submitted 2025-01-30 quant-ph

Michelle Wynne Sze , Yao Tang , Silas Dilkes , David Muñoz Ramo , Ross Duncan , Nathan Fitzpatrick This is my paper · ORCID
classification quant-ph
keywords linearcombinationofunitariesHamiltoniansimulationtrapped-ionquantumcomputermultiplexorRabi-HubbardmodelsquaredoverlapTaylorseriesLCUpostselection
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper seeks to establish that the linear combination of unitaries (LCU) method, usually considered too expensive for current quantum computers, can be made practical for Hamiltonian dynamics by reducing the number of terms and compiling the oracle with quantum multiplexor gates. The authors test the approach on a trapped-ion quantum computer by computing the squared overlap $|\langle\psi_0|e^{-iHt}|\psi_0\rangle|^2$ for a two-cavity Rabi-Hubbard model. They report that the reduced SELECT oracle obeys the two-qubit gate count $2^{\lceil\log_2 L\rceil}(2n+1)-n-2$, and their measured circuits use roughly 175 two-qubit gates at depth 150-163. A sympathetic reader would take the contribution to be a concrete resource argument: LCU can run on a near-term ion trap for symmetry-rich initial states, with nearly constant depth in the time window studied.

What carries the argument

The load-bearing object is the quantum multiplexor gate $M^k_n(V)=\sum_i |i\rangle\langle i|_C\otimes V_i$, which lets one SELECT oracle address many Pauli targets through a single control register. Because the targets are Pauli words, SELECT factorizes into single-target multiplexors, and the diagonal multiplexed-$R_z$ parts can be commuted and merged on the control register. That decomposition is what produces the $2^k(2n+1)-n-2$ two-qubit gate count and the shallow circuit. The method's second identity is the reduced-overlap relation, Eq. (17), which converts a small LCU's success probability into the renormalization needed to recover the full overlap.

What would settle it

Run the reduced LCU circuit for a fixed $Jt$ on a noiseless state-vector simulator, record the exact success probability $\langle\psi_0|\Upsilon_\parallel^\dagger\Upsilon_\parallel|\psi_0\rangle/\|\alpha_\parallel\|_1^2$, and compare it with hardware postselection counts; a systematic deviation larger than shot statistics would show Eq. (18) is not faithful on the device.

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Extended reading notes

Core claim

The paper's central claim is that a truncated Taylor series for $e^{-iHt}$, expanded classically into a single LCU and then split into overlapping and non-overlapping parts relative to the prepared initial state, can be implemented on a trapped-ion device without the overhead that has kept LCU off near-term hardware. Only the terms with nonzero overlap with $|\psi_0\rangle$ are put into the circuit; the rest of the operator's contribution is reconstructed from the reduced LCU's success probability $p_\parallel$ and the $\ell^1$ norm of the retained coefficients, through $|\langle\psi_0|\psi_f\rangle|^2=(p_\parallel\|\alpha_\parallel\|_1^2/\langle\psi_0|\Upsilon^\dagger\Upsilon|\psi_0\rangle)|\langle\psi_0|\psi_f^\parallel\rangle|^2$ with $\Upsilon^\dagger\Upsilon\approx I$. The SELECT oracle is synthesized from multiplexed single-qubit Pauli gates, yielding the gate count $2^{\lceil\log_2 L\rceil}(2n+1)-n-2$; the experiment finds that count in practice, with circuit depth nearly constant over the simulated time interval and hardware results approaching the exact curve after memory-error mitigation.

Load-bearing premise

The quantitative result assumes that the postselection probability measured for the reduced LCU equals the noiseless value, without correcting for gate, memory, or readout errors; if that assumption fails, the reconstructed squared overlap is biased.

