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Calculation of generating function in many-body systems with quantum computers: technical challenges and use in hybrid quantum-classical methods
Edgar Andres Ruiz Guzman, Denis Lacroix
TL;DR
The paper addresses how quantum computers can provide useful information about many-body Hamiltonians despite limited qubits and NISQ constraints. It computes a generating function with one ancillary qubit and post-processes its moments classically, illustrating ground- and excited-state estimation and long-time dynamics. The approach is useful when the generating function and its moments can be obtained accurately, but moment precision degrades at higher orders.
Problem
NISQ noise, decoherence, and limited qubit counts constrain quantum simulation of complex many-body systems.
Source: §I · p. 1Method
The paper computes a Hamiltonian generating function over short times with one ancillary qubit and uses its moments in classical t-expansion, Padé, and Krylov-based post-processing.
Source: §I · p. 1, §II · p. 2, §IV · p. 11Results
The t-expansion with Padé approximation gives rather accurate ground-state energies, while moment-based classical post-processing also targets excited states and long-time evolution.
Source: A. t-expansion approach for the ground… · p. 7, §1 · p. 9, §IV · p. 11Takeaways & Limitations
A quantum computer can be assigned the generating-function calculation while classical processing extracts static and dynamical information from the resulting moments.
Source: §I · p. 1, §IV · p. 11Finite-difference extraction of moments loses precision as K increases, limiting reliable moment estimates even for noiseless quantum computers.
Source: C. Critical discussion of the… · p. 5
Abstract
from arXiv · showhide
The generating function of a Hamiltonian $H$ is defined as $F(t)=\langle e^{-itH}\rangle$, where $t$ is the time and where the expectation value is taken on a given initial quantum state. This function gives access to the different moments of the Hamiltonian $\langle H^{K}\rangle$ at various orders $K$. The real and imaginary parts of $F(t)$ can be respectively evaluated on quantum computers using one extra ancillary qubit with a set of measurement for each value of the time $t$. The low cost in terms of qubits renders it very attractive in the near term period where the number of qubits is limited. Assuming that the generating function can be precisely computed using quantum devices, we show how the information content of this function can be used a posteriori on classical computers to solve quantum many-body problems. Several methods of classical post-processing are illustrated with the aim to predict approximate ground or excited state energies and/or approximate long-time evolutions. This post-processing can be achieved using methods based on the Krylov space and/or on the $t$-expansion approach that is closely related to the imaginary time evolution. Hybrid quantum-classical calculations are illustrated in many-body interacting systems using the pairing and Fermi-Hubbard models.
I. INTRODUCTION
The paper proposes computing a Hamiltonian generating function on a quantum computer with one ancillary qubit, then using its moments for classical post-processing of many-body problems.
I. INTRODUCTION
- NISQ noise and decoherence constrain quantum algorithms, motivating hybrid strategies that divide computation between quantum and classical devices.Source: §I · p. 1
II. GENERATING FUNCTION ON QUANTUM COMPUTERS AND ⟨HK⟩ESTIMATES
- The generating function can be evaluated over a restricted time interval with one ancillary qubit and provides access to Hamiltonian moments for later classical analysis.Source: §I · p. 1, §II · p. 2
- The generating function is passed to a classical computer for post-processing methods that study static and dynamical properties.Source: §I · p. 1, §II · p. 2
- For Hermitian O, choosing γ = −it makes e^−itO unitary, enabling quantum estimation of expectation values associated with non-unitary operators such as H and its powers.Source: §II · p. 2
- The real and imaginary parts of the generating function are obtained from ancillary-qubit measurements using standard and modified Hadamard tests.Source: Figure 1 · p. 2, §II · p. 2
A. Illustration of the method
The paper illustrates fermion-to-qubit mapping for many-body Hamiltonians using the Jordan-Wigner transformation and Pauli-operator representations.
A. Illustration of the method
- The examples map second-quantized pairing and Fermi-Hubbard Hamiltonians to interacting qubits using the Jordan-Wigner transformation.Source: A. Illustration of the method · p. 2
- The mapping expresses fermionic creation and annihilation operators through qubit Pauli operators and associated number operators.Source: A. Illustration of the method · pp. 2–3
1. Fermi-Hubbard model
The Fermi-Hubbard example models interacting lattice fermions and implements its hopping and interaction dynamics through Trotterized qubit circuits.
