Practical Quantum Computation of Chemical and Nuclear Energy Levels Using Quantum Imaginary Time Evolution and Lanczos Algorithms

I. INTRODUCTION

• Short coherence times make time-evolution calculations exceedingly difficult on NISQ devices, despite their promise for studying excited and non-equilibrium states.Source: §I · p. 1
• QLanczos combines QITE with Lanczos optimization to obtain time-evolved phenomena in many-body systems.Source: §I · p. 1
• The authors calculate the deuteron ground-state energy and molecular hydrogen ground and excited states on cloud-accessible IBM Q devices, comparing results with exact diagonalization.They report a deuteron ground-state energy without nonlinear optimization or ancillae, and molecular-hydrogen energies close to or within chemical accuracy (1.6×10−3 Hartree).Source: §I · p. 1
• QITE approximates imaginary-time evolution, whose operator suppresses states other than the ground state, using unitary updates without requiring ancillae in the method of Motta et al..Choosing larger accurate imaginary-time steps can reduce Trotter steps; for the deuteron, the resulting unitary matched the familiar unitary coupled cluster ansatz.Source: §I · pp. 1–2
• QLanczos builds a Krylov space with QITE-generated vectors and uses their Hamiltonian matrix elements in a generalized eigenvalue equation to obtain ground and excited states.The single-step QITE method reduces circuit depth and improves compatibility with NISQ devices.Source: §I · p. 2

1. Quantum Imaginary Time Evolution (QITE)

• The QITE setup decomposes the Hamiltonian into local terms, including non-commuting terms handled with a first-order Lie-Trotter-Suzuki decomposition.The calculation assumes small ∆τ and approximates imaginary-time evolution to O(∆τ).Source: §1 · p. 2
• For small ∆τ, QITE determines Pauli-operator coefficients by solving a linear system at each imaginary-time step, producing unitary updates that approximate imaginary-time evolution.The updates use operators written in terms of Pauli operators up to D + 1 qubits.Source: §1 · p. 3
• The deuteron and molecular-hydrogen QITE updates recover the unitary coupled cluster ansatz proposed for those systems.Source: §1 · p. 3
• QITE can yield an excited-state energy when the initial state is non-orthogonal to that eigenstate, because evolution depends on the initial-state choice.Source: §1 · p. 3

2. Quantum Lanczos (QLanczos) Algorithm

• QLanczos fills a Krylov subspace with powers of e^−2∆τH applied to a state, with QITE producing the vectors.Source: §2 · p. 3
• After constructing the Krylov subspace, QLanczos calculates overlap and Hamiltonian matrix elements from experimentally accessible expectation values, then solves a generalized eigenvalue equation.Source: §2 · p. 3
• Using equation (18) is numerically more stable and gives much better quantum-computation results than using the eigenvalues of generalized eigenvalue equation (16).Source: §2 · p. 3
• QLanczos converges faster than QITE, but each added Krylov vector requires more QITE measurements and increases circuit depth.The single-step method reduces circuit depth and possible gate noise.Source: §2 · pp. 3–4

B. Quantum Program

• The standard two-qubit QITE circuit repeats its boxed update after the second step, while circuit depth grows with additional time steps.Large-depth circuits are described as noisy and not necessarily amenable to error mitigation.Source: Figure 1 · p. 4, B. Quantum Program · p. 4
• The single-step approach uses a calculated coefficient to build one circuit for a chosen initial state, then varies its angle parameters until the energy converges.For the two-qubit initial state |10⟩, the circuit includes one CNOT gate.Source: B. Quantum Program · p. 4
• For the three-qubit deuteron example, the initial state |100⟩ leads to unitary updates approximated by the circuit in Fig. 2(b).Figure 2 contrasts that circuit with the two-qubit circuit in panel (a).Source: B. Quantum Program · p. 4, Figure 2 · p. 4
• The authors also apply error-mitigation strategies alongside the single-step QITE approach to obtain energy expectation values.The strategies are described in Sec. B.Source: B. Quantum Program · p. 4

C. Results and Discussion

• Experimental results for QITE and QLanczos were obtained using IBM Q hardware.Information about the experiments and hardware is given in supplementary Table IV.Source: C. Results and Discussion · p. 4