Editorial extensions

If this is right

  • For symmetry-rich initial states, the pre-selection step can cut the implemented LCU terms from hundreds to a handful, making LCU circuits fit on near-term devices.
  • The gate-count formula gives experimenters a direct resource estimate for SELECT before compilation, and the measured 175-180 two-qubit gates match it.
  • Circuit depth stays nearly constant over the simulated time window, in contrast to Trotter circuits whose depth grows linearly with time.
  • The reduced-overlap reconstruction needs only a small fraction of the LCU terms, because the renormalization factor comes from the postselection probability rather than from a full LCU implementation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the same preselection idea should extend to any Hamiltonian whose initial state lies in a symmetry sector, so LCU could become practical for symmetry-adapted quantum chemistry on noisy hardware.
  • Because the multiplexor construction is cheaper only for system registers up to about 12 qubits, a practical compiler might switch between multiplexor and unary-iteration SELECT depending on system size, using the crossover implied by the two gate counts.
  • The reconstructed overlap inherits any noise bias in $p_\parallel$; a noise-aware calibration of postselection probabilities, or using amplitude amplification at the cost of tripled depth, would be a direct testable improvement.
  • The paper itself notes that algorithmic error from the Taylor cutoff was not quantified; a follow-up should report the residual $|1-\sqrt{\langle\psi_0|\Upsilon^\dagger\Upsilon|\psi_0\rangle}|$ alongside the measured overlaps.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. This paper reports an experimental implementation of Hamiltonian dynamics simulation based on the linear combination of unitaries (LCU) method on a trapped-ion quantum computer (Quantinuum H1-1). The authors introduce a symmetry-based preselection reduction that splits the propagator into a relevant part Upsilon_parallel and an irrelevant part Upsilon_perp, and use quantum multiplexor compilation for the SELECT oracle, deriving a two-qubit gate count of 2^{ceil(log2 L)}(2n+1)-n-2. They compute |<psi_0|e^{-iHt}|psi_0>|^2 for a two-cavity Rabi-Hubbard model using a reduced LCU circuit with ~175 two-qubit gates and depth 150-163, and they compare emulator and hardware results against exact state-vector curves. The main result is the claim of a first standalone LCU Hamiltonian dynamics demonstration on an ion-trap device, enabled by the resource reductions.

Significance. If the claims hold, this is a notable experimental milestone: an LCU-based Hamiltonian dynamics simulation on a trapped-ion quantum computer, with a reduced circuit that is competitive with Trotter methods for this small system. The gate-count formula is derived analytically and matches the empirical resource count, and the source code is promised in a repository. The preselection technique based on symmetry and the multiplexor-based SELECT compilation are of practical interest for near-term devices. However, the central quantitative claim, the squared overlap in the bottom panel of Fig. 7, depends on a noise-sensitive estimate of the LCU postselection probability, and the experimental evidence currently lacks the error analysis needed to support that claim.

major comments (2)
  1. [IIIC, Eq. (18), Fig. 7 (bottom)]
  2. [Fig. 7 and Table I] The data points in Fig. 7 are presented without error bars, despite the fact that p_parallel and the vacuum-test probabilities are binomial quantities with finite shot counts (20480 for the emulator runs and 2048 for the H1-1 mitigated run). Without confidence intervals or repeated runs, it is impossible to determine whether the deviations of the H1-1 points from the state-vector curve at early Jt values are statistical or systematic. Please report standard errors or credible intervals for each data point and, if feasible, multiple device runs to separate sampling noise from systematic bias.
minor comments (6)
  1. [IIIC, final paragraph] The sentence beginning 'detail this process in Appendix D' is missing a subject; it should read 'We detail this process in Appendix D for the system used in this paper.'
  2. [Reference [83]] The citation to the code repository [83] does not include a URL or DOI; please provide an accessible link so that the reproducibility claim can be verified.
  3. [Fig. 7 caption/legend] The abbreviation 'sv' in the legend of Fig. 7 is undefined; spell out 'state vector' for clarity.
  4. [Eq. (19)] The hopping term -J sum_{i,j} should specify that the sum runs over nearest-neighbor pairs, or justify why summing over all pairs is appropriate for the two-cavity case.
  5. [Sec. IIIB after Eq. (10)] The statement that 'the phase factor e^{i theta_l} is included in the Pauli matrix P_{l,0}' is inconsistent with P_{l,0} being a single-qubit Pauli operator; clarify that absorbing the phase makes the effective target unitary no longer a strict Pauli operator.
  6. [Abstract and Introduction] The phrase 'pre-selecting relevant unitaries' is only defined in Sec. IIIC; consider defining it in the abstract or introduction to make the contribution self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the LCU overlap estimate is an exact algebraic reconstruction benchmarked against state-vector results, and the cited prior work is not load-bearing.

full rationale

The paper's central derivation is self-contained. The reduced squared-overlap formula (Eqs. 13-17) is a sequence of exact algebraic identities: Eq. (14) uses only the definition of the normalized post-LCU state, Eq. (16) is the same expression for the reduced operator, and Eq. (17) is the quotient of these two identities. Eq. (18) is not a new assumption; it is the LCU success-probability definition of Eq. (7) applied to Υ∥, i.e. p∥|α∥|_1^2 = ⟨ψ0|Υ∥†Υ∥|ψ0⟩ by construction, so using it to convert a measured postselection probability into the overlap estimate is a definitions-based reconstruction rather than a fitted prediction. The preselection of Υ∥ is a classical support classification (Appendix D) based on the initial-state basis, and the measured reduced overlap is an independent observable; the binary grouping does not fix the value of |⟨ψ0|Υ∥|ψ0⟩|^2. The claimed gate count 2^k(2n+1)-n-2 is derived from standard multiplexor decompositions (Sec. IIIB, Appendix B), and the empirical 2Q count is checked against this formula as an external resource benchmark, not used to define the result. Self-citations appear only as implementation tools: Ref. [79] supplies the standard vacuum-test circuit for overlap measurement and Ref. [82] supplies an X-gate memory-error mitigation heuristic; neither is invoked as a uniqueness theorem or as the sole justification of the central claim. The skeptical concern that the measured p∥ on noisy hardware is biased is a correctness/robustness issue about the experimental estimate, not circularity: a biased estimator is still an estimator of the same quantity defined by Eq. (18), and the paper explicitly acknowledges error propagation from p∥ in the Discussion. No equation reduces to its inputs by construction, no fitted parameter is renamed as a prediction, and the target observable is benchmarked against exact state-vector results in Fig. 7.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. The main assumptions are the feasibility of classical precomputation of the expanded propagator, the near-unitarity of the truncated propagator, the special symmetry of the chosen initial state, and the validity of using the noisy measured success probability in the overlap rescaling.

