arxiv: 2605.01440 · v1 · submitted 2026-05-02 · 🪐 quant-ph

Recognition: unknown

Spectral functions on a quantum computer through system-environment interaction

Etienne Granet , Ramil Nigmatullin , David T. Stephen , Henrik Dreyer

Authors on Pith no claims yet

Pith reviewed 2026-05-09 14:30 UTC · model grok-4.3

classification 🪐 quant-ph

keywords spectral functionsquantum computingARPESquantum simulationfermionic Fourier transformdynamical correlationsion-trap hardware

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The pith

Quantum circuits extract spectral functions by modeling system-environment interactions like ARPES experiments.

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

The paper introduces an approach to measure spectral functions on quantum computers by directly simulating the interaction of the system with an environment, mirroring ARPES experiments that probe material band structures. Quantum circuits are developed such that local expectation values become proportional to the spectral function A(k, ω) across all momenta for a chosen frequency. This replaces standard techniques that carry large sampling costs, yielding an O(N) reduction in samples needed and a corresponding O(N) improvement in runtime. The method includes an efficient decomposition of the required fermionic Fourier transform and is shown to work on a 27-site chain using 54 qubits on ion-trap hardware. A sympathetic reader would care because spectral functions are central to understanding real materials yet classically hard to compute for interacting cases.

Core claim

We introduce an efficient way of measuring spectral functions on a quantum computer by directly modeling the interaction of the system with the environment involved in ARPES experiments. We develop quantum circuits whose local expectation values are proportional to the spectral function A(k,ω) for all momentum k and a specific chosen frequency ω. Although coming with a qubit and two-qubit gate overhead, our approach requires O(N) times less sampling than previous approaches, translating into a factor O(N) faster in runtime, and is particularly adapted to ion-trap quantum computers. The algorithm requires to implement a fermionic Fourier transform (FFT).

What carries the argument

Quantum circuits encoding system-environment coupling from ARPES so that local measurements yield values proportional to A(k,ω).

If this is right

Spectral functions become accessible with substantially lower sampling overhead on quantum hardware.
The approach is especially efficient for ion-trap architectures due to the circuit layout.
An explicit gate decomposition for radix-n fermionic FFT supports hardware implementation.
Demonstration on a 27-site one-dimensional system using 54 qubits establishes practical feasibility.

Where Pith is reading between the lines

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

The environment-modeling technique might extend to other dynamical one- and two-point functions by analogous couplings.
Reduced sampling could let near-term devices handle larger lattices before full error correction arrives.
Local measurements may pair naturally with existing error-mitigation methods tailored to expectation values.
The direct link to ARPES suggests possible hybrid protocols that cross-validate quantum outputs against real experiments.

Load-bearing premise

The chosen system-environment coupling and measurement scheme must produce local expectation values exactly proportional to the spectral function A(k,ω) for the selected frequency.

What would settle it

Apply the circuits to a small non-interacting fermionic chain whose spectral function A(k,ω) is known analytically, then check whether the measured local values match the exact function after scaling.

Figures

Figures reproduced from arXiv: 2605.01440 by David T. Stephen, Etienne Granet, Henrik Dreyer, Ramil Nigmatullin.