2. Fermi-Hubbard model
- The one-dimensional Fermi-Hubbard Hamiltonian describes spinful fermions on lattice sites and separates into hopping HJ and interaction HU terms.Source: §1 · p. 3
- Trotter-Suzuki evolution approximates each short time step by applying the hopping and interaction propagators sequentially.Source: §1 · p. 3
- The hopping propagator is decomposed into two-qubit operations acting on neighboring qubits.Source: §1 · p. 3
- The interaction propagator is implemented with a controlled phase-shift gate whose phase is φ = −∆tU.Source: §1 · p. 3, Figure 2 · p. 3
2. Pairing Hamiltonian
The pairing example uses a pair-based qubit encoding and Trotterized propagators for one-body and interaction terms.
2. Pairing Hamiltonian
- The pairing Hamiltonian is presented as a model relevant to nuclear physics and small superconducting systems.Source: §2 · p. 3
- The pairing model uses pair occupation and pair creation operators, with time-reversed single-particle states assigned shifted energies.Source: §2 · pp. 3–4
- In the seniority-zero scheme, each pair operator can be mapped directly to one qubit, reducing the qubit count for even-particle systems.Source: §2 · p. 4
- The interaction Hamiltonian is represented with XpXq and YpYq couplings and simulated using a propagator analogous to the Hubbard hopping circuit.Source: §2 · p. 4
- Trotter-Suzuki evolution is applied separately to the one-body and interaction propagators for short time steps.Source: §2 · p. 4
3. Illustration of generating function and Hamiltonian moments obtained by quantum computation
The generating function’s real and imaginary parts were extracted with one ancillary qubit and compared across pairing and Fermi-Hubbard models. In noiseless Qiskit simulations, the quantum results matched classical calculations when measurements and time discretization were sufficient.
3. Illustration of generating function and Hamiltonian moments obtained by quantum computation
- One extra ancillary qubit suffices to extract the real and imaginary parts of the generating function using Hadamard-test circuits.The procedure requires separate measurements for each time value.Source: §3 · p. 4
- The simulations evaluated the generating function for pairing and Fermi-Hubbard model cases using the circuits shown in Fig. 1.Figure 3 presents the real and imaginary components for both models.Source: §3 · p. 4, Figure 3 · p. 4
- For a perfect, noiseless quantum computer, quantum and classical results perfectly coincide; each plotted point is calculated by averaging 10^4 events using IBM Qiskit qasm.Source: §3 · p. 4
- The calculations used Δt·J = 0.02 for the Fermi-Hubbard model and Δt·Δe = 0.002 for the pairing model to maintain the Trotter-Suzuki approximation.Source: §3 · p. 5
B. Physical content of the generating function
The generating function encodes Hamiltonian spectral information and moments, making it useful for quantum measurement and classical post-processing. However, moment-extraction methods become accuracy- and resource-limited at high orders, while Fourier-based recovery trades improved convergence for longer quantum computations.
B. Physical content of the generating function
- Knowing the generating function over all times provides Hamiltonian eigenenergies and the initial state’s overlaps with the corresponding eigenstates.Its Fourier transform is related to strength or response functions.Source: B. Physical content of the generating… · p. 5
- The response function can be computed with quantum phase estimation using several ancillas or with the one-ancilla approach proposed here.Source: B. Physical content of the generating… · p. 5
- For γ = −it and O = H, the generating function gives access to expectation values of Hamiltonian moments ⟨H^K⟩ for the initial state.Source: B. Physical content of the generating… · p. 5
C. Critical discussion of the extraction of the moments from the generating function
- Finite differences provide reasonably precise low-order moments, but their precision degrades as K increases and small moment errors can strongly affect post-processing.In the models considered, reasonable accuracy was reported up to roughly K = 10–15 depending on interaction strength.Source: C. Critical discussion of the… · p. 5
- Polynomial interpolation, including Chebyshev interpolation, likewise achieves reasonable precision for early moments but insufficient accuracy at high K.Source: C. Critical discussion of the… · p. 5
- Fourier-based methods were the only tested approaches achieving globally converged moments with sufficient accuracy, provided time resolution and duration were adequate.They require many generating-function evaluations over long times, substantially increasing quantum-computation effort relative to the quantum-phase-estimation alternative.Source: C. Critical discussion of the… · p. 5
- The subsequent post-processing illustrations assume that the moments can be computed with good precision.Source: C. Critical discussion of the… · p. 5
III. ILLUSTRATION OF APPLICATIONS
The paper illustrates classical post-processing of limited Hamiltonian moments through t-expansion and Padé methods to estimate ground-state energies. Across pairing and Fermi-Hubbard examples, Padé extrapolation closely follows exact imaginary-time evolution, while convergence depends on truncation order and the initial state.