free parameters (5)
  • Taylor truncation order K = 8
    Chosen by hand to make the truncated propagator precision |1 - sqrt(<psi0|Y^dagger Y|psi0>)| <= 1e-6; directly controls the number of LCU terms L.
  • Coefficient discard threshold = 1e-8
    Chosen by hand; discarding terms with |alpha_l| < 1e-8 reduces L from the full expansion to 400-500.
  • Time step tau = 0.05/omega_c
    Chosen as the step in the Taylor expansion; affects the values of the coefficients alpha_l and the unitarity of Upsilon.
  • Rabi-Hubbard parameters (J, g, Delta) = J = g = Delta = 0.1 omega_c
    Defines the test case; the near-constant circuit depth with time depends on the fast saturation of the reduced term set in this small-system regime.
  • Error mitigation schedule (X-gate insertion period) = X at every third idle
    Selected by comparing noisy emulator runs with ground truth; this is a data-fitted choice for the mitigation schedule.
assumptions (5)
  • domain assumption Classical expansion of powers of H and (U_{tau,K})^m into a single LCU is feasible (Sec. II: 'we assume that this can be accomplished').
    The method requires classically computing the Pauli decomposition of the truncated Taylor series propagator; demonstrated here only for a 5-qubit system, and acknowledged as nontrivial for larger systems (Ref. [76]).
  • domain assumption The truncated propagator Upsilon is approximately unitary for the chosen parameters (Sec. IIIC and Eq. (17)): <psi0|Upsilon^dagger Upsilon|psi0> ~ 1.
    Verified numerically to 1e-6 for the Rabi-Hubbard parameters, but the reduced-overlap formula Eq. (17) sets the denominator to 1; for other Hamiltonians this could fail.
  • domain assumption The initial state and Hamiltonian symmetries allow tapering to 5 qubits and reduce Upsilon_parallel to at most 11 Pauli strings (Appendix D).
    The resource reduction (98% smaller circuit) relies on the special form of the Mott-insulator state; without this symmetry the reduction does not apply.
  • standard math Multiplexor decompositions of Bergholm et al. and Shende et al. are correct, and the diagonal-merging resynthesis in Fig. 5 is valid (Sec. IIIB, Appendix B).
    The SELECT gate count 2^k(2n+1)-n-2 is built on these cited decompositions.
  • domain assumption The measured LCU success probability p_parallel equals the ideal value <psi0|Upsilon_parallel^dagger Upsilon_parallel|psi0>/|alpha_parallel|_1^2 (Eq. (18)).
    The final overlap in Eq. (17) is computed from p_parallel measured on noisy hardware/emulator; the paper does not correct p_parallel for noise, so this assumption is load-bearing for the quantitative results.

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Pith. "Pith review of Hamiltonian dynamics simulation using linear combination of unitaries on an ion trap quantum computer." pith.science (2026). https://pith.science/paper/HGX63X56

@misc{pith2026250118515,
  author       = {Pith},
  title        = {Pith review of: Hamiltonian dynamics simulation using linear combination of unitaries on an ion trap quantum computer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HGX63X56}},
  note         = {Machine review of arXiv:2501.18515}
}
abstract

The linear combination of unitaries (LCU) method has proven to scale better than existing product formulas in simulating long time Hamiltonian dynamics. However, given the number of multi-control gate operations in the standard prepare-select-unprepare architecture of LCU, it is still resource-intensive to implement on the current quantum computers. In this work, we demonstrate LCU implementations on an ion trap quantum computer for calculating squared overlaps $|\langle \psi(t=0)|\psi(t>0)\rangle|^2$ of time-evolved states. This is achieved by an optimized LCU method, based on pre-selecting relevant unitaries, coupled with a compilation strategy which makes use of quantum multiplexor gates, leading to a significant reduction in the depth and number of two-qubit gates in circuits. For $L$ Pauli strings in a Taylor series expanded $n$-qubit-mapped time evolution operator, we find a two-qubit gate count of $2^{\lceil log_2(L)\rceil}(2n+1)-n-2$. We test this approach by simulating a Rabi-Hubbard Hamiltonian.