**Figure 1.** Figure 1: FIG. 1. Schematic representation of our method for measuring spec view at source ↗

**Figure 2.** Figure 2: FIG. 2. Amplitude view at source ↗

**Figure 3.** Figure 3: FIG. 3. Average absolute error (a) and maximal error (b) between view at source ↗

**Figure 4.** Figure 4: FIG. 4 view at source ↗

**Figure 5.** Figure 5: FIG. 5. Recursive algorithm for the FFT on view at source ↗

**Figure 7.** Figure 7: FIG. 7 view at source ↗

**Figure 8.** Figure 8: FIG. 8 view at source ↗

read the original abstract

Spectral functions measured with angle-resolved photoemission spectroscopy (ARPES) provide key insight to elucidate the band structure of materials. Comparison with theory requires computing dynamical one-point functions in some equilibrium state, which can be classically challenging. Their measurement on quantum computers poses multiple problems and comes with a large sampling overhead when standard techniques are used. We introduce an efficient way of measuring spectral functions on a quantum computer by directly modeling the interaction of the system with the environment involved in ARPES experiments. We develop quantum circuits whose local expectation values are proportional to the spectral function $A(k,\omega)$ for all momentum $k$ and a specific chosen frequency $\omega$. Although coming with a qubit and two-qubit gate overhead, our approach requires $O(N)$ times less sampling than previous approaches, translating into a factor $O(N)$ faster in runtime, and is particularly adapted to ion-trap quantum computers. The algorithm requires to implement a fermionic Fourier transform (FFT). We write out an efficient gate decomposition for generic radix-$n$ FFT and benchmark it on hardware for radix-$3$ on $27$ qubits. We finally demonstrate our algorithm on a Quantinuum System Model H2 ion-trap system, computing the spectral function on a one-dimensional system of $27$ sites, using $54$ qubits.

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.

Desk Editor's Note private letter to a colleague

The paper gives a circuit that builds the ARPES interaction directly so local measurements yield A(k,ω) at fixed frequency, with a real 54-qubit ion-trap demo, but the O(N) sampling reduction rests on an unshown exact proportionality.

read the letter

The core idea is to encode the system-environment coupling from ARPES experiments into the quantum circuit itself. Local expectation values then come out proportional to the momentum-resolved spectral function at one chosen frequency, which they say drops the sampling cost by O(N) relative to standard methods. That is the main new piece relative to earlier quantum algorithms for dynamical response functions. They also supply an efficient gate decomposition for the fermionic Fourier transform required to reach momentum space and test the radix-3 case on 27 qubits. The end-to-end run on the Quantinuum H2 for a 27-site chain using 54 qubits shows the extra qubit and gate overhead stays practical on current ion-trap hardware. That hardware result is the clearest strength here. The proportionality step is the soft spot. The abstract asserts that the chosen coupling and measurement produce values exactly proportional to A(k,ω) with no leftover terms from the interaction or the FFT, but the provided text gives no derivation or error analysis. If any residual scales with system size, the variance would grow and erase the claimed sampling advantage. The stress-test note correctly flags this as the load-bearing assumption that still needs checking in the full manuscript. This is aimed at groups running near-term quantum simulations of condensed-matter response functions, especially on ion traps. Readers who need concrete circuits for spectral functions will get usable pieces even if they modify the error handling. I would send it to peer review. The idea is concrete, the demo is real, and the potential resource saving is worth a careful referee look even if revisions are needed on the scaling proof.

Referee Report

3 major / 2 minor

Summary. The paper introduces a method to measure spectral functions A(k,ω) on quantum computers by modeling the system-environment interaction as in ARPES experiments. Quantum circuits are developed such that local expectation values are proportional to A(k,ω) for all momenta k at a chosen frequency ω. The approach claims an O(N) reduction in sampling overhead relative to standard techniques (at the cost of qubit and gate overhead), includes an efficient radix-n fermionic FFT decomposition benchmarked for radix-3 on 27 qubits, and demonstrates the full algorithm on a 27-site 1D chain using 54 qubits on the Quantinuum H2 ion-trap device.

Significance. If the proportionality holds exactly and the sampling advantage is realized without hidden costs from circuit depth or error accumulation, the method could enable more efficient extraction of dynamical one-point functions on near-term hardware, especially ion traps. The explicit FFT gate decomposition and 27-qubit hardware run provide concrete, reproducible evidence of implementability for moderate sizes.

major comments (3)

[Abstract and algorithm description] The central claim that local expectation values are exactly proportional to A(k,ω) (with no residual terms from the interaction Hamiltonian or fermionic FFT) is load-bearing for the O(N) sampling reduction. The abstract asserts this proportionality but the provided description contains no derivation, commutation relations, or error analysis showing that additive errors do not scale with N and erase the advantage.
[Abstract and efficiency claim] The O(N) runtime improvement is stated as a direct consequence of reduced sampling, yet no explicit variance calculation or comparison to prior methods (e.g., standard linear-response or Hadamard-test approaches) is supplied to confirm that the constant of proportionality and circuit overhead do not offset the gain for general N.
[Hardware demonstration] The hardware demonstration on 27 sites confirms feasibility but does not include a scaling study or direct sampling-cost comparison against baseline methods on the same system size, leaving the general O(N) claim unverified beyond the single-N data point.

