Orders of magnitude increased accuracy for quantum many-body problems on quantum computers via an exact transcorrelated method

Transcorrelated methods provide an efficient way of partially transferring the description of electronic correlations from the ground-state wave function directly into the underlying Hamiltonian. In particular, Dobrautz et al. [Phys. Rev. B 99, 075119 (2019)] have demonstrated that the use of momentum-space representation, combined with a nonunitary similarity transformation, results in a Hubbard Hamiltonian that possesses a significantly more “compact” ground-state wave function, dominated by a single Slater determinant. This compactness/single-reference character greatly facilitates electronic structure calculations. As a consequence, however, the Hamiltonian becomes non-Hermitian, posing problems for quantum algorithms based on the variational principle. We overcome these limitations with the Ansatz-based quantum imaginary-time evolution algorithm and apply the transcorrelated method in the context of digital quantum computing. We demonstrate that this approach enables up to four orders of magnitude more accurate and compact solutions in various instances of the Hubbard model at intermediate interaction strength ($U/t=4$), enabling the use of shallower quantum circuits for wave-function Ansätzes. In addition, we propose a more efficient implementation of the quantum imaginary-time evolution algorithm in quantum circuits that is tailored to non-Hermitian problems. To validate our approach, we perform hardware experiments on the ibmq_lima quantum computer. Our work paves the way for the use of exact transcorrelated methods for the simulations of ab initio systems on quantum computers.

Figure 1

(a) Hybrid quantum-classical procedure employed to perform VarQITE. First, the Gutzwiller Ansatz is optimized by VMC or projection on a classical computer. The resulting TC Hubbard Hamiltonian is represented as a sum of Pauli strings by a fermion-to-qubit mapping (e.g., Jordan-Wigner). The initial parameter values $\mathit{\theta }$ and the list of necessary Pauli-string measurements are sent to the quantum computer. Given a wave-function Ansatz $|\mathrm{\Phi }\left(\mathit{\theta }\right)\rangle =U\left(\mathit{\theta }\right)|{\mathrm{\Phi }}_0\rangle$, the matrix elements of the metric $A_{ij}^{\mathrm{new}}$ and the gradient $C_i^{\mathrm{new}}$ are measured using the differentiation of general gates via a linear combination of unitaries represented as $W\left(\mathit{\theta }\right)$ [124]. Finally, the linear system derived from McLachlan's variational principle is solved and by applying the Euler method, we obtain the parameters ${\mathit{\theta }}^{\mathrm{new}}$ of the next time step. The whole procedure is repeated until convergence (i.e., Euclidean norm of the gradient is below a set threshold). (b) Optimization of the Jastrow parameter $J$. We show that when $J\approx J_{\mathrm{proj}}$ or $J\approx J_{\mathrm{VMC}}$, the ground state of six-site TC momentum-space Hubbard model is “compact,” meaning almost the entire weight of the wave function is concentrated in the Hartree-Fock (HF) state. $J_{\mathrm{VMC}}$ and $J_{\mathrm{proj}}$ denote the results of a VMC simulation and the solution of the projective scheme, respectively. (c) Heuristic RY unitary operator $U\left(\mathit{\theta }\right)$ applied on the HF initial state $|{\mathrm{\Psi }}_0\rangle$ of the two-site Hubbard model. In purple are the single-qubit rotations and in blue are the CNOTs. The definitions of the single-qubit gates are given in Appendix pp1. Exactly this Ansatz and initial state are employed in the hardware experiments on the ibmq_lima chip [see its layout in (a) where numbers denote the qubits].

Figure 2

Quantum circuit $W_2\left(\mathit{\theta }\right)$ used to calculate the $2C_i$ term in the non-Hermitian case by separation of the TC Hamiltonian into Hermitian and anti-Hermitian parts. We define that $V=H$ and $V=R_x\left(\frac{\pi }{2}\right)$ for the Hermitian and anti-Hermitian parts of a TC Hamiltonian, respectively. This circuit should be repeated for every term of the Hamiltonian. Note that no controlled Hamiltonian term is present.