A. t-expansion approach for the ground state energy
- The approach uses an initial state with nonzero ground-state overlap, whose Hamiltonian moments determine the imaginary-time energy over a finite interval.The available interval depends on both the initial state and the number of moments.Source: A. t-expansion approach for the ground… · p. 6
- The Padé approximation enforces I + J = M and J − I ≥ 2, while the energy derivative must decrease toward zero sufficiently rapidly for the integrated energy to converge.The derivative is also constrained to remain negative during the approximation.Source: A. t-expansion approach for the ground… · pp. 6–7
- Padé-based t-expansion provides rather accurate ground-state energy extrapolations from a limited number of Hamiltonian moments in both tested models.The method replaces a truncated Taylor approximation of the energy derivative with a constrained Padé form and integrates the derivative.Source: A. t-expansion approach for the ground… · p. 7
- For the Fermi-Hubbard model, convergence toward the ground-state energy requires higher-order moments when only one Slater determinant replaces a mixture of six initial determinants.The figure compares exact, Taylor, and Padé imaginary-time evolutions under the model conditions.Source: Figure 4 · p. 6, Figure 5 · p. 7
- Increasing the truncation order rapidly improves convergence, whereas low-order Padé forms can produce poles and unphysical positive energy derivatives.For M = 4, the only Padé[1,3] choice has a pole; this issue disappears when M increases.Source: A. t-expansion approach for the ground… · p. 7
- With fixed M = 10, predictive accuracy degrades as g/∆e increases, although the estimate remains reasonably close at the largest tested coupling.The authors suggest symmetry-breaking, symmetry-restored initial states could improve strong-coupling predictions.Source: Figure 6 · p. 7, A. t-expansion approach for the ground… · p. 8
B. Excited states and time-dependent evolution
The paper constructs real-time evolution in a truncated Krylov subspace generated from powers of the Hamiltonian applied to the initial state. A time-dependent variational formulation or equivalent generalized eigenvalue problem uses initial Hamiltonian moments to approximate dynamics.
B. Excited states and time-dependent evolution
- The truncated Krylov basis consists of {|Φ0⟩, H|Φ0⟩, …, HM|Φ0⟩}, enabling evolution approximations in a subspace generated by Hamiltonian powers.Without truncation the Krylov expansion describes the evolution exactly for all time, while truncation preserves the expansion through order t^M.Source: B. Excited states and time-dependent… · p. 8
- A time-dependent variational principle yields coupled equations for the coefficients of the approximate state in the non-orthogonal Krylov basis.The overlap and Hamiltonian matrix elements define the resulting equations with the initial condition C_K(0) = δ_K0.Source: B. Excited states and time-dependent… · p. 8
- Solving the dynamics requires the first L = 2M + 1 Hamiltonian moments, and the variational solution matches exact evolution through order t^M.The equations may be integrated directly or transformed into an eigenvalue problem in the Krylov subspace.Source: B. Excited states and time-dependent… · p. 8
- The non-orthogonal Krylov problem is solved by diagonalizing the overlap matrix, orthonormalizing the basis, and then diagonalizing the Hamiltonian.This produces approximate eigenvalues and eigenstates associated with each truncation M.Source: B. Excited states and time-dependent… · p. 8
1. Excited states from moments
The truncated-Krylov diagonalization produces eigenvalues that converge toward exact pairing-Hamiltonian eigenvalues as the basis grows. Lower-energy states converge faster, while some excited states are missed when their initial-state overlaps are too small or the basis is insufficient.
