Less precise but less noisy: local circuits for momentum-space state preparation and measurement

The gist: Local circuits for momentum-space state preparation and measurement demonstrate that adiabatic evolution can outperform the Fermionic Fourier Transform (FFT) on noisy hardware beyond a certain system size, emphasizing that noise sensitivity should be considered alongside gate count or circuit depth.

Noise Sensitivity of FFT and Adiabatic Evolution

The paper investigates whether noise sensitivity should be a factor in circuit optimization, contrasting the Fermionic Fourier Transform (FFT) with Hamiltonian simulation via adiabatic evolution on the Quantinuum System Model H2 quantum computer. The FFT is described as a fully digital circuit with high precision, capable of distinguishing momenta by 1/N, which requires long-range couplings in real space and thus propagates errors faster than local circuits like adiabatic evolution. Conversely, although adiabatic evolution has coarser momentum resolution, it propagates errors more slowly. The study shows that for a tight-binding chain ground state preparation of size N beyond a certain threshold, the adiabatic evolution achieves significantly lower energies than the FFT with the same number of gates and circuit depth. This better performance on noisy hardware is attributed to the lower precision in momentum space of Hamiltonian simulation techniques compared to FFT.

Performance Comparison and Error Dilution

The comparison between circuits like FFT and adiabatic evolution reveals that at small system sizes, the FFT reaches lower energies than the adiabatic evolution, but beyond a system size N ≈ 20, there is a crossing where the adiabatic evolution achieves lower energies on noisy hardware. The energy error for the FFT strongly increases with system size, whereas for the adiabatic evolution it increases only mildly or is constant. This difference in noise sensitivity is explained by error dilution: errors in Hamiltonian simulation circuits typically impact local observables by only O(1/N), while they impact general circuits by O(1). The optimal system size N∗ at which adiabatic evolution outperforms FFT with the same number of gates scales as N∗ = O(1/(p log2 p)) for a given noise rate p.

Momentum Measurement Scheme (MDLM)

The authors propose an alternative to the FFT for momentum measurement, termed Momentum Distribution from Local Measurements (MDLM). This scheme uses local operators On and O′n to measure the number of particles in each momentum mode n(k), which can then be used to deduce momentum mode occupation. The expectation value of On is proportional to cos(kn)⟨n(k)⟩, while the expectation value of O′n is proportional to sin(kn)⟨n(k)⟩. By measuring M observables, one maximizes the Shannon entropy S(p) subject to these constraints to estimate the momentum densities ⟨n(k)⟩. Numerical tests on a system size N = 64 show that MDLM improves agreement with exact momentum densities, especially when including observables up to n = 5.

Classically-Optimized Parametrized Circuits (COP)

An alternative approach involves using Classically-Optimized Parametrized Circuits (COP) which rely on classical optimization of gate angles to minimize energy. For free fermions, a natural adiabatic-inspired ansatz is proposed, and in practice, COP circuits have been observed to yield significant improvements in energies compared to adiabatic evolution with the same number of gates. The sensitivity of these circuits to noise depends on the number of Trotter steps; for even numbers of steps, the optimal gate angles are more regular in time than for odd numbers of steps.

Conclusion and Future Directions

The work concludes that the high precision in momentum space offered by FFT comes at the cost of high sensitivity to errors due to long-range couplings. The paper demonstrates that for physical applications, this loss of precision is often acceptable if it drastically reduces noise sensitivity. Future research directions include devising adiabatic protocols for preparing arbitrary excited states and exploring smooth modifications to the FFT circuit structure to decrease noise sensitivity at the cost of momentum precision.

Appendix A provides hardware gate counts comparing FFT and adiabatic evolution for various system sizes N, showing that for N = 48, the adiabatic circuit has approximately twice more two-qubit gates than the FFT when aiming for a similar circuit depth. Appendix B details optimal angles of COP circuits as a function of the number of steps, highlighting differences between even and odd numbers of steps. Table I summarizes expectation values of On’s measured on Quantinuum H2-2 hardware, showing good agreement with noiseless and exact profiles for MDLM. The paper suggests that the structure of the circuit significantly influences sensitivity to noise beyond just the number of gates and depth.

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Less precise but less noisy: Etienne Granet, Henrik Dreyer 1 Quantinuum, Leopoldstrasse 180, 80804 Munich, Germany (Dated: October 2, 2026) Quantum algorithms are usually optimized for gate count or circuit depth. We find on Quantinuum System Model H2 quantum computer that for a tight-binding chain ground state preparation, there is a system size N beyond which the adiabatic evolution reaches significantly lower energies than the Fermionic Fourier Transform (FFT), with the same number of gates, and with the same circuit depth. We attribute this high noise sensitivity of the FFT to its high precision, being able to distinguish momenta by 1/N. This high resolution in momentum space requires long-range couplings in real space, which propagates errors faster. In contrast, although local and physical circuits such as the adiabatic evolution have a coarser momentum resolution, they also propagate errors more slowly. For physical applications, high momentum resolution is rarely required and is often worth trading for low noise sensitivity. We also introduce a momentum measurement scheme that although less precise than FFT, is less costly and less noisy. We show that it achieves better performance than FFT for spectral function measurement on Quantinuum System Model H2 quantum computer. Our work emphasizes the importance of reducing the noise sensitivity of quantum algorithms, beyond the number of gates or circuit depth.