Deuteron

• The deuteron QITE energies were −1.762 ± 0.2 MeV for N = 2 and −2.033 ± 0.1 MeV for N = 3, 0.76% and 0.64% from exact diagonalization, respectively.The energy expectation values are plotted against imaginary time in Fig. 3.Source: Deuteron · p. 4, Figure 3 · p. 5
• QITE reached chemical accuracy (1.6 × 10^-3 Hartree) for one or two steps when calculating molecular-hydrogen ground and first excited state energies.The ground-state calculation used |Ψ0⟩ = |00⟩ and the first excited-state calculation used |Ψ0⟩ = |10⟩.Source: Deuteron · p. 5, Figure 5 · p. 6

2. QLanczos Results

• Using Eq. (18), QLanczos gave values in good agreement with exact diagonalization, whereas stabilized generalized-eigenvalue values did not agree well for molecular hydrogen.Two Krylov vectors sufficed for the deuteron and molecular-hydrogen examples; the hydrogen spectrum was calculated with two initial states across bond lengths.Source: §2 · p. 6
• The deuteron bound-state energies were extrapolated to an infinite harmonic-oscillator basis using Lüscher’s formula, following the scheme used in.The extrapolated values are listed in Table II.Source: §2 · pp. 6–7, Table II · p. 7

III. CONCLUSION

• The study presents single-step QITE and QLanczos as a practical, potentially low-depth alternative, and reports that QLanczos agreed well with exact diagonalization without additional error mitigation.It also notes that QITE can calculate an excited-state energy whose eigenvector is non-orthogonal to the initial state.Source: §III · p. 7
• At next-to-leading order, deuteron bound-state energy errors were 0.5% (0.9%) for QITE and 2.2% (1.6%) for QLanczos at N = 2 (N = 3), compared with the experimental value −2.22 MeV.Source: §III · p. 7
• Ground and excited state energies of two-qubit molecular Hydrogen were calculated within chemical accuracy using QLanczos for a few bond lengths.Source: §III · p. 7

1. Deuteron

• The deuteron model implements pion-less EFT in a discrete variable representation based on the harmonic-oscillator basis.The model follows and [21], with the basis based on and.Source: §1 · p. 7
• Here, N is the maximum number of oscillator quanta in the harmonic-oscillator basis, and a_n and a†_n are fermionic annihilation and creation operators.Source: §1 · p. 7
• The deuteron model uses ℏω ≈ 7 MeV, V0 ≈ −5.686 MeV, and an ultraviolet cutoff Λ ≈ 152 MeV.Source: §1 · p. 7
• The creation and annihilation operators are mapped to Pauli matrices using the Jordan-Wigner transformation to obtain the N = 2 and N = 3 deuteron Hamiltonians.The passage also defines σ_j as a Pauli matrix at position j in a tensor product with identity matrices elsewhere.Source: §1 · pp. 7–8

2. Hydrogen Molecule

• The two-qubit Hydrogen Hamiltonian uses real coefficients h_i(R) that depend on bond length R, taken from STO-3G calculations in.Source: §2 · p. 8

Appendix B: Error Mitigation

• Readout-error mitigation estimates Hamiltonian-operator expectation values using qubit outcome probabilities and measured readout-flip probabilities.The authors obtained flip probabilities by preparing each qubit in computational-basis states and measuring it; simulations used 8192 shots.Source: Appendix B · p. 8
• Richardson extrapolation deliberately increases gate error by adding CNOT pairs, then extrapolates expectation values toward the noiseless value.The authors applied it to short-depth circuits and did not use it for QLanczos measurements because those results agreed well with exact diagonalization.Source: Appendix B · p. 8

Appendix C: Extrapolation to the infinite harmonic oscillator basis

• The finite-size corrections to the infinite-size harmonic-oscillator basis are stated in equation (C1), whose variables are defined in Table III.The right-hand-side terms are identified as LO, NLO, and N2LO; fitting the LO and NLO terms gives the binding momentum k∞ and asymptotic normalization coefficient γ for each order.Source: Appendix C · pp. 8–9, Table III · p. 9

• Fig. 5(b) lists 8192, 3, Poughkeepsie, and Rochester.Source: Figure 5 · p. 9
• Table IV is titled as information about the experimental runs on hardware.Source: Table IV · p. 9