1. Excited states from moments
- For g/∆e = 2.0, Krylov diagonalization converges to exact eigenvalues, with the lower-energy states converging fastest.The figure tracks red approximate eigenvalues against horizontal lines representing the lowest exact eigenvalues.Source: §1 · p. 9, Figure 7 · p. 9
- The ground state is already accurate at M = 3, corresponding to the first 7 moments, despite the pairing model being in the strong-coupling regime.The truncated Krylov space is much smaller than the full Hilbert space in the illustrated system.Source: §1 · p. 9
- Compared with t-expansion, Krylov diagonalization appears to converge more rapidly with increasing moments and additionally provides access to excited states.Some excited states remain absent because their overlaps with the initial state are too small or the Krylov basis needs enlargement.Source: §1 · p. 9
- At M = 5, Fourier transforming the survival probability already reproduces several dominant frequencies, consistent with low-order Krylov eigenvalue convergence.The frequency result connects spectral reconstruction with the eigenvalue behavior shown in the figure.Source: §1 · p. 9
2. Long-time evolution from moments
The truncated Krylov-space evolution reproduces the exact survival-probability dynamics up to an M-dependent time that grows with M, while M = 5 already captures several dominant strength-function frequencies.
Long-time evolution
- Figure 8 compares P (M) from TDCE equations at increasing M with the exact pairing-model evolution for g/∆e = 2.The black line denotes the exact evolution.Source: Figure 8 · p. 9
- The approximate survival-probability evolution matches the exact solution up to a time tmax(M) that increases with M and converges as more states are included.The method remains effective even when the retained Krylov space is much smaller than the complete Hilbert space.Source: §2 · p. 9
Strength function
- M = 5 already reproduces several dominant frequencies in the strength function obtained by Fourier transforming the survival probability.This agrees with the observation that some exact eigenvalues are well reproduced at relatively low M.Source: §2 · p. 9
C. Application on noisy quantum platforms
On noisy Santiago-based simulations, the generating function deviates substantially from the exact result, although combined error corrections partially recover it on FakeSantiago. The conclusion identifies moment accuracy and quantum-resource cost as major NISQ limitations.
C. Application on noisy quantum platforms
- The single-pair, two-level test uses two system qubits plus one ancilla, but raw Santiago-QPU results are strongly polluted by noise.FakeSantiago reproduces Santiago’s topology and noise processes for testing correction methods.Source: C. Application on noisy quantum… · pp. 9–10
- Both real and imaginary parts deviate significantly from the exact solution even at short times because of device noise and single-step Trotter-Suzuki discretization.Increasing the Trotter step count reduces discretization error but strongly increases circuit depth.Source: C. Application on noisy quantum… · p. 10, Figure 9 · p. 10
- For n > 1 Trotter steps, error-corrected results are worse than for n = 1, because increasing n strongly increases errors on the generating function.Source: C. Application on noisy quantum… · p. 10
- Combined read-out, post-selection, and reference corrections rather accurately correct the FakeSantiago generating-function error.The reference correction calibrates a matrix at t = 0 and assumes it remains applicable at later times.Source: C. Application on noisy quantum… · p. 10
IV. CONCLUSION
- The same protocol on the real Santiago device produced more noise than FakeSantiago and no reasonable corrected results, motivating combination with variational techniques.The conclusion also states that high-precision moment extraction becomes difficult as K increases and may require costly Fourier analysis.Source: C. Application on noisy quantum… · pp. 10–11, §IV · p. 11
Appendix A: Validity of the solution in the HM space in the TDCE approach
The TDCE solution in the truncated Krylov space is exact through order t^M in the Taylor expansion, with the first omitted contribution proportional to t^(M+1).
Appendix A: Validity of the solution in the H_M space in the TDCE approach
- The TDCE evolution in H_M is exact through order t^M in the Taylor series of the full-Hilbert-space evolution.The approximate state is denoted |Φ^(M)(t)⟩, while |Φ(t)⟩ denotes the exact state.Source: Appendix A · p. 11
- For K ≤ M, the projector P_M leaves every Krylov state unchanged, so the projected Hamiltonian H_M = P_M H P_M governs the truncated evolution.This projector property is used in deriving the TDCE equation and its formal solution.Source: Appendix A · p. 11
- All terms with K ≤ M vanish in the difference between exact and approximate evolutions, making the first nonzero term proportional to t^(M+1).This establishes the claimed Taylor-order accuracy of the truncated-space solution.Source: Appendix A · p. 11