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2 Beyond this observation and analytical explanation, we propose an alternative to FFT for momentum measurement. We implement our momentum measurement scheme on Quantinuum H2-2 hardware for a spectral function measurement problem, and observe significant improvement over the FFT.

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3 We present a hardware implementation on Quantinuum H2-2 quantum computer [33]. For different values of N, we implement the FFT with unitary circuits with a depth scaling as log2 N (see Ref [22, 31]). The adiabatic evolution for NTrott Trotter steps is implemented as U = N YTrott t=1 e ixt PN j=1 XjXj+1 e iyt PN j=1 YjYj+1, with yt/xt going from 0 to 1 along the path, and dt = 0.4 q 1 − t−0.5 NTrott.

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4 We observe that at small system sizes, the FFT reaches lower energies than the adiabatic evolution. However, at larger system sizes N ⪆ 20, there is a crossing between the two curves and the adiabatic evolution reaches lower energies than the FFT. The energy error strongly increases with system size for the FFT, whereas at large system sizes it increases only mildly (or even is constant) for the adiabatic evolution. We emphasize again that for a same system size, the two circuits, FFT and adiabatic, have approximately the same number of two-qubit gates. In terms of circuit depth, the adiabatic circuits are shallower than the FFT since they are denser. At N = 48, the two-qubit gate circuit depth of the adiabatic circuit is around twice smaller than the FFT circuit. We also ran on hardware at N = 48 an adiabatic evolution with approximately same circuit depth as the FFT, which corresponds to around twice more two-qubit gates. We obtain an energy −0.577 ± 0.017, which is even lower than the result for same number of two-qubit gates plotted in Fig 2. This shows that in size N = 48, the adiabatic evolution always performs better than the FFT, whether we impose the same number of two-qubit gates or the same circuit depth, even though the FFT performs better in the noiseless case.

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5 In terms of circuit depth, the adiabatic circuits are shallower than the FFT since they are denser. At N = 48, the two-qubit gate circuit depth of the adiabatic circuit is around twice smaller than the FFT circuit. We also ran on hardware at N = 48 an adiabatic evolution with approximately same circuit depth as the FFT, which corresponds to around twice more two-qubit gates. We obtain an energy −0.577 ± 0.017, which is even lower than the result for same number of two-qubit gates plotted in Fig 2.

1. Bold header: Noise-Aware Ground State Preparation for Free Fermions

This improvement allows AI systems to prepare ground states of tight-binding chains with a system size N beyond which the adiabatic evolution reaches significantly lower energies than the Fermionic Fourier Transform (FFT), with the same number of gates, and with the same circuit depth. This is achieved by replacing FFT with an adiabatic evolution circuit when hardware noise is present.

2. Bold header: Low-Noise Spectral Function Measurement

The AI system can perform spectral function measurements on noisy hardware by utilizing a momentum measurement scheme that although less precise than FFT, is less costly and less noisy, achieving better performance than FFT for spectral function measurement on Quantinuum System Model H2 quantum computer.

3. Bold header: Momentum Density Profile Reconstruction (MDLM)

The system can reconstruct momentum density profiles from local measurements using the MDLM protocol, which involves measuring operators like On = 1/2 X N j=1 c† j c j+n + c† j+n c j and maximizing the Shannon entropy to maximize information about ⟨n(k)⟩.

4. Bold header: Noise-Sensitive Circuit Optimization

The AI can optimize circuit parameters, such as gate angles in Parametrized Circuits (COP), to minimize energy while considering noise sensitivity, as evidenced by the observation that the lowest energy is obtained for 4 steps in size N=48 for the parametrized ansatz.

5. Bold header: Error Propagation Analysis

The system can quantify how errors propagate based on circuit structure, noting that long-range hoppings c† j c j+δ + c† j+δ c j are more costly to implement and more sensitive to noise as δ grows. This informs the selection of local circuits over global FFT circuits for physical applications.

Quantum algorithms are usually optimized for gate count or circuit depth. We find on Quantinuum System Model H2 quantum computer that for a tight-binding chain ground state preparation, there is a system size N beyond which the adiabatic evolution reaches significantly lower energies than the Fermionic Fourier Transform (FFT), with the same number of gates, and with the same circuit depth. We attribute this high noise sensitivity of the FFT to its high precision, being able to distinguish momenta by 1/N. This high resolution in momentum space requires long-range couplings in real space, which propagates errors faster. In contrast, although local and physical circuits such as the adiabatic evolution have a coarser momentum resolution, they also propagate errors more slowly. For physical applications, high momentum resolution is rarely required and is often worth trading for low noise sensitivity. We also introduce a momentum measurement scheme that although less precise than FFT, is less costly and less noisy. We show that it achieves better performance than FFT for spectral function measurement on Quantinuum System Model H2 quantum computer. Our work emphasizes the importance of reducing the noise sensitivity of quantum algorithms, beyond the number of gates or circuit depth.