Figure 3

Quantum circuit $W_1\left(\mathit{\theta }\right)$ used to calculate the $2C_i$ term in the non-Hermitian case when, for simplicity, the Hamiltonian is given by a single Pauli on the $i\mathrm{th}$ qubit, $\widehat{H}=1^{\left(0\right)}\otimes \cdots \otimes 1^{(i-1)}{Z}^{\left(i\right)}{1}^{(i+1)}\otimes \cdots \otimes 1^{(N_q-1)}$ (controlled-$Z$ gate) where $N_q$ denotes the number of qubits. This circuit should be repeated and adapted for every term of the Hamiltonian (controlled Pauli string) leading to $O\left(N^6\right)$ different circuits to be measured in TC cases.

Figure 4

Results of VarQITE state-vector simulations of repulsive Hubbard models with (a) two, (b) four, and (c) six sites at half-filling with $t=1$ and $U=4$, using the qUCCSD Ansatz with one layer. For every system, we report the evolution of the energy $E$ (top), absolute energy error $\left|\mathrm{\Delta }E\right|=|E-E_{\mathrm{exact}}|$ (middle), and infidelity $1-F$ (bottom) where the fidelity $F=|\langle \mathrm{\Phi }|{\mathrm{\Phi }}_{\mathrm{exact}}\rangle {|}^2$ is computed with respect to the exact ground state at half-filling $|{\mathrm{\Phi }}_{\mathrm{exact}}\rangle$ of the corresponding Hamiltonian. The dashed lines represent the exact ground-state energies $E_{\mathrm{exact}}(-2.472,-2.103,-3.669$, respectively) for the three systems. (For the six-site system, we only show every fifth data point for improved readability.) In most cases, the TC momentum-space method presents orders of magnitude improvement both in energy error and infidelity.

Figure 5

Results of VarQITE state-vector simulations of repulsive Hubbard models with (a) two, (b) four, and (c) six sites at half-filling with $t=1$ and $U=4$, using the qUCCSD Ansatz with $n$ layers. For every system, we report the absolute energy error $\left|\mathrm{\Delta }E\right|=|E-E_{\mathrm{exact}}|$ (top) and the infidelity $1-F$ (bottom) at convergence, where the fidelity $F=|\langle \mathrm{\Phi }|{\mathrm{\Phi }}_{\mathrm{exact}}\rangle {|}^2$ is computed with respect to the exact ground state at half-filling $|{\mathrm{\Phi }}_{\mathrm{exact}}\rangle$ of the corresponding Hamiltonian. The solid lines mark the converged outcomes of VQE state-vector simulations (VQE/SV) for the real and momentum-space Hamiltonians. The benefits of the transcorrelated approach can be seen, for instance, in the result of the six-site TC momentum-space Hamiltonian with $n=1$, which is significantly more accurate than the plain momentum-space outcome with $n=2$, especially in terms of fidelity.

Figure 6

Results of VarQITE simulations of the repulsive two-site Hubbard model at half-filling with $U/t=4$. The RY heuristic Ansatz (applied to the Hartree-Fock state), consisting of one layer with linear entanglement, is used [see Fig. 1]. We report the evolution of the energy $E$ (top) and absolute energy error $\left|\mathrm{\Delta }E\right|=|E-E_{\mathrm{exact}}|$ (bottom) computed with respect to the exact ground state at half-filling $|{\mathrm{\Phi }}_{\mathrm{exact}}\rangle$ of the corresponding Hamiltonian. The dashed line represents the exact ground-state energy $E_{\mathrm{exact}}=-2.472$. HW denotes the simulations performed on the ibmq_lima quantum computer. QASM marks the classical noisy simulations with the noise model of the ibmq_lima chip. (Only every second data point is shown to improve the readability.)