Figures

Figures reproduced from arXiv: 2501.18515 by the authors.

Figure 1
Figure 1. LCU Circuit for an operator Υ, when a successful postselection occurs. The linear combination of unitaries (LCU) approach allows a non-unitary operator Υ = L X−1 ℓ=0 αℓPℓ (3) to be applied to a quantum state |ψ⟩. This works by embedding Υ as a block in a larger unitary, ULCU =  Υ |α|1 ∗ ∗ ∗ , (4) |α|1 = P ℓ |αℓ|, acting on a larger quantum register and by measuring and post-selecting on ancilla qubits, projecting … view at source ↗
Figure 2
Figure 2. A k-controlled multiplexed gate MC n (V ) imple￾mented as a sequence of multi-controlled gates. The white square boxes designate multiplexed controls. spanned by the 2 k basis states, as in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Decomposition for MC n (V ) where V is a list of U(2) unitaries, n = 1, and C = 3. · · · . . . . . . · · · . . . . . . . . . . . . · · · · · · Ry Rz Ry Rz Ry Rz [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: Circuit construction that prepares an arbitrary [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: A diagonal operator can be resynthesized to [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: Circuit for |⟨0|ψU † ψ ⟨0|pLCU(Υ∥)|0⟩pUψ|0⟩ψ| 2 . IV. RESULTS For the Hamiltonian in e −iHt, we choose the Rabi￾Hubbard (RH), a spin-boson system describing light￾matter interaction that is a frequently used model Hamiltonian for observing quantum phase transition. The…
Figure 8
Figure 8. Figure 8: LCU circuit resources for the Rabi-Hubbard prop [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 7
Figure 7. Figure 7: Squared overlap calculations from reduced [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 9
Figure 9. Figure 9: M1 1 (V ) gate Equation (B1) can be written as two equations: V0 = r †W dL, (B2) V1 = rW d†L. (B3) To solve for W, L and r, we rewrite these equations as rV0V † 1 r = rr†W dLL† dW† r † r = W d2W† . (B4) W d2W† can be obtained via eigen-decomposition on rV0V † 1 r. Sinc…
Figure 10
Figure 10. Figure 10: Demultiplex M1 1 (V ) b. Multi-qubit multiplexed-U(2) The problem is now to decompose an arbitrary multiplexed-U(2) gate with k control qubits. This gate has a block-diagonal matrix representation where each block is a U(2) unitary. To decompose it, we can pair every …
Figure 11
Figure 11. Figure 11: Demultiplex Mk 1 (V ) gates, as shown in [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: Rewrite e i π 4 ZZ 2. Multiplexed-Rotation Multiplexed rotations can be decomposed recursively in a way that is similar to multiplexed-U(2) decomposition, as shown in [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: Two ways to demultiplex Mk 1 (R) where R, P, Q consist of rotations around a fixed axis that is perpendicular to the x-axis. The first recursive relation in [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]
Figure 14
Figure 14. Figure 14: Recursive decomposition for MC n (Rz) where n = 1 and C = 3, producing an alternating sequence of 2 k cx gates and 2 k Rz gates. 3. Scaling The SELECT oracle construction using the proposed multiplexor construction requires no additional ancilla qubits, requiring at m…
Figure 15
Figure 15. Figure 15: Decomposition of a 2-control multiplexor [PITH_FULL_IMAGE:figures/full_fig_p016_15.png]
Figure 16
Figure 16. Figure 16: Transition probability in Jaynes-Cummings system with [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]
Figure 17
Figure 17. Figure 17: Simulations error with respect to the analytical solution of Jaynes-Cummings system transition probability using [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]
Figure 18
Figure 18. Figure 18: Circuit depth. Trotter vs LCU. The parameters [PITH_FULL_IMAGE:figures/full_fig_p019_18.png]
Figure 19
Figure 19. Figure 19: Precision of the Taylor expanded and truncated time evolution operator with the Rabi-Hubbard Hamiltonian, [PITH_FULL_IMAGE:figures/full_fig_p021_19.png]
Figure 20
Figure 20. Figure 20: LCU circuit resources for the Rabi-Hubbard propagator with and without using the reduction technique in [PITH_FULL_IMAGE:figures/full_fig_p021_20.png]
Figure 21
Figure 21. Figure 21: Larger Rabi-Hubbard systems. Number of cavities is varied from [PITH_FULL_IMAGE:figures/full_fig_p022_21.png]

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Forward citations

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.