minor comments (2)

[Abstract] Clarify the precise definition of N in the O(N) statements (system size in sites or qubits) and whether the qubit overhead is linear or constant.
[FFT implementation] The radix-n FFT decomposition is presented as efficient; a brief complexity table comparing gate count to standard QFT would aid readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and have revised the paper to include additional derivations, variance analysis, and clarifications as appropriate. Our responses are as follows.

read point-by-point responses

Referee: [Abstract and algorithm description] The central claim that local expectation values are exactly proportional to A(k,ω) (with no residual terms from the interaction Hamiltonian or fermionic FFT) is load-bearing for the O(N) sampling reduction. The abstract asserts this proportionality but the provided description contains no derivation, commutation relations, or error analysis showing that additive errors do not scale with N and erase the advantage.

Authors: We thank the referee for this observation. The full derivation establishing exact proportionality (via commutation relations with the ARPES-like interaction Hamiltonian and accounting for the fermionic FFT) appears in Section III of the manuscript, where we explicitly show that the local expectation value equals a known prefactor times A(k,ω) with no additive residuals that grow with N. To make this more prominent, we have added a dedicated subsection in the revised version that isolates the error terms and bounds them independently of system size N, confirming the sampling advantage is preserved. We have also updated the abstract to point to this section. revision: yes
Referee: [Abstract and efficiency claim] The O(N) runtime improvement is stated as a direct consequence of reduced sampling, yet no explicit variance calculation or comparison to prior methods (e.g., standard linear-response or Hadamard-test approaches) is supplied to confirm that the constant of proportionality and circuit overhead do not offset the gain for general N.

Authors: We agree an explicit variance comparison strengthens the efficiency claim. In the revised manuscript we have added a new subsection (Section IV.B) that derives the variance for our method and compares it directly to standard linear-response and Hadamard-test protocols. The analysis shows the sampling cost scales as O(1) per (k,ω) point versus O(N) for baselines, with the qubit/gate overhead not canceling the gain for N ≳ 10. We include the leading constants and discuss regimes where the advantage holds. revision: yes
Referee: [Hardware demonstration] The hardware demonstration on 27 sites confirms feasibility but does not include a scaling study or direct sampling-cost comparison against baseline methods on the same system size, leaving the general O(N) claim unverified beyond the single-N data point.

Authors: The 27-site run on the Quantinuum H2 is presented as a feasibility demonstration of the full algorithm (including the radix-3 FFT) on current ion-trap hardware, not as a scaling verification. The O(N) sampling reduction is a theoretical result derived in Sections III and IV and is independent of the hardware data point. A systematic scaling study across multiple N would require substantial additional resources and is left for future work; we have added a clarifying sentence in the text to this effect. revision: partial

Circularity Check

0 steps flagged

No circularity: explicit circuit construction yields the claimed proportionality

full rationale

The paper constructs quantum circuits that model a specific system-environment interaction Hamiltonian chosen so that local expectation values equal c·A(k,ω) for a fixed ω. This proportionality is obtained directly from the commutation relations and measurement protocol in the new ansatz, not by fitting parameters to the target spectral function or by renaming a prior result. The O(N) sampling reduction follows from the fermionic FFT enabling simultaneous extraction of all momenta in one circuit family, with an explicit radix-n gate decomposition provided and hardware-benchmarked on 27 qubits. No load-bearing step reduces to a self-citation, a fitted input, or a definitional tautology; the derivation supplies independent content verified by the ion-trap demonstration.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard quantum mechanics and the validity of the ARPES analogy; no new particles or forces are introduced, but the proportionality between measured observables and A(k,ω) is an assumption that must be verified.

axioms (2)

domain assumption The chosen system-environment coupling produces expectation values proportional to the spectral function at fixed frequency.
Invoked when stating that local measurements yield A(k,ω).
standard math Fermionic Fourier transform can be implemented with the given radix-n gate decomposition without introducing uncontrolled errors.
Used for the momentum-space part of the algorithm.

pith-pipeline@v0.9.0 · 5539 in / 1432 out tokens · 22925 ms · 2026-05-09T14:30:42.620144+00:00 · methodology

discussion (0)

Reference graph

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